arXiv subjects

Pierre Le Doussal

Publications and source records attributed to Pierre Le Doussal.

At least 19 recordsLinked to original sources

arXiv2026

Random walks in Dirichlet random environment in dimension $d+1$

The atypical behaviour of random walks in time-dependent random environment was recently related to Kardar-Parisi-Zhang (KPZ) growth. While this is now well-understood in spatial dimension $d=1$, further efforts are necessary to better understand these connections in dimensions $d>1$. In this paper, we study this problem numerically for $d=1, 2$ and $3$, focusing on a discrete model with Dirichlet distributed transition probabilities. This model is a generalization of an integrable model in $d=1$, and it has the advantage of admitting an explicit, product-form, stationary measure. We verify that the growth of the variance of the logarithm of point-to-point probabilities, namely from the origin to position $x$ in time $t$, is compatible with KPZ growth in dimension $d=1$ and $d=2$. In spatial dimension $d=3$, we confirm the existence of a phase transition as the angle $\vert x\vert /t$ increases and we obtain a lower bound based on an exact second moment calculation. We find that in the weak disorder phase the point-to-point probability acquires a heavy tailed distribution, and that in the strong disorder phase the cumulants of its logarithm grow with time. Further, we show that for this model, we can compute exactly the sample to sample variance of the thermal average $\overline{ \langle x \rangle^2}$ and that it is related to the extreme diffusion coefficient introduced recently.

cond-mat.stat-mech↗

arXiv2026

Randomly twisted bilayer graphene -- the cascade transitions

Twisted bilayer graphene (TBG) is known to have disorder in its twist angle. We show that in terms of a Dirac equation with a random gauge potential ${\bf A}({\bf r})$ this disorder becomes huge when the average twist angle is near the magic angle where the Dirac velocity vanishes. The density of states (DOS) then diverges at the Dirac point as $ρ(E)\sim E^{(2/z)-1}$ with $z\gg 1$ and we deduce that all electrons occupy energies very near $E=0$. We prove a sum rule on the disorder averaged eigenfunctions from which we deduce that each added electron contributes equal intraband Coulomb interaction energy. The various bands in TBG are related by either ${\bf A}({\bf r})\rightarrow {\bf A}({-\bf r})$ or ${\bf A}({\bf r})\rightarrow -{\bf A}({\bf r})$ which affects the interband interaction energy. We find, within Hartree-Fock, jumps in the chemical potential at each integer filling, accounting for the cascade transitions.

cond-mat.mes-hall↗

arXiv2026

A transition in the hole probability at finite temperature for free fermions in $d$ dimensions

In a free Fermi gas at temperature $T$ much higher than the Fermi temperature one expects that the fluctuations of the number of particles in a given region has Poissonian/classical statistics. On the other hand at low temperature the Pauli exclusion principle leads to non trivial counting statistics. It is of great interest from a theoretical and experimental point of view to characterize the crossover between these two limits. Here we focus on the hole probability $P(R,T)$, i.e. the probability that a region of size $R$ is devoid of particles, in dimension $d$, and on the case of a spherical region of large radius $R$. We show that at low temperature it takes the scaling form $P(R,T)\sim \exp\big[-(k_F R)^{d+1}Φ_d(u=2R\,T/k_F)\big],$ where $k_F$ is the Fermi momentum. By mapping the problem to an effective Coulomb gas, we compute exactly the scaling function $Φ_d(u)$ in any dimension. Remarkably, it exhibits a transition of order $\tfrac{3}{2}(d+1)$ at the universal critical value $u_c=2/π$, signaling a sharp change in the mechanism of rare fluctuations, associated with the emergence of a macroscopic gap in the optimal density of the associated Coulomb gas. Our analytical predictions are supported by precise numerical evaluations of the corresponding Fredholm determinants.

cond-mat.stat-mech↗

arXiv2026

The Tracy-Widom distribution at large Dyson index

We study the Tracy-Widom (TW) distribution $f_β(a)$ in the limit of large Dyson index $β\to +\infty$. This distribution describes the fluctuations of the rescaled largest eigenvalue $a_1$ of the Gaussian (alias Hermite) ensemble (G$β$E) of (infinitely) large random matrices. We show that, at large $β$, its probability density function takes the large deviation form $f_β(a) \sim e^{-βΦ(a)}$. While the typical deviation of $a_1$ around its mean is Gaussian of variance $O(1/β)$, this large deviation form describes the probability of rare events with deviation $O(1)$, and governs the behavior of the higher cumulants. We obtain the rate function $Φ(a)$ as a solution of a Painlevé II equation. We derive explicit formula for its large argument behavior, and for the lowest cumulants, up to order 4. We compute $Φ(a)$ numerically for all $a$ and compare with exact numerical computations of the TW distribution at finite $β$. These results are obtained by applying saddle-point approximations to an associated problem of energy levels $E=-a$, for a random quantum Hamiltonian defined by the stochastic Airy operator (SAO). We employ two complementary approaches: (i) we use the optimal fluctuation method to find the most likely realization of the noise in the SAO, conditioned on its ground-state energy being $E$ (ii) we apply the weak-noise theory to the representation of the TW distribution in terms of a Ricatti diffusion process associated to the SAO. We extend our results to the full Airy point process $a_1>a_2>\dots$ which describes all edge eigenvalues of the G$β$E, and correspond to (minus) the higher energy levels of the SAO, obtaining large deviation forms for the marginal distribution of $a_i$, the joint distributions, and the gap distributions.

cond-mat.stat-mech↗

arXiv2026

Non-uniqueness of the steady state for run-and-tumble particles with a double-well interaction potential

We study $N$ run-and-tumble particles (RTPs) in one dimension interacting via a double-well potential $W(r)=-k_0 \, r^2/2+g \, r^4/4$, which is repulsive at short interparticle distance $r$ and attractive at large distance. At large time, the system forms a bound state where the density of particles has a finite support. We focus on the determination of the total density of particles in the stationary state $ρ_s(x)$, in the limit $N\to+\infty$. We obtain an explicit expression for $ρ_s(x)$ as a function of the ''renormalized" interaction parameter $k=k_0-3m_2$ where $m_2$ is the second moment of $ρ_s(x)$. Interestingly, this stationary solution exhibits a transition between a connected and a disconnected support for a certain value of $k$, which has no equivalent in the case of Brownian particles. Analyzing in detail the expression of the stationary density in the two cases, we find a variety of regimes characterized by different behaviors near the edges of the support and around $x=0$. Furthermore, we find that the mapping $k_0\to k$ becomes multi-valued below a certain value of the tumbling rate $γ$ of the RTPs for some range of values of $k_0$ near the transition, implying the existence of two stable solutions. Finally, we show that in the case of a disconnected support, it is possible to observe steady states where the density $ρ_s(x)$ is not symmetric. All our analytical predictions are in good agreement with numerical simulations already for systems of $N = 100$ particles. The non-uniqueness of the stationary state is a particular feature of this model in the presence of active (RTP) noise, which contrasts with the uniqueness of the Gibbs equilibrium for Brownian particles. We argue that these results are also relevant for a class of more realistic interactions with both an attractive and a repulsive part, but which decay at infinity.

cond-mat.stat-mech↗

arXiv2026

Mesoscopic fluctuation theory of particle systems driven by Poisson noise: study of the $q$-TASEP

We pursue our study of integrable weak noise theories of directed polymer and interacting particle stochastic models in the 1D KPZ universality class. Here we focus on the $q$-TASEP in either continuous or discrete time. Each particle on $\mathbb{Z}$ jumps independently by $+1$ with a rate (or probability) depending on the gap to the next particle on its right. We consider initial conditions (either step or random) which are empty of particles on $\mathbb{Z}^+$, and focus on the dynamics of the $N$ rightmost particles. In the limit $q \to 1$ and at large time (and large gaps) we identify a new intermediate "mesoscopic" (i.e. finite $N$) regime which corresponds to weak noise. In that regime Poisson noise remains important. We obtain the large deviations of the position of a given particle by two methods. The first derives asymptotics of $q$-TASEP Fredholm determinant formula. The second maps the weak noise limit to a system of semi-discrete or fully discrete, non linear differential equations. These are obtained as saddle point classical equations of a dynamical field theory, and their solutions represent the optimal configurations in the large deviation regime. We show the classical integrability of these two systems, and exhibit their explicit Lax pair. In the case of the continuous time $q$-TASEP it provides the first instance of classical integrability arising in a stochastic system, with signatures of the Poisson noise persisting in the weak noise limit. For this model, we solve the scattering problem associated to its Lax pair and fully characterize the large deviations associated to the weak noise theory. Finally, we supplement this work with an Appendix on the first cumulant method to obtain the large deviations of several lattice polymer models (Strict Weak, Log Gamma, Beta).

cond-mat.stat-mech↗

arXiv2026

A mean-field theory for heterogeneous random growth with redistribution

We study the competition between random multiplicative growth and redistribution/migration in the mean-field limit, when the number of sites is very large but finite. We find that for static random growth rates, migration should be strong enough to prevent localisation, i.e. extreme concentration on the fastest growing site. In the presence of an additional temporal noise in the growth rates, a third partially localised phase is predicted theoretically, using results from Derrida's Random Energy Model. Such temporal fluctuations mitigate concentration effects, but do not make them disappear. We discuss our results in the context of population growth and wealth inequalities.

cond-mat.dis-nn↗

arXiv2025

Ground state energy fluctuations of pinned elastic manifolds

We describe the atypical fluctuations of the ground state energy of the random elastic manifold, a disordered model defined on a lattice of linear size $L$ with internal dimension $0\leq d<4$ embedded in a medium of dimension $N\gg 1$. The ground-state energy results from a competition between confinement, elasticity and disorder. We obtain an exact description of the large deviation rate function with speed $NL^d$ and its different phases, corresponding to different patterns of replica symmetry breaking (RSB). Our results show that the ground-state energy satisfies a central limit theorem and we obtain an explicit expression for the rescaled variance. In the (massless) limit of zero confinement, this variance vanishes for short-range disorder and the ground-state energy displays super-concentration. From our results on the large deviation function, we characterise explicitly the left tail of the distribution of the typical fluctuations of the ground state energy. It displays an exponential tail for a one step RSB pattern while for a full RSB pattern it decays super-exponentially with a non trivial exponent $ξ$ that we compute explicitly.

cond-mat.stat-mech↗

arXiv2025

Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials

We study the moments of $\overline{|\det(H-E)|^q}$ and the associated large deviations of $\log |\det(H-E)|$ where $H$ are random matrix operators involving Laplace operators and random potentials. This includes as a special case Hessians of random elastic manifolds at a generic energy configuration. In one dimension $d=1$ these are $N \times N$ matrix valued random Schrödinger operators and $\log | \det(H-E) | $ is the sum of the $N$ associated Lyapunov exponents. Using a mapping to a stochastic matrix Ricatti equation we make a connection between the spectral properties of these operators and the total $N$ particle current of a Dyson Brownian motion (DBM) in a cubic potential. The latter model was studied by Allez and Dumaz [1] who showed that for $N=+\infty$ it exhibits a sharp transition between a phase with non-zero current and a confined (zero current) phase. We compute the barrier-crossing probability of the DBM at large but finite $N$, which gives an estimate of the exponential tail of the average density of states of a matrix Schrodinger operator below the edge of its spectrum. The barrier behaves as $\sim N (-E)^{3/2}$ at large negative energy and vanishes as $\sim N(E^*-E)^{5/4}$ near the edge. For $q=1$ the present work provides an independent derivation of the total complexity of stationary points for an elastic string embedded in $N$ dimension in presence of disorder.

math-ph↗

arXiv2025

Universal Random Matrix Behavior of a Fermionic Quantum Gas

The pursuit of universal governing principles is a foundational endeavor in physics, driving breakthroughs from thermodynamics to general relativity and quantum mechanics. In 1951, Wigner introduced the concept of a statistical description of energy levels of heavy atoms, which led to the rise of Random Matrix Theory (RMT) in physics. The theory successfully captured spectral properties across a wide range of atomic systems, circumventing the complexities of quantum many-body interactions. Rooted in the fundamental principles of stochasticity and symmetry, RMT has since found applications and revealed universal laws in diverse physical contexts, from quantum field theory to disordered systems and wireless communications. A particularly compelling application arises in describing the mathematical structure of the many-body wavefunction of non-interacting Fermi gases, which underpins a complex spatial organization driven by Pauli's exclusion principle. However, experimental validation of the counting statistics predicted in such systems has remained elusive. Here, we probe at the single-atom level ultracold atomic Fermi gases made of two interacting spin states, obtaining direct access to their counting statistics in situ. Our measurements show that, while the system is strongly attractive, each spin-component is extremely well described by RMT predictions based on Fredholm determinants. Our results constitutes the first experimental validation of the Fermi-sphere point process through the lens of RMT, and establishes its relevance for strongly-interacting systems.

cond-mat.quant-gas↗

arXiv2025

Large deviations of the largest eigenvalue for deformed GOE/GUE random matrices via replica

We study the probability distribution function $P(λ)$ of the largest eigenvalue $λ_{\rm max}$ of $N \times N$ random matrices of the form $H + V$, where $H$ belongs to the GOE/GUE ensemble and $V$ is a full rank deterministic diagonal perturbation. This model is related to spherical spin glasses and semi-discrete directed polymers. In the large $N$ limit, using the replica method introduced in Ref. \cite{TrivializationUs2014}, we obtain the rate function ${\cal L}(λ)$ which describes the upper large deviation tail $P(λ) \sim e^{- βN {\cal L}(λ) }$. We also obtain the moment generating function $\langle e^{N s λ_{\max} } \rangle \sim e^{N ϕ(s)}$ and the overlap of the optimal eigenvector with the perturbation $V$. For suitable $V$, a transition generically occurs in the rate functions. For the GUE it has a direct interpretation as a localisation transition for tilted directed polymers with competing columnar and point disorder. Although in a different form, our results are consistent with those obtained recently by Mc Kenna in \cite{McKenna2021}. Finally, we consider briefly the quadratic optimisation problem in presence of an additional random field and obtain its large deviation rate function, although only within the replica symmetric phase.

cond-mat.stat-mech↗

arXiv2025

Large time cumulants of the KPZ equation on an interval

We consider the Kardar-Parisi-Zhang equation on the interval $[0,L]$ with Neumann type boundary conditions and boundary parameters $u,v$. We show that the $k$-th order cumulant of the height behaves as $c_k(L,u,v)\, t$ in the large time limit $t \to +\infty$, and we compute the coefficients $c_k(L,u,v)$. We obtain an expression for the upper tail large deviation function of the height. We also consider the limit of large $L$, with $u=\tilde u/\sqrt{L}$, $u=\tilde v/\sqrt{L}$, which should give the same quantities for the two parameter family $(\tilde u, \tilde v)$ KPZ fixed point on the interval. We employ two complementary methods. On the one hand we adapt to the interval the replica Bethe ansatz method pioneered by Brunet and Derrida for the periodic case. On the other hand, we perform a scaling limit using previous results available for the open ASEP. The latter method allows to express the cumulants of the KPZ equation in terms a functional equation involving an integral operator.

math-ph↗

arXiv2025

Dynamic scaling of growing interfaces

We give a brief overview of the seminal paper which introduced the Kardar-Parisi-Zhang equation as a paradigmatic model for random growth in 1986. We describe some of the developments to which it gave rise in mathematics and physics over the years, and some examples of applications.

cond-mat.dis-nn↗

arXiv2025

Measurement-Induced Phase Transition in State Estimation of Chaotic Systems and the Directed Polymer

We introduce a solvable model of a measurement-induced phase transition (MIPT) in a deterministic but chaotic dynamical system with a positive Lyapunov exponent. In this setup, an observer only has a probabilistic description of the system but mitigates chaos-induced uncertainty through repeated measurements. Using a minimal representation via a branching tree, we map this problem to the directed polymer (DP) model on the Cayley tree, although in a regime dominated by rare events. By studying the Shannon entropy of the probability distribution estimated by the observer, we demonstrate a phase transition distinguishing a chaotic phase with reduced Lyapunov exponent from a strong-measurement phase where uncertainty remains bounded. Remarkably, the location of the MIPT transition coincides with the freezing transition of the DP, although the critical properties differ. We provide an exact, universal scaling function describing the entropy growth in the critical regime. Numerical simulations confirm our theoretical predictions, highlighting a simple yet powerful framework to explore measurement-induced transitions in classical chaotic systems.

cond-mat.stat-mech↗

arXiv2025

Crystal to liquid cross-over for active particles with inverse-square power-law interaction

We consider a one-dimensional system comprising of $N$ run-and-tumble particles confined in a harmonic trap interacting via a repulsive inverse-square power-law interaction. We numerically compute the global density profile in the steady state which shows interesting crossovers between three different regimes: as the activity increases, we observe a change from a density with sharp peaks characteristic of a crystal region to a smooth bell-shaped density profile, passing through the intermediate stage of a smooth Wigner semi-circle characteristic of a liquid phase. We also investigate analytically the crossover between the crystal and the liquid regions by computing the covariance of the positions of these particles in the steady state in the weak noise limit. It is achieved by using the method introduced in Touzo {\it et al.} [Phys. Rev. E {\bf 109}, 014136 (2024)] to study the active Dyson Brownian motion. Our analytical results are corroborated by thorough numerical simulations.

cond-mat.stat-mech↗

arXiv2025

Integrability and exact large deviations of the weakly-asymmetric exclusion process

The weakly asymmetric exclusion process (WASEP) in one dimension is a paradigmatic system of interacting particles described by the macroscopic fluctuation theory (MFT) in the presence of driving. We consider an initial condition with densities $ρ_1,ρ_2$ on either side of the origin, so that for $ρ_1=ρ_2$ the gas is stationary. Starting from the microscopic description, we obtain exact formulae for the cumulant generating functions, and large deviation rate functions of the time-integrated current and the position of a tracer. As the asymmetry/driving is increased, these describe the crossover between the symmetric exclusion process (SSEP) and the weak noise regime of the Kardar-Parisi-Zhang (KPZ) equation: we recover the two limits and describe the crossover from the WASEP cubic tail to the $5/2$ and $3/2$ KPZ tail exponents. Finally, we show that the MFT of the WASEP is classically integrable, by exhibiting the explicit Lax pairs, which are obtained through a novel mapping between the MFT of the WASEP and a complex extension of the classical anisotropic Landau-Lifshitz spin chain. This shows integrability of all MFTs of asymmetric models with quadratic mobility as well as their dual versions.

cond-mat.stat-mech↗

arXiv2025

Linear statistics at the microscopic scale for the 2D Coulomb gas

We consider the classical Coulomb gas in two dimensions at the inverse temperature $β=2$, confined within a droplet of radius $R$ by a rotationally invariant potential $U(r)$. For $U(r)\sim r^2$ this describes the eigenvalues of the complex Ginibre ensemble of random matrices. We study linear statistics of the form ${\cal L}_N = \sum_{i=1}^N f(|{\bf x}_i|)$, where ${\bf x}_i$'s are the positions of the $N$ particles, in the large $N$ limit with $R=O(1)$. It is known that for smooth functions $f(r)$ the variance ${\rm Var} \,{\cal L}_N= O(1)$, while for an indicator function relevant for the disk counting statistics, all cumulants of ${\cal L}_N$ of order $q \geq 2$ behave as $\sim \sqrt{N}$. In addition, for smooth functions, it was shown that the cumulants of ${\cal L}_N$ of order $q \geq 3$ scale as $\sim N^{2-q}$. Surprisingly it was found that they depend only on $f'(|\bf x|)$ and its derivatives evaluated exactly at the boundary of the droplet. To understand this property, and interpolate between the two behaviors (smooth versus step-like), we study the microscopic linear statistics given by $f(r) \to f_N(r) = ϕ((r-\hat r) \sqrt{N}/ξ)$, which probes the fluctuations at the scale of the inter-particle distance. We compute the cumulants of ${\cal L}_N$ at large $N$ for a fixed $ϕ(u)$ at arbitrary $ξ$. For large $ξ$ they match the predictions for smooth functions which shows that the leading contribution in that case comes from a boundary layer of size $1/\sqrt{N}$ near the boundary of the droplet. Finally we show that the full probability distribution of ${\cal L}_N$ take two distinct large deviation forms, in the regime ${\cal L}_N \sim \sqrt{N}$ and ${\cal L}_N \sim N$ respectively. We also discuss applications of our results to fermions in a rotating harmonic trap and to the Ginibre symplectic ensemble.

cond-mat.stat-mech↗

arXiv2025

Run-and-tumble particles with 1D Coulomb interaction: the active jellium model and the non-reciprocal self-gravitating gas

Recently we studied $N$ run-and-tumble particles in one dimension - which switch with rate $γ$ between driving velocities $\pm v_0$ - interacting via the long range 1D Coulomb potential (also called rank interaction), both in the attractive and in the repulsive case, with and without a confining potential. We extend this study in two directions. First we consider the same system, but inside a harmonic confining potential, which we call "active jellium". We obtain a parametric representation of the particle density in the stationary state at large $N$, which we analyze in detail. Contrary to the linear potential, there is always a steady-state where the density has a bounded support. However, we find that the model still exhibits transitions between phases with different behaviors of the density at the edges, ranging from a continuous decay to a jump, or even a shock (i.e. a cluster of particles, which manifests as a delta peak in the density). Notably, the interactions forbid a divergent density at the edges, which may occur in the non-interacting case. In the second part, we consider a non-reciprocal version of the rank interaction: the $+$ particles (of velocity $+v_0$) are attracted towards the $-$ particles (of velocity $-v_0$) with a constant force $b/N$, while the $-$ particles are repelled by the $+$ particles with a force of same amplitude. In order for a stationary state to exist we add a linear confining potential. We derive an explicit expression for the stationary density at large $N$, which exhibits an explicit breaking of the mirror symmetry with respect to $x=0$. This again shows the existence of several phases, which differ by the presence or absence of a shock at $x=0$, with one phase even exhibiting a vanishing density on the whole region $x>0$. Our analytical results are complemented by numerical simulations for finite $N$.

cond-mat.stat-mech↗