# LPTMS Publications

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## Extremes of $2d$ Coulomb gas: universal intermediate deviation regime

### Bertrand Lacroix-A-Chez-Toine

^{1}Aurélien Grabsch^{1}Satya N. Majumdar^{1}Gregory Schehr^{1}*Journal of Statistical Mechanics: Theory and Experiment*, IOP Science, 2018, 〈10.06222〉In this paper, we study the extreme statistics in the complex Ginibre ensemble of $N \times N$ random matrices with complex Gaussian entries, but with no other symmetries. All the $N$ eigenvalues are complex random variables and their joint distribution can be interpreted as a $2d$ Coulomb gas with a logarithmic repulsion between any pair of particles and in presence of a confining harmonic potential $v(r) \propto r^2$. We study the statistics of the eigenvalue with the largest modulus $r_{\max}$ in the complex plane. The typical and large fluctuations of $r_{\max}$ around its mean had been studied before, and they match smoothly to the right of the mean. However, it remained a puzzle to understand why the large and typical fluctuations to the left of the mean did not match. In this paper, we show that there is indeed an intermediate fluctuation regime that interpolates smoothly between the large and the typical fluctuations to the left of the mean. Moreover, we compute explicitly this "intermediate deviation function" (IDF) and show that it is universal, i.e. independent of the confining potential $v(r)$ as long as it is spherically symmetric and increases faster than $\ln r^2$ for large $r$ with an unbounded support. If the confining potential $v(r)$ has a finite support, i.e. becomes infinite beyond a finite radius, we show via explicit computation that the corresponding IDF is different. Interestingly, in the borderline case where the confining potential grows very slowly as $v(r) \sim \ln r^2$ for $r \gg 1$ with an unbounded support, the intermediate regime disappears and there is a smooth matching between the central part and the left large deviation regime.

- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques

## Higher-order corrections to the effective potential close to the jamming transition in the perceptron model

### Ada Altieri

^{1, 2}*Physical Review E*, American Physical Society (APS), 2018, 97 (1), 〈10.1103/PhysRevE.97.012103〉We analyze the perceptron model performing a Plefka-like expansion of the free energy. This model falls in the same universality class as hard spheres near jamming, allowing to get exact predictions in high dimensions for more complex systems. Our method enables to define an effective potential (or TAP free energy), namely a coarse-grained functional depending on the contact forces and the effective gaps between the particles. The derivation is performed up to the third order, with a particular emphasis on the role of third order corrections to the TAP free energy. These corrections, irrelevant in a mean-field framework in the thermodynamic limit, might instead play a fundamental role when considering finite-size effects. We also study the typical behavior of the forces and we show that two kinds of corrections can occur. The first contribution arises since the system is analyzed at a finite distance from jamming, while the second one is due to finite-size corrections. In our analysis, third order contributions vanish in the jamming limit, both for the potential and the generalized forces, in agreement with the argument proposed by Wyart and coworkers invoking isostaticity. Finally, we analyze the scalings emerging close to the jamming line, which define a crossover regime connecting the control parameters of the model to an effective temperature.

- 1. SMC/INFM - Department of Physics, Sapienza University of Rome
- 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques

## Origin of the correlations between exit times in pedestrian flows through a bottleneck

### Alexandre Nicolas

^{1}Ioannis Touloupas^{1}*Journal of Statistical Mechanics*, 2018, 1, pp.013402. 〈http://iopscience.iop.org/article/10.1088/1742-5468/aa9dcd〉Robust statistical features have emerged from the microscopic analysis of dense pedestrian flows through a bottleneck, notably with respect to the time gaps between successive passages. We pinpoint the mechanisms at the origin of these features thanks to simple models that we develop and analyse quantitatively. We disprove the idea that anticorrelations between successive time gaps (i.e., an alternation between shorter ones and longer ones) are a hallmark of a zipper-like intercalation of pedestrian lines and show that they simply result from the possibility that pedestrians from distinct 'lines' or directions cross the bottleneck within a short time interval. A second feature concerns the bursts of escapes, i.e., egresses that come in fast succession. Despite the ubiquity of exponential distributions of burst sizes, entailed by a Poisson process, we argue that anomalous (power-law) statistics arise if the bottleneck is nearly congested, albeit only in a tiny portion of parameter space. The generality of the proposed mechanisms implies that similar statistical features should also be observed for other types of particulate flows.

- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques

## Planar screening by charge polydisperse counterions

### M. Trulsson

^{1, 2}E. Trizac^{1}L. Samaj^{3}*Journal of Physics: Condensed Matter*, IOP Publishing, 2018, 30 (3), 〈10.1088/1361-648X/aa9a79〉We study how a neutralising cloud of counterions screens the electric field of a uniformly charged planar membrane plate, when the counterions are characterised by a distribution of charges (or valence), $n(q)$. We work out analytically the one-plate and two-plate cases, at the level of non-linear Poisson-Boltzmann theory. The (essentially asymptotic) predictions are successfully compared to numerical solutions of the full Poisson-Boltzmann theory, but also to Monte Carlo simulations. The counterions with smallest valence control the long-distance features of interactions, and may qualitatively change the results pertaining to the classic monodisperse case where all counterions have the same charge. Emphasis is put on continuous distributions $n(q)$, for which new power-laws can be evidenced, be it for the ionic density or the pressure, in the one- and two-plates situations respectively. We show that for discrete distributions, more relevant for experiments, these scaling laws persist in an intermediate but yet observable range. Furthermore, it appears that from a practical point of view, hallmarks of the continuous $n(q)$ behaviour is already featured by discrete mixtures with a relatively small number of constituents.

- 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
- 2. Lund University [Lund]
- 3. Institute of Physics

## Topological Zak Phase in Strongly-Coupled LC Circuits

### Tal Goren

^{1}Kirill Plekhanov^{1, 2}Félicien Appas^{1}Karyn Le Hur^{1}*Physical Review B : Condensed matter and materials physics*, American Physical Society, 2018We show the emergence of topological Bogoliubov bosonic excitations in the relatively strong coupling limit of an LC (inductance-capacitance) one-dimensional quantum circuit. This dimerized chain model reveals a ${\cal Z}_2$ local symmetry as a result of the counter-rotating wave (pairing) terms. The topology is protected by the sub-lattice symmetry, represented by an anti-unitary transformation. We present a methos to measure the winding of the topological Zak phase across the Brillouin zone by a reflection measurement of (microwave) light. Our method probes bulk quantities and can be implemented even in small systems. We study the robustness of edge modes towards disorder.

- 1. CPHT - Centre de Physique Théorique [Palaiseau]
- 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques