The density of eigenvalues seen from the soft edge of random matrices in the Gaussian beta-ensembles

We characterize the phenomenon of "crowding" near the largest eigenvalue $\lambda_{\max}$ of random $N \times N$ matrices belonging to the Gaussian $\beta$-ensemble of random matrix theory, including in particular the Gaussian orthogonal ($\beta=1$), unitary ($\beta=2$) and symplectic ($\beta = 4$) ensembles. We focus on two distinct quantities: (i) the density of states (DOS) near $\lambda_{\max}$, $\rho_{\rm DOS}(r,N)$, which is the average density of eigenvalues located at a distance $r$ from $\lambda_{\max}$ (or the density of eigenvalues seen from $\lambda_{\max}$) and (ii) the probability density function of the gap between the first two largest eigenvalues, $p_{\rm GAP}(r,N)$. Using heuristic arguments as well as well numerical simulations, we generalize our recent exact analytical study of the Hermitian case (corresponding to $\beta = 2$). We also discuss some applications of these two quantities to statistical physics models.Comment: 16 pages, 5 figures, contribution to the proceedings of the Workshop "Random Matrix Theory: Foundations and Applications" in Cracow, July 1-6 201

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

We study the fluctuations of the largest eigenvalue $\lambda_{\max}$ of $N \times N$ random matrices in the limit of large $N$. The main focus is on Gaussian $\beta$-ensembles, including in particular the Gaussian orthogonal ($\beta=1$), unitary ($\beta=2$) and symplectic ($\beta = 4$) ensembles. The probability density function (PDF) of $\lambda_{\max}$ 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 $\lambda_{\max}$ -- 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.Comment: 32 pages, 8 figures, contribution to Statphys25 (Seoul, 2013) proceedings. Revised version where references have been added and typos correcte

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.Comment: 5 pages, 3 figures, published versio

Dynamic crossover in the persistence probability of manifolds at criticality

We investigate the persistence properties of critical d-dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d'-dimensional manifold. The persistence probability P(t) shows three distinct long-time decays depending on the value of the parameter \zeta = (D-2+\eta)/z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d-d' and of the critical exponents z and \eta. We find that the asymptotic behavior of P(t) is exponential for \zeta > 1, stretched exponential for 0 <= \zeta <= 1, and algebraic for \zeta < 0. Whereas the exponential and stretched exponential relaxations are not affected by the initial value of the order parameter, we predict and observe a crossover between two different power-law decays when the algebraic relaxation occurs, as in the case d'=d of the global order parameter. We confirm via Monte Carlo simulations our analytical predictions by studying the magnetization of a line and of a plane of the two- and three-dimensional Ising model, respectively, with Glauber dynamics. The measured exponents of the ultimate algebraic decays are in a rather good agreement with our analytical predictions for the Ising universality class. In spite of this agreement, the expected scaling behavior of the persistence probability as a function of time and of the initial value of the order parameter remains problematic. In this context, the non-equilibrium dynamics of the O(n) model in the limit n->\infty and its subtle connection with the spherical model is also discussed in detail.Comment: 23 pages, 6 figures; minor changes, added one figure, (old) fig.4 replaced by the correct fig.

Survival Probability of Random Walks and L\'evy Flights on a Semi-Infinite Line

We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, $f(\eta)$, characterized by a L\'evy index $\mu \in (0,2]$, which includes standard random walks ($\mu=2$) and L\'evy flights ($0<\mu<2$). We study the survival probability, $q(x_0,n)$, representing the probability that the RW stays non-negative up to step $n$, starting initially at $x_0 \geq 0$. Our main focus is on the $x_0$-dependence of $q(x_0,n)$ for large $n$. We show that $q(x_0,n)$ displays two distinct regimes as $x_0$ varies: (i) for $x_0= O(1)$ ("quantum regime"), the discreteness of the jump process significantly alters the standard scaling behavior of $q(x_0,n)$ and (ii) for $x_0 = O(n^{1/\mu})$ ("classical regime") the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for $\mu =2$ this corresponds to the standard Brownian scaling limit). The purpose of this paper is to study how precisely the crossover in $q(x_0,n)$ occurs between the quantum and the classical regime as one increases $x_0$.Comment: 20 pages, 3 figures, revised and accepted versio

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 $p_i(m|{\bf{u}}) \sim {N^2B_N\mathcal{F}_{N}({\bf u})}/{m^{\nu_N}}$ with exponent $\nu_N =N^2+1$. 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.Comment: 28 pages, 9 figure