arXiv subjects

Benjamin De Bruyne

Publications and source records attributed to Benjamin De Bruyne.

16 recordsLinked to original sources

arXiv2025

The Eigenvector Bead Process

We investigate the overlap matrix between the eigenvectors of a Wigner matrix $H_{N+K}$ of size $(N+K)\times(N+K)$ and those of its principal minor $H_N$ of size $N\times N$, for both the real symmetric ($β=1$) and complex Hermitian ($β=2$) ensembles, in the regime where $N \to \infty$ while $K$ remains fixed. Our analysis yields two main results. (i) In the \emph{bulk} of the spectrum, an eigenvector of $H_{N+K}$ associated with an eigenvalue at energy level $E$ projects primarily onto eigenvectors of $H_N$ located at the same local spectral level. This phenomenon, which we call \emph{local projection}, highlights a robust stability of the eigenbasis under matrix growth. (ii) At the \emph{spectral edge}, the change of basis between the leading eigenspaces of consecutive minors is asymptotically governed by a random antisymmetric perturbation of order $N^{-1/3}$. In both cases, we provide the asymptotic law of the overlaps expressed in terms of the Airy and Sine kernels. We further extend our analysis to the case of Wishart matrices, that is, sample covariance matrices of the form $W = X^{\!\top} X$, where $X \in \mathbb{R}^{T \times N}$ is a matrix with i.i.d.\ random entries. We establish analogous results for the overlaps between eigenvectors of consecutive minors of $W$, both in the bulk and at the spectral edges (soft and hard). The limiting laws share the same universal structure as in the Wigner case, up to explicit constants depending on the aspect ratio $q = N/T$. This demonstrates the universality of the eigenvector overlap process across distinct random matrix ensembles.

math.PR↗

arXiv2023

Linear statistics for Coulomb gases: higher order cumulants

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

math-ph↗

arXiv2023

PhD thesis "Extreme value statistics and optimization problems in stochastic processes"

This thesis is devoted to the study of extreme value statistics in stochastic processes and their applications. In the first part, we obtain exact analytical results on the extreme value statistics of both discrete-time and continuous-time random walks. In particular, we focus on the gap statistics of random walks and exhibit their asymptotic universality with respect to the jump distribution in the limit of a large number of steps. In addition, we compute the asymptotic behavior of the expected maximum of random walks in the presence of a bridge constraint and reveal a rich behavior in their finite-size correction. Moreover, we compute the expected length of the convex hull of Brownian motion confined in a disk and show that it converges slowly to the perimeter of the disk with a stretched exponential decay. In the second part, we focus on numerically sampling rare trajectories of stochastic processes. We introduce an efficient method to sample bridge discrete-time random walks. We illustrate it and apply it to various examples. We further extend the method to other stochastic processes, both Markovian and non-Markovian. We apply our method to sample surviving particles in the presence of a periodic trapping environment. Finally, we discuss several optimization problems in stochastic processes involving extreme value statistics. In particular, we introduce a new technique to optimally control dynamical systems undergoing a resetting policy.

cond-mat.stat-mech↗

arXiv2023

Resetting in Stochastic Optimal Control

``When in a difficult situation, it is sometimes better to give up and start all over again''. While this empirical truth has been regularly observed in a wide range of circumstances, quantifying the effectiveness of such a heuristic strategy remains an open challenge. In this paper, we combine the notions of optimal control and stochastic resetting to address this problem. The emerging analytical framework allows not only to measure the performance of a given restarting policy but also to obtain the optimal strategy for a wide class of dynamical systems. We apply our technique to a system with a final reward and show that the reward value must be larger than a critical threshold for resetting to be effective. Our approach, analogous to the celebrated Hamilton-Jacobi-Bellman paradigm, provides the basis for the investigation of realistic restarting strategies across disciplines. As an application, we show that the framework can be applied to an epidemic model to predict the optimal lockdown policy.

cond-mat.stat-mech↗

arXiv2022

Transport properties of diffusive particles conditioned to survive in trapping environments

We consider a one-dimensional Brownian motion with diffusion coefficient $D$ in the presence of $n$ partially absorbing traps with intensity $β$, separated by a distance $L$ and evenly spaced around the initial position of the particle. We study the transport properties of the process conditioned to survive up to time $t$. We find that the surviving particle first diffuses normally, before it encounters the traps, then undergoes a period of transient anomalous diffusion, after which it reaches a final diffusive regime. The asymptotic regime is governed by an effective diffusion coefficient $D_\text{eff}$, which is induced by the trapping environment and is typically different from the original one. We show that when the number of traps is \emph{finite}, the environment enhances diffusion and induces an effective diffusion coefficient that is systematically equal to $D_\text{eff}=2D$, independently of the number of the traps, the trapping intensity $β$ and the distance $L$. On the contrary, when the number of traps is \emph{infinite}, we find that the environment inhibits diffusion with an effective diffusion coefficient that depends on the traps intensity $β$ and the distance $L$ through a non-trivial scaling function $D_\text{eff}=D \mathcal{F}(βL/D)$, for which we obtain a closed-form. Moreover, we provide a rejection-free algorithm to generate surviving trajectories by deriving an effective Langevin equation with an effective repulsive potential induced by the traps. Finally, we extend our results to other trapping environments.

cond-mat.stat-mech↗

arXiv2022

Universal order statistics for random walks & Lévy flights

We consider one-dimensional discrete-time random walks (RWs) of $n$ steps, starting from $x_0=0$, with arbitrary symmetric and continuous jump distributions $f(η)$, including the important case of Lévy flights. We study the statistics of the gaps $Δ_{k,n}$ between the $k^\text{th}$ and $(k+1)^\text{th}$ maximum of the set of positions $\{x_1,\ldots,x_n\}$. We obtain an exact analytical expression for the probability distribution $P_{k,n}(Δ)$ valid for any $k$ and $n$, and jump distribution $f(η)$, which we then analyse in the large $n$ limit. For jump distributions whose Fourier transform behaves, for small $q$, as $\hat f (q) \sim 1 - |q|^μ$ with a Lévy index $0< μ\leq 2$, we find that, the distribution becomes stationary in the limit of $n\to \infty$, i.e. $\lim_{n\to \infty} P_{k,n}(Δ)=P_k(Δ)$. We obtain an explicit expression for its first moment $\mathbb{E}[Δ_{k}]$, valid for any $k$ and jump distribution $f(η)$ with $μ>1$, and show that it exhibits a universal algebraic decay $ \mathbb{E}[Δ_{k}]\sim k^{1/μ-1} Γ\left(1-1/μ\right)/π$ for large $k$. Furthermore, for $μ>1$, we show that in the limit of $k\to\infty$ the stationary distribution exhibits a universal scaling form $P_k(Δ) \sim k^{1-1/μ} \mathcal{P}_μ(k^{1-1/μ}Δ)$ which depends only on the Lévy index $μ$, but not on the details of the jump distribution. We compute explicitly the limiting scaling function $\mathcal{P}_μ(x)$ in terms of Mittag-Leffler functions. For $1< μ<2$, we show that, while this scaling function captures the distribution of the typical gaps on the scale $k^{1/μ-1}$, the atypical large gaps are not described by this scaling function since they occur at a larger scale of order $k^{1/μ}$.

cond-mat.stat-mech↗

arXiv2022

Optimal Resetting Brownian Bridges

We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time $t_f$ is finite and the searcher returns to its starting point at $t_f$. This is simply a Brownian motion with a Poissonian resetting rate $r$ to the origin which is constrained to start and end at the origin at time $t_f$. We first provide a rejection-free algorithm to generate such resetting bridges in all dimensions by deriving an effective Langevin equation with an explicit space-time dependent drift $\tilde μ({\bf x},t)$ and resetting rate $\tilde r({\bf x}, t)$. We also study the efficiency of the search process in one-dimension by computing exactly various observables such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. Surprisingly, we find that there exists an optimal resetting rate $r^*$ that maximizes the search efficiency, even in the presence of a bridge constraint. We show however that the physical mechanism responsible for this optimal resetting rate for bridges is entirely different from resetting Brownian motions without the bridge constraint.

cond-mat.stat-mech↗

arXiv2021

Statistics of the maximum and the convex hull of a Brownian motion in confined geometries

We consider a Brownian particle with diffusion coefficient $D$ in a $d$-dimensional ball of radius $R$ with reflecting boundaries. We study the maximum $M_x(t)$ of the trajectory of the particle along the $x$-direction at time $t$. In the long time limit, the maximum converges to the radius of the ball $M_x(t) \to R$ for $t\to \infty$. We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations $Δ(t) = [R-M_x(t)]/R$ in the limit of large $t$ in all dimensions. We find that the distribution of $Δ(t)$ exhibits a rich variety of behaviors depending on the dimension $d$. These results are obtained by establishing a connection between this problem and the narrow escape time problem. We apply our results in $d=2$ to study the convex hull of the trajectory of the particle in a disk of radius $R$ with reflecting boundaries. We find that the mean perimeter $\langle L(t)\rangle$ of the convex hull exhibits a slow convergence towards the perimeter of the circle $2πR$ with a stretched exponential decay $2πR-\langle L(t)\rangle \propto \sqrt{R}(Dt)^{1/4} \,e^{-2\sqrt{2Dt}/R}$. Finally, we generalise our results to other confining geometries, such as the ellipse with reflecting boundaries. Our results are corroborated by thorough numerical simulations.

cond-mat.stat-mech↗

arXiv2021

Generating stochastic trajectories with global dynamical constraints

We propose a method to exactly generate Brownian paths $x_c(t)$ that are constrained to return to the origin at some future time $t_f$, with a given fixed area $A_f = \int_0^{t_f}dt\, x_c(t)$ under their trajectory. We derive an exact effective Langevin equation with an effective force that accounts for the constraint. In addition, we develop the corresponding approach for discrete-time random walks, with arbitrary jump distributions including Lévy flights, for which we obtain an effective jump distribution that encodes the constraint. Finally, we generalise our method to other types of dynamical constraints such as a fixed occupation time on the positive axis $T_f=\int_0^{t_f}dt\, Θ\left[x_c(t)\right]$ or a fixed generalised quadratic area $\mathcal{A}_f=\int_0^{t_f}dt \,x_c^2(t)$.

cond-mat.stat-mech↗

arXiv2021

Survival probability of random walks leaping over traps

We consider one-dimensional discrete-time random walks (RWs) in the presence of finite size traps of length $\ell$ over which the RWs can jump. We study the survival probability of such RWs when the traps are periodically distributed and separated by a distance $L$. We obtain exact results for the mean first-passage time and the survival probability in the special case of a double-sided exponential jump distribution. While such RWs typically survive longer than if they could not leap over traps, their survival probability still decreases exponentially with the number of steps. The decay rate of the survival probability depends in a non-trivial way on the trap length $\ell$ and exhibits an interesting regime when $\ell\rightarrow 0$ as it tends to the ratio $\ell/L$, which is reminiscent of strongly chaotic deterministic systems. We generalize our model to continuous-time RWs, where we introduce a power-law distributed waiting time before each jump. In this case, we find that the survival probability decays algebraically with an exponent that is independent of the trap length. Finally, we derive the diffusive limit of our model and show that, depending on the chosen scaling, we obtain either diffusion with uniform absorption, or diffusion with periodically distributed point absorbers.

cond-mat.stat-mech↗

arXiv2021

Generating discrete-time constrained random walks and Lévy flights

We introduce a method to exactly generate bridge trajectories for discrete-time random walks, with arbitrary jump distributions, that are constrained to initially start at the origin and return to the origin after a fixed time. The method is based on an effective jump distribution that implicitly accounts for the bridge constraint. It is illustrated on various jump distributions and is shown to be very efficient in practice. In addition, we show how to generalize the method to other types of constrained random walks such as generalized bridges, excursions, and meanders.

cond-mat.stat-mech↗

arXiv2021

Generating constrained run-and-tumble trajectories

We propose a method to exactly generate bridge run-and-tumble trajectories that are constrained to start at the origin with a given velocity and to return to the origin after a fixed time with another given velocity. The method extends the concept of effective Langevin equations, valid for Markovian stochastic processes such as Brownian motion, to a non-Markovian stochastic process driven by a telegraphic noise, with exponentially decaying correlations. We obtain effective space-time dependent tumbling rates that implicitly accounts for the bridge constraint. We extend the method to other types of constrained run-and-tumble particles such as excursions and meanders. The method is implemented numerically and is shown to be very efficient.

cond-mat.stat-mech↗

arXiv2021

Expected maximum of bridge random walks & Lévy flights

We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions $f(η)$, including the case of Lévy flights. We study the expected maximum ${\mathbb E}[M_n]$ of bridge RWs, i.e., RWs starting and ending at the origin after $n$ steps. We obtain an exact analytical expression for ${\mathbb E}[M_n]$ valid for any $n$ and jump distribution $f(η)$, which we then analyze in the large $n$ limit up to second leading order term. For jump distributions whose Fourier transform behaves, for small $k$, as $\hat f(k) \sim 1 - |a\, k|^μ$ with a Lévy index $0<μ\leq 2$ and an arbitrary length scale $a>0$, we find that, at leading order for large $n$, ${\mathbb E}[M_n]\sim a\, h_1(μ)\, n^{1/μ}$. We obtain an explicit expression for the amplitude $h_1(μ)$ and find that it carries the signature of the bridge condition, being different from its counterpart for the free random walk. For $μ=2$, we find that the second leading order term is a constant, which, quite remarkably, is the same as its counterpart for the free RW. For generic $0< μ< 2$, this second leading order term is a growing function of $n$, which depends non-trivially on further details of $\hat f (k)$, beyond the Lévy index $μ$. Finally, we apply our results to compute the mean perimeter of the convex hull of the $2d$ Rouse polymer chain and of the $2d$ run-and-tumble particle, as well as to the computation of the survival probability in a bridge version of the well-known "lamb-lion" capture problem.

cond-mat.stat-mech↗

arXiv2021

Wigner function for noninteracting fermions in hard wall potentials

The Wigner function $W_N({\bf x}, {\bf p})$ is a useful quantity to characterize the quantum fluctuations of an $N$-body system in its phase space. Here we study $W_N({\bf x}, {\bf p})$ for $N$ noninteracting spinless fermions in a $d$-dimensional spherical hard box of radius $R$ at temperature $T=0$. In the large $N$ limit, the local density approximation (LDA) predicts that $W_N({\bf x}, {\bf p}) \approx 1/(2 π\hbar)^d$ inside a finite region of the $({\bf x}, {\bf p})$ plane, namely for $|{\bf x}| < R$ and $|{\bf p}| < k_F$ where $k_F$ is the Fermi momentum, while $W_N({\bf x}, {\bf p})$ vanishes outside this region, or "droplet", on a scale determined by quantum fluctuations. In this paper we investigate systematically, in this quantum region, the structure of the Wigner function along the edge of this droplet, called the Fermi surf. In one dimension, we find that there are three distinct edge regions along the Fermi surf and we compute exactly the associated nontrivial scaling functions in each regime. We also study the momentum distribution $\hat ρ_N(p)$ and find a striking algebraic tail for very large momenta $\hat ρ_N(p) \propto 1/p^4$, well beyond $k_F$, reminiscent of a similar tail found in interacting quantum systems (discussed in the context of Tan's relation). We then generalize these results to higher $d$ and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension~$d$.

cond-mat.stat-mech↗

arXiv2021

Survival probability of a run-and-tumble particle in the presence of a drift

We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate $γ$, the (new) velocity of the particle is drawn randomly from a distribution $W(v)$. We study the survival probability $S(x,t)$ of a particle starting from $x \geq 0$ up to time $t$ and obtain an explicit expression for its double Laplace transform (with respect to both $x$ and $t$) for an arbitrary velocity distribution $W(v)$, not necessarily symmetric. This result is obtained as a consequence of Spitzer's formula, which is well known in the theory of random walks and can be viewed as a generalization of the Sparre Andersen theorem. We then apply this general result to the specific case of a two-state particle with velocity $\pm v_0$, the so-called persistent random walk (PRW), and in the presence of a constant drift $μ$ and obtain an explicit expression for $S(x,t)$, for which we present more detailed results. Depending on the drift $μ$, we find a rich variety of behaviours for $S(x,t)$, leading to three distinct cases: (i) subcritical drift $-v_0\!<\!μ\!<\! v_0$, (ii) supercritical drift $μ< -v_0$ and (iii) critical drift $μ=-v_0$. In these three cases, we obtain exact analytical expressions for the survival probability $S(x,t)$ and establish connections with existing formulae in the mathematics literature. Finally, we discuss some applications of these results to record statistics and to the statistics of last-passage times.

cond-mat.stat-mech↗

arXiv2019

Inferring crystal electronic properties from experimental data sets through Semidefinite Programming

Constructing a quantum description of crystals from scattering experiments is of paramount importance to explain their macroscopic properties and to evaluate the pertinence of theoretical ab-initio models. While reconstruction methods of the one-electron reduced density matrix have already been proposed, they are usually tied to strong assumptions that limit and may introduce bias in the model. The goal of this paper is to infer a one-electron reduced density matrix (1-RDM) with minimal assumptions. We have found that the mathematical framework of Semidefinite Programming can achieve this goal. Additionally, it conveniently addresses the nontrivial constraints on the 1-RDM which were major hindrances for the existing models. The framework established in this work can be used as a reference to interpret experimental results. This method has been applied to the crystal of dry ice and provides very satisfactory results when compared with periodic ab-initio calculations.

cond-mat.mtrl-sci↗