arXiv subjects

Gregory Schehr

Publications and source records attributed to Gregory Schehr.

At least 19 recordsLinked to original sources

arXiv2026

Effects of confinement in a Brownian gas with simultaneous stochastic resetting and dynamically emergent correlations

We study $N$ non-interacting Brownian particles in an external potential under simultaneous stochastic resetting to the origin. Although they do not interact directly, common resets generate strong dynamically emergent correlations (DEC). We analyze how confinement modifies these correlations and the nonequilibrium stationary state for $V(x)=\kappa |x|^\alpha$, $\alpha\geq0$, focusing mainly on two analytically tractable cases: harmonic confinement (HC), $\alpha=2$, and box confinement (BC), $\alpha\to\infty$. In both cases the stationary state is controlled by the competition between confinement and resetting lengths. We derive exact results for the stationary joint distribution, density, correlations, extreme value statistics (EVS), and gap statistics. While the density behaves similarly in HC and BC, the normalized correlation coefficient differs sharply. In BC it is non-monotonic and overshoots the unconfined value, as hard walls suppress decorrelating trajectories. In HC it instead increases monotonically toward the unconfined limit. For general $\alpha$, the behavior is monotonic for $0<\alpha<\alpha_c=1+\sqrt{5}$ and non-monotonic for $\alpha>\alpha_c$. The difference between HC and BC is also visible in edge observables. In HC, the maximum scales as $M_1=O(\sqrt{\ln N})$ and has a limiting distribution with bounded support and a shape transition controlled by the ratio of the two length scales. In BC, the maximum is at distance $O(1/N)$ from the boundary, as in equilibrium, but its fluctuations have a broad power-law tail with logarithmic corrections. The first gap shows a similar contrast: BC gives a smaller typical gap but stronger anomalous fluctuations than HC. Finally, we extend the EVS analysis to general $\alpha$ and identify, via simulations and scaling arguments, three universality classes: $0\leq\alpha\leq1$, $1<\alpha<\infty$, and the singular limit $\alpha\to\infty$.

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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 $P(R,T)\sim \exp\big[-(k_F R)^{d+1}\Phi_d(u=2R\,T/k_F)\big],$ where $k_F$ is the Fermi momentum. By mapping the problem to an effective Coulomb gas, we compute exactly the scaling function $\Phi_d(u)$ in any dimension. Remarkably, it exhibits a transition of order $\tfrac{3}{2}(d+1)$ at the universal critical value $u_c=2/\pi$, 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.

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arXiv2026

Extreme value statistics and some applications in statistical physics

These notes are based on lectures delivered by G. Schehr at the XVIth School on Fundamental Problems in Statistical Physics (FPSP), held in Oropa (Italy) from 30 June to 11 July 2025. After a brief introduction to extreme value statistics (EVS) for independent and identically distributed (IID) random variables, we discuss several paradigmatic examples of strongly correlated systems where classical extreme value theory no longer applies. In particular, we focus on time series generated by random walks and Brownian motion, as well as on eigenvalue statistics in random matrix theory. Emphasis is placed on applications of EVS to fundamental problems in statistical physics and disordered systems, including the Random Energy Model, stochastic search problems, as well as fluctuating interfaces, and directed polymers in random media within the Kardar-Parisi-Zhang universality class.

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arXiv2026

Dynamically Emergent Correlations

In this perspective article, we discuss the scenario of dynamically emergent correlation (DEC) arising in classical and quantum noninteracting systems when they are subjected to a common fluctuating stochastic environment. The key property of such systems is that the strong correlations between different particles emerge from the dynamics and not from built-in interactions. In many cases, these strong correlations persist even at long times in the stationary state. Computing observables explicitly for such strongly correlated states in general is very hard. Remarkably, the stationary states in several models of DEC exhibit an interesting analytical structure that allows to compute physical observables, despite being strongly correlated. Recent experiments on trapped colloidal particles have established that these DEC in the stationary state can in fact be measured. DEC is a rapidly emerging domain of strongly correlated out-of-equilibrium statistical physics, with both theoretical and experimental, as well as classical and quantum, components.

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arXiv2026

Equilibrium measures for higher dimensional rotationally symmetric Riesz gases

We study equilibrium measures for Riesz gases in dimension $d$ with pairwise interaction kernel $|x-y|^{-s}$, subject to radially symmetric external fields. We characterise broad classes of confining potentials for which the equilibrium measure is supported on the unit ball and admits an explicit density. Our main contribution is a converse construction: starting from a prescribed radially symmetric equilibrium density given as a power series in the squared radius, we determine the associated external potential and establish the corresponding Euler-Lagrange variational conditions. A key ingredient in the proof is an identity between two ${}_3F_2$ hypergeometric functions evaluated at unit argument, which is of independent interest. As applications, we identify the external potentials corresponding to equilibrium densities proportional to $(1-|x|^2)^\alpha$, $\alpha>-1$, and show that these potentials can be expressed in terms of Gauss hypergeometric functions ${}_2F_1$, reducing to polynomials for special values of $\alpha$. We also determine the equilibrium measure associated with purely power-type external potentials, often referred to as Freud or Mittag--Leffler potentials in the context of log gases, for which the equilibrium density admits an explicit ${}_2F_1$ representation. Furthermore, we apply our framework to a Coulomb gas in dimension $d+1$ confined by a harmonic potential to the half-space. We derive a necessary condition under which the equilibrium measure is fully supported on the boundary hyperplane of dimension $d$, with the induced density corresponding to that of a Riesz gas with exponent $s=d-1$.

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arXiv2026

Tuning the strength of emergent correlations in a Brownian gas via batch resetting

We study a gas of $N$ diffusing particles on the line subject to batch resetting: at rate $r$, a uniformly random subset of $m$ particles is reset to the origin. Despite the absence of interactions, the dynamics generates a nonequilibrium stationary state (NESS) with long-range correlations. We obtain exact results, both for the NESS and for the time dependence of the correlations, which are valid for arbitrary $m$ and $N$. By varying $m$, the system interpolates between an uncorrelated regime ($m=1$) and the fully synchronous resetting case ($m=N$). For all $1<m<N$, correlations exhibit a non-monotonic time dependence due to the emergence of an intrinsic decorrelation mechanism. In the stationary state, the correlation strength can be tuned by varying $m$, and it displays a transition at a critical value $N_c=6$. Our predictions extend straightforwardly to any spatial dimension $d$ and the critical value $N_c=6$ remains the same in all dimensions. Our predictions are testable in existing experimental setups on optically trapped colloidal particles.

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arXiv2025

First Passage Resetting Gas

We study a one-dimensional gas of $N$ Brownian particles that diffuse independently but are simultaneously reset whenever any of them reaches a fixed threshold located at $L > 0$. For any $N > 2$, the system reaches a non-equilibrium stationary state (NESS) at long-times with strong long-range correlations. These correlations emerge purely from the dynamics, and not from built-in interactions. Despite being strongly correlated, the NESS has a solvable structure that allows for an exact computation of several physical observables, both global and local. These include the average density profile, the distribution of the position of the $k$-th ordered particles, the distribution of the gap between two consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in a finite interval around the origin.

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arXiv2025

The distribution of the moment of inertia for harmonically trapped noninteracting Bosons at finite temperature: large deviations

We compute the full probability distribution of the moment of inertia $I \propto \sum_{i=1}^N \vec{r}_i^2$ of a gas of $N$ noninteracting bosons trapped in a harmonic potential $V(r) = (1/2) m \omega^2 r^2$, in all dimensions and at all temperatures. The appropriate thermodynamic limit in a trapped Bose gas consists in taking $N\to\infty$ and $\omega\to 0$ with their product $\rho = N\omega^d$ fixed, where $\rho$ plays a role analogous to density in a translationally invariant system. In this thermodynamic limit and in dimensions $d>1$, the harmonically trapped Bose gas undergoes a Bose--Einstein condensation (BEC) transition as $\rho$ crosses a critical value $\rho_c(\beta)$, where $\beta$ denotes the inverse temperature. We show that the nature of the condensation is different for $1 2$. Near the condensation transition, for simplicity, we provide the analysis only for $d>2$. We show that the probability distribution $P_\beta(I,N)$ of $I$ admits a large deviation form $P_\beta(I,N) \sim e^{-V\Phi(I/V)}$, where $V = \omega^{-d} \gg 1$. We compute explicitly the rate function $\Phi(z)$ and show that it exhibits a singularity at a critical value $z=z_c$, where its second derivative undergoes a discontinuous jump. In addition, on the condensed side, $\Phi(z)$ becomes independent of $\rho$ for $z 1$. This provides a real-space diagnostic for the BEC transition in the noninteracting Bose gas.

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arXiv2025

Dynamically emergent correlations in Brownian particles subject to simultaneous non-Poissonian resetting protocols

We consider a one-dimensional gas of $N$ independent Brownian particles subject to simultaneous stochastic resetting, with inter-reset times drawn from a general waiting-time distribution $\psi(\tau)$. This includes the well-known Poissonian case, where $\psi(\tau)=re^{-r\tau}$, and extends to more general classes of resetting, such as heavy-tailed and bounded distributions. We show that the simultaneous resetting generates correlations between particles dynamically. These correlations grow with time and eventually drive the system into a strongly correlated non-equilibrium stationary state (NESS). Exploiting the renewal structure of the resetting dynamics, we derive explicit analytical expressions for the joint distribution of the positions of the particles in the NESS. We show that the NESS has a conditionally independent and identically distributed (CIID) structure that enables us to compute various physical observables exactly for arbitrary $\psi(\tau)$. These observables include the average density, extreme value and order statistics, the spacing distribution between consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in a given interval centered at the origin. We discuss the universal features of the large $N$ scaling behaviors of these observables for different choices of the resetting protocol $\psi(\tau)$. Our results provide an interesting example of a stochastic control whereby, by tuning the inter-reset distribution $\psi(\tau)$, one can generate a class of tunable, and yet solvable, strongly correlated NESS in a many-body system.

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arXiv2025

Experimental evidence for strong emergent correlations between particles in a switching trap

We experimentally study a system of $N = 4$ two-dimensional Brownian particles, each confined in a harmonic trap with identical stiffness. The stiffness switches simultaneously between two values at random Poissonian times. This collective switching drives the system into a non-equilibrium stationary state (NESS) with strong long-range correlations between the positions of the particles. Remarkably, we find that, despite the presence of hydrodynamic interactions between the particles mediated by the surrounding fluid, the statistics of some observables are insensitive to hydrodynamic interactions and are well described by the noninteracting theory. Comparing with exact theoretical predictions for noninteracting particles, we observe excellent agreement between theory and experiments for three such observables, namely the correlations between particles, extreme value and order statistics (maxima, minima and ranked positions) and the full counting statistics (i.e., the distribution of the number of particles in a finite interval $[-L, L]$ around the trap center).

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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 $\beta=2$, 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 $f(r) \to f_N(r) = \phi((r-\hat r) \sqrt{N}/\xi)$, 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 $\phi(u)$ at arbitrary $\xi$. For large $\xi$ 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.

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arXiv2025

Resetting Dyson Brownian motion

In this paper, we introduce a new stochastic process of $N$ interacting particles on the line that evolve via Dyson Brownian motion (DBM) with Dyson's index $\beta > 0$ and undergo simultaneous resetting to their initial positions at a constant rate $r$. We call this process the resetting Dyson Brownian motion (RDBM) -- in short the $\beta$-RDBM. For $\beta = 1,2,4$, the positions of the particles in the RDBM can be interpreted as the eigenvalues of a random matrix ensemble where the entries of an $N x N$ Gaussian matrix evolve as simultaneously resetting Brownian motions (with rate $r$) in the presence or absence of a harmonic trap. For $r=0$ and in the presence of a harmonic trap, this system reaches an equilibrium Gibbs-Boltzmann state of the so called Dyson log-gas. However, the stochastic resetting drives the system at long time to a nonequilibrium stationary state (NESS). We compute exactly the joint distribution of the positions of the particles in this NESS for all $\beta>0$ and calculate several observables for large $N$: the average density profile of the gas, the extreme value statistics, the spacing between two consecutive particles and the full counting statistics. We show that a nonzero resetting rate $r>0$ drastically changes the nature of the fluctuations in the stationary state: while the log-gas is rather rigid, the $\beta$-RDBM in its NESS becomes fluffy, i.e., the fluctuations of different observables are of the same order as their mean. In the absence of a harmonic trap, our results for the $\beta = 2$-RDBM can be related to nonintersecting Brownian motions in the presence of resetting. Our model demonstrates interesting effects arising from the interplay between the eigenvalue repulsion and the all-to-all attraction (generated by stochastic resetting) in an interacting particle system. Numerical simulations are in excellent agreement with our analytical results.

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arXiv2025

Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions

We study the diffusion of a particle with a time-dependent diffusion constant $D(t)$ that switches between random values drawn from a distribution $W(D)$ at a fixed rate $r$. Using a renewal approach, we compute exactly the moments of the position of the particle $\langle x^{2n}(t) \rangle$ at any finite time $t$, and for any $W(D)$ with finite moments $\langle D^n \rangle$. For $t \gg 1$, we demonstrate that the cumulants $\langle x^{2n}(t) \rangle_c$ grow linearly with $t$ and are proportional to the free cumulants of a random variable distributed according to $W(D)$. For specific forms of $W(D)$, we compute the large deviations of the position of the particle, uncovering rich behaviors and dynamical transitions of the rate function $I(y=x/t)$. Our analytical predictions are validated numerically with high precision, achieving accuracy up to $10^{-2000}$.

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arXiv2024

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.

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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 $\rho$. They are characterized by a dynamical exponent $z_s=\min(1+s,2)$. We also obtain the gap statistics, described by a roughness exponent $\zeta_s=\frac{1}{2} \min(s,1)$. 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 $1/\gamma$, with $\gamma$ 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 $\gamma$ the results of our earlier work.

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arXiv2024

Run-and-tumble particle in one-dimensional potentials: mean first-passage time and applications

We study a one-dimensional run-and-tumble particle (RTP), which is a prototypical model for active system, moving within an arbitrary external potential. Using backward Fokker-Planck equations, we derive the differential equation satisfied by its mean first-passage time (MFPT) to an absorbing target, which, without any loss of generality, is placed at the origin. Depending on the shape of the potential, we identify four distinct ``phases'', with a corresponding expression for the MFPT in every case, which we derive explicitly. To illustrate these general expressions, we derive explicit formulae for two specific cases which we study in detail: a double-well potential and a logarithmic potential. We then present different applications of these general formulae to (i) the generalization of the Kramer's escape law for an RTP in the presence of a potential barrier, (ii) the ``trapping'' time of an RTP moving in a harmonic well and (iii) characterizing the efficiency of the optimal search strategy of an RTP subjected to stochastic resetting. Our results reveal that the MFPT of an RTP in an external potential exhibits a far more complex and, at times, counter-intuitive behavior compared to that of a passive particle (e.g., Brownian) in the same potential.

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arXiv2024

The number of minima in random landscapes generated by constrained random walk and L\'evy flights: universal properties

We provide a uniform framework to compute the exact distribution of the number of minima/maxima in three different random walk landscape models in one dimension. The landscape is generated by the trajectory of a discrete-time continuous space random walk with arbitrary symmetric and continuous jump distribution at each step. In model I, we consider a ``free'' random walk of $N$ steps. In model II, we consider a ``meander landscape'' where the random walk, starting at the origin, stays non-negative up to $N$ steps. In model III, we study a ``first-passage landscape'' which is generated by the trajectory of a random walk that starts at the origin and stops when it crosses the origin for the first time. We demonstrate that while the exact distribution of the number of minima is different in the three models, for each model it is universal for all $N$, in the sense that it does not depend on the jump distribution as long as it is symmetric and continuous. In the last two cases we show that this universality follows from a non trivial mapping to the Sparre Andersen theorem known for the first-passage probability of discrete-time random walks with symmetric and continuous jump distribution. Our analytical results are in excellent agreement with our numerical simulations.

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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.

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