# LPTMS Publications

Archives :

• ## A Bijection between classes of Fully Packed Loops and Plane Partitions

### P. Di Francesco 1, Paul Zinn-Justin 2, J. -B. Zuber 1

#### Electronic Journal of Combinatories 11 (2004) R64

It has recently been observed empirically that the number of FPL configurations with 3 sets of a, b and c nested arches equals the number of plane partitions in a box of size a x b x c. In this note, this result is proved by constructing explicitly the bijection between these FPL and plane partitions.

• 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

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• ## Algebraic Bethe Ansatz for the FPL^2 model

### Jesper-Lykke Jacobsen 1, Paul Zinn-Justin 1

#### Journal of Physics A 37 (2004) 7213-7225

An exact solution of the model of fully packed loops of two colors on a square lattice has recently been proposed by Dei Cont and Nienhuis using the coordinate Bethe Ansatz approach. We point out here a simpler alternative, in which the transfer matrix is directly identified as a product of R-matrices; this allows to apply the (nested) algebraic Bethe Ansatz, which leads to the same Bethe equations. We comment on some of the applications of this result.

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

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• ## An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution

### Satya Majumdar 1, Sergei K. Nechaev 2, 3

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 69 (2004) 011103

We compute exactly the asymptotic distribution of scaled height in a (1+1)--dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in $(1+1)$ dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all share the same universal Tracy-Widom distribution for the suitably scaled height variables.

• 1. Laboratoire de Physique Théorique - IRSAMC (LPT), CNRS : UMR5152 – Université Paul Sabatier - Toulouse III
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. Landau Institute for Theoretical Physics, Landau Institute for Theoretical Physics

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• ## Classical intermittency and quantum Anderson transition

### Antonio M. Garcia-Garcia 1

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 69 (2004) 066216

We investigate the quantum properties of 1D quantum systems whose classical counterpart presents intermittency. The spectral correlations are expressed in terms of the eigenvalues of an anomalous diffusion operator by using recent semiclassical techniques. For certain values of the parameters the spectral properties of our model show similarities with those of a disordered system at the Anderson transition. In Hamiltonian systems, intermittency is closely related to the presence of cantori in the classical phase space. We suggest, based on this relation, that our findings may be relevant for the description of the spectral correlations of (non-KAM) Hamiltonians with a classical phase space filled by cantori. Finally we discuss the extension of our results to higher dimensions and their relation to Anderson models with long range hopping.

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

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• ## Conformal field theory of the Flory model of polymer melting

### Jesper Lykke Jacobsen 1, Jane' Kondev 2

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 69 (2004) 066108

We study the scaling limit of a fully packed loop model in two dimensions, where the loops are endowed with a bending rigidity. The scaling limit is described by a three-parameter family of conformal field theories, which we characterize via its Coulomb-gas representation. One choice for two of the three parameters reproduces the critical line of the exactly solvable six-vertex model, while another corresponds to the Flory model of polymer melting. Exact central charge and critical exponents are calculated for polymer melting in two dimensions. Contrary to predictions from mean-field theory we show that polymer melting, as described by the Flory model, is continuous. We test our field theoretical results against numerical transfer matrix calculations.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Physics Department, MS057, Brandeis University

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• ## Construction of the factorized steady state distribution in models of mass transport

### Royce K.P. Zia 1, Martin R. Evans 2, Satya N. Majumdar 3

#### Journal of Statistical Mechanics: Theory and Experiment 1 (2004) L10001

For a class of one-dimensional mass transport models we present a simple and direct test on the chipping functions, which define the probabilities for mass to be transferred to neighbouring sites, to determine whether the stationary distribution is factorized. In cases where the answer is affirmative, we provide an explicit method for constructing the single-site weight function. As an illustration of the power of this approach, previously known results on the Zero-range process and Asymmetric random average process are recovered in a few lines. We also construct new models, namely a generalized Zero-range process and a binomial chipping model, which have factorized steady states.

• 1. Department of Physics and Center for Stochastic Processes in Science and Engineering, Virgina Tech
• 2. School of Physics, University of Edinburgh
• 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

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• ## Cooperativity in two-state protein folding kinetics

### Thomas R. Weikl 1, 2, Matteo Palassini 1, 3, Ken A. Dill 1

#### Protein Science 13 (2004) 822-829

We present a solvable model that predicts the folding kinetics of two-state proteins from their native structures. The model is based on conditional chain entropies. It assumes that folding processes are dominated by small-loop closure events that can be inferred from native structures. For CI2, the src SH3 domain, TNfn3, and protein L, the model reproduces two-state kinetics, and it predicts well the average Phi-values for secondary structures. The barrier to folding is the formation of predominantly local structures such as helices and hairpins, which are needed to bring nonlocal pairs of amino acids into contact.

• 1. Department of Pharmaceutical Chemistry, University of California, San Francisco
• 2. Max-Planck-institut für Kolloid - und Grenzflächenforschung, Max-Planck-Institut
• 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

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• ## Effect of a magnetic flux on the critical behavior of a system with long range hopping

### Antonio M. Garcia-Garcia 1

#### Physical Review B 69 (2004) 245121

We study the effect of a magnetic flux in a 1D disordered wire with long range hopping. It is shown that this model is at the metal-insulator transition (MIT) for all disorder values and the spectral correlations are given by critical statistics. In the weak disorder regime a smooth transition between orthogonal and unitary symmetry is observed as the flux strength increases. By contrast, in the strong disorder regime the spectral correlations are almost flux independent. It is also conjectured that the two level correlation function for arbitrary flux is given by the dynamical density-density correlations of the Calogero-Sutherland (CS) model at finite temperature. Finally we describe the classical dynamics of the model and its relevance to quantum chaos.

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

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• ## Electron-Electron Interactions in Isolated and Realistic Quantum Dots: A Density Functional Theory Study

### Hong Jiang 1, 2, 3, Denis Ullmo 2, 4, Weitao Yang 1, Harold U. Baranger 2

#### Physical Review B 69 (2004) 235326

We use Kohn-Sham spin-density-functional theory to study the statistics of ground-state spin and the spacing between conductance peaks in the Coulomb blockade regime for both 2D isolated and realistic quantum dots. We make a systematic investigation of the effects of electron-electron interaction strength and electron number on both the peak spacing and spin distributions. A direct comparison between the distributions from isolated and realistic dots shows that, despite the difference in the boundary conditions and confining potential, the statistical properties are qualitatively the same. Strong even/odd pairing in the peak spacing distribution is observed only in the weak e-e interaction regime and vanishes for moderate interactions. The probability of high spin ground states increases for stronger e-e interaction and seems to saturate around $r_s \sim 4$. The saturated value is larger than previous theoretical predictions. Both spin and conductance peak spacing distributions show substantial variation as the electron number increases, not saturating until $N \sim 150$. To interpret our numerical results, we analyze the spin distribution in the even $N$ case using a simple two-level model.

• 1. Department of Chemistry, Duke University
• 2. Duke Physics, Duke University
• 3. College of Chemistry and Molecular Engineering, Peking University
• 4. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

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• ## Exact Maximal Height Distribution of Fluctuating Interfaces

### Satya Majumdar 1, Alain Comtet 2, 3

#### Physical Review Letters 92 (2004) 225501

We present an exact solution for the distribution P(h_m,L) of the maximal height h_m (measured with respect to the average spatial height) in the steady state of a fluctuating Edwards-Wilkinson interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L where the function f(x) is the Airy distribution function that describes the probability density of the area under a Brownian excursion over a unit interval. For the free boundary case, the same scaling holds but the scaling function is different from that of the periodic case. Numerical simulations are in excellent agreement with our analytical results. Our results provide an exactly solvable case for the distribution of extremum of a set of strongly correlated random variables.

• 1. Laboratoire de Physique Théorique - IRSAMC (LPT), CNRS : UMR5152 – Université Paul Sabatier - Toulouse III
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. IHP, Institut Henri Poincaré

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• ## Exact Potts Model Partition Functions for Strips of the Triangular Lattice

### Shu-Chiuan Chang 1, 2, Jesper-Lykke Jacobsen 3, Jesus Salas 4, 5, Robert Shrock 1

#### Journal of Statistical Physics 114 (2004) 763-823

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}} c_{Z,G,j}(\lambda_{Z,G,j})^{m-1}. We give general formulas for N_{Z,G,j} and its specialization to v=-1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus {\cal B}, arising as the accumulation set of partition function zeros as m\to\infty, in the q plane for fixed v and in the v plane for fixed q.

• 1. C. N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook
• 2. Department of Applied Physics, Faculty of Science, Tokyo University of Science
• 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 4. Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza

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• ## Factorised Steady States in Mass Transport Models

### Martin R. Evans 1, Satya Majumdar 2, 3, Royce K.P. Zia 4, 5

#### Journal of Physics A 37 (2004) L275-L280

We study a class of mass transport models where mass is transported in a preferred direction around a one-dimensional periodic lattice and is globally conserved. The model encompasses both discrete and continuous masses and parallel and random sequential dynamics and includes models such as the Zero-range process and Asymmetric random average process as special cases. We derive a necessary and sufficient condition for the steady state to factorise, which takes a rather simple form.

• 1. School of Physics, University of Edinburgh
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. Laboratoire de Physique Théorique - IRSAMC (LPT), CNRS : UMR5152 – Université Paul Sabatier - Toulouse III
• 4. Department of Physics and Center for Stochastic Processes in Science and Engineering, Virginia Tech
• 5. FB-Physik, Universität Duisburg-Essen

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• ## Family of generalized random matrix ensembles

### A. C. Bertuola 1, O. Bohigas 2, M. P. Pato 1

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 70 (2004) 065102

Using the Generalized Maximium Entropy Principle based on the nonextensive q entropy a new family of random matrix ensembles is generated. This family unifies previous extensions of Random Matrix Theory and gives rise to an orthogonal invariant stable Levy ensemble with new statistical properties. Some of them are analytically derived.

• 1. Instituto de Fisica, Universidade de São Paulo
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

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• ## Fermi-Edge Singularities in the Mesoscopic X-Ray Edge Problem

### Martina Hentschel 1, Denis Ullmo 1, 2, Harold U. Baranger 1

#### Physical Review Letters 93 (2004) 176807

We study the x-ray edge problem for a chaotic quantum dot or nanoparticle displaying mesoscopic fluctuations. In the bulk, x-ray physics is known to produce deviations from the naively expected photoabsorption cross section in the form of a peaked or rounded edge. For a coherent system with chaotic dynamics, we find substantial changes and in particular that a photoabsorption cross section showing a rounded edge in the bulk will change to a slightly peaked edge on average as the system size is reduced to a mesoscopic (coherent) scale.

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

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• ## Fermionic field theory for trees and forests

### Sergio Caracciolo 1, Jesper-Lykke Jacobsen 2, Hubert Saleur 3, 4, Alan D. Sokal 5, Andrea Sportiello 1

#### Physical Review Letters 93 (2004) 080601

We prove a generalization of Kirchhoff's matrix-tree theorem in which a large class of combinatorial objects are represented by non-Gaussian Grassmann integrals. As a special case, we show that unrooted spanning forests, which arise as a q \to 0 limit of the Potts model, can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. We show that this latter model can be mapped, to all orders in perturbation theory, onto the N-vector model at N=-1 or, equivalently, onto the sigma-model taking values in the unit supersphere in R^{1|2}. It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free.

• 1. Dipartimento di Fisica (Milano), Università degli studi di Milano
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT
• 4. Department of Physics and Astronomy, University of Southern California
• 5. Department of Physics, New York University

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• ## Glass models on Bethe lattices

### Olivier Rivoire 1, Giulio Biroli 2, Olivier C. Martin 1, Marc Mézard 1

#### European Physical Journal B 37 (2004) 55-78

We consider lattice glass models'' in which each site can be occupied by at most one particle, and any particle may have at most l occupied nearest neighbors. Using the cavity method for locally tree-like lattices, we derive the phase diagram, with a particular focus on the vitreous phase and the highest packing limit. We also study the energy landscape via the configurational entropy, and discuss different equilibrium glassy phases. Finally, we show that a kinetic freezing, depending on the particular dynamical rules chosen for the model, can prevent the equilibrium glass transitions.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT

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• ## Glassy phases in Random Heteropolymers with correlated sequences

### Markus Muller 1, Marc Mézard 1, Andrea Montanari 2

#### Journal of Chemical Physics 120 (2004) 11233

We develop a new analytic approach for the study of lattice heteropolymers, and apply it to copolymers with correlated Markovian sequences. According to our analysis, heteropolymers present three different dense phases depending upon the temperature, the nature of the monomer interactions, and the sequence correlations: (i) a liquid phase, (ii) a soft glass'' phase, and (iii) a frozen glass'' phase. The presence of the new intermediate soft glass'' phase is predicted for instance in the case of polyampholytes with sequences that favor the alternation of monomers. Our approach is based on the cavity method, a refined Bethe Peierls approximation adapted to frustrated systems. It amounts to a mean field treatment in which the nearest neighbor correlations, which are crucial in the dense phases of heteropolymers, are handled exactly. This approach is powerful and versatile, it can be improved systematically and generalized to other polymeric systems.

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

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• ## Landau Fermi Liquid Picture of Spin Density Functional Theory: Strutinsky Approach to Quantum Dots

### Denis Ullmo 1, 2, Hong Jiang 2, 3, Weitao Yang 3, Harold U. Baranger 2

#### Physical Review B 70 (2004) 205309

We analyze the ground state energy and spin of quantum dots obtained from spin density functional theory (SDFT) calculations. First, we introduce a Strutinsky-type approximation, in which quantum interference is treated as a correction to a smooth Thomas-Fermi description. For large irregular dots, we find that the second-order Strutinsky expressions have an accuracy of about 5 percent compared to the full SDFT and capture all the qualitative features. Second, we perform a random matrix theory/random plane wave analysis of the Strutinsky SDFT expressions. The results are statistically similar to the SDFT quantum dot statistics. Finally, we note that the second-order Strutinsky approximation provides, in essence, a Landau Fermi liquid picture of spin density functional theory. For instance, the leading term in the spin channel is simply the familiar exchange constant. A direct comparison between SDFT and the perturbation theory derived universal Hamiltonian'' is thus made possible.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Duke Physics, Duke University
• 3. Department of Chemistry, Duke University

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• ## Landau theory of glassy dynamics

### Satya Majumdar 1, 2, Dibyendu Das 3, Jane' Kondev 4, Bulbul Chakraborty 4

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 70 (2004) 060501

An exact solution of a Landau model of an order-disorder transition with activated critical dynamics is presented. The model describes a funnel-shaped topography of the order parameter space in which the number of energy lowering trajectories rapidly diminishes as the ordered ground-state is approached. This leads to an asymmetry in the effective transition rates which results in a non-exponential relaxation of the order-parameter fluctuations and a Vogel-Fulcher-Tammann divergence of the relaxation times, typical of a glass transition. We argue that the Landau model provides a general framework for studying glassy dynamics in a variety of systems.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Laboratoire de Physique Théorique - IRSAMC (LPT), CNRS : UMR5152 – Université Paul Sabatier - Toulouse III
• 3. Department of Physics, Indian Institute of Technology Bombay
• 4. Martin Fisher School of Physics, Brandeis University

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• ## Large deviations in spin-glass ground-state energies

### A. Andreanov, Francesca Barbieri, Olivier C. Martin 1

#### European Physical Journal B 41 (2004) 365

The ground-state energy E_0 of a spin glass is an example of an extreme statistic. We consider the large deviations of this energy for a variety of models when the number of spins N goes to infinity. In most cases, the behavior can be understood qualitatively, in particular with the help of semi-analytical results for hierarchical lattices. Particular attention is paid to the Sherrington-Kirkpatrick model; after comparing to the Tracy-Widom distribution which follows from the spherical approximation, we find that the large deviations give rise to non-trivial scaling laws with N.

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

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• ## Long range disorder and Anderson transition in systems with chiral symmetry

### Antonio M. Garcia-Garcia 1, Kazutaka Takahashi 2

#### Nuclear Physics B 700 (2004) 361

We study the spectral properties of a chiral random banded matrix (chRBM) with elements decaying as a power-law ${{\cal H}_{ij}}\sim |i-j|^{-\alpha}$. This model is equivalent to a chiral 1D Anderson Hamiltonian with long range power-law hopping. In the weak disorder limit we obtain explicit nonperturbative analytical results for the density of states and the two-level correlation function by mapping the chRBM onto a nonlinear sigma model. We also put forward, by exploiting the relation between the chRBM at $\alpha=1$ and a generalized chiral random matrix model, an exact expression for the above correlation functions. We give compelling analytical and numerical evidence that for this value the chRBM reproduces all the features of an Anderson transition. Finally we discuss possible applications of our results to QCD.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Theoretische Physik III, Ruhr-Universitaet Bochum, Universität Bochum

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• ## Low-dimensional trapped gases

### D. Petrov 1, 2, Dimitri M. Gangardt 3, 4, Gora V. Shlyapnikov 1, 2, 3, 4

#### Journal de Physique IV Colloque 116 (2004) 5-44

Recent developments in the physics of ultracold gases provide wide possibilities for reducing the dimensionality of space for magnetically or optically trapped atoms. The goal of these lectures is to show that regimes of quantum degeneracy in two-dimensional (2D) and one-dimensional (1D) trapped gases are drastically different from those in three dimensions and to stimulate an interest in low-dimensional systems. Attention is focused on the new physics appearing in currently studied low-dimensional trapped gases and related to finite-size and finite-temperature effects.

• 1. FOM Institute, FOM Institute
• 2. National Research Centre "Kurchatov Institute" (NRC KI), University of Moscow
• 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 4. Laboratoire Kastler Brossel (LKB (Lhomond)), CNRS : UMR8552 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris

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• ## Matrix Integrals and the Generation and Counting of Virtual Tangles and Links

### Paul Zinn-Justin 1, Jean-Bernard Zuber 2

#### Journal of Knot Theory and Its Ramifications 13 (2004) 325-355

Virtual links are generalizations of classical links that can be represented by links embedded in a thickened'' surface $\Sigma\times I$, product of a Riemann surface of genus $h$ with an interval. In this paper, we show that virtual alternating links and tangles are naturally associated with the $1/N^2$ expansion of an integral over $N\times N$ complex matrices. We suggest that it is sufficient to count the equivalence classes of these diagrams modulo ordinary (planar) flypes. To test this hypothesis, we use an algorithm coding the corresponding Feynman diagrams by means of permutations that generates virtual diagrams up to 6 crossings and computes various invariants. Under this hypothesis, we use known results on matrix integrals to get the generating functions of virtual alternating tangles of genus 1 to 5 up to order 10 (i.e.\ 10 real crossings). The asymptotic behavior for $n$ large of the numbers of links and tangles of genus $h$ and with $n$ crossings is also computed for $h=1,2,3$ and conjectured for general $h$.

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

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• ## On metric structure of ultrametric spaces

### Oleg A. Vasilyev 1, Sergei K. Nechaev 1, 2

#### Journal of Physics A 37 (2004) 3783-3804

In our work we have reconsidered the old problem of diffusion at the boundary of ultrametric tree from a 'number theoretic' point of view. Namely, we use the modular functions (in particular, the Dedekind eta-function) to construct the 'continuous' analog of the Cayley tree isometrically embedded in the Poincare upper half-plane. Later we work with this continuous Cayley tree as with a standard function of a complex variable. In the frameworks of our approach the results of Ogielsky and Stein on dynamics on ultrametric spaces are reproduced semi-analytically/semi-numerically. The speculation on the new 'geometrical' interpretation of replica n->0 limit is proposed.

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

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• ## On the Asymptotic Number of Plane Curves and Alternating Knots

### Gilles Schaeffer 1, Paul Zinn-Justin 2

#### Experimental Mathematics 13 (2004) 4

We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of plane curves, a similar conjecture is made for the asymptotic number of prime alternating knots. The rationale leading to these conjectures is given by quantum field theory. Plane curves are viewed as configurations of loops on a random planar lattices, that are in turn interpreted as a model of 2d quantum gravity with matter. The identification of the universality class of this model yields the conjecture. Since approximate counting or sampling planar curves with more than a few dozens of intersections is an open problem, direct confrontation with numerical data yields no convincing indication on the correctness of our conjectures. However, our physical approach yields a more general conjecture about connected systems of curves. We take advantage of this to design an original and feasible numerical test, based on recent perfect samplers for large planar maps. The numerical datas strongly support our identification with a conformal field theory recently described by Read and Saleur.

• 1. Laboratoire d'informatique de l'école polytechnique (LIX), CNRS : UMR7161 – Polytechnique - X
• 2. Laboratoire de Physique Théorique et Hautes Energies (LPTHE), CNRS : UMR7589 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot

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• ## Optimization and Physics: On the satisfiability of random Boolean formulae

### Marc Mézard 1

#### Annales de l'Institut Henri Poincare Physique Theorique 4 (2004) S475-S488

LECTURE GIVEN AT TH2002. Given a set of Boolean variables, and some constraints between them, is it possible to find a configuration of the variables which satisfies all constraints? This problem, which is at the heart of combinatorial optimization and computational complexity theory, is used as a guide to show the convergence between these fields and the statistical physics of disordered systems. New results on satisfiability, both on the theoretical and practical side, can be obtained thanks to the use of physics concepts and methods.

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

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• ## Periodic orbit spectrum in terms of Ruelle–Pollicott resonances

### Patricio Leboeuf 1

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 69 (2004) 026204

Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g. a trajectory p'' returns to its initial conditions after some fixed time tau_p. Our aim is to investigate the spectrum tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle--Pollicott resonances). For large periods, corrections to the well--known exponential growth of the smooth part of the density are obtained. An alternative formula for rho(tau) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random matrix theory and discrete maps are also considered. In particular, a random matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards.

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

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• ## Persistence Exponents and the Statistics of Crossings and Occupation Times for Gaussian Stationary Processes

### George M. C. A. Ehrhardt 1, Satya Majumdar 2, 3, Alan J. Bray 1

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 69 (2004) 016106

We consider the persistence probability, the occupation-time distribution and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian Stationary Processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the 'partial survival\' probability. We then elaborate on the correlator expansion developed in an earlier paper [G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2001)] to calculate discretely sampled persistence exponents of GSPs of known correlator by means of a series expansion in the correlator. We apply this method to the processes d^n x/dt^n=\\eta(t) with n > 2, incorporating an extrapolation of the series to the limit of continuous sampling. We extend the correlator method to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. We apply these general methods to the d^n x/dt^n=\\eta(t) processes for n=1 (random walk), n=2 (random acceleration) and larger n, and to diffusion from random initial conditions in 1-3 dimensions. The results for discrete sampling are extrapolated to the continuum limit where possible.

• 1. Department of Physics and Astronomy, University of Manchester
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. Laboratoire de Physique Théorique - IRSAMC (LPT), CNRS : UMR5152 – Université Paul Sabatier - Toulouse III

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• ## Persistence in nonequilibrium surface growth

### M. Constantin 1, 2, Chandan Dasgupta 1, 3, Punyindu Chatraphorn 1, 4, Satya Majumdar 5, 6, S. Das Sarma 1

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 69 (2004) 061608

Persistence probabilities of the interface height in (1+1)- and (2+1)-dimensional atomistic, solid-on-solid, stochastic models of surface growth are studied using kinetic Monte Carlo simulations, with emphasis on models that belong to the molecular beam epitaxy (MBE) universality class. Both the initial transient and the long-time steady-state regimes are investigated. We show that for growth models in the MBE universality class, the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady-state regimes. For the MBE universality class, the positive and negative persistence exponents in the steady-state are found to be $\theta^S_{+} = 0.66 \pm 0.02$ and $\theta^S_{-} = 0.78 \pm 0.02$, respectively, in (1+1) dimensions, and $\theta^S_{+} = 0.76 \pm 0.02$ and $\theta^S_{-} =0.85 \pm 0.02$, respectively, in (2+1) dimensions. The noise reduction technique is applied on some of the (1+1)-dimensional models in order to obtain accurate values of the persistence exponents. We show analytically that a relation between the steady-state persistence exponent and the dynamic growth exponent, found earlier to be valid for linear models, should be satisfied by the smaller of the two steady-state persistence exponents in the nonlinear models. Our numerical results for the persistence exponents are consistent with this prediction. We also find that the steady-state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state. The dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form.

• 1. Condensed Matter Theory Center, Department of Physics, University of Maryland at College Park
• 2. Materials Research Science and Engineering Center, Department of Physics, University of Maryland at College Park
• 3. Department of Physics, Indian Institute od Science
• 4. Department of Physics, Faculty of Science, Chulalongkorn University
• 5. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 6. Laboratoire de Physique Théorique - IRSAMC (LPT), CNRS : UMR5152 – Université Paul Sabatier - Toulouse III

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• ## Pinning and Sliding of Driven Elastic Systems: from Domain Walls to Charge Density Waves

### Serguei Brazovskii 1, Thomas Nattermann 1

#### Advances In Physics 53 (2004) 177-252

The review is devoted to the theory of collective and it local pinning effects in various disordered non-linear driven systems. Although the emphasis is put on charge and spin density waves and magnetic domain walls, the theory has also applications to flux lines and lattices thereof, dislocation lines, adsorbed mono-layers and related systems. In the first part we focus on the theory of the collective pinning which includes the equilibrium properties of elastic systems with frozen-in disorder as well as the features close to the dynamic depinning transition enforced by an external driving force and at finite temperatures. Thermal fluctuations smear out this transition and allow for a creep motion of the elastic objects even at small forces. An ac-driving force also destroys the sharp transition which is replaced by a velocity hysteresis. The second part is devoted to the local pinning picture and its applications. Inclusion of plastic deformations results in a rich cross-over behavior of the force-velocity relation as well as of the frequency dependence of the dynamic response. The local pinning recovers and exploits new elements of the energy landscape such as termination points of metastable branches or irreversibility of other ones related to generation of topological defects in the course of sliding. It also gives access to the quantum creep described as a tunneling between retarded and advanced configurations.

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

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• ## Quantum Mechanics of a Particle with Two Magnetic Impurities

### Stefan Mashkevich 1, Jan Myrheim 2, Stéphane Ouvry 3

#### Physics Letters A 330 (2004) 41-47

A two-dimensional quantum mechanical system consisting of a particle coupled to two magnetic impurities of different strengths, in a harmonic potential, is considered. Topological boundary conditions at impurity locations imply that the wave functions are linear combinations of two-dimensional harmonics. A number of low-lying states are computed numerically, and the qualitative features of the spectrum are analyzed.

• 1. Physics Department, Taras Shevchenko Kiev National University
• 2. Department of Physics, The Norwegian University of Science and Technology
• 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

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• ## Spectral statistics of a quantum interval-exchange map

### E. Bogomolny 1, C. Schmit 1

#### Physical Review Letters 93 (2004) 254102

Curious spectral properties of an ensemble of random unitary matrices appearing in the quantization of a map p -> p+alpha, q -> q+f(p+alpha) in [Giraud et al. nlin.CD/0403033] are investigated. When alpha=m/n with integer co-prime m,n and matrix dimension N -> infinity is such that mN = 1 or -1 mod n, local spectral statistics of this ensemble tends to the semi-Poisson distribution [Bogomolny et al. Eur. Phys. J. B 19, 121 (2001)] with arbitrary integer or half-integer level repulsion at small distances: R(s)-> s^{beta} when s -> 0 and beta=n-1 or n/2-1 depending on time-reversal symmetry of the map.

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

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• ## Summary for Theory of the Ferroelectric Phase in Organic Conductors in Relation to Experiments

### Serguei Brazovskii 1

#### Journal de Physique IV Colloque 114 (2004) 9

Mysterious high temperature structureless transitions in (TMTTF)2X compounds have been discovered in mid 80's (Coulon, Lawersanne,vet al), but left unexplained and abandoned, together with other warnings from structural effects (Moret, Pouget, et al), with dramatic consequences for the whole field. Recently their nature has been identified as the ferroelectricity (Nad, Monceau, et al) and, more generally, the charge disproportionation (Brown, et al). New phenomena unify an unusual variety of concepts: ferroelectricity of good conductors, structural instability towards Mott- Hubbard state, Wigner crystallization in a dense electronic system, ordered 4kF density wave, richness of physics of solitons, interplay of structural and electronic symmetries. The ferroelectric state gives rise to several types of solitons carrying electronic charge, a noninteger charge, spin or both spin and charge in special cases. They are clearly observed via conductivity, electric and magnetic susceptibilities. Solitons are challenging for optics where they already seem to determine the pseudogap in absorption. Various features also appear, or are expected, from collective electronic and coupled electron-phonon modes. REFERENCES: cond-mat/0012237 cond-mat/0304076 cond-mat/0304483 cond-mat/0306006

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

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• ## Survival in equilibrium step fluctuations

### C. Dasgupta 1, M. Constantin 1, 2, S. Das Sarma 1, Satya N. Majumdar 3

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 69 (2004) 022101

We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability $S(t)$ in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. $S(t)$ is shown to exhibit simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of experimental dynamical STM data on step fluctuations on Al/Si(111) and Ag(111) surfaces.

• 1. Condensed Matter Theory Center, Department of Physics, University of Maryland at College Park
• 2. Materials Research Science and Engineering Center, Department of Physics, University of Maryland at College Park
• 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

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• ## The Phase Diagram of Random Heteropolymers

### Andrea Montanari 1, Markus Muller 2, Marc Mézard 2

#### Physical Review Letters 92 (2004) 185509

We propose a new analytic approach to study the phase diagram of random heteropolymers, based on the cavity method. For copolymers we analyze the nature and phenomenology of the glass transition as a function of sequence correlations. Depending on these correlations, we find that two different scenarios for the glass transition can occur. We show that, beside the much studied possibility of an abrupt freezing transition at low temperature, the system can exhibit, upon cooling, a first transition to a soft glass phase with fully broken replica symmetry and a continuously growing degree of freezing as the temperature is lowered.

• 1. Laboratoire de Physique Théorique de l'ENS (LPTENS), CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud

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• ## The traveling salesman problem, conformal invariance, and dense polymers

### Jesper-Lykke Jacobsen 1, Nicholas Read 2, Hubert Saleur 3, 4

#### Physical Review Letters 93 (2004) 038701

We propose that the statistics of the optimal tour in the planar random Euclidean traveling salesman problem is conformally invariant on large scales. This is exhibited in power-law behavior of the probabilities for the tour to zigzag repeatedly between two regions, and in subleading corrections to the length of the tour. The universality class should be the same as for dense polymers and minimal spanning trees. The conjectures for the length of the tour on a cylinder are tested numerically.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Department of Physics, Yale University
• 3. Department of Physics and Astronomy, University of Southern California
• 4. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT

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• ## Universal correlations of trapped one-dimensional impenetrable bosons

### Dimitri M. Gangardt 1, 2

#### Journal of Physics A 37 (2004) 9335-9356

We calculate the asymptotic behaviour of the one body density matrix of one-dimensional impenetrable bosons in finite size geometries. Our approach is based on a modification of the Replica Method from the theory of disordered systems. We obtain explicit expressions for oscillating terms, similar to fermionic Friedel oscillations. These terms are universal and originate from the strong short-range correlations between bosons in one dimension.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Laboratoire Kastler Brossel (LKB (Lhomond)), CNRS : UMR8552 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris

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• ## Weakly bound dimers of fermionic atoms

### D. Petrov 1, 2, Christophe Salomon 3, Gora V. Shlyapnikov 1, 2, 3

#### Physical Review Letters 93 (2004) 090404

We discuss the behavior of weakly bound bosonic dimers formed in a cold Fermi gas at a large positive scattering length $a$ for the interspecies interaction. We find the exact solution for the dimer-dimer elastic scattering and obtain a strong decrease of their collisional relaxation and decay with increasing $a$. The large ratio of the elastic to inelastic rate is promising for achieving Bose-Einstein condensation of the dimers and cooling the condensed gas to very low temperatures.

• 1. FOM Institute for Atomic and Molecular Physics, Institut for Atomic and Molecular Physics
• 2. National Research Centre "Kurchatov Institute" (NRC KI), University of Moscow
• 3. Laboratoire Kastler Brossel (LKB (Lhomond)), CNRS : UMR8552 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris

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• ## Hydrodynamics within the Electric Double Layer on slipping surfaces

### Laurent Joly 1 Christophe Ybert 1 Emmanuel Trizac 2 Lyderic Bocquet 1

#### Physical Review Letters, American Physical Society, 2004, 93 (25), pp.257805. 〈10.1103/PhysRevLett.93.257805〉

We show, using extensive Molecular Dynamics simulations, that the dynamics of the electric double layer (EDL) is very much dependent on the wettability of the charged surface on which the EDL develops. For a wetting surface, the dynamics, characterized by the so-called Zeta potential, is mainly controlled by the electric properties of the surface, and our work provides a clear interpretation for the traditionally introduced immobile Stern layer. In contrast, the immobile layer disappears for non-wetting surfaces and the Zeta potential deduced from electrokinetic effects is considerably amplified by the existence of a slippage at the solid substrate.

• 1. LPMCN - Laboratoire de Physique de la Matière Condensée et Nanostructures
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques