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Ofer Zeitouni

Publications and source records attributed to Ofer Zeitouni.

At least 127 records · Page 7Linked to original sources
arXiv2010

Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three

We consider the quenched and the averaged (or annealed) large deviation rate functions $I_q$ and $I_a$ for space-time and (the usual) space-only RWRE on $\mathbb{Z}^d$. By Jensen's inequality, $I_a\leq I_q$. In the space-time case, when $d\geq3+1$, $I_q$ and $I_a$ are known to be equal on an open set containing the typical velocity $ξ_o$. When $d=1+1$, we prove that $I_q$ and $I_a$ are equal only at $ξ_o$. Similarly, when d=2+1, we show that $I_a<I_q$ on a punctured neighborhood of $ξ_o$. In the space-only case, we provide a class of non-nestling walks on $\mathbb{Z}^d$ with d=2 or 3, and prove that $I_q$ and $I_a$ are not identically equal on any open set containing $ξ_o$ whenever the walk is in that class. This is very different from the known results for non-nestling walks on $\mathbb{Z}^d$ with $d\geq4$.

math.PR↗
arXiv2009

Consistent Minimal Displacement of Branching Random Walks

Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in \mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $Λ(\cdot)$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by $\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a constant, which for convenience may be taken to be 0. With $\bar S_v=\max\{S_w:{\rm $w$ is on the geodesic connecting the root to $v$}\}$, define $L_n=\min_{v\in \mathbb{D}_n} \bar S_v$. We prove that $L_n/n^{1/3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.

math.PR↗
arXiv2009

Deterministic approximation for the cover time of trees

We present a deterministic algorithm that given a tree T with n vertices, a starting vertex v and a slackness parameter epsilon > 0, estimates within an additive error of epsilon the cover and return time, namely, the expected time it takes a simple random walk that starts at v to visit all vertices of T and return to v. The running time of our algorithm is polynomial in n/epsilon, and hence remains polynomial in n also for epsilon = 1/n^{O(1)}. We also show how the algorithm can be extended to estimate the expected cover (without return) time on trees.

cs.DS↗
arXiv2009

Tightness for a family of recursion equations

In this paper we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of a branching random walk in one dimension and the cover time of a symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings.

math.PR↗
arXiv2009

A CLT for regularized sample covariance matrices

We consider the spectral properties of a class of regularized estimators of (large) empirical covariance matrices corresponding to stationary (but not necessarily Gaussian) sequences, obtained by banding. We prove a law of large numbers (similar to that proved in the Gaussian case by Bickel and Levina), which implies that the spectrum of a banded empirical covariance matrix is an efficient estimator. Our main result is a central limit theorem in the same regime, which to our knowledge is new, even in the Gaussian setup.

math.PR↗
arXiv2008

Searching for a trail of evidence in a maze

Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables have common distribution N(0,1) while under the alternative, there is an unknown path along which random variables have distribution $N(μ,1)$, $μ> 0$, and distribution N(0,1) away from it. For which values of the mean shift $μ$ can one reliably detect and for which values is this impossible? Consider, for example, the usual regular lattice with vertices of the form \[\{(i,j):0\le i,-i\le j\le i and j has the parity of i\}\] and oriented edges $(i,j)\to (i+1,j+s)$, where $s=\pm1$. We show that for paths of length $m$ starting at the origin, the hypotheses become distinguishable (in a minimax sense) if $μ_m\gg1/\sqrt{\log m}$, while they are not if $μ_m\ll1/\log m$. We derive equivalent results in a Bayesian setting where one assumes that all paths are equally likely; there, the asymptotic threshold is $μ_m\approx m^{-1/4}$. We obtain corresponding results for trees (where the threshold is of order 1 and independent of the size of the tree), for distributions other than the Gaussian and for other graphs. The concept of the predictability profile, first introduced by Benjamini, Pemantle and Peres, plays a crucial role in our analysis.

math.ST↗
arXiv2008

On Information Rates of the Fading Wyner Cellular Model via the Thouless Formula for the Strip

We apply the theory of random Schrödinger operators to the analysis of multi-users communication channels similar to the Wyner model, that are characterized by short-range intra-cell broadcasting. With $H$ the channel transfer matrix, $HH^\dagger$ is a narrow-band matrix and in many aspects is similar to a random Schrödinger operator. We relate the per-cell sum-rate capacity of the channel to the integrated density of states of a random Schrödinger operator; the latter is related to the top Lyapunov exponent of a random sequence of matrices via a version of the Thouless formula. Unlike related results in classical random matrix theory, limiting results do depend on the underlying fading distributions. We also derive several bounds on the limiting per-cell sum-rate capacity, some based on the theory of random Schrödinger operators, and some derived from information theoretical considerations. Finally, we get explicit results in the high-SNR regime for some particular cases.

cs.IT↗
arXiv2008

On Certain Large Random Hermitian Jacobi Matrices with Applications to Wireless Communications

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-SNR, when an intra-cell 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.

cs.IT↗
arXiv2008

A quenched invariance principle for certain ballistic random walks in i.i.d. environments

We prove that every random walk in i.i.d. environment in dimension greater than or equal to 2 that has an almost sure positive speed in a certain direction, an annealed invariance principle and some mild integrability condition for regeneration times also satisfies a quenched invariance principle. The argument is based on intersection estimates and a theorem of Bolthausen and Sznitman.

math.PR↗
arXiv2007

A law of large numbers for finite-range dependent random matrices

We consider random hermitian matrices in which distant above-diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that the limit has algebraic Stieltjes transform by an argument based on dimension theory of noetherian local rings.

math.PR↗
arXiv2006

Multiscale analysis of exit distributions for random walks in random environments

We present a multiscale analysis for the exit measures from large balls in Z^d, d\geq 3, of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont and Kupianien. The analysis is based on propagating estimates on the variational distance between the exit measure and that of simple random walk, in addition to estimates on the variational distance between smoothed versions of these quantities.

math.PR↗
arXiv2006

Shortest spanning trees and a counterexample for random walks in random environments

We construct forests that span $\mathbb{Z}^d$, $d\geq2$, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For $d\geq3$, two independent copies of such forests, pointing in opposite directions, can be pruned so as to become disjoint. From this, we construct in $d\geq3$ a stationary, polynomially mixing and uniformly elliptic environment of nearest-neighbor transition probabilities on $\mathbb{Z}^d$, for which the corresponding random walk disobeys a certain zero--one law for directional transience.

math.PR↗
arXiv2006

A Central Limit Theorem for biased random walks on Galton-Watson trees

Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $λ$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $λ/(λ+d_v)$, and moves to each of the offspring with probability $1/(λ+d_v)$. It is known that this walk has an a.s. constant speed $\v=\lim_n |X_n|/n$ (where $|X_n|$ is the distance of $X_n$ from the root), with $\v>0$ for $ 0<λ m$ the walk is positive recurrent, and there is no CLT.) The most interesting case by far is $λ=m$, where the CLT has the following form: for almost every ${\cal T}$, the ratio $|X_{[nt]}|/\sqrt{n}$ converges in law as $n \to \infty$ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle (previously, such a measure was explicitly known only for $λ=1$) and the construction of appropriate harmonic coordinates.

math.PR↗
arXiv2006

Random Walk in Dynamic Markovian Random Environment

We consider a model, introduced by Boldrighini, Minlos and Pellegrinotti, of random walks in dynamical random environments on the integer lattice Z^d with d>=1. In this model, the environment changes over time in a Markovian manner, independently across sites, while the walker uses the environment at its current location in order to make the next transition. In contrast with the cluster expansions approach of Boldrighini, Minlos and Pellegrinotti, we follow a probabilistic argument based on regeneration times. We prove an annealed SLLN and invariance principle for any dimension, and provide a quenched invariance principle for dimension d > 7, providing for d>7 an alternative to the analytical approach of Boldrighini, Minlos and Pellegrinotti, with the added benefit that it is valid under weaker assumptions. The quenched results use, in addition to the regeneration times already mentioned, a technique introduced by Bolthausen and Sznitman.

math.PR↗
arXiv2006

Late points for random walks in two dimensions

Let $\mathcal{T}_n(x)$ denote the time of first visit of a point $x$ on the lattice torus $\mathbb {Z}_n^2=\mathbb{Z}^2/n\mathbb{Z}^2$ by the simple random walk. The size of the set of $α$, $n$-late points $\mathcal{L}_n(α)=\{x\in \mathbb {Z}_n^2:\mathcal{T}_n(x)\geq α\frac{4}π(n\log n)^2\}$ is approximately $n^{2(1-α)}$, for $α\in (0,1)$ [$\mathcal{L}_n(α)$ is empty if $α>1$ and $n$ is large enough]. These sets have interesting clustering and fractal properties: we show that for $β\in (0,1)$, a disc of radius $n^β$ centered at nonrandom $x$ typically contains about $n^{2β(1-α/β^2)}$ points from $\mathcal{L}_n(α)$ (and is empty if $β<\sqrtα $), whereas choosing the center $x$ of the disc uniformly in $\mathcal{L}_n(α)$ boosts the typical number of $α, n$-late points in it to $n^{2β(1-α)}$. We also estimate the typical number of pairs of $α$, $n$-late points within distance $n^β$ of each other; this typical number can be significantly smaller than the expected number of such pairs, calculated by Brummelhuis and Hilhorst [Phys. A 176 (1991) 387--408]. On the other hand, our results show that the number of ordered pairs of late points within distance $n^β$ of each other is larger than what one might predict by multiplying the total number of late points, by the number of late points in a disc of radius $n^β$ centered at a typical late point.

math.PR↗
arXiv2004

A CLT for a band matrix model

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincare type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices.

math.PR↗
arXiv2004

Concentration of permanent estimators for certain large matrices

Let A_n=(a_{ij})_{i,j=1}^n be an n\times n positive matrix with entries in [a,b], 0<a\le b. Let X_n=(\sqrta_{ij}x_{ij})_{i,j=1}^n be a random matrix, where {x_{ij}} are i.i.d. N(0,1) random variables. We show that for large n, \det (X_n^TX_n) concentrates sharply at the permanent of A_n, in the sense that n^{-1}\log (\det(X_n^TX_n)/perA_n)\to_{n\to\infty}0 in probability.

math.PR↗