arXiv subjects

Gregory Schehr

Publications and source records attributed to Gregory Schehr.

At least 145 records · Page 8Linked to original sources

arXiv2015

Top eigenvalue of a random matrix: large deviations and third order phase transition

We study the fluctuations of the largest eigenvalue $λ_{\max}$ of $N \times N$ random matrices in the limit of large $N$. The main focus is on Gaussian $β$-ensembles, including in particular the Gaussian orthogonal ($β=1$), unitary ($β=2$) and symplectic ($β= 4$) ensembles. The probability density function (PDF) of $λ_{\max}$ consists, for large $N$, of a central part described by Tracy-Widom distributions flanked, on both sides, by two large deviations tails. While the central part characterizes the typical fluctuations of $λ_{\max}$ -- of order ${\cal O}(N^{-2/3})$ --, the large deviations tails are instead associated to extremely rare fluctuations -- of order ${\cal O}(1)$. Here we review some recent developments in the theory of these extremely rare events using a Coulomb gas approach. We discuss in particular the third-order phase transition which separates the left tail from the right tail, a transition akin to the so-called Gross-Witten-Wadia phase transition found in 2-d lattice quantum chromodynamics. We also discuss the occurrence of similar third-order transitions in various physical problems, including non-intersecting Brownian motions, conductance fluctuations in mesoscopic physics and entanglement in a bipartite system.

cond-mat.stat-mech↗

arXiv2015

Dynamical transition in the temporal relaxation of stochastic processes under resetting

A stochastic process, when subject to resetting to its initial condition at a constant rate, generically reaches a non-equilibrium steady state. We study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems. We show that as time progresses, an inner core region around the resetting point reaches the steady state, while the region outside the core is still transient. The boundaries of the core region grow with time as power laws at late times. Alternatively, at a fixed spatial point, the system undergoes a dynamical transition from the transient to the steady state at a characteristic space dependent timescale $t^*(x)$. We calculate analytically in several examples the large deviation function associated with this spatio-temporal fluctuation and show that generically it has a second order discontinuity at a pair of critical points characterizing the edges of the inner core. Our results are verified in the numerical simulations of several models, such as simple diffusion and fluctuating one-dimensional interfaces.

cond-mat.stat-mech↗

arXiv2015

Spatial Extent of Branching Brownian Motion

We study the one dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost ($X_{\max}\geq 0$) and leftmost ($X_{\min} \leq 0$) visited sites up to time $t$. At each time step the existing particles in the system either diffuse (with diffusion constant $D$), die (with rate $a$) or split into two particles (with rate $b$). We focus on the regime $b \leq a$ where these two extreme values $X_{\max}$ and $X_{\min}$ are strongly correlated. We show that at large time $t$, the joint probability distribution function (PDF) of the two extreme points becomes stationary $P(X,Y,t \to \infty) \to p(X,Y)$. Our exact results for $p(X,Y)$ demonstrate that the correlation between $X_{\max}$ and $X_{\min}$ is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF $p(ζ)$ of the (dimensionless) span $ζ= {(X_{\max} - X_{\min})}/{\sqrt{D/b}}$, which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior ${p}(ζ) \sim \frac{1}{2} \left(1 + Δ\right) ζ$ for small spans, with $Δ= \left(\frac{a}{b} -1\right)$. In the critical case ($Δ= 0$) this distribution has a non-trivial power law tail ${p}(ζ) \sim 8 π\sqrt{3} /ζ^3$ for large spans. On the other hand, in the subcritical case ($Δ> 0$), we show that the span distribution decays exponentially as ${p}(ζ) \sim (A^2/2) ζ\exp \left(- \sqrtΔ~ζ\right)$ for large spans, where $A$ is a non-trivial function of $Δ$ which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between $X_{\max}$ and $X_{\min}$. Finally we verify our results via direct Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2015

On certain functionals of the maximum of Brownian motion and their applications

We consider a Brownian motion (BM) $x(τ)$ and its maximal value $x_{\max} = \max_{0 \leq τ\leq t} x(τ)$ on a fixed time interval $[0,t]$. We study functionals of the maximum of the BM, of the form ${\cal O}_{\max}(t)=\int_0^t\, V(x_{\max} - x(τ)) {\rm d} τ$ where $V(x)$ can be any arbitrary function and develop various analytical tools to compute their statistical properties. These tools rely in particular on (i) a "counting paths" method and (ii) a path-integral approach. In particular, we focus on the case where $V(x) = δ(x-r)$, with $r$ a real parameter, which is relevant to study the density of near-extreme values of the BM (the so called density of states), $ρ(r,t)$, which is the local time of the BM spent at given distance $r$ from the maximum. We also provide a thorough analysis of the family of functionals ${T}_α(t)=\int_0^t (x_{\max} - x(τ))^α\, {\rm d}τ$, corresponding to $V(x) = x^α$, with $α$ real. As $α$ is varied, $T_α(t)$ interpolates between different interesting observables. For instance, for $α=1$, $T_{α= 1}(t)$ is a random variable of the "area", or "Airy", type while for $α=-1/2$ it corresponds to the maximum time spent by a ballistic particle through a Brownian random potential. On the other hand, for $α= -1$, it corresponds to the cost of the optimal algorithm to find the maximum of a discrete random walk, proposed by Odlyzko. We revisit here, using tools of theoretical physics, the statistical properties of this algorithm which had been studied before using probabilistic methods. Finally, we extend our methods to constrained BM, including in particular the Brownian bridge, i.e., the Brownian motion starting and ending at the origin.

cond-mat.stat-mech↗

arXiv2015

Statistics of the longest interval in renewal processes

We consider renewal processes where events, which can for instance be the zero crossings of a stochastic process, occur at random epochs of time. The intervals of time between events, $τ_{1},τ_{2},...$, are independent and identically distributed (i.i.d.) random variables with a common density $ρ(τ)$. Fixing the total observation time to $t$ induces a global constraint on the sum of these random intervals, which accordingly become interdependent. Here we focus on the largest interval among such a sequence on the fixed time interval $(0,t)$. Depending on how the last interval is treated, we consider three different situations, indexed by $α=$ I, II and III. We investigate the distribution of the longest interval $\ell^α_{\max}(t)$ and the probability $Q^α(t)$ that the last interval is the longest one. We show that if $ρ(τ)$ decays faster than $1/τ^2$ for large $τ$, then the full statistics of $\ell^α_{\max}(t)$ is given, in the large $t$ limit, by the standard theory of extreme value statistics for i.i.d. random variables, showing in particular that the global constraint on the intervals $τ_i$ does not play any role at large times in this case. However, if $ρ(τ)$ exhibits heavy tails, $ρ(τ)\simτ^{-1-θ}$ for large $τ$, with index $0 <θ<1$, we show that the fluctuations of $\ell^α_{\max}(t)/t$ are governed, in the large $t$ limit, by a stationary universal distribution which depends on both $θ$ and $α$, which we compute exactly. On the other hand, $Q^α(t)$ is generically different from its counterpart for i.i.d. variables (both for narrow or heavy tailed distributions $ρ(τ)$). In particular, in the case $0<θ<1$, the large $t$ behaviour of $Q^α(t)$ gives rise to universal constants (depending also on both $θ$ and $α$) which we compute exactly.

cond-mat.stat-mech↗

arXiv2014

First order transition for the optimal search time of Lévy flights with resetting

We study analytically an intermittent search process in one dimension. There is an immobile target at the origin and a searcher undergoes a discrete time jump process starting at $x_0\geq0$, where successive jumps are drawn independently from an arbitrary jump distribution $f(η)$. In addition, with a probability $0\leq r \leq1$ the position of the searcher is reset to its initial position $x_0$. The efficiency of the search strategy is characterized by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump distribution $f(η)$, initial position $x_0$ and resetting probability $r$, we compute analytically the MFPT. For the heavy-tailed Lévy stable jump distribution characterized by the Lévy index $0<μ< 2$, we show that, for any given $x_0$, the MFPT has a global minimum in the $(μ,r)$ plane at $(μ^*(x_0),r^*(x_0))$. We find a remarkable first-order phase transition as $x_0$ crosses a critical value $x_0^*$ at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2014

Diffusion in periodic, correlated random forcing landscapes

We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically-extended (with period $L$) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent $H \in (0,1)$. While the periodicity ensures that the ultimate long-time behavior is diffusive, the generalised Sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times. These two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient $D_L$: Although one has the typical value $D^{\rm typ}_L \sim \exp(-βL^H)$, we show via an exact analytical approach that the positive moments ($k>0$) scale like $\langle D^k_L \rangle \sim \exp{[-c' (k βL^{H})^{1/(1+H)}]}$, and the negative ones as $\langle D^{-k}_L \rangle \sim \exp(a' (k βL^{H})^2)$, $c'$ and $a'$ being numerical constants and $β$ the inverse temperature. These results demonstrate that $D_L$ is strongly non-self-averaging. We further show that the probability distribution of $D_L$ has a log-normal left tail and a highly singular, one-sided log-stable right tail reminiscent of a Lifshitz singularity.

cond-mat.stat-mech↗

arXiv2014

Branching Brownian Motion Conditioned on Particle Numbers

We study analytically the order and gap statistics of particles at time $t$ for the one dimensional branching Brownian motion, conditioned to have a fixed number of particles at $t$. The dynamics of the process proceeds in continuous time where at each time step, every particle in the system either diffuses (with diffusion constant $D$), dies (with rate $d$) or splits into two independent particles (with rate $b$). We derive exact results for the probability distribution function of $g_k(t) = x_k(t) - x_{k+1}(t)$, the distance between successive particles, conditioned on the event that there are exactly $n$ particles in the system at a given time $t$. We show that at large times these conditional distributions become stationary $P(g_k, t \to \infty|n) = p(g_k|n)$. We show that they are characterised by an exponential tail $p(g_k|n) \sim \exp[-\sqrt{\frac{|b - d|}{2 D}} ~g_k]$ for large gaps in the subcritical ($b < d$) and supercritical ($b > d$) phases, and a power law tail $p(g_k) \sim 8\left(\frac{D}{b}\right){g_k}^{-3}$ at the critical point ($b = d$), independently of $n$ and $k$. Some of these results for the critical case were announced in a recent letter [K. Ramola, S. N. Majumdar and G. Schehr, Phys. Rev. Lett. 112, 210602 (2014)].

cond-mat.stat-mech↗

arXiv2014

On the Gap and Time Interval between the First Two Maxima of Long Random Walks

In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker, starting from $x_0 = 0$, after $n$ time steps (taking the $x$-axis vertical). The jumps $η_i = x_i - x_{i-1}$ are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), $f(η)$, the Fourier transform of which has the small $k$ behavior $1 - \hat f(k) \propto |k|^μ$, with $0 < μ\leq 2$. For $μ=2$, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For $0<μ<2$, the RW is a Lévy flight of index $μ$. We show that the joint PDF of $G_n$ and $L_n$ converges to a well defined stationary bi-variate distribution $p(g,l)$ as the RW duration $n$ goes to infinity. We present a thorough analytical study of the limiting joint distribution $p(g,l)$, as well as of its associated marginals $p_{\rm gap}(g)$ and $p_{\rm time}(l)$, revealing a rich variety of behaviors depending on the tail of $f(η)$ (from slow decreasing algebraic tail to fast decreasing super-exponential tail). We also address the problem for a random bridge where the RW starts and ends at the origin after $n$ time steps. We show that in the large $n$ limit, the PDF of $G_n$ and $L_n$ converges to the {\it same} stationary distribution $p(g,l)$ as in the case of the free-end RW. Finally, we present a numerical check of our analytical predictions. Some of these results were announced in a recent letter [S. N. Majumdar, Ph. Mounaix, G. Schehr, Phys. Rev. Lett. {\bf 111}, 070601 (2013)].

cond-mat.stat-mech↗

arXiv2014

Universal statistics of longest lasting records of random walks and Lévy flights

We study the record statistics of random walks after $n$ steps, $x_0, x_1,\ldots, x_n$, with arbitrary symmetric and continuous distribution $p(η)$ of the jumps $η_i = x_i - x_{i-1}$. We consider the age of the records, i.e. the time up to which a record survives. Depending on how the age of the current last record is defined, we propose three distinct sequences of ages (indexed by $α$ = I, II, III) associated to a given sequence of records. We then focus on the longest lasting record, which is the longest element among this sequence of ages. To characterize the statistics of these longest lasting records, we compute: (i) the probability that the record of the longest age is broken at step $n$, denoted by $Q^α(n)$, which we call the probability of record breaking and: (ii) the duration of the longest lasting record, $\ell_{\max}^α(n)$. We show that both $Q^α(n)$ and the full statistics of $\ell_{\max}^α(n)$ are universal, i.e. independent of the jump distribution $p(η)$. We compute exactly the large $n$ asymptotic behaviors of $Q^α(n)$ as well as $\langle \ell_{\max}^α(n)\rangle$ (when it exists) and show that each case gives rise to a different universal constant associated to random walks (including Lévy flights). While two of them appeared before in the excursion theory of Brownian motion, for which we provide here a simpler derivation, the third case gives rise to a non-trivial new constant $C^{\rm III} = 0.241749 \ldots$ associated to the records of random walks. Other observables characterizing the ages of the records, exhibiting an interesting universal behavior, are also discussed.

cond-mat.stat-mech↗

arXiv2014

Universal Order and Gap Statistics of Critical Branching Brownian Motion

We study the order statistics of one dimensional branching Brownian motion in which particles either diffuse (with diffusion constant $D$), die (with rate $d$) or split into two particles (with rate $b$). At the critical point $b=d$ which we focus on, we show that, at large time $t$, the particles are collectively bunched together. We find indeed that there are two length scales in the system: (i) the diffusive length scale $\sim \sqrt{Dt}$ which controls the collective fluctuations of the whole bunch and (ii) the length scale of the gap between the bunched particles $\sim \sqrt{D/b}$. We compute the probability distribution function $P(g_k,t|n)$ of the $k$th gap $g_k = x_k - x_{k+1}$ between the $k$th and $(k+1)$th particles given that the system contains exactly $n>k$ particles at time $t$. We show that at large $t$, it converges to a stationary distribution $P(g_k,t\to \infty|n) = p(g_k|n)$ with an algebraic tail $p(g_k|n) \sim 8(D/b) g_k^{-3}$, for $g_k \gg 1$, independent of $k$ and $n$. We verify our predictions with Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2014

Maximal distance travelled by N vicious walkers till their survival

We consider $N$ Brownian particles moving on a line starting from initial positions ${\bf{u}}\equiv \{u_1,u_2,\dots u_N\}$ such that $0<u_1 < u_2 < \cdots < u_N$. Their motion gets stopped at time $t_s$ when either two of them collide or when the particle closest to the origin hits the origin for the first time. For $N=2$, we study the probability distribution function $p_1(m|{\bf{u}})$ and $p_2(m|{\bf{u}})$ of the maximal distance travelled by the $1^{\text{st}}$ and $2^{\text{nd}}$ walker till $t_s$. For general $N$ particles with identical diffusion constants $D$, we show that the probability distribution $p_N(m|{\bf u})$ of the global maximum $m_N$, has a power law tail $p_i(m|{\bf{u}}) \sim {N^2B_N\mathcal{F}_{N}({\bf u})}/{m^{ν_N}}$ with exponent $ν_N =N^2+1$. We obtain explicit expressions of the function $\mathcal{F}_{N}({\bf u})$ and of the $N$ dependent amplitude $B_N$ which we also analyze for large $N$ using techniques from random matrix theory. We verify our analytical results through direct numerical simulations.

cond-mat.stat-mech↗

arXiv2013

Fluctuating interfaces subject to stochastic resetting

We study one-dimensional fluctuating interfaces of length $L$ where the interface stochastically resets to a fixed initial profile at a constant rate $r$. For finite $r$ in the limit $L \to \infty$, the system settles into a nonequilibrium stationary state with non-Gaussian interface fluctuations, which we characterize analytically for the Kardar-Parisi-Zhang and Edwards-Wilkinson universality class. Our results are corroborated by numerical simulations. We also discuss the generality of our results for a fluctuating interface in a generic universality class.

cond-mat.stat-mech↗

arXiv2013

Near-extreme eigenvalues and the first gap of Hermitian random matrices

We study the phenomenon of "crowding" near the largest eigenvalue $λ_{\max}$ of random $N \times N$ matrices belonging to the Gaussian Unitary Ensemble (GUE) of random matrix theory. We focus on two distinct quantities: (i) the density of states (DOS) near $λ_{\max}$, $ρ_{\rm DOS}(r,N)$, which is the average density of eigenvalues located at a distance $r$ from $λ_{\max}$ and (ii) the probability density function of the gap between the first two largest eigenvalues, $p_{\rm GAP}(r,N)$. In the edge scaling limit where $r = {\cal O}(N^{-1/6})$, which is described by a double scaling limit of a system of unconventional orthogonal polynomials, we show that $ρ_{\rm DOS}(r,N)$ and $p_{\rm GAP}(r,N)$ are characterized by scaling functions which can be expressed in terms of the solution of a Lax pair associated to the Painlevé XXXIV equation. This provides an alternative and simpler expression for the gap distribution, which was recently studied by Witte, Bornemann and Forrester in Nonlinearity 26, 1799 (2013). Our expressions allow to obtain precise asymptotic behaviors of these scaling functions both for small and large arguments.

math-ph↗

arXiv2013

Asymmetric Lévy flights in the presence of absorbing boundaries

We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution ϕ(η) displaying asymmetric power law tails (i.e. ϕ(η) \sim c/η^{α+1} for large positive jumps and ϕ(η) \sim c/(γ|η|^{α+1}) for large negative jumps, with 0 < α< 2). In absence of boundaries and after a large number of steps n, the probability density function (PDF) of the walker position, x_n, converges to an asymmetric Lévy stable law of stability index αand skewness parameter β=(γ-1)/(γ+1). In particular the right tail of this PDF decays as c n/x_n^{1+α}. Much less is known when the walker is confined, or partially confined, in a region of the space. In this paper we first study the case of a walker constrained to move on the positive semi-axis and absorbed once it changes sign. In this case, the persistence exponent θ_+, which characterizes the algebraic large time decay of the survival probability, can be computed exactly and we show that the tail of the PDF of the walker position decays as c \, n/[(1-θ_+) \, x_n^{1+α}]. This last result can be generalized in higher dimensions such as a planar Lévy walker confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations.

cond-mat.stat-mech↗

arXiv2013

Near-extreme statistics of Brownian motion

We study the statistics of near-extreme events of Brownian motion (BM) on the time interval [0,t]. We focus on the density of states (DOS) near the maximum ρ(r,t) which is the amount of time spent by the process at a distance r from the maximum. We develop a path integral approach to study functionals of the maximum of BM, which allows us to study the full probability density function (PDF) of ρ(r,t) and obtain an explicit expression for the moments, \langle [ρ(r,t)]^k \rangle, for arbitrary integer k. We also study near-extremes of constrained BM, like the Brownian bridge. Finally we also present numerical simulations to check our analytical results.

cond-mat.stat-mech↗

arXiv2013

Exact record and order statistics of random walks via first-passage ideas

While records and order statistics of independent and identically distributed (i.i.d.) random variables X_1, ..., X_N are fully understood, much less is known for strongly correlated random variables, which is often the situation encountered in statistical physics. Recently, it was shown, in a series of works, that one-dimensional random walk (RW) is an interesting laboratory where the influence of strong correlations on records and order statistics can be studied in detail. We review here recent exact results which have been obtained for these questions about RW, using techniques borrowed from the study of first-passage problems. We also present a brief review of the well known (and not so well known) results for records and order statistics of i.i.d. variables.

cond-mat.stat-mech↗

arXiv2013

Anomalous fluctuations of currents in Sinai-type random chains with strongly correlated disorder

We study properties of a random walk in a generalized Sinai model, in which a quenched random potential is a trajectory of a fractional Brownian motion with arbitrary Hurst parameter H, 0< H <1, so that the random force field displays strong spatial correlations. In this case, the disorder-average mean-square displacement grows in proportion to log^{2/H}(n), n being time. We prove that moments of arbitrary order k of the steady-state current J_L through a finite segment of length L of such a chain decay as L^{-(1-H)}, independently of k, which suggests that despite a logarithmic confinement the average current is much higher than its Fickian counterpart in homogeneous systems. Our results reveal a paradoxical behavior such that, for fixed n and L, the mean square displacement decreases when one varies H from 0 to 1, while the average current increases. This counter-intuitive behavior is explained via an analysis of representative realizations of disorder.

cond-mat.dis-nn↗