# LPTMS Publications

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## A Toy Model of the Rat Race

### D. ben-Avraham

^{1}, Satya N. Majumdar^{2}, S. Redner^{3}#### Journal of Statistical Mechanics: Theory and Experiment (2007) L04002

We introduce a toy model of the 'rat race' in which individuals try to better themselves relative to the rest of the population. An individual is characterized by a real-valued fitness and each advances at a constant rate by an amount that depends on its standing in the population. The leader advances to remain ahead of its nearest neighbor, while all others advance by an amount that is set by the distance to the leader. A rich dynamics occurs as a function of the mean jump size of the trailing particles. For small jumps, the leader maintains its position, while for large jumps, there are long periods of stasis that are punctuated by episodes of explosive advancement and many lead changes. Intermediate to these two regimes, agents reach a common fitness and evolution grinds to a halt.

- 1. Department of Physics [Potsdam NY], Clarkson University
- 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
- 3. Center for Polymer Studies (CPS), Boston University

## Al’tshuler-Aronov correction to the conductivity of a large metallic square network

### Christophe Texier

^{1}, Gilles Montambaux^{2}#### Physical Review B

**76**(2007) 094202We consider the correction $\Delta\sigma_\mathrm{ee}$ due to electron-electron interaction to the conductivity of a weakly disordered metal (Al'tshuler-Aronov correction). The correction is related to the spectral determinant of the Laplace operator. The case of a large square metallic network is considered. The variation of $\Delta\sigma_\mathrm{ee}(L_T)$ as a function of the thermal length $L_T$ is found very similar to the variation of the weak localization $\Delta\sigma_\mathrm{WL}(L_\phi)$ as a function of the phase coherence length. Our result for $\Delta\sigma_\mathrm{ee}$ interpolates between the known 1d and 2d results, but the interaction parameter entering the expression of $\Delta\sigma_\mathrm{ee}$ keeps a 1d behaviour. Quite surprisingly, the result is very close to the 2d logarithmic behaviour already for $L_T\sim{a}/2$, where $a$ is the lattice parameter.

- 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
- 2. Laboratoire de Physique des Solides (LPS), CNRS : UMR8502 – Université Paris XI - Paris Sud

**2**)## Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials

### Sanchez-Palencia, L., Clément, D., Lugan, P., Bouyer, P., Shlyapnikov, G.V., Aspect, A.

#### Physical Review Letters 98 (2007) 210401

Details## Asymmetric quantum error correcting codes

### Ioffe, L., Mezard, M.

#### Physical Review A 75 (2007) 032345

Details## BCS-BEC crossover in a random external potential

### G. Orso

^{1}#### Physical Review Letters

**99**(2007) 250402We investigate the ground state properties of a disordered superfluid Fermi gas across the BCS-BEC (Bose Einstein condensate) crossover. We show that, for weak disorder, both the depletion of the condensate fraction of pairs and the normal fluid density exhibit a nonmonotonic behavior as a function of the interaction parameter $1/k_Fa$, reaching their minimum value near unitarity. We find that, moving away from the weak coupling BCS regime, Anderson's theorem ceases to apply and the superfluid order parameter is more and more affected by the random potential.

- 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

**20**)## Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction

### E. Trizac

^{1}, A. Barrat^{2}, M. H. Ernst^{3}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 031305Combining analytical and numerical methods, we study within the framework of the homogeneous non-linear Boltzmann equation, a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We use the new method presented in a previous paper [J. Stat. Phys. 124, 549 (2006)] and extend our results to a different heating mechanism, namely a deterministic non-linear friction force. We derive analytically the high energy tail of the velocity distribution and compare the theoretical predictions with high precision numerical simulations. Stretched exponential forms are obtained when the non-equilibrium steady state is stable. We derive sub-leading corrections and emphasize their relevance. In marginal stability cases, power-law behaviors arise, with exponents obtained as the roots of transcendental equations. We also consider some simple BGK (Bhatnagar, Gross, Krook) models, driven by similar heating devices, to test the robustness of our predictions.

- 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
- 2. Laboratoire de Physique Théorique d'Orsay (LPT), CNRS : UMR8627 – Université Paris XI - Paris Sud
- 3. Instituut voor Theoretische Fysica, Universiteit Utrecht

**2**)## Casimir force driven ratchets

### Thorsten Emig

^{1}#### Physical Review Letters

**98**(2007) 160801We explore the non-linear dynamics of two parallel periodically patterned metal surfaces that are coupled by the zero-point fluctuations of the electromagnetic field between them. The resulting Casimir force generates for asymmetric patterns with a time-periodically driven surface-to-surface distance a ratchet effect, allowing for directed lateral motion of the surfaces in sizeable parameter ranges. It is crucial to take into account inertia effects and hence chaotic dynamics which are described by Langevin dynamics. Multiple velocity reversals occur as a function of driving, mean surface distance, and effective damping. These transport properties are shown to be stable against weak ambient noise.

**41**)## Casimir forces between arbitrary compact objects

### T. Emig

^{1}, N. Graham^{2, 3}, R. L. Jaffe^{3}, M. Kardar^{2}#### Physical Review Letters

**99**(2007) 170403We develop an exact method for computing the Casimir energy between arbitrary compact objects, either dielectrics or perfect conductors. The energy is obtained as an interaction between multipoles, generated by quantum current fluctuations. The objects' shape and composition enter only through their scattering matrices. The result is exact when all multipoles are included, and converges rapidly. A low frequency expansion yields the energy as a series in the ratio of the objects' size to their separation. As an example, we obtain this series for two dielectric spheres and the full interaction at all separations for perfectly conducting spheres.

- 2. Department of Physics, Aucune
- 3. Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Aucune

**147**)## Combinatorial point for higher spin loop models

### Paul Zinn-Justin

^{1}#### Communications in Mathematical Physics

**272**(2007) 661-682Integrable loop models associated with higher representations (spin k/2) of U_q(sl(2)) are investigated at the point q=-e^{i\pi/(k+2)}. The ground state eigenvalue and eigenvectors are described. Introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries.

- 1. Laboratoire de Physique Théorique et Hautes Energies (LPTHE), CNRS : UMR7589 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot

**12**)## Correlation functions and excitation spectrum of the frustrated ferromagnetic spin-1/2 chain in an external magnetic fieldPhysical Review B 76 (2007) 174420

### Vekua, T., Honecker, A., Mikeska, H.J., Heidrich-Meisner, F.

#### Physical Review B76 (2007) 174420

Details## Crystalline phase of strongly interacting Fermi mixtures

### D. S. Petrov

^{1, 2}, G. E. Astrakharchik^{3}, D. J. Papoular^{1}, C. Salomon^{4}, G. V. Shlyapnikov^{1, 5}#### Physical Review Letters

**99**(2007) 130407We show that the system of weakly bound molecules of heavy and light fermionic atoms is characterized by a long-range intermolecular repulsion and can undergo a gas-crystal quantum transition if the mass ratio exceeds a critical value. For the critical mass ratio above 100 obtained in our calculations, this crystalline order can be observed as a superlattice in an optical lattice for heavy atoms with a small filling factor. We also find that this novel system is sufficiently stable with respect to molecular relaxation into deep bound states and to the process of trimer formation.

- 2. National Research Centre "Kurchatov Institute" (NRC KI), University of Moscow
- 3. Departament de Fisica i Enginyeria Nuclear, Campus Nord B4-B5, Universitat Politécnica de Catalunya
- 4. Laboratoire Kastler Brossel (LKB (Lhomond)), CNRS : UMR8552 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris
- 5. Van der Waals-Zeeman Institute, University of Amsterdam

**44**)## Density of near-extreme events

### Sanjib Sabhapandit

^{1}, Satya N. Majumdar^{1}#### Physical Review Letters

**98**(2007) 140201We provide a quantitative analysis of the phenomenon of crowding of near-extreme events by computing exactly the density of states (DOS) near the maximum of a set of independent and identically distributed random variables. We show that the mean DOS converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than, faster than, or as a pure exponential function. We argue that some of these results would remain valid even for certain {\em correlated} cases and verify it for power-law correlated stationary Gaussian sequences. Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.

**16**)## Dimensional crossover in quantum networks: from macroscopic to mesoscopic Physics

### Félicien Schopfer

^{1}, François Mallet^{1}, D. Mailly^{2}, C. Texier^{3, 4}, G. Montambaux^{4}, Christopher Bauerle^{1}, Laurent Saminadayar^{1, 5, 6}#### Physical Review Letters

**98**(2007) 026807We report on magnetoconductance measurements of metallic networks of various sizes ranging from 10 to $10^{6}$ plaquettes, with anisotropic aspect ratio. Both Altshuler-Aronov-Spivak (AAS) $h/2e$ periodic oscillations and Aharonov-Bohm (AB) $h/e$ periodic oscillations are observed for all networks. For large samples, the amplitude of both oscillations results from the incoherent superposition of contributions of phase coherent regions. When the transverse size becomes smaller than the phase coherent length $L_\phi$, one enters a new regime which is phase coherent (mesoscopic) along one direction and macroscopic along the other, leading to a new size dependence of the quantum oscillations.

- 1. Institut Néel (NEEL), CNRS : UPR2940 – Université Joseph Fourier - Grenoble I – Institut National Polytechnique de Grenoble (INPG)
- 2. Laboratoire de photonique et de nanostructures (LPN), CNRS : UPR20
- 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
- 4. Laboratoire de Physique des Solides (LPS), CNRS : UMR8502 – Université Paris XI - Paris Sud
- 5. Université Joseph Fourier (Grenoble 1 UJF), Université Joseph Fourier - Grenoble I
- 6. Institut Universitaire de France (IUF), Ministère de l'Enseignement Supérieur et de la Recherche Scientifique

## Directional emission of stadium-shaped micro-lasers

### Mélanie Lebental

^{1, 2}, Jean-Sébastien Lauret^{1}, Joseph Zyss^{1}, C. Schmit^{2}, E. Bogomolny^{2}#### Physical Review A: Atomic, Molecular and Optical Physics

**75**(2007) 033806The far-field emission of two dimensional (2D) stadium-shaped dielectric cavities is investigated. Micro-lasers with such shape present a highly directional emission. We provide experimental evidence of the dependance of the emission directionality on the shape of the stadium, in good agreement with ray numerical simulations. We develop a simple geometrical optics model which permits to explain analytically main observed features. Wave numerical calculations confirm the results.

- 1. Laboratoire de Photonique Quantique et Moléculaire (LPQM), CNRS : UMR8537 – École normale supérieure de Cachan - ENS Cachan
- 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

**24**)## Discrete-time and continuous-time modelling : some bridges and gaps

### Krivine, H., Lesne, A., Treiner, J.

#### Mathematical Structures in Computer Science 17 (2007) 261-276

## Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

### Randon-Furling, J., Majumdar, S.N.

#### Journal of Statistical Mechanics(2007) P10008

Details## Effect of connecting wires on the decoherence due to electron-electron interaction in a metallic ring

### Christophe Texier

^{1}#### Physical Review B

**76**(2007) 153312We consider the weak localization in a ring connected to reservoirs through leads of finite length and submitted to a magnetic field. The effect of decoherence due to electron-electron interaction on the harmonics of AAS oscillations is studied, and more specifically the effect of the leads. Two results are obtained for short and long leads regimes. The scale at which the crossover occurs is discussed. The long leads regime is shown to be more realistic experimentally.

**2**)## Eigenvalues amplitudes of the Potts model on a torus

### Richard, J.F., Jacobsen, J.L.

#### Nuclear Physics B 769 (2007) 256-274

Details## Equilibrium and dynamics of a trapped superfluid Fermi gas with unequal masses

### G. Orso

^{1}, L. P. Pitaevskii^{2, 3}, S. Stringari^{2}#### Physical Review A: Atomic, Molecular and Optical Physics

**77**(2007) 033611Interacting Fermi gases with equal populations but unequal masses are investigated at zero temperature using local density approximation and the hydrodynamic theory of superfluids in the presence of harmonic trapping. We derive the conditions of energetic stability of the superfluid configuration with respect to phase separation and the frequencies of the collective oscillations in terms of the mass ratio and the trapping frequencies of the two components. We discuss the behavior of the gas after the trapping potential of a single component is switched off and show that, near a Feshbach resonance, the released component can still remain trapped due to many-body interaction effects. Explicit predictions are presented for a mixture of $^6$Li and $^{40}$K with resonant interaction.

- 2. Dipartimento di Fisica, UNIVERSITÀ DEGLI STUDI DI TRENTO
- 3. Kapitza Institute for Physical Problems, Kapitza Institute for Physical Problems

**17**)## Exact enumeration of Hamiltonian circuits, walks, and chains in two and three dimensions

### Jacobsen, J.

#### Journal of Physics A40 (2007) 14667-14678

Details## Exact results for the Spectra of Bosons and Fermions with Contact Interaction

### Stefan Mashkevich

^{1}, Sergey I. Matveenko^{2}, Stéphane Ouvry^{3}#### Nuclear Physics B

**763**(2007) 431-444An N-body bosonic model with delta-contact interactions projected on the lowest Landau level is considered. For a given number of particles in a given angular momentum sector, any energy level can be obtained exactly by means of diagonalizing a finite matrix: they are roots of algebraic equations. A complete solution of the three-body problem is presented, some general properties of the N-body spectrum are pointed out, and a number of novel exact analytic eigenstates are obtained. The FQHE N-fermion model with Laplacian-delta interactions is also considered along the same lines of analysis. New exact eigenstates are proposed, along with the Slater determinant, whose eigenvalues are shown to be related to Catalan numbers.

- 1. Schrodinger, Schrodinger
- 2. Landau Institute for Theoretical Physics, Landau Institute for Theoretical Physics
- 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

**5**)## Exotic phases in geometrically frustrated triangular Ising magnets

### Jiang, Y., Emig, T.

#### Journal of Physics : Condensed Matter 19 (2007) 145234

## Fermi Edge Singularities in the Mesoscopic Regime: II. Photo-absorption Spectra

### M. Hentschel

^{1, 2}, D. Ullmo^{2, 3}, H. U. Baranger^{2}#### Physical Review B

**76**(2007) 245419We study Fermi edge singularities in photo-absorption spectra of generic mesoscopic systems such as quantum dots or nanoparticles. We predict deviations from macroscopic-metallic behavior and propose experimental setups for the observation of these effects. The theory is based on the model of a localized, or rank one, perturbation caused by the (core) hole left behind after the photo-excitation of an electron into the conduction band. The photo-absorption spectra result from the competition between two many-body responses, Anderson's orthogonality catastrophe and the Mahan-Nozieres-DeDominicis contribution. Both mechanisms depend on the system size through the number of particles and, more importantly, fluctuations produced by the coherence characteristic of mesoscopic samples. The latter lead to a modification of the dipole matrix element and trigger one of our key results: a rounded K-edge typically found in metals will turn into a (slightly) peaked edge on average in the mesoscopic regime. We consider in detail the effect of the 'bound state' produced by the core hole.

- 1. Max Planck Institute for Physics of Complex Systems, Max-Planck-Institut
- 2. Duke Physics, Duke University
- 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

**5**)## Finite-size effects on the dynamics of the zero-range process

### Shamik Gupta

^{1}, Mustansir Barma^{1}, Satya N. Majumdar^{2}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 060101We study finite-size effects on the dynamics of a one-dimensional zero-range process which shows a phase transition from a low-density disordered phase to a high-density condensed phase. The current fluctuations in the steady state show striking differences in the two phases. In the disordered phase, the variance of the integrated current shows damped oscillations in time due to the motion of fluctuations around the ring as a dissipating kinematic wave. In the condensed phase, this wave cannot propagate through the condensate, and the dynamics is dominated by the long-time relocation of the condensate from site to site.

- 1. Department of Theoretical Physics, Tata institute of Fundamental Research
- 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

**4**)## Fractional Laplacian in Bounded Domains

### A. Zoia

^{1}, A. Rosso^{2, 3}, M. Kardar^{3}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 021116The fractional Laplacian operator, $-(-\triangle)^{\frac{\alpha}{2}}$, appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalues spectrum are also obtained.

- 1. Department of Nuclear Engineering,, Polytechnic of Milan
- 3. Department of Physics Massachusetts Institute of Technology, Massachusetts Institute of Technology

**43**)## General flux to a trap in one and three dimensions

### Robert M. Ziff

^{1}, Satya N. Majumdar^{2}, Alain Comtet^{2, 3}#### Journal of Physics: Condensed Matter

**19**(2007) 065102The problem of the flux to a spherical trap in one and three dimensions, for diffusing particles undergoing discrete-time jumps with a given radial probability distribution, is solved in general, verifying the Smoluchowski-like solution in which the effective trap radius is reduced by an amount proportional to the jump length. This reduction in the effective trap radius corresponds to the Milne extrapolation length.

- 1. Michigan Center for Theoretical Physics and Department of chemical Engineering, University of Michigan-Ann Arbor
- 3. Unite mixte de service de l'institut Henri Poincaré (UMSIHP), CNRS : UMS839 – Université Paris VI - Pierre et Marie Curie

**5**)## Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems

### Florent Krzakala

^{1}, Andrea Montanari^{2}, Federico Ricci-Tersenghi, Guilhem Semerjian^{2}, Lenka Zdeborova^{3}#### Proceeding of the national academy of sciences

**104**, 25 (2007) 10318An instance of a random constraint satisfaction problem defines a random subset S (the set of solutions) of a large product space (the set of assignments). We consider two prototypical problem ensembles (random k-satisfiability and q-coloring of random regular graphs), and study the uniform measure with support on S. As the number of constraints per variable increases, this measure first decomposes into an exponential number of pure states ('clusters'), and subsequently condensates over the largest such states. Above the condensation point, the mass carried by the n largest states follows a Poisson-Dirichlet process. For typical large instances, the two transitions are sharp. We determine for the first time their precise location. Further, we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem. The degree of correlation naturally affects the performances of many search/sampling algorithms. Empirical evidence suggests that local Monte Carlo Markov Chain strategies are effective up to the clustering phase transition, and belief propagation up to the condensation point. Finally, refined message passing techniques (such as survey propagation) may beat also this threshold.

- 1. Laboratoire de Physico-Chimie Théorique (LPCT), CNRS : UMR7083 – ESPCI ParisTech
- 2. Laboratoire de Physique Théorique de l'ENS (LPTENS), CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris

**92**)## Incipient Wigner Localization in Circular Quantum Dots

### Amit Ghosal

^{1, 2}, A.D. Guclu^{1, 3}, C.J. Umrigar^{3}, Denis Ullmo^{1, 4}, Harold U. Baranger^{1}#### Physical Review B

**76**(2007) 085341We study the development of electron-electron correlations in circular quantum dots as the density is decreased. We consider a wide range of both electron number, N<=20, and electron gas parameter, r_s<18, using the diffusion quantum Monte Carlo technique. Features associated with correlation appear to develop very differently in quantum dots than in bulk. The main reason is that translational symmetry is necessarily broken in a dot, leading to density modulation and inhomogeneity. Electron-electron interactions act to enhance this modulation ultimately leading to localization. This process appears to be completely smooth and occurs over a wide range of density. Thus there is a broad regime of ``incipient'' Wigner crystallization in these quantum dots. Our specific conclusions are: (i) The density develops sharp rings while the pair density shows both radial and angular inhomogeneity. (ii) The spin of the ground state is consistent with Hund's (first) rule throughout our entire range of r_s for all 4

- 1. Duke Physics, Duke University
- 2. Physics Department, University of California, Los Angeles
- 3. Laboratory of Atomic and Solid State Physics (LASSP), Cornell University
- 4. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

**24**)## Inferring periodic orbits from spectra of simple shaped micro-lasers

### Mélanie Lebental

^{1, 2}, Nadia Djellali^{1}, Carole Arnaud^{1}, Jean-Sébastien Lauret^{1}, Joseph Zyss^{1}, R. Dubertrand^{2}, C. Schmit^{2}, E. Bogomolny^{2}#### Physical Review A: Atomic, Molecular and Optical Physics

**76**(2007) 023830Dielectric micro-cavities are widely used as laser resonators and characterizations of their spectra are of interest for various applications. We experimentally investigate micro-lasers of simple shapes (Fabry-Perot, square, pentagon, and disk). Their lasing spectra consist mainly of almost equidistant peaks and the distance between peaks reveals the length of a quantized periodic orbit. To measure this length with a good precision, it is necessary to take into account different sources of refractive index dispersion. Our experimental and numerical results agree with the superscar model describing the formation of long-lived states in polygonal cavities. The limitations of the two-dimensional approximation are briefly discussed in connection with micro-disks.

- 1. Laboratoire de Photonique Quantique et Moléculaire (LPQM), CNRS : UMR8537 – École normale supérieure de Cachan - ENS Cachan

**24**)## Innovation and robustness in complex regulatory gene networks

### Ciliberti, S., Martin, O.C., Wagner, A.

#### PNAS104 (2007) 13591-13596

## Integer Partitions and Exclusion Statistics

### Alain Comtet

^{1, 2}, Satya N. Majumdar^{1}, Stephane Ouvry^{1}#### Journal of Physics A General Physics

**40**(2007) 11255-11269We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic point $ p=0$ is a repulsive fixed point for which the limiting distribution has a Gumbel form. For all positive $p$ the distribution is shown to be Gaussian.

- 2. IHP, Institut Henri Poincaré

**10**)## Integer partitions and exclusion statistics: Limit shapes and the largest part of Young diagrams

### Alain Comtet

^{1, 2}, Satya N. Majumdar^{1}, Stephane Ouvry^{1}, Sanjib Sabhapandit^{1}#### Journal of statistical mechanics-theory and experiment (2007) P10001

We compute the limit shapes of the Young diagrams of the minimal difference $p$ partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young diagram and show that the scaled distribution has a Gumbel form for all $p$. This Gumbel statistics for the largest part remains unchanged even for general partitions of the form $E=\sum_i n_i i^{1/\nu}$ with $\nu>0$ where $n_i$ is the number of times the part $i$ appears.

- 2. Unite mixte de service de l'institut Henri Poincaré (UMSIHP), CNRS : UMS839 – Université Paris VI - Pierre et Marie Curie

**2**)## Interaction-induced crossover versus finite-size condensation in a weakly interacting trapped one-dimensional Bose gas

### Bouchoule, I., Kheruntsyan, K.V., Shlyapnikov, G.V.

#### Physical Review A 75 (2007) 031606

Details## Large Deviations and Random Matrices

### Vivo, P., Majumdar, S.N., Bohigas, O.

#### Acta Physica Polonica38 (2007) 4129-4151

## Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices

### Pierpaolo Vivo

^{1}, Satya N. Majumdar^{2}, Oriol Bohigas^{2}#### Journal of Physics A: Mathematical and General

**40**(2007) 4317-4337We compute analytically the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N x N) Wishart matrix W=X^T X (where X is a rectangular M x N matrix with independent Gaussian entries) are smaller than the mean value <\lambda>=N/c decreases for large N as $\sim \exp[-\frac{\beta}{2}N^2 \Phi_{-}(\frac{2}{\sqrt{c}}+1;c)]$, where \beta=1,2 correspond respectively to real and complex Wishart matrices, c=N/M < 1 and \Phi_{-}(x;c) is a large deviation function that we compute explicitly. The result for the Anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of constrained Wishart matrices whose eigenvalues are forced to be smaller than a fixed barrier. The numerical simulations are in excellent agreement with the analytical predictions.

- 1. School of Information Systems, Computing & Mathematics, Brunel University

**38**)## Level Density of a Bose Gas and Extreme Value Statistics

### A. Comtet

^{1, 2}, P. Leboeuf^{1}, Satya N. Majumdar^{1}#### Physical Review Letters

**98**(2007) 070404We establish a connection between the level density of a gas of non-interacting bosons and the theory of extreme value statistics. Depending on the exponent that characterizes the growth of the underlying single-particle spectrum, we show that at a given excitation energy the limiting distribution function for the number of excited particles follows the three universal distribution laws of extreme value statistics, namely Gumbel, Weibull and Fréchet. Implications of this result, as well as general properties of the level density at different energies, are discussed.

- 2. Unite mixte de service de l'institut Henri Poincaré (UMSIHP), CNRS : UMS839 – Université Paris VI - Pierre et Marie Curie

**17**)## Longest increasing subsequence as expectation of a simple nonlinear stochastic PDE with a low noise intensity

### E. Katzav

^{1}, S. Nechaev^{2}, O. Vasilyev^{3, 4}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**75**(2007) 061113We report some new observation concerning the statistics of Longest Increasing Subsequences (LIS). We show that the expectation of LIS, its variance, and apparently the full distribution function appears in statistical analysis of some simple nonlinear stochastic partial differential equation (SPDE) in the limit of very low noise intensity.

- 1. Laboratoire de Physique Statistique de l'ENS (LPS), CNRS : UMR8550 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot – Ecole Normale Supérieure de Paris - ENS Paris
- 3. Max-Planck-Institut für Metallforschung, Max-Planck-Institut
- 4. Institut für Theoretische und Angewandte Physik, Universität Stuttgart

**1**)## Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices

### Paul Zinn-Justin

^{1}#### Journal of Statistical Mechanics: Theory and Experiment (2007) P01007

The integrable loop model with mixed boundary conditions based on the 1-boundary extended Temperley--Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L+3. A refined counting is also considered.

- 1. Laboratoire de Physique Théorique et Hautes Energies (LPTHE), CNRS : UMR7589 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot

**12**)## Magnetic exponents of two-dimensional Ising spin glasses

### F. Liers

^{1}, O. C. Martin^{2}#### Physical Review B

**76**(2007) 060405The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using exact ground state determination, we extract the properties of clusters flipped when increasing continuously a uniform field. We show that these clusters have many holes but otherwise have statistical properties similar to those of zero-field droplets. A detailed analysis gives for the magnetization exponent delta = 1.30 +/- 0.02 using lattice sizes up to 80x80; this is compatible with the droplet model prediction delta = 1.282. The reason for previous disagreements stems from the need to analyze both singular and analytic contributions in the low-field regime.

- 1. Institut für Informatik, Universität zu Köln

**1**)## Many-body effects in the mesoscopic x-ray edge problem

### Martina Hentschel

^{1}, Georg Roeder^{1}, Denis Ullmo^{2}#### Progress of Theoretical Physics Supplement

**166**(2007) 143-151Many-body phenomena, a key interest in the investigation of bulk solid state systems, are studied here in the context of the x-ray edge problem for mesoscopic systems. We investigate the many-body effects associated with the sudden perturbation following the x-ray excitation of a core electron into the conduction band. For small systems with dimensions at the nanoscale we find considerable deviations from the well-understood metallic case where Anderson orthogonality catastrophe and the Mahan-Nozieres-DeDominicis response cause characteristic deviations of the photoabsorption cross section from the naive expectation. Whereas the K-edge is typically rounded in metallic systems, we find a slightly peaked K-edge in generic mesoscopic systems with chaotic-coherent electron dynamics. Thus the behavior of the photoabsorption cross section at threshold depends on the system size and is different for the metallic and the mesoscopic case.

- 1. Max Planck Institute for Physics of Complex Systems, Max-Planck-Institut

**3**)## Mosaic length and finite interaction-range effects in a one dimensional random energy model

### Silvio Franz

^{1}, Giorgio Parisi^{2, 3, 4}, Federico Ricci-Tersenghi^{2}#### Journal of Physics A: Mathematical and General

**41**(2007) 324011In this paper we study finite interaction range corrections to the mosaic picture of the glass transition as emerges from the study of the Kac limit of large interaction range for disordered models. To this aim we consider point to set correlation functions, or overlaps, in a one dimensional random energy model as a function of the range of interaction. In the Kac limit, the mosaic length defines a sharp first order transition separating a high overlap phase from a low overlap one. Correspondingly we find that overlap curves as a function of the window size and different finite interaction ranges cross roughly at the mosaic lenght. Nonetheless we find very slow convergence to the Kac limit and we discuss why this could be a problem for measuring the mosaic lenght in realistic models.

- 2. Dipartimento di Fisica and INFM, Università degli studi di Roma I - La Sapienza
- 3. Dipartimento di Fisica, SMC, INFM, and INFN, Università degli studi di Roma I - La Sapienza
- 4. Dipartimento di Fisica, Sezione INFN, SMC and UdRm1 of INFM, Università degli studi di Roma I - La Sapienza

**3**)## Necklace-cloverleaf transition in associating RNA-like diblock copolymers

### Tamm, M.V., Nechaev, S.K.

#### Physical Review E 75 (2007) 031904

## Network of inherent structures in spin glasses: scaling and scale-free distributions

### Z. Burda

^{1}, A. Krzywicki^{2}, O. C. Martin^{3}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 051107The local minima (inherent structures) of a system and their associated transition links give rise to a network. Here we consider the topological and distance properties of such a network in the context of spin glasses. We use steepest descent dynamics, determining for each disorder sample the transition links appearing within a given barrier height. We find that differences between linked inherent structures are typically associated with local clusters of spins; we interpret this within a framework based on droplets in which the characteristic ``length scale'' grows with the barrier height. We also consider the network connectivity and the degrees of its nodes. Interestingly, when the spin glass is of the mean-field type, the degree distribution of the network of inherent structures exhibits a non-trivial scale-free behavior.

- 1. Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagellonian University
- 2. Laboratoire de Physique Théorique d'Orsay (LPT), CNRS : UMR8627 – Université Paris XI - Paris Sud

**8**)## New structural variation in evolutionary searches of RNA neutral networks

### Sumedha, Martin, O.C., Wagner, A.

#### BioSystems90 (2) (2007) 475-485

## Non-Abelian states with negative flux: a new series of quantum Hall states

### Thierry Jolicoeur

^{1}#### Physical Review Letters

**99**(2007) 036805By applying the idea of parafermionic clustering to composite bosons with positive as well as negative flux, a new series of trial wavefunctions to describe fractional quantum Hall states is proposed. These non-Abelian states compete at filling factors k/(3k +/- 2) with other ground states like stripes or composite fermion states. These two series contain all the states recently discovered by Pan et al. [Phys. Rev. Lett. 90, 016801 (2003)] including the even denominator cases. Exact diagonalization studies on the sphere and torus point to their possible relevance for filling factors 3/7, 3/11, and 3/8.

**8**)## Non-intersection exponents of fully packed trails on the square lattice

### Ikhlef, Y., Jacobsen, J.L., Saleur, H.

#### Journal of Statistical Mechanics (2007) P05005

Details## Numerical Calculation of the Functional renormalization group fixed-point functions at the depinning transition

### Alberto Rosso

^{1}, Pierre Le Doussal^{2}, Kay Joerg Wiese^{2}#### Physical Review B

**75**(2007) 22020aWe compute numerically the sequence of successive pinned configurations of an elastic line pulled quasi-statically by a spring in a random bond (RB) and random field (RF) potential. Measuring the fluctuations of the center of mass of the line allows to obtain the functional renormalization group (FRG) functions at the depinning transition. The universal form of the second cumulant Delta(u) is found to have a linear cusp at the origin, to be identical for RB and RF, different from the statics, and in good agreement with 2-loop FRG. The cusp is due to avalanches, which we visualize. Avalanches also produce a cusp in the third cumulant, whose universal form is obtained, as predicted by FRG.

- 2. Laboratoire de Physique Théorique de l'ENS (LPTENS), CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris

**27**)## Occupation times for planar and higher dimensional Brownian motion

### Desbois, J.

#### Journal of Physics A 40 (2007) 2251-2262

## On the distribution of surface extrema in several one- and two-dimensional random landscapes

### Florent Hivert

^{1}, S. Nechaev^{2, 3}, G. Oshanin^{4, 5, 6}, O. Vasilyev^{4, 7}#### Journal of Statistical Physics

**126**(2007) 243-279We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one- and two-dimensional substrates focusing our analysis on the probability distribution function $P(M,L)$ of the number $M$ of maximal points (i.e., local ``peaks'') of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one--dimensional ballistic growth process in the steady state can be mapped onto ''rise-and-descent'' sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. \cite{ov}, that different characteristics of ``rise-and-descent'' patterns in random permutations can be interpreted in terms of a certain continuous--space Hammersley--type process. For one--dimensional system we compute $P(M,L)$ exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long--ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as $L \to \infty$.

- 1. Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes (LITIS), Institut National des Sciences Appliquées (INSA) - Rouen – Université du Havre – Université de Rouen : EA4108
- 3. P. N. Lebedev Physical Institute, Russian Academy of Science
- 4. Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), CNRS : UMR7600 – Université Paris VI - Pierre et Marie Curie
- 5. Max-Planck-Institut fur Metallforschung, Max-Planck-Institut
- 6. Institut fur Theoretische und Angewandte Physik, Universität Stuttgart
- 7. Center for Molecular Modelling, Materia Nova, Université de Mons-Hainaut

## On the Feasibility of Portfolio Optimization under Expected Shortfall

### Stéfano Ciliberti

^{1}, Imre Kondor^{2}, Marc Mézard^{1}#### Quantitative Finance

**7**(2007) 389-396We address the problem of portfolio optimization under the simplest coherent risk measure, i.e. the expected shortfall. As it is well known, one can map this problem into a linear programming setting. For some values of the external parameters, when the available time series is too short, the portfolio optimization is ill posed because it leads to unbounded positions, infinitely short on some assets and infinitely long on some others. As first observed by Kondor and coworkers, this phenomenon is actually a phase transition. We investigate the nature of this transition by means of a replica approach.

- 2. Collegium Bupadest, Collegium Budapest

## On the ground–state energy of finite Fermi systems

### Jérôme Roccia

^{1}, Patricio Leboeuf^{1}#### Physical Review C

**76**(2007) 014301We study the ground--state shell correction energy of a fermionic gas in a mean--field approximation. Considering the particular case of 3D harmonic trapping potentials, we show the rich variety of different behaviors (erratic, regular, supershells) that appear when the number--theoretic properties of the frequency ratios are varied. For self--bound systems, where the shape of the trapping potential is determined by energy minimization, we obtain accurate analytic formulas for the deformation and the shell correction energy as a function of the particle number $N$. Special attention is devoted to the average of the shell correction energy. We explain why in self--bound systems it is a decreasing (and negative) function of $N$.

**5**)## On the Inelastic Collapse of a Ball Bouncing on a Randomly Vibrating Platform

### Satya N. Majumdar

^{1}, Michael J. Kearney^{2}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 031130We study analytically the dynamics of a ball bouncing inelastically on a randomly vibrating platform, as a simple toy model of inelastic collapse. Of principal interest are the distributions of the number of flights n_f till the collapse and the total time \tau_c elapsed before the collapse. In the strictly elastic case, both distributions have power law tails characterised by exponents which are universal, i.e., independent of the details of the platform noise distribution. In the inelastic case, both distributions have exponential tails: P(n_f) ~ exp[-\theta_1 n_f] and P(\tau_c) ~ exp[-\theta_2 \tau_c]. The decay exponents \theta_1 and \theta_2 depend continuously on the coefficient of restitution and are nonuniversal; however as one approches the elastic limit, they vanish in a universal manner that we compute exactly. An explicit expression for \theta_1 is provided for a particular case of the platform noise distribution.

- 2. School of Electronics and Physical Sciences, University of Surrey

**6**)## On the macroion virial contribution to the osmotic pressure in charge-stabilized colloidal suspensions

### E. Trizac

^{1, 2}, L. Belloni^{3}, J. Dobnikar^{4, 5}, H. H. von Grunberg^{4}, R. Castaneda-Priego^{6}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**75**(2007) 011401Our interest goes to the different virial contributions to the equation of state of charged colloidal suspensions. Neglect of surface effects in the computation of the colloidal virial term leads to spurious and paradoxical results. This pitfall is one of the several facets of the danger of a naive implementation of the so called One Component Model, where the micro-ionic degrees of freedom are integrated out to only keep in the description the mesoscopic (colloidal) degrees of freedom. On the other hand, due incorporation of wall induced forces dissolves the paradox brought forth in the naive approach, provides a consistent description, and confirms that for salt-free systems, the colloidal contribution to the pressure is dominated by the micro-ionic one. Much emphasis is put on the no salt case but the situation with added electrolyte is also discussed.

- 2. Center for Theoretical Biological Physics (CTBP), University of San Diego
- 3. Laboratoire Interdisciplinaire sur l'Organisation Nanométrique et Supramoléculaire, Service de Chimie Moléculaire, CEA
- 4. Institut für Chemie, Karl-Franzens-Universität
- 5. Jozef Stefan Institute, Jozef Stefan Institute
- 6. Instituto de Fisica, Universidad de Guanajuato

**10**)## Parafermionic states in rotating Bose-Einstein condensates

### Nicolas Regnault

^{1}, Thierry Jolicoeur^{1, 2}#### Physical Review B

**76**(2007) 235324We investigate possible parafermionic states in rapidly rotating ultracold bosonic atomic gases at lowest Landau level filling factor nu=k/2. We study how the system size and interactions act upon the overlap between the true ground state and a candidate Read-Rezayi state. We also consider the quasihole states which are expected to display non-Abelian statistics. We numerically evaluate the degeneracy of these states and show agreement with a formula given by E. Ardonne. We compute the overlaps between low-lying exact eigenstates and quasihole candidate wavefunctions. We discuss the validity of the parafermion description as a function of the filling factor.

- 1. Laboratoire Pierre Aigrain (LPA), CNRS : UMR8551 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot – Ecole Normale Supérieure de Paris - ENS Paris

**13**)## Patterns of recombination and MLH1 foci density along mouse chromosomes: modeling effetcs of interference and obligate chiasma

### Falque, M., Mercier, R., Mezard, M., de Vienne, D., Martin, O.C.

#### Genetics176 (2007) 1453-1467

## Persistence of a Rouse polymer chain under transverse shear flow

### Somnath Bhattacharya

^{1}, Dibyendu Das^{1}, Satya N. Majumdar^{2}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**75**(2007) 061122We consider a single Rouse polymer chain in two dimensions in presence of a transverse shear flow along the $x$ direction and calculate the persistence probability $P_0(t)$ that the $x$ coordinate of a bead in the bulk of the chain does not return to its initial position up to time $t$. We show that the persistence decays at late times as a power law, $P_0(t)\sim t^{-\theta}$ with a nontrivial exponent $\theta$. The analytical estimate of $\theta=0.359...$ obtained using an independent interval approximation is in excellent agreement with the numerical value $\theta\approx 0.360\pm 0.001$.

- 1. Department of Physics, Indian Institute of Technology

**5**)## Phase diagram of the chromatic polynomial on a torus

### Jesper Lykke Jacobsen

^{1, 2}, Jesus Salas^{3}#### Nuclear Physics B - Proceedings Supplements

**783**(2007) 238-296We study the zero-temperature partition function of the Potts antiferromagnet (i.e., the chromatic polynomial) on a torus using a transfer-matrix approach. We consider square- and triangular-lattice strips with fixed width L, arbitrary length N, and fully periodic boundary conditions. On the mathematical side, we obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the square and triangular lattices. On the physical side, we obtain the exact ``phase diagrams'' for these strips of width L and infinite length, and from these results we extract useful information about the infinite-volume phase diagram of this model: in particular, the number and position of the different phases.

- 2. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT
- 3. Grupo de Modelizacion, Simulacion Numerica y Matematica Industrial, Universidad Carlos III de Madrid

## Phase Transitions in the Coloring of Random Graphs

### Lenka Zdeborová

^{1}, Florent Krzakala^{2}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 031131We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdos-Renyi and regular random graphs and determine their asymptotic values for large number of colors. Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.

- 2. Laboratoire de Physico-Chimie Théorique (LPCT), CNRS : UMR7083 – ESPCI ParisTech

**66**)## Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2

### A. V. Razumov

^{1}, Yu. G. Stroganov^{1}, Paul Zinn-Justin^{2}#### Journal of Physics A Mathematical and Theoretical

**40**(2007) 11827-11847Integral formulae for polynomial solutions of the quantum Knizhnik-Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to e^{+- 2 pi i/3} and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit it is a ground state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Delta equal to -1/2 and odd number of sites. The obtained integral representations for the components of this eigenvector allow to prove some conjectures on its properties formulated earlier. A new statement relating the ground state components of XXZ spin chains and Temperley-Lieb loop models is formulated and proved.

- 1. Division of Theoretical Physics, Institut for High Energy Physics
- 2. Laboratoire de Physique Théorique et Hautes Energies (LPTHE), CNRS : UMR7589 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot

**10**)## Population size effects in evolutionary dynamics on neutral networks and toy landscapes

### Olivier C Martin

^{1, 2}, Luca Peliti^{3}, S. Sumedha^{1}#### Journal of Statistical Mechanics: Theory and Experiment (2007) P05011

We study the dynamics of a population subject to selective pressures, evolving either on RNA neutral networks or in toy fitness landscapes. We discuss the spread and the neutrality of the population in the steady state. Different limits arise depending on whether selection or random drift are dominant. In the presence of strong drift we show that observables depend mainly on $M \mu$, $M$ being the population size and $\mu$ the mutation rate, while corrections to this scaling go as 1/M: such corrections can be quite large in the presence of selection if there are barriers in the fitness landscape. Also we find that the convergence to the large $M \mu$ limit is linear in $1/M \mu$. Finally we introduce a protocol that minimizes drift; then observables scale like 1/M rather than $1/(M\mu)$, allowing one to determine the large $M$ limit faster when $\mu$ is small; furthermore the genotypic diversity increases from $O(\ln M)$ to $O(M)$.

- 2. Génétique Végétale (GV), CNRS : UMR8120 – Institut national de la recherche agronomique (INRA) : UMR0320 – Université Paris XI - Paris Sud – Institut National Agronomique Paris-Grignon
- 3. Dipartimento di Scienze Fisiche, Università degli studi di Napoli Federico II

**1**)## Preferential interaction coefficient for nucleic acids and other cylindrical poly-ions

### E. Trizac

^{1, 2}, G. Tellez^{3}#### Macromolecules

**40**(2007) 1305-1310The thermodynamics of nucleic acid processes is heavily affected by the electric double-layer of micro-ions around the polyions. We focus here on the Coulombic contribution to the salt-polyelectrolyte preferential interaction (Donnan) coefficient and we report extremely accurate analytical expressions valid in the range of low salt concentration (when polyion radius is smaller than the Debye length). The analysis is performed at Poisson-Boltzmann level, in cylindrical geometry, with emphasis on highly charged poly-ions (beyond ``counter-ion condensation''). The results hold for any electrolyte of the form $z_-$:$z_+$. We also obtain a remarkably accurate expression for the electric potential in the vicinity of the poly-ion.

- 2. Center for Theoretical Biological Physics (CTBP), University of San Diego
- 3. Departamento de Fisica, Universidad de Los Andes

**1**)## Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics

### P. Di Francesco

^{1}, Paul Zinn-Justin^{2}#### Journal of statistical mechanics-theory and experiment (2007) P12009

We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley--Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of Cyclically Symmetric Transpose Complement Plane Partitions and related combinatorial objects.

- 1. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT
- 2. Laboratoire de Physique Théorique et Hautes Energies (LPTHE), CNRS : UMR7589 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot

**12**)## Quasi-elastic solutions to the nonlinear Boltzmann equation for dissipative gases

### Barrat, A., Trizac, E., Ernst, M.H.

#### Journal of Physics A 40 (2007) 4057-4073

Details## Random patterns generated by random permutations of natural numbers

### G. Oshanin

^{1, 2}, R. Voituriez^{1}, S. Nechaev^{3}, O. Vasilyev^{2}, Florent Hivert^{4}#### The European Physical Journal Special Topics

**143**(2007) 143-157We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time $n$, whose moves to the right or to the left are induced by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site $X$ at time $n$, obtain the probability measure of different excursions and define the asymptotic distribution of the number of 'U-turns' of the trajectories - permutation 'peaks' and 'through'. In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers $1,2,3, >...,L$ on sites of a 1d or 2d square lattices containing $L$ sites. We calculate the distribution function of the number of local 'peaks' - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss some surprising collective behavior emerging in this model.

- 1. Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), CNRS : UMR7600 – Université Paris VI - Pierre et Marie Curie
- 2. Department of Inhomogeneous Condensed Matter Theory, Max-Planck-Institut
- 4. Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes (LITIS), Institut National des Sciences Appliquées (INSA) - Rouen – Université du Havre – Université de Rouen : EA4108

**4**)## Random wavefunctions and percolation

### Bogomolny, E., Schmit, C.

#### Journal of Physics A40 (2007) 14033-14043

## Relation between directed polymers in random media and random bond dimer models

### Ying Jiang

^{1}, Thorsten Emig^{2}#### Physical Review B

**75**(2007) 134413We reassess the relation between classical lattice dimer models and the continuum elastic description of a lattice of fluctuating polymers. In the absence of randomness we determine the density and line tension of the polymers in terms of the bond weights of hard-core dimers on the square and the hexagonal lattice. For the latter, we demonstrate the equivalence of the canonical ensemble for the dimer model and the grand-canonical description for polymers by performing explicitly the continuum limit. Using this equivalence for the random bond dimer model on a square lattice, we resolve a previously observed discrepancy between numerical results for the random dimer model and a replica approach for polymers in random media. Further potential applications of the equivalence are briefly discussed.

- 1. Department de Physique, Université de Fribourg

## Riemann Zeta Function and Quantum Chaos

### Bogomolny, E.

#### Progress of Theoretical Physics Supplement 166 (2007) 19-36

## Risk Minimization through Portfolio Replication

### Stefano Ciliberti

^{1, 2}, Marc Mezard^{1}#### European Physical Journal B

**57**, 2 (2007) 175-180We use a replica approach to deal with portfolio optimization problems. A given risk measure is minimized using empirical estimates of asset values correlations. We study the phase transition which happens when the time series is too short with respect to the size of the portfolio. We also study the noise sensitivity of portfolio allocation when this transition is approached. We consider explicitely the cases where the absolute deviation and the conditional value-at-risk are chosen as a risk measure. We show how the replica method can study a wide range of risk measures, and deal with various types of time series correlations, including realistic ones with volatility clustering.

- 2. Science & Finance, Capital Fund Management

**3**)## Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology

### Ciliberti, S., Martin, O.C., Wagner, A.

#### PLOS Computational Biology (2007) 3 (2) : e15

## Role of conformational entropy in force-induced bio-polymer unfolding

### Sanjay Kumar

^{1}, Iwan Jensen^{2}, Jesper L. Jacobsen^{3}, Anthony J. Guttmann^{2}#### Physical Review Letters

**98**(2007) 128101A statistical mechanical description of flexible and semi-flexible polymer chains in a poor solvent is developed in the constant force and constant distance ensembles. We predict the existence of many intermediate states at low temperatures stabilized by the force. A unified response to pulling and compressing forces has been obtained in the constant distance ensemble. We show the signature of a cross-over length which increases linearly with the chain length. Below this cross-over length, the critical force of unfolding decreases with temperature, while above, it increases with temperature. For stiff chains, we report for the first time 'saw-tooth' like behavior in the force-extension curves which has been seen earlier in the case of protein unfolding.

- 1. Department of Physics, Banaras Hindu University
- 2. ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne

## Sinusoidal swinging dynamics of the telomere repair and cell growth activation functions of telomerase in rat liver cancer cells

### Wolfrom, C., Martin, O.C., Laurent, M., Deschatrette, J.

#### FEBS Letters581 (2007) 125-130

## SLE description of the nodal lines of random wave functions

### E. Bogomolny

^{1}, R. Dubertrand^{1}, C. Schmit^{1}#### Journal of Physics A: Mathematical and General

**40**(2007) 381-395The nodal lines of random wave functions are investigated. We demonstrate numerically that they are well approximated by the so-called SLE_6 curves which describe the continuum limit of the percolation cluster boundaries. This result gives an additional support to the recent conjecture that the nodal domains of random (and chaotic) wave functions in the semi classical limit are adequately described by the critical percolation theory. It is also shown that using the dipolar variant of SLE reduces significantly finite size effects.

**17**)## Soliton-soliton scattering in dipolar Bose-Einstein condensates

### R. Nath

^{1}, P. Pedri^{2}, L. M.N.B.F. Santos^{1}#### Physical Review A: Atomic, Molecular and Optical Physics

**76**(2007) 013606We analyze the scattering of bright solitons in dipolar Bose-Einstein condensates placed in unconnected layers. Whereas for short-range interactions unconnected layers are independent, a remarkable consequence of the dipole interaction is the appearance of novel nonlocal interlayer effects. In particular, we show that the interlayer interaction leads to an effective molecular potential between disconnected solitons, inducing a complex scattering physics between them, which includes inelastic fusion into soliton-molecules, and strong symmetric and asymmetric inelastic resonances. In addition, a fundamentally new 2D scattering scenario in matter-wave solitons is possible, in which inelastic spiraling occurs, resembling phenomena in photorrefractive materials. Finally, we consider the scattering of unconnected 1D solitons and discuss the feasibility in current on going experiments.

- 1. Institut fur Theoretische Physik, Leibniz Universität Hannover

## Statistical Mechanics of the Hyper Vertex Cover Problem

### M. Mézard

^{1}, M. Tarzia^{1}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 041124We introduce and study a new optimization problem called Hyper Vertex Cover. This problem is a generalization of the standard vertex cover to hypergraphs: one seeks a configuration of particles with minimal density such that every hyperedge of the hypergraph contains at least one particle. It can also be used in important practical tasks, such as the Group Testing procedures where one wants to detect defective items in a large group by pool testing. Using a Statistical Mechanics approach based on the cavity method, we study the phase diagram of the HVC problem, in the case of random regualr hypergraphs. Depending on the values of the variables and tests degrees different situations can occur: The HVC problem can be either in a replica symmetric phase, or in a one-step replica symmetry breaking one. In these two cases, we give explicit results on the minimal density of particles, and the structure of the phase space. These problems are thus in some sense simpler than the original vertex cover problem, where the need for a full replica symmetry breaking has prevented the derivation of exact results so far. Finally, we show that decimation procedures based on the belief propagation and the survey propagation algorithms provide very efficient strategies to solve large individual instances of the hyper vertex cover problem.

**7**)## Statistical properties of single-file diffusion front

### Sanjib Sabhapandit

^{1}#### Journal of Statistical Mechanics: Theory and Experiment (2007) L05002

Statistical properties of the front of a semi-infinite system of single-file diffusion (one dimensional system where particles cannot pass each other, but in-between collisions each one independently follow diffusive motion) are investigated. Exact as well as asymptotic results are provided for the probability density function of (a) the front-position, (b) the maximum of the front-positions, and (c) the first-passage time to a given position. The asymptotic laws for the front-position and the maximum front-position are found to be governed by the Fisher-Tippett-Gumbel extreme value statistics. The asymptotic properties of the first-passage time is dominated by a stretched-exponential tail in the distribution. The farness of the front with the rest of the system is investigated by considering (i) the gap from the front to the closest particle, and (ii) the density profile with respect to the front-position, and analytical results are provided for late time behaviors.

**1**)## Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation

### Schehr, G., Majumdar, S.N.

#### Physical Review Letters99 (2007) 060603

Details## Superfluidity versus Anderson localization in a dilute Bose gas

### T. Paul

^{1}, P. Schlagheck^{2}, P. Leboeuf^{1}, N. Pavloff^{1}#### Physical Review Letters

**98**(2007) 210602We consider the motion of a quasi one dimensional beam of Bose-Einstein condensed particles in a disordered region of finite extent. Interaction effects lead to the appearance of two distinct regions of stationary flow. One is subsonic and corresponds to superfluid motion. The other one is supersonic, dissipative and shows Anderson localization. We compute analytically the interaction-dependent localization length. We also explain the disappearance of the supersonic stationary flow for large disordered samples.

- 2. Institut für Theoretische Physik, Universitat Regensburg

**59**)## Tagged Particle Correlations in the Asymmetric Simple Exclusion Process: Finite Size Effects

### Shamik Gupta

^{1}, Satya N. Majumdar^{2}, Claude Godrèche^{3}, Mustansir Barma^{1}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 021112We study finite size effects in the variance of the displacement of a tagged particle in the stationary state of the Asymmetric Simple Exclusion Process (ASEP) on a ring of size $L$. The process involves hard core particles undergoing stochastic driven dynamics on a lattice. The variance of the displacement of the tagged particle, averaged with respect to an initial stationary ensemble and stochastic evolution, grows linearly with time at both small and very large times. We find that at intermediate times, it shows oscillations with a well defined size-dependent period. These oscillations arise from sliding density fluctuations (SDF) in the stationary state with respect to the drift of the tagged particle, the density fluctuations being transported through the system by kinematic waves. In the general context of driven diffusive systems, both the Edwards-Wilkinson (EW) and the Kardar-Parisi-Zhang (KPZ) fixed points are unstable with respect to the SDF fixed point, a flow towards which is generated on adding a gradient term to the EW and the KPZ time-evolution equation. We also study tagged particle correlations for a fixed initial configuration, drawn from the stationary ensemble, following earlier work by van Beijeren. We find that the time dependence of this correlation is determined by the dissipation of the density fluctuations. We show that an exactly solvable linearized model captures the essential qualitative features seen in the finite size effects of the tagged particle correlations in the ASEP. Moreover, this linearized model also provides an exact coarse-grained description of two other microscopic models.

- 1. Department of Theoretical Physics, Tata institute of Fundamental Research
- 3. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT

**17**)## Tetromino tilings and the Tutte polynomial

### Jacobsen, J.L.

#### Journal of Physics A 40(2007) 1439-1446

Details## The first-passage area for drifted Brownian motion and the moments of the Airy distribution

### Michael J. Kearney

^{1}, Satya N. Majumdar^{2}, Richard J. Martin^{3}#### Journal of Physics A Mathematical and Theoretical

**40**(2007) F863An exact expression for the distribution of the area swept out by a drifted Brownian motion till its first-passage time is derived. A study of the asymptotic behaviour confirms earlier conjectures and clarifies their range of validity. The analysis also leads to a simple closed-form solution for the moments of the Airy distribution.

- 1. Faculty of Engineering and Physical Sciences, University of Surrey
- 3. Quantitative Credit Strategy Group, Credit suisse

**8**)## The Phase Diagram of 1-in-3 Satisfiability Problem

### Jack Raymond

^{1}, Andrea Sportiello^{2}, Lenka Zdeborová^{3}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**76**(2007) 011101We study the typical case properties of the 1-in-3 satisfiability problem, the boolean satisfaction problem where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause. Random 1-in-3 Satisfiability and Exact 3-Cover are special cases of this ensemble. We interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard, in some interval of connectivities. We derive several rigorous results in the first region, and develop the one-step--replica-symmetry-breaking cavity analysis in the second one. We discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase, and other structural properties of the phase diagram, in light of cavity method results.

- 1. NCRG, Aston University
- 2. Università degli Studi di Milano, Università degli studi di Milano

**13**)## The renormalized jellium model for spherical and cylindrical colloids

### Salete Pianegonda

^{1, 2}, Emmanuel Trizac^{1}, Yan Levin^{2}#### Journal of Chemical Physics

**126**(2007) 014702Starting from a mean-field description for a dispersion of highly charged spherical or (parallel) rod-like colloids, we introduce the simplification of a homogeneous background to include the contribution of other polyions to the static field created by a tagged polyion. The charge of this background is self-consistently renormalized to coincide with the polyion effective charge, the latter quantity thereby exhibiting a non-trivial density dependence, which directly enters into the equation of state through a simple analytical expression. The good agreement observed between the pressure calculated using the renormalized jellium and Monte Carlo simulations confirms the relevance of the {renormalized} jellium model for theoretical and experimental purposes and provides an alternative to the Poisson-Boltzmann cell model since it is free of some of the intrinsic limitations of this approach.

- 2. Instituto de Física, Universidade Federal do Rio de Janeiro

**20**)## The Rotor Model with spectral parameters and enumerations of Alternating Sign Matrices

### Luigi Cantini

^{1}#### Journal of statistical mechanics-theory and experiment (2007) P08012

In this paper we study the Rotor Model of Martins and Nienhuis. After introducing spectral parameters, a combined use of integrability, polynomiality of the ground state wave function and a mapping into the fully-packed O(1)-model allows us to determine the sum rule and a family of maximally nested components for different boundary conditions. We see in this way the appearance of 3-enumerations of Alternating Sign Matrices.

## Universal Extremal Statistics in a Freely Expanding Jepsen Gas

### Ioana Bena

^{1}, Satya N. Majumdar^{2}#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

**75**(2007) 051103We study the extremal dynamics emerging in an out-of-equilibrium one-dimensional Jepsen gas of $(N+1)$ hard-point particles. The particles undergo binary elastic collisions, but move ballistically in-between collisions. The gas is initally uniformly distributed in a box $[-L,0]$ with the 'leader' (or the rightmost particle) at X=0, and a random positive velocity, independently drawn from a distribution $\phi(V)$, is assigned to each particle. The gas expands freely at subsequent times. We compute analytically the distribution of the leader's velocity at time $t$, and also the mean and the variance of the number of collisions that are undergone by the leader up to time $t$. We show that in the thermodynamic limit and at fixed time $t\gg 1$ (the so-called 'growing regime'), when interactions are strongly manifest, the velocity distribution exhibits universal scaling behavior of only three possible varieties, depending on the tail of $\phi(V)$. The associated scaling functions are novel and different from the usual extreme-value distributions of uncorrelated random variables. In this growing regime the mean and the variance of the number of collisions of the leader up to time $t$ increase logarithmically with $t$, with universal prefactors that are computed exactly. The implications of our results in the context of biological evolution modeling are pointed out.

- 1. Département de Physique Théorique, University of Geneva

**10**)## Universal width distributions in non-Markovian Gaussian processes

### Raoul Santachiara

^{1, 2}, Alberto Rosso^{3}, Werner Krauth^{4}#### Journal of Statistical Mechanics: Theory and Experiment (2007) P02009

We study the influence of boundary conditions on self-affine random functions u(t) in the interval t/L \in [0,1], with independent Gaussian Fourier modes of variance ~ 1/q^{\alpha}. We consider the probability distribution of the mean square width of u(t) taken over the whole interval or in a window t/L \in [x, x+\delta]. Its characteristic function can be expressed in terms of the spectrum of an infinite matrix. This distribution strongly depends on the boundary conditions of u(t) for finite \delta, but we show that it is universal (independent of boundary conditions) in the small-window limit. We compute it directly for all values of \alpha, using, for \alpha<3, an asymptotic expansion formula that we derive. For \alpha > 3, the limiting width distribution is independent of \alpha. It corresponds to an infinite matrix with a single non-zero eigenvalue. We give the exact expression for the width distribution in this case. Our analysis facilitates the estimation of the roughness exponent from experimental data, in cases where the standard extrapolation method cannot be used

- 1. Laboratoire de Physique Théorique (LPTH), CNRS : UMR7085 – Université Louis Pasteur - Strasbourg I
- 2. Laboratoire de Physique Théorique de l'ENS (LPTENS), CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris
- 4. Laboratoire de Physique Statistique de l'ENS (LPS), CNRS : UMR8550 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot – Ecole Normale Supérieure de Paris - ENS Paris

**6**)## Archive ouverte HAL – A congruence index for testing topological similarity between trees

### Damien M VienneTatiana Giraud

^{1}Olivier C Martin^{2, 3, 4, 5}*Bioinformatics*, Oxford University Press (OUP), 2007, 23 (23), pp.3119--3124MOTIVATION: Phylogenetic trees are omnipresent in evolutionary biology and the comparison of trees plays a central role there. Tree congruence statistics are based on the null hypothesis that two given trees are not more congruent (topologically similar) than expected by chance. Usually, one searches for the most parsimonious evolutionary scenario relating two trees and then one tests the null hypothesis by generating a high number of random trees and comparing these to the one between the observed trees. However, this approach requires a lot of computational work (human and machine) and the results depend on the evolutionary assumptions made. RESULTS: We propose an index, I(cong), for testing the topological congruence between trees with any number of leaves, based on maximum agreement subtrees (MAST). This index is straightforward, simple to use, does not rely on parametrizing the likelihood of evolutionary events, and provides an associated confidence level. AVAILABILITY: A web site has been created that allows rapid and easy online computation of this index and of the associated P-value at http://www.ese.u-psud.fr/bases/upresa/pages/devienne/index.html

- 1. ESE - Ecologie Systématique et Evolution
- 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. BIOSP - Biostatistique et Processus Spatiaux
- 5. IS2 - Statistical Inference for Industry and Health

## Sex-Specific Crossover Distributions and Variations in Interference Level along Arabidopsis thaliana Chromosome 4

### Jan Drouaud

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. <10.1371/journal.pgen.0030106.eor>In many species, sex-related differences in crossover (CO) rates have been described at chromosomal and regional levels. In this study, we determined the CO distribution along the entire Arabidopsis thaliana Chromosome 4 (18 Mb) in male and female meiosis, using high density genetic maps built on large backcross populations (44 markers, .1,300 plants). We observed dramatic differences between male and female map lengths that were calculated as 88 cM and 52 cM, respectively. This difference is remarkably parallel to that between the total synaptonemal complex lengths measured in male and female meiocytes by immunolabeling of ZYP1 (a component of the synaptonemal complex). Moreover, CO landscapes were clearly different: in particular, at both ends of the map, male CO rates were higher (up to 4-fold the mean value), whereas female CO rates were equal or even below the chromosomal average. This unique material gave us the opportunity to perform a detailed analysis of CO interference on Chromosome 4 in male and female meiosis. The number of COs per chromosome and the distances between them clearly departs from randomness. Strikingly, the interference level (measured by coincidence) varied significantly along the chromosome in male meiosis and was correlated to the physical distance between COs. The significance of this finding on the relevance of current CO interference models is discussed.

- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE - Génétique Quantitative et Evolution (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

## A congruence index for testing topological similarity between trees

### Damien M VienneTatiana Giraud

^{1}Olivier C Martin^{2, 3, 4, 5}*Bioinformatics*, Oxford University Press (OUP), 2007, 23 (23), pp.3119--3124- 1. ESE - Ecologie Systématique et Evolution
- 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. BIOSP - Biostatistique et Processus Spatiaux
- 5. IS2 - Statistical Inference for Industry and Health

## Archive ouverte HAL – Sex-Specific Crossover Distributions and Variations in Interference Level along Arabidopsis thaliana Chromosome 4

### Jan Drouaud

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}#### Jan Drouaud, Raphaël Mercier, Liudmila Chelysheva, Aurélie Bérard, Matthieu Falque, et al.. Sex-Specific Crossover Distributions and Variations in Interference Level along Arabidopsis thaliana Chromosome 4.

*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩In many species, sex-related differences in crossover (CO) rates have been described at chromosomal and regional levels. In this study, we determined the CO distribution along the entire Arabidopsis thaliana Chromosome 4 (18 Mb) in male and female meiosis, using high density genetic maps built on large backcross populations (44 markers, .1,300 plants). We observed dramatic differences between male and female map lengths that were calculated as 88 cM and 52 cM, respectively. This difference is remarkably parallel to that between the total synaptonemal complex lengths measured in male and female meiocytes by immunolabeling of ZYP1 (a component of the synaptonemal complex). Moreover, CO landscapes were clearly different: in particular, at both ends of the map, male CO rates were higher (up to 4-fold the mean value), whereas female CO rates were equal or even below the chromosomal average. This unique material gave us the opportunity to perform a detailed analysis of CO interference on Chromosome 4 in male and female meiosis. The number of COs per chromosome and the distances between them clearly departs from randomness. Strikingly, the interference level (measured by coincidence) varied significantly along the chromosome in male meiosis and was correlated to the physical distance between COs. The significance of this finding on the relevance of current CO interference models is discussed.

- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

## Archive ouverte HAL – Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}#### S. Ciliberti, O. Martin, A. Wagner. Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology.

*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩The topology of cellular circuits (the who-interacts-with-whom) is key to understand their robustness to both mutations and noise. The reason is that many biochemical parameters driving circuit behavior vary extensively and are thus not fine-tuned. Existing work in this area asks to what extent the function of any one given circuit is robust. But is high robustness truly remarkable, or would it be expected for many circuits of similar topology? And how can high robustness come about through gradual Darwinian evolution that changes circuit topology gradually, one interaction at a time? We here ask these questions for a model of transcriptional regulation networks, in which we explore millions of different network topologies. Robustness to mutations and noise are correlated in these networks. They show a skewed distribution, with a very small number of networks being vastly more robust than the rest. All networks that attain a given gene expression state can be organized into a graph whose nodes are networks that differ in their topology. Remarkably, this graph is connected and can be easily traversed by gradual changes of network topologies. Thus, robustness is an evolvable property. This connectedness and evolvability of robust networks may be a general organizational principle of biological networks. In addition, it exists also for RNA and protein structures, and may thus be a general organizational principle of all biological systems.

- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

## Archive ouverte HAL – Sex-Specific Crossover Distributions and Variations in Interference Level along Arabidopsis thaliana Chromosome 4

### Jan Drouaud

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}#### Jan Drouaud, Raphaël Mercier, Liudmila Chelysheva, Aurélie Bérard, Matthieu Falque, et al.. Sex-Specific Crossover Distributions and Variations in Interference Level along Arabidopsis thaliana Chromosome 4.

*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩In many species, sex-related differences in crossover (CO) rates have been described at chromosomal and regional levels. In this study, we determined the CO distribution along the entire Arabidopsis thaliana Chromosome 4 (18 Mb) in male and female meiosis, using high density genetic maps built on large backcross populations (44 markers, .1,300 plants). We observed dramatic differences between male and female map lengths that were calculated as 88 cM and 52 cM, respectively. This difference is remarkably parallel to that between the total synaptonemal complex lengths measured in male and female meiocytes by immunolabeling of ZYP1 (a component of the synaptonemal complex). Moreover, CO landscapes were clearly different: in particular, at both ends of the map, male CO rates were higher (up to 4-fold the mean value), whereas female CO rates were equal or even below the chromosomal average. This unique material gave us the opportunity to perform a detailed analysis of CO interference on Chromosome 4 in male and female meiosis. The number of COs per chromosome and the distances between them clearly departs from randomness. Strikingly, the interference level (measured by coincidence) varied significantly along the chromosome in male meiosis and was correlated to the physical distance between COs. The significance of this finding on the relevance of current CO interference models is discussed.

- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

## Archive ouverte HAL – Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}#### S. Ciliberti, O. Martin, A. Wagner. Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology.

*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩The topology of cellular circuits (the who-interacts-with-whom) is key to understand their robustness to both mutations and noise. The reason is that many biochemical parameters driving circuit behavior vary extensively and are thus not fine-tuned. Existing work in this area asks to what extent the function of any one given circuit is robust. But is high robustness truly remarkable, or would it be expected for many circuits of similar topology? And how can high robustness come about through gradual Darwinian evolution that changes circuit topology gradually, one interaction at a time? We here ask these questions for a model of transcriptional regulation networks, in which we explore millions of different network topologies. Robustness to mutations and noise are correlated in these networks. They show a skewed distribution, with a very small number of networks being vastly more robust than the rest. All networks that attain a given gene expression state can be organized into a graph whose nodes are networks that differ in their topology. Remarkably, this graph is connected and can be easily traversed by gradual changes of network topologies. Thus, robustness is an evolvable property. This connectedness and evolvability of robust networks may be a general organizational principle of biological networks. In addition, it exists also for RNA and protein structures, and may thus be a general organizational principle of all biological systems.

- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

## Archive ouverte HAL – Sex-Specific Crossover Distributions and Variations in Interference Level along Arabidopsis thaliana Chromosome 4

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}#### Jan Drouaud, Raphaël Mercier, Liudmila Chelysheva, Aurélie Bérard, Matthieu Falque, et al.. Sex-Specific Crossover Distributions and Variations in Interference Level along Arabidopsis thaliana Chromosome 4.

*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

## Archive ouverte HAL – Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}#### S. Ciliberti, O. Martin, A. Wagner. Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology.

*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩The topology of cellular circuits (the who-interacts-with-whom) is key to understand their robustness to both mutations and noise. The reason is that many biochemical parameters driving circuit behavior vary extensively and are thus not fine-tuned. Existing work in this area asks to what extent the function of any one given circuit is robust. But is high robustness truly remarkable, or would it be expected for many circuits of similar topology? And how can high robustness come about through gradual Darwinian evolution that changes circuit topology gradually, one interaction at a time? We here ask these questions for a model of transcriptional regulation networks, in which we explore millions of different network topologies. Robustness to mutations and noise are correlated in these networks. They show a skewed distribution, with a very small number of networks being vastly more robust than the rest. All networks that attain a given gene expression state can be organized into a graph whose nodes are networks that differ in their topology. Remarkably, this graph is connected and can be easily traversed by gradual changes of network topologies. Thus, robustness is an evolvable property. This connectedness and evolvability of robust networks may be a general organizational principle of biological networks. In addition, it exists also for RNA and protein structures, and may thus be a general organizational principle of all biological systems.

- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry

^{1}Raphaël Mercier^{1}Liudmila Chelysheva^{1}Aurélie Bérard^{2}Matthieu Falque^{3}Olivier Martin^{3, 4}Vanessa Zanni^{5}Dominique Brunel^{2}Christine Mezard^{1}*PLoS Genetics*, Public Library of Science, 2007, 3 (6), pp.1096-1107. ⟨10.1371/journal.pgen.0030106.eor⟩. ⟨hal-01203942⟩- 1. IJPB - Institut Jean-Pierre Bourgin
- 2. UR Etude du Polymorphisme des Génomes végétaux
- 3. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 4. Laboratoire de Physique Théorique et Modèles Statistiques
- 5. UR254 - Unité de Recherche en Génétique et Amélioration des Plantes

### S. Ciliberti

^{1}O. Martin^{1, 2}A. Wagner^{3}*PLoS Computational Biology*, Public Library of Science, 2007, 3 (2) (2), pp.e15. ⟨10.1371/journal.pcbi.0030015⟩. ⟨hal-00143879⟩- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. GQE-Le Moulon - Génétique Quantitative et Evolution - Le Moulon (Génétique Végétale)
- 3. Department of Biochemistry