arXiv subjects

Bertrand Lacroix-A-Chez-Toine

Publications and source records attributed to Bertrand Lacroix-A-Chez-Toine.

At least 19 recordsLinked to original sources

arXiv2026

Parametric correlations in non-Hermitian quantum chaos: random matrix approach

Motivated by the surge of interest in statistics of non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, we address the problem of characterizing the parametric correlations of spectral densities. Considering parameter-dependent ensemble of complex Ginibre matrices we derive an explicit, closed-form expression for the parametric number covariance in the systems of symmetry class $\mathbf{A}$ for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk. This behavior is expected to be universal, as further supported by numerical evidence for the real Ginibre ensemble, non-Hermitian Bernoulli Wigner matrices and bi-unitarily invariant ensembles. We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years.

quant-ph↗

arXiv2026

One-sided large deviations for the ground-state energy of spin glasses

We describe the large deviations above its typical value of the maximal energy of a spin glass with +/-1 spins. Thanks to the relatively explicit description of the rate function we identify, we then show that the latter is asymptotically quadratic near its minimum if and only if an external magnetic field is present. The proof starts from a Parisi-type formula for the fractional moments of the partition function, which we then leverage to obtain the limit of the Laplace transform of the maximum energy. Using convex-duality arguments, we then rewrite this Laplace transform as a supremum over martingales, and thereby deduce the large-deviation principle with explicit rate function.

math.PR↗

arXiv2025

Ground state energy fluctuations of pinned elastic manifolds

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

cond-mat.stat-mech↗

arXiv2024

Superposition of plane waves in high spatial dimensions: from landscape complexity to the deepest minimum value

In this article, we introduce and analyse some statistical properties of a class of models of random landscapes of the form ${\cal H}({\bf x})=\fracμ{2}{\bf x}^2+\sum_{l=1}^M ϕ_l({\bf k}_l\cdot {\bf x}), \, \, {\bf x}\in \mathbb{R}^N,\,\, μ>0 $ where both the functions $ϕ_l(z)$ and vectors ${\bf k}_l$ are random. An important example of such landscape describes superposition of $M$ plane waves with random amplitudes, directions of the wavevectors, and phases, further confined by a parabolic potential of curvature $μ$. Our main efforts are directed towards analysing the landscape features in the limit $N\to \infty, M\to \infty$ keeping $α=M/N$ finite. In such a limit we find (i) the rates of asymptotic exponential growth with $N$ of the mean number of all critical points and of local minima known as the annealed complexities and (ii) the expression for the mean value of the deepest landscape minimum (the ground-state energy). In particular, for the latter we derive the Parisi-like optimisation functional and analyse conditions for the optimiser to reflect various phases for different values of $μ$ and $α$: replica-symmetric, one-step and full replica symmetry broken, as well as criteria for continuous, Gardner and random first order transitions between different phases.

cond-mat.dis-nn↗

arXiv2023

Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass

We derive, within the replica formalism, a generalisation of the Crisanti-Sommers formula to describe the large deviation function (LDF) ${\cal L}(e)$ for the speed-$N$ atypical fluctuations of the intensive ground-state energy $e$ of a generic spherical spin-glass in the presence of a random external magnetic field of variance $Γ$. We then analyse our exact formula for the LDF in much detail for the Replica symmetric, single step Replica Symmetry Breaking (1-RSB) and Full Replica Symmetry Breaking (FRSB) situations. Our main qualitative conclusion is that the level of RSB governing the LDF may be different from that for the typical ground state. We find that while the deepest ground-states are always controlled by a LDF of replica symmetric form, beyond a finite threshold $e\geq e_{t}$ a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the $p=2$ spherical spin-glass and the rigorous results by Dembo and Zeitouni which we are able to reproduce invoking an 1-RSB pattern. Finally at an even larger critical energy $e_{c}\geq e_{t}$, acting as a "wall", the LDF diverges logarithmically, which we interpret as a change in the large deviation speed from $N$ to a faster growth. In addition, we show that in the limit $Γ\to 0$ the LDF takes non-trivial scaling forms (i) ${\cal L}(e) \sim G((e-e_c)/Γ)$ in the vicinity of the wall (ii) ${\cal L}(e) \sim Γ^{ην} F((e-e_{\rm typ})/Γ^ν)$ in the vicinity of the typical energy, characterised by two new exponents $η\geq 1$ and $ν$ characterising universality classes. Via matching the latter allows us to formulate several conjectures concerning the regime of {\it typical fluctuations}, identified as $e-e_{\rm typ} \sim N^{-1/η}$ and $Γ\sim N^{-1/(ην)}$.

cond-mat.stat-mech↗

arXiv2023

On-Site Potential Creates Complexity in Systems with Disordered Coupling

We calculate the average number of critical points $\overline{\mathcal{N}}$ of the energy landscape of a many-body system with disordered two-body interactions and a weak on-site potential. We find that introducing a weak nonlinear on-site potential dramatically increases $\overline{\mathcal{N}}$ to exponential in system size and give a complete picture of the organization of critical points. Our results extend solvable spin-glass models to physically more realistic models and are of relevance to glassy systems, nonlinear oscillator networks and many-body interacting systems.

cond-mat.dis-nn↗

arXiv2022

Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates

We consider a nonlinear autonomous random dynamical system of $N$ degrees of freedom coupled by Gaussian random interactions and characterized by a continuous spectrum $n_μ(λ)$ of real positive relaxation rates. Using Kac-Rice formalism, the computation of annealed complexities (both of stable equilibria and of all types of equilibria) is reduced to evaluating the averages involving the modulus of the determinant of the random Jacobian matrix. In the limit of large system $N\gg 1$ we derive exact analytical results for the complexities for short-range correlated coupling fields, extending results previously obtained for the "homogeneous" relaxation spectrum characterised by a single relaxation rate. We show the emergence of a "topology trivialisation" transition from a complex phase with exponentially many equilibria to a simple phase with a single equilibrium as the magnitude of the random field is decreased. Within the complex phase the complexity of stable equilibria undergoes an additional transition from a phase with exponentially small probability to find a single equilibrium to a phase with exponentially many stable equilibria as the fraction of gradient component of the field is increased. The behaviour of the complexity at the transition is found only to depend on the small $λ$ behaviour of the spectrum of relaxation rates $n_μ(λ)$ and thus conjectured to be universal. We also provide some insights into a counting problem motivated by a paper by B. Spivak and A. Zyuzin of 2004 about wave scattering in a disordered nonlinear medium.

cond-mat.stat-mech↗

arXiv2022

Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity

Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in $\mathbb{R}^N$ obtained by superimposing $M>N$ plane waves of random wavevectors and amplitudes. For this landscape we show how to compute the "annealed complexity" controlling the asymptotic growth rate of the mean number of stationary points as $N\to \infty$ at fixed ratio $α=M/N>1$. The framework of this computation requires us to study spectral properties of $N\times N$ matrices $W=KTK^T$, where $T$ is diagonal with $M$ mean zero i.i.d. real normally distributed entries, and all $MN$ entries of $K$ are also i.i.d. real normal random variables. We suggest to call the latter Gaussian Marchenko-Pastur Ensemble, as such matrices appeared in the seminal 1967 paper by those authors. We compute the associated mean spectral density and evaluate some moments and correlation functions involving products of characteristic polynomials for such and related matrices.

cond-mat.stat-mech↗

arXiv2021

Spectral Analysis of Current Fluctuations in Periodically Driven Stochastic Systems

Current fluctuations play an important role in non-equilibrium statistical mechanics, and are a key object of interest in both theoretical studies and in practical applications. So far, most of the studies were devoted to the fluctuations in the time-averaged current -- the zero frequency Fourier component of the time dependent current. However, in many practical applications the fluctuations at other frequencies are of equal importance. Here we study the full frequency dependence of current statistics in periodically driven stochastic systems. First, we show a general method to calculate the current statistics, valid even when the current's frequency is incommensurate with the driving frequency, breaking the time periodicity of the system. Somewhat surprisingly, we find that the cumulant generating function (CGF), that encodes all the statistics of the current, is composed of a continuous background at any frequency accompanied by either positive or negative peaks at current's frequencies commensurate with the driving frequency. We show that cumulants of increasing orders display peaks at an increasing number of locations but with decreasing amplitudes that depend on the commensurate ratio of frequencies. All these peaks are then transcribed in the behaviour of the CGF. As the measurement time increases, these peaks become sharper but keep the same amplitude and eventually lead to discontinuities of the CGF at all the frequencies that are commensurate with the driving frequency in the limit of infinitely long measurement. We demonstrate our formalism and its consequences on three types of models: an underdamped Brownian particle in a periodically driven harmonic potential; a periodically driven run-and-tumble particle; and a two-state system.

cond-mat.stat-mech↗

arXiv2020

Universal survival probability for a correlated random walk and applications to records

We consider a model of space-continuous one-dimensional random walk with simple correlation between the steps: the probability that two consecutive steps have same sign is $q$ with $0\leq q\leq 1$. The parameter $q$ allows thus to control the persistence of the random walk. We compute analytically the survival probability of a walk of $n$ steps, showing that it is independent of the jump distribution for any finite $n$. This universality is a consequence of the Sparre-Andersen theorem for random walks with uncorrelated and symmetric steps. We then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk, which show the same universality. In particular, we show that the distribution of the number of records for a walk of $n\gg 1$ steps is the same as for a random walk with $n_{\rm eff}(q)=n/(2(1-q))$ uncorrelated and symmetrically distributed steps. We also show that in the regime where $n\to \infty$ and $q\to 1$ with $y=n(1-q)$, this model converges to the run-and-tumble particle, a persistent random walk often used to model the motion of bacteria. Our theoretical results are confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2020

Extreme value statistics for branching run-and-tumble particles

The extreme value statistics of active matter offer significant insight into their unique properties. A phase transition has recently been reported in a model of branching run-and-tumble particles, describing the spatial spreading of an evolving colony of active matter in one-dimension. In a "persistent" phase, the particles form macroscopic robust clusters that ballistically propagate as a whole while in an "intermittent" phase, particles are isolated instead. We focus our study on the fluctuations of the rightmost position $x_{\max}(t)$ reached by time $t$ for this model. At long time, as the colony progressively invades the unexplored region, the cumulative probability of $x_{\max}(t)$ is described by a travelling front. The transition has a remarkable impact on this front. In the intermittent phase it is qualitatively similar to the front satisfying the Fisher-KPP equation, which famously describes the extreme value statistics of the non-active branching Brownian motion. A dramatically different behaviour appears in the persistent phase, where activity imparts the front with unexpected and unusual features which we compute exactly.

cond-mat.stat-mech↗

arXiv2019

PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"

In this thesis, we study three physically relevant models of strongly correlated random variables: trapped fermions, random matrices and random walks. In the first part, we show several exact mappings between the ground state of a trapped Fermi gas and ensembles of random matrix theory. The Fermi gas is inhomogeneous in the trapping potential and in particular there is a finite edge beyond which its density vanishes. Going beyond standard semi-classical techniques (such as local density approximation), we develop a precise description of the spatial statistics close to the edge. This description holds for a large universality class of hard edge potentials. We apply these results to compute the statistics of the position of the fermion the farthest away from the centre of the trap, the number of fermions in a given domain (full counting statistics) and the related bipartite entanglement entropy. Our analysis also provides solutions to open problems of extreme value statistics in random matrix theory. We obtain for instance a complete description of the fluctuations of the largest eigenvalue in the complex Ginibre ensemble. In the second part of the thesis, we study extreme value questions for random walks. We consider the gap statistics, which requires to take explicitly into account the discreteness of the process. This question cannot be solved using the convergence of the process to its continuous counterpart, the Brownian motion. We obtain explicit analytical results for the gap statistics of the walk with a Laplace distribution of jumps and provide numerical evidence suggesting the universality of these results.

cond-mat.stat-mech↗

arXiv2019

Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble

We study the Ginibre ensemble of $N \times N$ complex random matrices and compute exactly, for any finite $N$, the full distribution as well as all the cumulants of the number $N_r$ of eigenvalues within a disk of radius $r$ centered at the origin. In the limit of large $N$, when the average density of eigenvalues becomes uniform over the unit disk, we show that for $0<r<1$ the fluctuations of $N_r$ around its mean value $\langle N_r \rangle \approx N r^2$ display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order ${\cal O}(N^{1/4})$, (ii) an intermediate regime where $N_r - \langle N_r \rangle = {\cal O}(\sqrt{N})$, and (iii) a large deviation regime where $N_r - \langle N_r \rangle = {\cal O}({N})$. This intermediate behaviour (ii) had been overlooked in previous studies and we show here that it ensures a smooth matching between the typical and the large deviation regimes. In addition, we demonstrate that this intermediate regime controls all the (centred) cumulants of $N_r$, which are all of order ${\cal O}(\sqrt{N})$, and we compute them explicitly. Our analytical results are corroborated by precise "importance sampling" Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2019

Distribution of Brownian coincidences

We study the probability distribution, $P_N(T)$, of the coincidence time $T$, i.e. the total local time of all pairwise coincidences of $N$ independent Brownian walkers. We consider in details two geometries: Brownian motions all starting from $0$, and Brownian bridges. Using a Feynman-Kac representation for the moment generating function of this coincidence time, we map this problem onto some observables in three related models (i) the propagator of the Lieb Liniger model of quantum particles with pairwise delta function interactions (ii) the moments of the partition function of a directed polymer in a random medium (iii) the exponential moments of the solution of the Kardar-Parisi-Zhang equation. Using these mappings, we obtain closed formulae for the probability distribution of the coincidence time, its tails and some of its moments. Its asymptotics at large and small coincidence time are also obtained for arbitrary fixed endpoints. The universal large $T$ tail, $P_N(T) \sim \exp(- 3 T^2/(N^3-N))$ is obtained, and is independent of the geometry. We investigate the large deviations in the limit of a large number of walkers through a Coulomb gas approach. Some of our analytical results are compared with numerical simulations.

cond-mat.stat-mech↗

arXiv2019

Extremes of $2d$ Coulomb gas: universal intermediate deviation regime

In this paper, we study the extreme statistics in the complex Ginibre ensemble of $N \times N$ random matrices with complex Gaussian entries, but with no other symmetries. All the $N$ eigenvalues are complex random variables and their joint distribution can be interpreted as a $2d$ Coulomb gas with a logarithmic repulsion between any pair of particles and in presence of a confining harmonic potential $v(r) \propto r^2$. We study the statistics of the eigenvalue with the largest modulus $r_{\max}$ in the complex plane. The typical and large fluctuations of $r_{\max}$ around its mean had been studied before, and they match smoothly to the right of the mean. However, it remained a puzzle to understand why the large and typical fluctuations to the left of the mean did not match. In this paper, we show that there is indeed an intermediate fluctuation regime that interpolates smoothly between the large and the typical fluctuations to the left of the mean. Moreover, we compute explicitly this "intermediate deviation function" (IDF) and show that it is universal, i.e. independent of the confining potential $v(r)$ as long as it is spherically symmetric and increases faster than $\ln r^2$ for large $r$ with an unbounded support. If the confining potential $v(r)$ has a finite support, i.e. becomes infinite beyond a finite radius, we show via explicit computation that the corresponding IDF is different. Interestingly, in the borderline case where the confining potential grows very slowly as $v(r) \sim \ln r^2$ for $r \gg 1$ with an unbounded support, the intermediate regime disappears and there is a smooth matching between the central part and the left large deviation regime.

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

Entanglement Entropy and Full Counting Statistics for $2d$-Rotating Trapped Fermions

We consider $N$ non-interacting fermions in a $2d$ harmonic potential of trapping frequency $ω$ and in a rotating frame at angular frequency $Ω$, with $0<ω- Ω\ll ω$. At zero temperature, the fermions are in the non-degenerate lowest Landau level and their positions are in one to one correspondence with the eigenvalues of an $N\times N$ complex Ginibre matrix. For large $N$, the fermion density is uniform over the disk of radius $\sqrt{N}$ centered at the origin and vanishes outside this disk. We compute exactly, for any finite $N$, the Rényi entanglement entropy of order $q$, $S_q(N,r)$, as well as the cumulants of order $p$, $\langle{N_r^{p}}\rangle_c$, of the number of fermions $N_r$ in a disk of radius $r$ centered at the origin. For $N \gg 1$, in the (extended) bulk, i.e., for $0 < r/\sqrt{N} < 1$, we show that $S_q(N,r)$ is proportional to the number variance ${\rm Var}\,(N_r)$, despite the non-Gaussian fluctuations of $N_r$. This relation breaks down at the edge of the fermion density, for $r \approx \sqrt{N}$, where we show analytically that $S_q(N,r)$ and ${\rm Var}\,(N_r)$ have a different $r$-dependence.

cond-mat.stat-mech↗

arXiv2018

Non-interacting fermions in hard-edge potentials

We consider the spatial quantum and thermal fluctuations of non-interacting Fermi gases of $N$ particles confined in $d$-dimensional non-smooth potentials. We first present a thorough study of the spherically symmetric pure hard-box potential, with vanishing potential inside the box, both at $T=0$ and $T>0$. We find that the correlations near the wall are described by a "hard edge" kernel, which depend both on $d$ and $T$, and which is different from the "soft edge" Airy kernel, and its higher $d$ generalizations, found for smooth potentials. We extend these results to the case where the potential is non-uniform inside the box, and find that there exists a family of kernels which interpolate between the above "hard edge" kernel and the "soft edge" kernels. Finally, we consider one-dimensional singular potentials of the form $V(x)\sim |x|^{-γ}$ with $γ>0$. We show that the correlations close to the singularity at $x=0$ are described by this "hard edge" kernel for $1\leqγ<2$ while they are described by a broader family of "hard edge" kernels known as the Bessel kernel for $γ=2$ and, finally by the Airy kernel for $γ>2$. These one-dimensional kernels also appear in random matrix theory, and we provide here the mapping between the $1d$ fermion models and the corresponding random matrix ensembles. Part of these results were announced in a recent Letter, EPL 120, 10006 (2017).

cond-mat.stat-mech↗