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Gregory Schehr

Publications and source records attributed to Gregory Schehr.

At least 19 recordsLinked to original sources
arXiv2026

Density and Current Fluctuations for Run-and-Tumble Particles With and Without Interactions

We study density and current fluctuations in a one-dimensional gas of run-and-tumble particles (RTPs), whose intrinsic velocities switch between two opposite values at a constant rate. We consider particles with and without pair interactions, including regularized Riesz gases, and also obtain corresponding results for Brownian particles. By linearizing the Dean-Kawasaki (DK) equations around a homogeneous stationary state, we compute density correlations and the variance of the integrated current through the origin. For repulsive Riesz interactions decaying as $1/|x|^s$ with $s>0$, this variance grows at long times as $t^{1/2}$ for $s>1$, but as $t^{s/(1+s)}$ for $0<s<1$. Starting from the same DK description, but retaining the Poisson statistics of direction reversals, we derive a macroscopic fluctuation theory (MFT) for RTPs with smooth pair interaction potential, which allows to compute the large time large deviation of the time integrated current. We show that, in the absence of interactions, the MFT saddle point equations can be solved explicitly using a generalized Cole-Hopf transform. This allows to recover and generalize previous results for the current distribution obtained by microscopic calculations, for RTPs with annealed or quenched step initial conditions. Beyond these distributions, MFT allows to compute the optimal density histories producing a prescribed current through the origin. We find that these density profiles, for both RTPs and Brownian particles, exhibit a nontrivial space-time structure. We also study stationary current fluctuations on an interval connected to reservoirs. Finally, we analyze the MFT equations perturbatively in the interaction. In both the infinite-line and reservoir geometries, we obtain explicit first-order corrections to the current generating functions for short-range interacting RTPs and Brownian particles.

cond-mat.stat-mech↗
arXiv2026

Linear statistics in the Coulomb gas: cumulants beyond rotational invariance

We consider a classical Coulomb gas of $N$ particles in dimension $d\geq2$, confined by a general external potential $U$ in the large-$N$ limit. In that limit the density of particles has a compact support called the droplet. We study smooth linear statistics ${\mathcal L}_N=\sum_{i=1}^N f({\bf x}_i)$ at large $N$. While the leading variance of ${\mathcal L}_N$ is by now well understood in many settings, much less is known about higher cumulants. When $f({\bf x})=f(|{\bf x}|)$ and the potential is rotationally invariant, these cumulants were computed recently at leading order. We derive explicit formulas for the leading third and fourth cumulants for general smooth droplets, without assuming that either the potential or the test function is radial. The answers are expressed as intrinsic boundary integrals over the equilibrium droplet. The third cumulant depends on the cube of the jump of the normal derivative of the harmonic extension of $f$, while the fourth cumulant also involves the exterior Dirichlet-to-Neumann operator, the mean curvature of the boundary, and normal derivatives of the background density. Our derivation uses the tilted potential $U_s=U+s f/β$, the associated deformation of the droplet, and shape derivatives of the Dirichlet energy. This deformation has a natural interpretation as the infinitesimal motion of the free boundary in the obstacle-problem formulation of the equilibrium measure. Our formulas extend previous studies by Wiegmann and Zabrodin in two dimensions for $β=2$. In their setting, we provide a rigorous proof of the limiting third and fourth cumulants, supporting the proposed formulas for general Coulomb gases. We specialize the general formulas to the disk, the half-plane and the sphere, and check them against exactly solvable non-radial deformations, including translated elliptic droplets in $d=2$ and spheroidal deformations in $d=3$.

math-ph↗
arXiv2026

Large Deviations in Switching Diffusion: Dynamical Condensation and Diffusivity BBP Transition

We consider a Brownian particle with position $x(t)$ whose diffusivity changes at Poissonian switching times, with each new value drawn independently from a bounded distribution $W(D)$. Assuming $W(D)\sim C(D_{\max}-D)^ν$ near its upper edge, we study the large-deviation regime $x(t)\simeq yt$ with $y = O(1)$ at large times, far beyond the typical diffusive scale. In a recent work Guéneau et al., Phys. Rev. Lett. 135, 067102 (2025), the rate function of the displacement was obtained through a renewal approach and it was shown to display several singularities. In this paper, we carry out a detailed study of the corresponding dynamical phase transitions and trajectory condensation. For moderate $y$, the imposed displacement is shared among $O(t)$ displacement increments accumulated during successive intervals between switches. Beyond a first threshold $y_-$, one interval produces an increment of order $t$ carrying a finite fraction of the total displacement, signaling dynamical condensation. For $ν>1$, a second threshold $y_+>y_-$ is reached when this macroscopic increment carries asymptotically the entire imposed displacement, whereas for $0<ν\leq1$ the two thresholds merge, $y_-=y_+$. In turn, the moment-generating function of $x(t)$ displays a complementary transition from collective contributions of many diffusivities to selection of states near $D_{\max}$. We show that this transition is governed by the same edge mechanism as the Baik-Ben Arous-Péché (BBP) transition, indicating that diffusivity selection and trajectory condensation are two manifestations of the same phenomenon. We validate our predictions for the dynamical condensation against numerical results from rare event sampling.

cond-mat.stat-mech↗
arXiv2026

Emergence of Correlations Between Records and Maximum in Random Time Series

We compute exactly the covariance between the maximum $M$ and the number of records $K$ in a discrete time series of $n$ entries as a quantitative probe of the history leading to an extreme event. For continuous IID series, with a common probability distribution function (PDF) $p(x)$, we find the striking result that $M$ and $K$ are uncorrelated for any $n \geq 1$. We show that this independence is rather fragile and the covariance $C_q(n)$ between $M$ and $K$ becomes nonzero when $p(x)$ becomes discontinuous due to a delta-component at $x=0$ with an amplitude $0< q < 1$. This time series models the occurrence of dry days with probability $q$ in a precipitation time series when there is no rainfall. We show that there is an interesting shape transition at a critical value $q_c$ where the covariance $C_q(n)$, as a function of $n$, changes from monotonic ($q q_c$). We compute the critical strength $q_c$ exactly. Furthermore, for large $n$, we show that $C_q(n)$ decays as a power law, with an exponent that depends on the tail of $p(x)$. We then compare our theoretical predictions with real precipitation data from several weather stations across Germany, finding a very good agreement.

cond-mat.stat-mech↗
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. Since standard renewal methods fail for $m<N$, we introduce a new general Fokker--Planck framework with an exact recursive closure of marginal densities. 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 to any spatial dimension $d$, where the critical value $N_c=6$ remains unchanged, and they are experimentally testable in existing optical-trap setups.

cond-mat.stat-mech↗
arXiv2026

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 $V(r) = (1/2) m ω^2 r^2$, in all dimensions and at all temperatures. The appropriate thermodynamic limit in a trapped Bose gas consists in taking $N\to\infty$ and $ω\to 0$ with their product $ρ= Nω^d$ fixed, where $ρ$ 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 $ρ$ crosses a critical value $ρ_c(β)$, where $β$ 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 $P_β(I,N)$ of $I$ admits a large deviation form $P_β(I,N) \sim e^{-VΦ(I/V)}$, where $V = ω^{-d} \gg 1$. We compute explicitly the rate function $Φ(z)$ 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, $Φ(z)$ becomes independent of $ρ$ for $z 1$. This provides a real-space diagnostic for the BEC transition in the noninteracting Bose gas.

cond-mat.stat-mech↗
arXiv2026

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).

cond-mat.stat-mech↗
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 $V(x)=κ|x|^α$, $α\geq0$, focusing mainly on two analytically tractable cases: harmonic confinement (HC), $α=2$, and box confinement (BC), $α\to\infty$. 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 $α$, the behavior is monotonic for $0<α<α_c=1+\sqrt{5}$ and non-monotonic for $α>α_c$. 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 $α$ and identify, via simulations and scaling arguments, three universality classes: $0\leqα\leq1$, $1<α<\infty$, and the singular limit $α\to\infty$.

cond-mat.stat-mech↗
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

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.

cond-mat.stat-mech↗
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.

cond-mat.stat-mech↗
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 $(1-|x|^2)^α$, $α>-1$, and show that these potentials can be expressed in terms of Gauss hypergeometric functions ${}_2F_1$, reducing to polynomials for special values of $α$. 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 $s=d-1$.

math-ph↗
arXiv2026

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 $ψ(τ)$. This includes the well-known Poissonian case, where $ψ(τ)=re^{-rτ}$, 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 $ψ(τ)$. 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 $ψ(τ)$. Our results provide an interesting example of a stochastic control whereby, by tuning the inter-reset distribution $ψ(τ)$, one can generate a class of tunable, and yet solvable, strongly correlated NESS in a many-body system.

cond-mat.stat-mech↗
arXiv2026

The distribution of the maximum of independent resetting Brownian motions

The probability distribution of the maximum $M_t$ of a single resetting Brownian motion (RBM) of duration $t$ and resetting rate $r$, properly centred and scaled, is known to converge to the standard Gumbel distribution of the classical extreme value theory. This Gumbel law describes the typical fluctuations of $M_t$ around its average $\sim \ln (r t)$ for large $t$ on a scale of $O(1)$. Here we compute the large-deviation tails of this distribution when $M_t = O(t)$ and show that the large-deviation function has a singularity where the second derivative is discontinuous, signalling a dynamical phase transition. Then we consider a collection of independent RBMs with initial (and resetting) positions uniformly distributed with a density $ρ$ over the negative half-line. We show that the fluctuations in the initial positions of the particles modify the distribution of $M_t$. The average over the initial conditions can be performed in two different ways, in analogy with disordered systems: (i) the annealed case where one averages over all possible initial conditions and (ii) the quenched case where one considers only the contributions coming from typical initial configurations. We show that in the annealed case, the limiting distribution of the maximum is characterized by a new scaling function, different from the Gumbel law but the large-deviation function remains the same as in the single particle case. In contrast, for the quenched case, the limiting (typical) distribution remains Gumbel but the large-deviation behaviors are new and nontrivial. Our analytical results, both for the typical as well as for the large-deviation regime of $M_t$, are verified numerically with extremely high precision, down to $10^{-250}$ for the probability density of $M_t$.

cond-mat.stat-mech↗
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.

cond-mat.stat-mech↗
arXiv2025

Extreme statistics and spacing distribution in a Brownian gas correlated by resetting

We study a one-dimensional gas of $N$ Brownian particles that diffuse independently, but are {\it simultaneously} reset to the origin at a constant rate $r$. The system approaches a non-equilibrium stationary state (NESS) with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the $k$-th rightmost particle and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realisation of this resetting gas using optical traps.

cond-mat.stat-mech↗
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 $10^{-2000}$.

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.

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