arXiv subjects

Grégory Schehr

Publications and source records attributed to Grégory Schehr.

At least 19 recordsLinked to original sources

arXiv2026

Largest eigenvalue statistics of wavefront shaping in complex scattering media

In wavefront shaping, light, sound, and other waves are focused through complex scattering media onto one or more target positions, and the resulting intensity enhancement is quantified by the enhancement factor. While reproducible enhancement is crucial in experiments, the fluctuations of the enhancement factor remain largely unexplored. Here, we combine experiments, numerical simulations, and exact random-matrix theory to determine its full distribution for multi-target focusing. Exact finite-size random-matrix predictions accurately describe both the mean enhancement factor and its fluctuations beyond the asymptotic Marčenko-Pastur regime, whenever long-range mesoscopic correlations are negligible (e.g., in weakly scattering media or when only a limited number of input channels is controlled). In contrast, strongly scattering media exhibit giant enhancement-factor fluctuations that increasingly exceed these parameter-free predictions as the number of controlled input channels increases. These findings establish the enhancement factor not only as a measure of focusing performance, but also as a sensitive statistical observable that provides a simple and experimentally accessible probe of long-range mesoscopic correlations.

physics.optics↗

arXiv2026

Exact Stationary State of a $d$-dimensional Run-and-Tumble Particle in a Harmonic Potential

We derive the exact nonequilibrium steady state of a run-and-tumble particle (RTP) in $d$ dimensions confined in an isotropic harmonic trap $V(\mathbf r)=μr^{2}/2$, with $r=\|\mathbf r\|$. Rotational invariance reduces the problem to the stationary single-coordinate marginal $p_X(x)$, from which the radial distribution $p_R(r)$ and the full joint stationary density follow by explicit integral transforms. We first focus on a generalized trapped RTP in one dimension, where post-tumble velocities are drawn from an arbitrary distribution $W(v)$. Using a Kesten-type recursion, we represent its stationary position in terms of a stick-breaking (or Dirichlet) process, yielding closed-form expressions for its distribution and its moments. Specializing $W(v)$ to the projected velocity law of an isotropic RTP, we reconstruct $p_R(r)$ and the full joint distribution of all the coordinates in $d=1,2,3$. In $d=1$ and $d=2$, the radial law simplifies to a beta distribution, while in $d=3$, we derive closed-form expressions for $p_R(r)$ and the stationary joint distribution $P(x,y,z)$, which differ from a beta distribution. In all cases, we characterize a persistence-controlled shape transition at the turning surface $r=v_0/μ$, where $v_0$ is the self-propulsion speed. We further include thermal noise characterized by a diffusion coefficient $D>0$, showing that the stationary law is a Gaussian convolution of the $D=0$ result, which regularizes turning-point singularities and controls the crossover between persistence- and diffusion-dominated regimes as $D \to 0$ and $D \to \infty$ respectively. All analytical predictions are systematically validated against numerical simulations.

cond-mat.stat-mech↗

arXiv2026

Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices

We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble.

math.PR↗

arXiv2026

The Wishart--Rosenzweig--Porter random matrix ensemble

In recent years the Rosenzweig--Porter (RP) ensemble, obtained by adding a diagonal matrix with independent and identically distributed elements to a Gaussian random matrix, has been widely used as a minimal model for the emergence of fractal eigenstates in complex many-body systems. A key open question concerns the robustness of its phase diagram when the assumption of independent and uncorrelated entries is relaxed -- an assumption that simplifies its analysis, but is generally violated in realistic quantum systems. In this work, we take a first step in this direction by considering a deformed Wishart (rather than Gaussian) random matrix, which we dub the ``Wishart--RP'' ensemble. Using perturbation theory, as well as the cavity and replica methods and the Dyson Brownian motion approach, we characterize its phase diagram and localization properties. Remarkably, we show that the level compressibility, which quantifies spectral correlations in the fractal phase, coincides with that of the Gaussian RP model, thereby extending the universality conjectured in [SciPost Phys. 14, 110 (2023)] beyond the fully uncorrelated setting. We confirm our results with numerical tests.

cond-mat.stat-mech↗

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↗

arXiv2024

Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs

We study the spectral properties of the adjacency matrix in the giant connected component of Erdös-Rényi random graphs, with average connectivity $p$ and randomly distributed hopping amplitudes. By solving the self-consistent cavity equations satisfied by the matrix elements of the resolvent, we compute the probability distribution of the local density of states, which governs the scaling with the system size of the moments of the eigenvectors' amplitudes, as well as several other observables related to the spectral statistics. For small values of $p>1$ above the percolation threshold, we unveil the presence of an exotic delocalized but (weakly) multifractal phase in a broad region of the parameter space, which separates the localized phase found for $p\le1$ from the fully-delocalized GOE-like phase expected for $p\to \infty$. We explore the fundamental physical mechanism underlying the emergence of delocalized multifractal states, rooted in the pronounced heterogeneity in the topology of the graph. This heterogeneity arises from the interplay between strong fluctuations in local degrees and hopping amplitudes, and leads to an effective fragmentation of the graph. We further support our findings by characterizing the level statistics and the two-point spatial correlations within the multifractal phase, and address the ensuing anomalous transport and relaxation properties affecting the quantum dynamical evolution.

cond-mat.dis-nn↗

arXiv2024

Full counting statistics of 1d short-range Riesz gases in confinement

We investigate the full counting statistics (FCS) of a harmonically confined 1d short-range Riesz gas consisting of $N$ particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent $k>1$ which includes the Calogero-Moser model for $k=2$. We examine the probability distribution of the number of particles in a finite domain $[-W, W]$ called number distribution, denoted by $\mathcal{N}(W, N)$. We analyze the probability distribution of $\mathcal{N}(W, N)$ and show that it exhibits a large deviation form for large $N$ characterised by a speed $N^{\frac{3k+2}{k+2}}$ and by a large deviation function of the fraction $c = \mathcal{N}(W, N)/N$ of the particles inside the domain and $W$. We show that the density profiles that create the large deviations display interesting shape transitions as one varies $c$ and $W$. This is manifested by a third-order phase transition exhibited by the large deviation function that has discontinuous third derivatives. Monte-Carlo (MC) simulations show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of $\mathcal{N}(W, N)$, obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as $N^{ν_k}$, with $ν_k = (2-k)/(2+k)$. We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite $(-\infty, W])$, linear statistics (the variance), thermodynamic pressure and bulk modulus.

cond-mat.stat-mech↗

arXiv2024

Optimizing the random search of a finite-lived target by a Lévy flight

In many random search processes of interest in chemistry, biology or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a non-trivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a non-conventional tricritical point. These results pave the way to the optimization of search processes under time constraints.

cond-mat.stat-mech↗

arXiv2024

Exact extreme, order and sum statistics in a class of strongly correlated system

Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations.. We consider a set of $N$ independent and identically distributed (i.i.d) random variables $\{X_1,\, X_2,\ldots, X_N\}$ whose common distribution has a parameter $Y$ (or a set of parameters) which itself is random with its own distribution. For a fixed value of this parameter $Y$, the $X_i$ variables are independent and we call them conditionally independent and identically distributed (c.i.i.d). However, once integrated over the distribution of the parameter $Y$, the $X_i$ variables get strongly correlated, yet retaining a solvable structure for various observables, such as for the sum and the extremes of $X_i$'s. This provides a simple procedure to generate a class of solvable strongly correlated systems. We illustrate how this procedure works via three physical examples where $N$ particles on a line perform independent (i) Brownian motions, (ii) ballistic motions with random initial velocities, and (iii) Lévy flights, but they get strongly correlated via {\it simultaneous resetting} to the origin. Our results are verified in numerical simulations. This procedure can be used to generate an endless variety of solvable strongly correlated systems.

cond-mat.stat-mech↗

arXiv2023

Replica approach to the generalized Rosenzweig-Porter model

The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries arguably constitutes the simplest random matrix ensemble displaying a phase with fractal eigenstates, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size $N$ of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. Interestingly, the same scaling function is found to describe the level compressibility of the complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results with numerical tests.

cond-mat.dis-nn↗

arXiv2022

From point processes to quantum optics and back

Some fifty years ago, in her seminal PhD thesis, Odile Macchi introduced permanental and determinantal point processes. Her initial motivation was to provide models for the set of detection times in fundamental bosonic or fermionic optical experiments, respectively. After two rather quiet decades, these point processes have quickly become standard examples of point processes with nontrivial, yet tractable, correlation structures. In particular, determinantal point processes have been since the 1990s a technical workhorse in random matrix theory and combinatorics, and a standard model for repulsive point patterns in machine learning and spatial statistics since the 2010s. Meanwhile, our ability to experimentally probe the correlations between detection events in bosonic and fermionic optics has progressed tremendously. In Part I of this survey, we provide a modern introduction to the concepts in Macchi's thesis and their physical motivation, under the combined eye of mathematicians, physicists, and signal processers. Our objective is to provide a shared basis of knowledge for later cross-disciplinary work on point processes in quantum optics, and reconnect with the physical roots of permanental and determinantal point processes.

math-ph↗

arXiv2022

Run-and-tumble particle in one-dimensional confining potential: Steady state, relaxation and first passage properties

We study the dynamics of a one-dimensional run and tumble particle subjected to confining potentials of the type $V(x) = α\, |x|^p$, with $p>0$. The noise that drives the particle dynamics is telegraphic and alternates between $\pm 1$ values. We show that the stationary probability density $P(x)$ has a rich behavior in the $(p, α)$-plane. For $p>1$, the distribution has a finite support in $[x_-,x_+]$ and there is a critical line $α_c(p)$ that separates an active-like phase for $α> α_c(p)$ where $P(x)$ diverges at $x_\pm$, from a passive-like phase for $α< α_c(p)$ where $P(x)$ vanishes at $x_\pm$. For $p<1$, the stationary density $P(x)$ collapses to a delta function at the origin, $P(x) = δ(x)$. In the marginal case $p=1$, we show that, for $α< α_c$, the stationary density $P(x)$ is a symmetric exponential, while for $α> α_c$, it again is a delta function $P(x) = δ(x)$. For the special cases $p=2$ and $p=1$, we obtain exactly the full time-dependent distribution $P(x,t)$, that allows us to study how the system relaxes to its stationary state. In addition, in these two cases, we also study analytically the full distribution of the first-passage time to the origin. Numerical simulations are in complete agreement with our analytical predictions.

cond-mat.stat-mech↗

arXiv2022

An exactly solvable predator prey model with resetting

We study a simple model of a diffusing particle (the prey) that on encounter with one of a swarm of diffusing predators can either perish or be reset to its original position at the origin. We show that the survival probability of the prey up to time $t$ decays algebraically as $\sim t^{-θ(p, γ)}$ where the exponent $θ$ depends continuously on two parameters of the model, with $p$ denoting the probability that a prey survives upon encounter with a predator and $γ= D_A/(D_A+D_B)$ where $D_A$ and $D_B$ are the diffusion constants of the prey and the predator respectively. We also compute exactly the probability distribution $P(N|t_c)$ of the total number of encounters till the capture time $t_c$ and show that it exhibits an anomalous large deviation form $P(N|t_c)\sim t_c^{- Φ\left(\frac{N}{\ln t_c}=z\right)}$ for large $t_c$. The rate function $Φ(z)$ is computed explicitly. Numerical simulations are in excellent agreement with our analytical results.

cond-mat.stat-mech↗

arXiv2020

Intermittent resetting potentials

We study the non-equilibrium steady states and first passage properties of a Brownian particle with position $X$ subject to an external confining potential of the form $V(X)=μ|X|$, and that is switched on and off stochastically. Applying the potential intermittently generates a physically realistic diffusion process with stochastic resetting toward the origin, a topic which has recently attracted a considerable interest in a variety of theoretical contexts but has remained challenging to implement in lab experiments. The present system exhibits rich features, not observed in previous resetting models. The mean time needed by a particle starting from the potential minimum to reach an absorbing target located at a certain distance can be minimized with respect to the switch-on and switch-off rates. The optimal rates undergo continuous or discontinuous transitions as the potential strength $μ$ is varied across non-trivial values. A discontinuous transition with metastable behavior is also observed for the optimal strength at fixed rates.

cond-mat.stat-mech↗

arXiv2020

Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids

We present a numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids established in [Phys. Rev. Lett. 116, 015902 (2016)]. For soft sphere interactions, we obtain the numerical solution by an iterative algorithm and a straightforward discretization of time. We also discuss the case of hard spheres, for which we first derive analytically the dynamical mean field theory as a non-trivial limit of the soft sphere one. We present numerical results for the memory function and the mean square displacement. Our results reproduce and extend kinetic theory in the dilute or short-time limit, while they also describe dynamical arrest towards the glass phase in the dense strongly-interacting regime.

cond-mat.stat-mech↗

arXiv2019

The convex hull of the run-and-tumble particle in a plane

We study the statistical properties of the convex hull of a planar run-and-tumble particle (RTP), also known as the "persistent random walk", where the particle/walker runs ballistically between tumble events at which it changes its direction randomly. We consider two different statistical ensembles where we either fix (i) the total number of tumblings $n$ or (ii) the total duration $t$ of the time interval. In both cases, we derive exact expressions for the average perimeter of the convex hull and then compare to numerical estimates finding excellent agreement. Further, we numerically compute the full distribution of the perimeter using Markov chain Monte Carlo techniques, in both ensembles, probing the far tails of the distribution, up to a precision smaller than $10^{-100}$. This also allows us to characterize the rare events that contribute to the tails of these distributions.

cond-mat.stat-mech↗

arXiv2018

Gap statistics close to the quantile of a random walk

We consider a random walk of $n$ steps starting at $x_0=0$ with a double exponential (Laplace) jump distribution. We compute exactly the distribution $p_{k,n}(Δ)$ of the gap $d_{k,n}$ between the $k^{\rm th}$ and $(k+1)^{\rm th}$ maxima in the limit of large $n$ and large $k$, with $α=k/n$ fixed. We show that the typical fluctuations of the gaps, which are of order $O( n^{-1/2})$, are described by a universal $α$-dependent distribution, which we compute explicitly. Interestingly, this distribution has an inverse cubic tail, which implies a non-trivial $n$-dependence of the moments of the gaps. We also argue, based on numerical simulations, that this distribution is universal, i.e. it holds for more general jump distributions (not only the Laplace distribution), which are continuous, symmetric with a well defined second moment. Finally, we also compute the large deviation form of the gap distribution $p_{αn,n}(Δ)$ for $Δ=O(1)$, which turns out to be non-universal.

cond-mat.stat-mech↗

arXiv2018

Ground state energy of noninteracting fermions with a random energy spectrum

We derive analytically the full distribution of the ground-state energy of $K$ non-interacting fermions in a disordered environment, modelled by a Hamiltonian whose spectrum consists of $N$ i.i.d.~random energy levels with distribution $p(\varepsilon)$ (with $\varepsilon \geq 0$), in the same spirit as the `Random Energy Model'. We show that for each fixed $K$, the distribution $P_{K,N}(E_0)$ of the ground-state energy $E_0$ has a universal scaling form in the limit of large $N$. We compute this universal scaling function and show that it depends only on $K$ and the exponent $α$ characterizing the small $\varepsilon$ behaviour of $p(\varepsilon) \sim \varepsilon^α$. We compared the analytical predictions with results from numerical simulations. For this purpose we employed a sophisticated importance-sampling algorithm that allowed us to obtain the distributions over a large range of the support down to probabilities as small as $10^{-160}$. We found asymptotically a very good agreement between analytical predictions and numerical results.

cond-mat.dis-nn↗