arXiv subjects

Pierre Le Doussal

Publications and source records attributed to Pierre Le Doussal.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2025

The stochastic porous medium equation in one dimension

We study the porous medium equation (PME) in one space dimension in presence of additive non-conservative white noise, and interpreted as a stochastic growth equation for the height field of an interface. We predict the values of the two growth exponents $α$ and $β$ using the functional RG. Extensive numerical simulations show agreement with the predicted values for these exponents, however they also show anomalous scaling with an additional "local" exponent $α_{\rm loc}$, as well as multiscaling originating from broad distributions of local height differences. The stationary measure of the stochastic PME is found to be well described by a random walk model, related to a Bessel process. This model allows for several predictions about the multiscaling properties.

cond-mat.stat-mech↗

arXiv2024

Spatio-temporal fluctuations in the passive and active Riesz gas on the circle

We consider a periodic Riesz gas consisting of $N$ classical particles on a circle, interacting via a two-body repulsive potential which behaves locally as a power law of the distance, $\sim g/|x|^s$ for $s>-1$. Long range (LR) interactions correspond to $s<1$, short range (SR) interactions to $s>1$, while the cases $s=0$ and $s=2$ describe the well-known log-gas and the Calogero-Moser (CM) model respectively. We study the fluctuations of the positions around the equally spaced crystal configuration, both for Brownian and run-and-tumble particles (RTP). Focusing on the regime of weak noise, we obtain exact expressions for the space-time correlations, both at the macroscopic and microscopic scale, for $N \gg 1$ and at fixed mean density $ρ$. They are characterized by a dynamical exponent $z_s=\min(1+s,2)$. We also obtain the gap statistics, described by a roughness exponent $ζ_s=\frac{1}{2} \min(s,1)$. For $s>0$ in the Brownian case, we find that in a broad time window, the mean square displacement of a particle is sub-diffusive as $t^{1/2}$ for SR as in single-file diffusion, and $t^{s/(1+s)}$ for LR interactions. Remarkably, this coincides, including the amplitude, with a recent prediction obtained using macroscopic fluctuation theory. These results also apply to RTPs beyond a time-scale $1/γ$, with $γ$ the tumbling rate, and a characteristic length-scale. Instead, for either shorter times or shorter distances, the active noise leads to a rich variety of static and dynamical regimes, with distinct exponents. For $-1<s<0$, the displacements are bounded, leading to true crystalline order at weak noise. The melting transition, recently observed numerically, is discussed in light of our calculation. Finally, we extend our method to the active Dyson Brownian motion and active CM model in a harmonic trap, generalizing to finite $γ$ the results of our earlier work.

cond-mat.stat-mech↗

arXiv2024

Integrable Matrix Probabilistic Diffusions and the Matrix Stochastic Heat Equation

We introduce a matrix version of the stochastic heat equation, the MSHE, and obtain its explicit invariant measure in spatial dimension $D=1$. We show that it is classically integrable in the weak-noise regime, in terms of the matrix extension of the imaginary-time $1D$ nonlinear Schrodinger equation which allows us to study its short-time large deviations through inverse scattering. The MSHE can be viewed as a continuum limit of the matrix log Gamma polymer on the square lattice introduced recently. We also show classical integrability of that discrete model, as well as of other extensions such as of the semi-discrete matrix O'Connell-Yor polymer and the matrix strict-weak polymer. For all these models, we obtain the Lax pairs of their weak-noise regime, as well as the invariant measure, using a fluctuation--dissipation transformation on the dynamical action.

cond-mat.stat-mech↗

arXiv2024

Large deviations in statistics of the local time and occupation time for a run and tumble particle

We investigate the statistics of the local time $\mathcal{T} = \int_0^T δ(x(t)) dt$ that a run and tumble particle (RTP) $x(t)$ in one dimension spends at the origin, with or without an external drift. By relating the local time to the number of times the RTP crosses the origin, we find that the local time distribution $P(\mathcal{T})$ satisfies the large deviation principle $P(\mathcal{T}) \sim \, e^{-T \, I(\mathcal{T} / T)} $ in the large observation time limit $T \to \infty$. Remarkably, we find that in presence of drift the rate function $I(ρ)$ is nonanalytic: We interpret its singularity as dynamical phase transitions of first order. We then extend these results by studying the statistics of the amount of time $\mathcal{R}$ that the RTP spends inside a finite interval (i.e., the occupation time), with qualitatively similar results. In particular, this yields the long-time decay rate of the probability $P(\mathcal{R} = T)$ that the particle does not exit the interval up to time $T$. We find that the conditional endpoint distribution exhibits an interesting change of behavior from unimodal to bimodal as a function of the size of the interval. To study the occupation time statistics, we extend the Donsker-Varadhan large-deviation formalism to the case of RTPs, for general dynamical observables and possibly in the presence of an external potential.

cond-mat.stat-mech↗

arXiv2024

Cumulants and large deviations for the linear statistics of the one-dimensional trapped Riesz gas

We consider the classical trapped Riesz gas, i.e., $N$ particles at positions $x_i$ in one dimension with a repulsive power law interacting potential $\propto 1/|x_i-x_j|^{k}$, with $k>-2$, in an external confining potential of the form $V(x) \sim |x|^n$. We focus on the equilibrium Gibbs state of the gas, for which the density has a finite support $[-\ell_0/2,\ell_0/2]$. We study the fluctuations of the linear statistics ${\cal L}_N = \sum_{i=1}^N f(x_i)$ in the large $N$ limit for smooth functions $f(x)$. We obtain analytic formulae for the cumulants of ${\cal L}_N$ for general $k>-2$. For long range interactions, i.e. $k<1$, which include the log-gas ($k \to 0$) and the Coulomb gas ($k =-1$) these are obtained for monomials $f(x)= |x|^m$. For short range interactions, i.e. $k>1$, which include the Calogero-Moser model, i.e. $k=2$, we compute the third cumulant of ${\cal L}_N$ for general $f(x)$ and arbitrary cumulants for monomials $f(x)= |x|^m$. We also obtain the large deviation form of the probability distribution of ${\cal L}_N$, which exhibits an "evaporation transition" where the fluctuation of ${\cal L}_N$ is dominated by the one of the largest $x_i$. In addition, in the short range case, we extend our results to a (non-smooth) indicator function $f(x)$, obtaining thereby the higher order cumulants for the full counting statistics of the number of particles in an interval $[-L/2,L/2]$. We show in particular that they exhibit an interesting scaling form as $L/2$ approaches the edge of the gas $L/\ell_0 \to 1$, which we relate to the large deviations of the emptiness probability of the complementary interval on the real line.

cond-mat.stat-mech↗

arXiv2024

A Dyson Brownian motion model for weak measurements in chaotic quantum systems

We consider a toy model for the study of monitored dynamics in a many-body quantum systems. We study the stochastic Schrodinger equation resulting from the continuous monitoring with a rate $Γ$ of a random hermitian operator chosen at every time from the gaussian unitary ensemble (GUE). Due to invariance by unitary transformations, the dynamics of the eigenvalues $\{λ_α\}_{α=1}^n$ of the density matrix can be decoupled from that of the eigenvectors. Thus, stochastic equations are derived that exactly describe the dynamics of $λ$'s. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large sizes the distribution of $λ$'s is described by an inverse Marchenko Pastur distribution. In the case of perfect measurements instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of $λ$'s at each time $t$ and for each size $n$. Two relevant regimes emerge: at small times $tΓ= O(1)$, the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the limit of $n\to\infty$. In that case, all moments of the density matrix become self-averaging and it is possible to characterize the entanglement spectrum exactly. In the limit of large times $t Γ= O(n)$ one enters instead a regime in which the eigenvalues are exponentially separated $\log(λ_α/λ_β) = O(Γt/n)$, but fluctuations $\sim O(\sqrt{Γt/n})$ play an essential role. We are still able to characterize the asymptotic behaviors of entanglement entropy in this regime.

cond-mat.stat-mech↗

arXiv2024

Timeliness criticality in complex systems

In complex systems, external parameters often determine the phase in which the system operates, i.e., its macroscopic behavior. For nearly a century, statistical physics has extensively studied systems' transitions across phases, (universal) critical exponents, and related dynamical properties. Here we consider the functionality of systems, notably operations in socio-technical ones, production in economic ones and, more generally, any schedule-based system, where timing is of crucial importance. We introduce a stylized model of delay propagation on temporal networks, where the magnitude of delay-mitigating buffer acts as a control parameter. The model exhibits {\it timeliness criticality}, a novel form of critical behavior. We characterize fluctuations near criticality, commonly referred to as ``avalanches'', and identify the corresponding critical exponents. The model exhibits timeliness criticality also when run on real-world temporal systems such as production networks. Additionally, we explore potential connections with the Mode-Coupling Theory of glasses, the depinning transition and the directed polymer problem.

physics.soc-ph↗

arXiv2024

Exact first-order effect of interactions on the ground-state energy of harmonically-confined fermions

We consider a system of $N$ spinless fermions, interacting with each other via a power-law interaction $ε/r^n$, and trapped in an external harmonic potential $V(r) = r^2/2$, in $d=1,2,3$ dimensions. For any $0 < n < d+2$, we obtain the ground-state energy $E_N$ of the system perturbatively in $ε$, $E_{N}=E_{N}^{\left(0\right)}+εE_{N}^{\left(1\right)}+O\left(ε^{2}\right)$. We calculate $E_{N}^{\left(1\right)}$ exactly, assuming that $N$ is such that the "outer shell" is filled. For the case of $n=1$ (corresponding to a Coulomb interaction for $d=3$), we extract the $N \gg 1$ behavior of $E_{N}^{\left(1\right)}$, focusing on the corrections to the exchange term with respect to the leading-order term that is predicted from the local density approximation applied to the Thomas-Fermi approximate density distribution. The leading correction contains a logarithmic divergence, and is of particular importance in the context of density functional theory. We also study the effect of the interactions on the fermions' spatial density. Finally, we find that our result for $E_{N}^{\left(1\right)}$ significantly simplifies in the case where $n$ is even.

cond-mat.stat-mech↗

arXiv2024

Anomalous scaling of heterogeneous elastic lines: a new picture from sample to sample fluctuations

We study a discrete model of an heterogeneous elastic line with internal disorder, submitted to thermal fluctuations. The monomers are connected through random springs with independent and identically distributed elastic constants drawn from $p(k)\sim k^{μ-1}$ for $k\to0$. When $μ>1$, the scaling of the standard Edwards-Wilkinson model is recovered. When $μ<1$, the elastic line exhibits an anomalous scaling of the type observed in many growth models and experiments. Here we derive and use the exact expression for the exact probability distribution of the line shape at equilibrium, as well as the spectral properties of the matrix containing the random couplings, to fully characterize the sample to sample fluctuations. Our results lead to novel scaling predictions that partially disagree with previous works, but which are corroborated by numerical simulations. We also provide a novel interpretation of the anomalous scaling in terms of the abrupt jumps in the line's shape that dominate the average value of the observable.

cond-mat.stat-mech↗

arXiv2023

Fluctuations in the active Dyson Brownian motion and the overdamped Calogero-Moser model

Recently, we introduced the active Dyson Brownian motion model (DBM), in which $N$ run-and-tumble particles interact via a logarithmic repulsive potential in the presence of a harmonic well. We found that in a broad range of parameters the density of particles converges at large $N$ to the Wigner semi-circle law, as in the passive case. In this paper, we provide an analytical support for this numerical observation, by studying the fluctuations of the positions of the particles in the nonequilibrium stationary state of the active DBM, in the regime of weak noise and large persistence time. In this limit, we obtain an analytical expression for the covariance between the particle positions for any $N$ from the exact inversion of the Hessian matrix of the system. We show that, when the number of particles is large $N \gg 1$, the covariance matrix takes scaling forms that we compute explicitly both in the bulk and at the edge of the support of the semi-circle. In the bulk, the covariance scales as $N^{-1}$, while at the edge, it scales as $N^{-2/3}$. Remarkably, we find that these results can be transposed directly to an equilibrium model, the overdamped Calogero-Moser model in the low temperature limit, providing an analytical confirmation of the numerical results by Agarwal, Kulkarni and Dhar. For this model, our method also allows us to obtain the equilibrium two-time correlations and their dynamical scaling forms both in the bulk and at the edge. Our predictions at the edge are reminiscent of a recent result in the mathematics literature by Gorin and Kleptsyn on the (passive) DBM. That result can be recovered by the present methods, and also, as we show, using the stochastic Airy operator. Finally, our analytical predictions are confirmed by precise numerical simulations, in a wide range of parameters.

cond-mat.stat-mech↗

arXiv2023

Interacting, running and tumbling: the active Dyson Brownian motion

We introduce and study a model in one dimension of $N$ run-and-tumble particles (RTP) which repel each other logarithmically in the presence of an external quadratic potential. This is an "active'' version of the well-known Dyson Brownian motion (DBM) where the particles are subjected to a telegraphic noise, with two possible states $\pm$ with velocity $\pm v_0$. We study analytically and numerically two different versions of this model. In model I a particle only interacts with particles in the same state, while in model II all the particles interact with each other. In the large time limit, both models converge to a steady state where the stationary density has a finite support. For finite $N$, the stationary density exhibits singularities, which disappear when $N \to +\infty$. In that limit, for model I, using a Dean-Kawasaki approach, we show that the stationary density of $+$ (respectively $-$) particles deviates from the DBM Wigner semi-circular shape, and vanishes with an exponent $3/2$ at one of the edges. In model II, the Dean-Kawasaki approach fails but we obtain strong evidence that the density in the large $N$ limit retains a Wigner semi-circular shape.

cond-mat.stat-mech↗

arXiv2023

"Tattered" membrane

Ideal crystalline membranes, realized by graphene and other atomic monolayers, exhibit rich physics - a universal anomalous elasticity of the critical "flat" phase characterized by a negative Poisson ratio, universally singular elastic moduli, order-from-disorder and a crumpling transition. We formulate a generalized $D$-dimensional field theory, parameterized by an $O(d)\times O(D)$ tensor field with an {\it energetic} longitudinal constraint. For a soft constraint the resulting field theory describes a new class of a fluctuating "tattered" membranes, exhibiting a nonzero density of topological connectivity defects - slits, cracks and faults at an effective medium level. For hard, infinite-coupling constraint, the model reproduces the conventional crystalline membrane and its crumpling transition, and thereby demonstrates the essence of the difference between an elastic membrane and conventional field theories. Two additional fixed points emerge within the critical manifold, (i) globally attractive, "isotropic" $O(d)\times O(D)$, and (ii) "transverse", which in $D=2$ is the exact "dual" of the elastic membrane. Their properties are obtained in general $D,d$ from the renormalization group and the self-consistent screening analyses.

cond-mat.stat-mech↗

arXiv2023

Linear statistics for Coulomb gases: higher order cumulants

We consider $N$ classical particles interacting via the Coulomb potential in spatial dimension $d$ and in the presence of an external trap, at equilibrium at inverse temperature $β$. In the large $N$ limit, the particles are confined within a droplet of finite size. We study smooth linear statistics, i.e. the fluctuations of sums of the form ${\cal L}_N = \sum_{i=1}^N f({\bf x}_i)$, where ${\bf x}_i$'s are the positions of the particles and where $f({\bf x}_i)$ is a sufficiently regular function. There exists at present standard results for the first and second moments of ${\cal L}_N$ in the large $N$ limit, as well as associated Central Limit Theorems in general dimension and for a wide class of confining potentials. Here we obtain explicit expressions for the higher order cumulants of ${\cal L}_N$ at large $N$, when the function $f({\bf x})=f(|{\bf x}|)$ and the confining potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of $f'(|{\bf x}|)$ and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a $d$-dimensional sphere. In the particular two-dimensional case $d=2$ at the special value $β=2$, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of ${\cal L}_N$.

math-ph↗

arXiv2023

Dynamics at the edge for independent diffusing particles

We study the dynamics of the outliers for a large number of independent Brownian particles in one dimension. We derive the multi-time joint distribution of the position of the rightmost particle, by two different methods. We obtain the two time joint distribution of the maximum and second maximum positions, and study the counting statistics at the edge. Finally we derive the multi-time joint distribution of the running maximum, as well as the joint distribution of the arrival time of the first particle at several space points.

cond-mat.stat-mech↗

arXiv2023

Non-equilibrium phase transitions in active rank diffusions

We consider N run and tumble particles in one dimension interacting via a linear 1D Coulomb potential, an active version of the rank diffusion problem. It was solved previously for N = 2 leading to a stationary bound state in the attractive case. Here the evolution of the density fields is obtained in the large N limit in terms of two coupled Burger's type equations. In the attractive case the exact stationary solution describes a non-trivial N-particle bound state, which exhibits transitions between a phase where the density is smooth with infinite support, a phase where the density has finite support and exhibits "shocks", i.e. clusters of particles, at the edges, and a fully clustered phase. In presence of an additional linear potential, the phase diagram, obtained for either sign of the interaction, is even richer, with additional partially expanding phases, with or without shocks. Finally, a general self-consistent method is introduced to treat more general interactions. The predictions are tested through extensive numerical simulations.

cond-mat.stat-mech↗

arXiv2023

Probing the large deviations for the Beta random walk in random medium

We consider a discrete-time random walk on a one-dimensional lattice with space and time-dependent random jump probabilities, known as the Beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time $t=0$ is at position beyond $ξ\sqrt{T/2}$ at time $T$. This probability fluctuates from sample to sample and we study the large-deviation rate function which characterizes the tails of its distribution at large time $T \gg 1$. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g. in the so-called KMP model of heat transfer. An extensive numerical simulation of the Beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.

cond-mat.stat-mech↗