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We study the directed polymer model for general graphs (beyond Z d ) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an L 2 region, and of very strong disorder, in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on various trees including the Galton–Watson trees, and provide a range of other examples that illustrate counter-examples to intuitive extensions of the Z d /SRW result. [ABSTRACT FROM AUTHOR]

Let TN denote an N × N Toeplitz matrix with finite, N independent symbol a. For EN a noise matrix satisfying mild assumptions (ensuring, in particular, that N−1/2||EN||HS → N → ∞ 0 at a polynomial rate), we prove that the empirical measure of eigenvalues of TN + EN converges to the law of a(U), where U is uniformly distributed on the unit circle in the complex plane. This extends results from [Forum Math. Sigma 7 (2019)] to the non-triangular setup and non-complex Gaussian noise, and confirms predictions obtained in [Linear Algebra Appl. 162/164 (1992), pp. 153-185] using the notion of pseudospectrum. [ABSTRACT FROM AUTHOR]

Lemma 9.1 in the paper is incorrect as stated. [ABSTRACT FROM AUTHOR]

Let C ϵ , S 2 ∗ denote the cover time of the two dimensional sphere by a Wiener sausage of radius ϵ . We prove that C ϵ , S 2 ∗ - 2 A S 2 π log ϵ - 1 - 1 4 log log ϵ - 1 is tight, where A S 2 = 4 π denotes the Riemannian area of S 2 . [ABSTRACT FROM AUTHOR]

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies. [ABSTRACT FROM AUTHOR]

We consider the heat equation with a multiplicative Gaussian potential in dimensions d ≥ 3. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance. [ABSTRACT FROM AUTHOR]

We consider the following detection problem: given a realization of a symmetric matrix X of dimension n , distinguish between the hypothesis that all upper triangular variables are independent and identically distributed (i.i.d). Gaussians variables with mean 0 and variance 1 and the hypothesis, where X is the sum of such matrix and an independent rank-one perturbation. This setup applies to the situation, where under the alternative, there is a planted principal submatrix B of size L for which all upper triangular variables are i.i.d. Gaussians with mean 1 and variance 1, whereas all other upper triangular elements of X not in B are i.i.d. Gaussians variables with mean 0 and variance 1. We refer to this as the “Gaussian hidden clique problem.” When L=(1+\epsilon )\sqrt n ( \epsilon >0 ), it is possible to solve this detection problem with probability 1-on(1) by computing the spectrum of X and considering the largest eigenvalue of X. We prove that this condition is tight in the following sense: when L<(1-\epsilon )\sqrt n no algorithm that examines only the eigenvalues of X can detect the existence of a hidden Gaussian clique, with error probability vanishing as $n\to \infty $ . We prove this result as an immediate consequence of a more general result on rank-one perturbations of $k$ -dimensional Gaussian tensors. In this context, we establish a lower bound on the critical signal-to-noise ratio below which a rank-one signal cannot be detected. [ABSTRACT FROM PUBLISHER]

We derive a large deviations principle for the two-dimensional two-component plasma in a box. As a consequence, we obtain a variational representation for the free energy, and also show that the macroscopic empirical measure of either positive or negative charges converges to the uniform measure. An appendix, written by Wei Wu, discusses applications to the subcritical complex Gaussian multiplicative chaos. [ABSTRACT FROM AUTHOR]

We give a new proof of a recent resolution [ 18 ] by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei [ 19 ] that the moduli of roots of Gaussian Kac polynomials of degree |$n$|⁠ , centered at |$1$| and rescaled by |$n^2$|⁠ , should form a Poisson point process. We use this new approach to verify a conjecture from [ 18 ] that the Poisson statistics are in fact universal. [ABSTRACT FROM AUTHOR]

Let η*n denote the maximum, at time n, of a nonlattice one-dimensional branching random walk ηn possessing (enough) exponential moments. In a seminal paper, Aïdekon (Ann. Probab. 41 (2013) 1362-1426) demonstrated convergence of η*n in law, after centering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing reasoning motivated by Bramson, Ding and Zeitouni (Convergence in law of the maximum of the two-dimensional discrete Gaussian free field; preprint). Instead of spine methods and a careful analysis of the renewal measure for killed random walks, our approach employs a modified version of the second moment method that may be of independent interest. We indicate the modifications needed in order to handle lattice random walks. [ABSTRACT FROM AUTHOR]

We discuss regularization by noise of the spectrum of large random non-normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in *-moments to a regular element a by the addition of a polynomially vanishing Gaussian Ginibre matrix forces the empirical measure of eigenvalues to converge to the Brown measure of a. [ABSTRACT FROM AUTHOR]

We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment in the case that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer [Comp. Math. 30 (1975) 145–168] for random walks in i.i.d. environments. The basic assumption is that the underlying Markov chain is irreducible, either with finite state space or with transition kernel dominated above and below by a probability measure. [Copyright &y& Elsevier]

Curvature-driven flows have been extensively considered from a deterministic point of view. Besides their mathematical interest, they have been shown to be useful for a number of applications including crystal growth, flame propagation, and computer vision. In this paper, we describe a random particle system, evolving on the discretized unit circle, whose profile converges toward the Gauss–Minkowsky transformation of solutions of curve-shortening flows initiated by convex curves. Our approach may be considered as a type of stochastic crystalline algorithm. Our proofs are based on certain techniques from the theory of hydrodynamical limits. [Copyright &y& Elsevier]

We study in this work a special class of multidimensional random walks in random environment for which we are able to prove in a non-perturbative fashion both a law of large numbers and a functional central limit theorem. As an application we provide new examples of diffusive random walks in random environment. In particular we construct examples of diffusive walks which evolve in an environment for which the static expectation of the drift does not vanish. [Copyright &y& Elsevier]

Suppose that the integers are assigned random variables ω[sub i] (taking values in the unit interval), which serve as an environment. This environment defines a random walk X[sub n] (called a RWRE) which, when at i, moves one step to the right with probability ω[sub i] , and one step to the left with probability 1 - ω[sub i] . When the ω[sub i] sequence is i.i.d., Greven and den Hollander (1994) proved a large deviation principle for X[sub n] /n, conditional upon the environment, with deterministic rate function. We consider in this paper large deviations, both conditioned on the environment (quenched) and averaged on the environment (annealed), for the RWRE, in the ergodic environment case. The annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy. We also give a detailed qualitative description of the resulting rate functions. Our techniques differ from those of Greven and den Hollander, and allow us to present also a trajectorial (quenched) large deviation principle. [ABSTRACT FROM AUTHOR]

We consider the stochastic heat equation ∂ s u = 1 2 Δ u + (β V (s , y) - λ) u , with a smooth space-time-stationary Gaussian random field V(s, y), in dimensions d ≥ 3 , with an initial condition u (0 , x) = u 0 (ε x) and a suitably chosen λ ∈ R . It is known that, for β small enough, the diffusively rescaled solution u ε (t , x) = u (ε - 2 t , ε - 1 x) converges weakly to a scalar multiple of the solution u ¯ (t , x) of the heat equation with an effective diffusivity a, and that fluctuations converge, also in a weak sense, to the solution of the Edwards-Wilkinson equation with an effective noise strength ν and the same effective diffusivity. In this paper, we derive a pointwise approximation w ε (t , x) = u ¯ (t , x) Ψ ε (t , x) + ε u 1 ε (t , x) , where Ψ ε (t , x) = Ψ (t / ε 2 , x / ε) , Ψ is a solution of the SHE with constant initial conditions, and u 1 ε is an explicit corrector. We show that Ψ (t , x) converges to a stationary process Ψ ~ (t , x) as t → ∞ , that E | u ε (t , x) - w ε (t , x) | 2 converges pointwise to 0 as ε → 0 , and that ε - d / 2 + 1 (u ε - w ε) converges weakly to 0 for fixed t. As a consequence, we derive new representations of the diffusivity a and effective noise strength ν . Our approach uses a Markov chain in the space of trajectories introduced in [17], as well as tools from homogenization theory. The corrector u 1 ε (t , x) is constructed using a seemingly new approximation scheme on a mesoscopic time scale. [ABSTRACT FROM AUTHOR]

Abstract. Suppose that the integers are assigned i.i.d, random variables {omega[sub x]} (taking values in the interval [1/2, 1)), which serve as an environment. This environment defines a random walk {X[sub k]} (called a RWRE) which, when at x, moves one step to the right with probability omega[sub x], and one step to the left with probability 1 - omega[sub x]. Solomon (1975) determined the almost-sure asymptotic speed (= rate of escape) of a RWRE, in a more general set-up. Dembo, Peres and Zeitouni (1996), following earlier work by Greven and den Hollander (1994) on the quenched case, have computed rough tail asymptotics for the empirical mean of the annealed RWRE. They conjectured the form of the rate function in a full LDP. We prove in this paper their conjecture. The proof is based on a "coarse graining scheme" together with comparison techniques. [ABSTRACT FROM AUTHOR]

We show that for any centered stationary Gaussian process of absolutely integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in |$[0,T]$| is within |$\eta T$| of its mean value, up to an exponentially small in |$T$| probability. [ABSTRACT FROM AUTHOR]

We prove, using probabilistic techniques and analysis on the Wiener space, that the large scale fluctuations of the KPZ equation in d ≥ 3 with a small coupling constant, driven by a white in time and colored in space noise, are given by the Edwards-Wilkinson model. This gives an alternative proof, that avoids perturbation expansions, to the results of Magnen and Unterberger (J Stat Phys 171:543–598, 2018). [ABSTRACT FROM AUTHOR]

We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton–Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrarily slow traps. [ABSTRACT FROM AUTHOR]

In this paper we study the spectrum of certain large random Hermitian Jacobi matrices. These matrices are known to describe certain communication setups. In particular, we are interested in an uplink cellular channel which models mobile users experiencing a soft-handoff situation under joint multicell decoding. Considering rather general fading statistics we provide a closed-form expression for the per-cell sum—rate of this channel in high signal-to-noise ratio (SNR), when an intra-cell time-division multiple-access (TDMA) protocol is employed. Since the matrices of interest are tridiagonal, their eigenvectors can be considered as sequences with second-order linear recurrence. Therefore, the problem is reduced to the study of the exponential growth of products of two-by-two matrices. For the case where K users are simultaneously active in each cell, we obtain a series of lower and upper bound on the high-SNR power offset of the per-cell sum-rate, which are considerably tighter than previously known bounds. [ABSTRACT FROM AUTHOR]

Studies large deviations for random walks in random environment. Key distinction between quenched asymptotics, conditional on the environment and annealed asymptotics, obtained from averaging over environments; Simple random walk on Galton-Watson family tree, arising from a supercritical branching process; Uncertainty in the location of the hitting point.