arXiv subjects

Amir Dembo

Publications and source records attributed to Amir Dembo.

At least 73 records · Page 4Linked to original sources

arXiv2012

Persistence of iterated partial sums

Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1, X_2,....,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{(2)}:=\PP(\max_{1\le i \le n}S_i^{(2)}< 0) \le c\sqrt{\frac{\EE|S_{n+1}|}{(n+1)\EE|X_1|}},$$ with $c \le 6 \sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $\min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{(2)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0 < γ< 1/4$ there exist integrable, zero-mean random variables for which the rate of decay of $p_n^{(2)}$ is $n^{-γ}$.

math.PR↗

arXiv2012

Central limit theorem for biased random walk on multi-type Galton-Watson trees

Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability lambda/[lambda+d(v)], and to each of the offspring with probability 1/[lambda+d(v)]. This walk is recurrent for lambda>=rho and transient for 0 4 for the offspring distributions, we prove the following quenched CLT for lambda-biased random walk at the critical value lambda=rho: for almost every T, the process |X_{floor(nt)}|/sqrt{n} converges in law as n tends to infinity to a reflected Brownian motion rescaled by an explicit constant. This result was proved under some stronger assumptions by Peres-Zeitouni (2008) for single-type Galton-Watson trees. Following their approach, our proof is based on a new explicit description of a reversing measure for the walk from the point of view of the particle (generalizing the measure constructed in the single-type setting by Peres-Zeitouni), and the construction of appropriate harmonic coordinates. In carrying out this program we prove moment and conductance estimates for MGW trees, which may be of independent interest. In addition, we extend our construction of the reversing measure to a biased random walk with random environment (RWRE) on MGW trees, again at a critical value of the bias. We compare this result against a transience-recurrence criterion for the RWRE generalizing a result of Faraud (2011) for Galton-Watson trees.

math.PR↗

arXiv2011

Persistence of iterated partial sums

Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zero-mean, i.i.d. random variables X_k, are negative. We show that p_n is bounded above up to universal constant by the square root of the expected absolute value of the empirical average of {X_k}. A converse bound holds whenever P(-X_1>t) is up to constant exp(-b t) for some b>0 or when P(-X_1>t) decays super-exponentially in t. Consequently, for such random variables we have that p_n decays as n^{-1/4} if X_1 has finite second moment. In contrast, we show that for any 0 < c < 1/4 there exist integrable, zero-mean random variables for which the rate of decay of p_n is n^{-c}.

math.PR↗

arXiv2010

Ising models on locally tree-like graphs

We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the "cavity" prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.

math.PR↗

arXiv2010

Markovian perturbation, response and fluctuation dissipation theorem

We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of "linear response function" in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is, if the initial measure is invariant for the given Markov semi-group, then for any pair of times s<t and nice functions f,g, the dissipation, that is, the derivative in s of the covariance of g(X(t)) and f(X(s)) equals the infinitesimal response at time t and direction g to any Markovian perturbation that alters the invariant measure of X(.) in the direction of f at time s. The same applies in the so called FDT regime near equilibrium, i.e. in the limit s going to infinity with t-s fixed, provided X(s) converges in law to an invariant measure for its dynamics. We provide the response function of two generic Markovian perturbations which we then compare and contrast for pure jump processes on a discrete space, for finite dimensional diffusion processes, and for stochastic spin systems.

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arXiv2009

Gibbs Measures and Phase Transitions on Sparse Random Graphs

Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We review this approach and provide some results towards a rigorous treatment of these problems.

math.PR↗

arXiv2009

Spectral measure of heavy tailed band and covariance random matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $μ$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^σ$ whose (i,j) entry is $σ(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $α$-stable law, $0<α<2$, and $σ$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^σ$ and with appropriate rescaling $a_N$, we prove that the distribution $μ$ of $a_N^{-1}Y_N^σ+ D_N$ converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.

math.PR↗

arXiv2008

Finite size scaling for the core of large random hypergraphs

The (two) core of a hypergraph is the maximal collection of hyperedges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF[2], or iterative decoding of low-density parity-check codes used over the binary erasure channel. Similar structures emerge in a variety of NP-hard combinatorial optimization and decision problems, from vertex cover to satisfiability. For a uniformly chosen random hypergraph of $m=nρ$ vertices and $n$ hyperedges, each consisting of the same fixed number $l\geq3$ of vertices, the size of the core exhibits for large $n$ a first-order phase transition, changing from $o(n)$ for $ρ>ρ_{\mathrm{c}}$ to a positive fraction of $n$ for $ρ<ρ_{\mathrm{c}}$, with a transition window size $Θ(n^{-1/2})$ around $ρ_{\mathrm{c}}>0$. Analyzing the corresponding ``leaf removal'' algorithm, we determine the associated finite-size scaling behavior. In particular, if $ρ$ is inside the scaling window (more precisely, $ρ=ρ_{\mathrm{c}}+rn^{-1/2}$), the probability of having a core of size $Θ(n)$ has a limit strictly between 0 and 1, and a leading correction of order $Θ(n^{-1/6})$. The correction admits a sharp characterization in terms of the distribution of a Brownian motion with quadratic shift, from which it inherits the scaling with $n$. This behavior is expected to be universal for a wide collection of combinatorial problems.

math.PR↗

arXiv2008

A Lower Bound on the Disconnection Time of a Discrete Cylinder

We study the asymptotic behavior for large N of the disconnection time T_N of simple random walk on a discrete cylinder with base a d-dimensional discrete torus of side-length N. When d is sufficiently large, we are able to substantially improve the lower bounds obtained by the authors in a previous article when d is bigger or equal to 2. We show here that the laws of N^(2d)/T_N are tight.

math.PR↗

arXiv2007

How large a disc is covered by a random walk in n steps?

We show that the largest disc covered by a simple random walk (SRW) on $\mathbb{Z}^2$ after n steps has radius n^{1/4+o(1)}, thus resolving an open problem of Révész [Random Walk in Random and Non-Random Environments (1990) World Scientific, Teaneck, NJ]. For any fixed $\ell$, the largest disc completely covered at least $\ell$ times by the SRW also has radius n^{1/4+o(1)}. However, the largest disc completely covered by each of $\ell$ independent simple random walks on $\mathbb{Z}^2$ after $n$ steps is only of radius $n^{1/(2+2\sqrt{\ell})+o(1)}$. We complement this by showing that the radius of the largest disc completely covered at least a fixed fraction $α$ of the maximum number of visits to any site during the first $n$ steps of the SRW on $\mathbb{Z}^2$, is $n^{(1-\sqrtα)/4+o(1)}$. We also show that almost surely, for infinitely many values of $n$ it takes about $n^{1/2+o(1)}$ steps after step n for the SRW to reach the first previously unvisited site (and the exponent 1/2 is sharp). This resolves a problem raised by Révész [Ann. Probab. 21 (1993) 318--328].

math.PR↗

arXiv2006

Limiting dynamics for spherical models of spin glasses at high temperature

We analyze the coupled non-linear integro-differential equations whose solutions is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p-spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds.

math.PR↗

arXiv2006

Valleys and the maximum local time for random walk in random environment

Let $ξ(n, x)$ be the local time at $x$ for a recurrent one-dimensional random walk in random environment after $n$ steps, and consider the maximum $ξ^*(n) = \max_x ξ(n,x)$. It is known that $\limsup ξ^*(n)/n$ is a positive constant a.s. We prove that $\liminf_n (\log\log\log n)ξ^*(n)/n$ is a positive constant a.s.; this answers a question of P. Révész (1990). The proof is based on an analysis of the {\em valleys /} in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time $n$ large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.

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↗

arXiv2006

Spectral measure of large random Hankel, Markov and Toeplitz matrices

We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables $\{X_k\}$ of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables $\{X_{ij}\}_{j>i}$ of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions $γ_H$, $γ_M$ and $γ_T$ of unbounded support. The moments of $γ_H$ and $γ_T$ are the sum of volumes of solids related to Eulerian numbers, whereas $γ_M$ has a bounded smooth density given by the free convolution of the semicircle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables $\{X_{ij}\}_{j>i}$ of mean $m$ and finite variance, scaling the eigenvalues by ${n}$ we prove the almost sure, weak convergence of the spectral measures to the atomic measure at $-m$. If $m=0$, and the fourth moment is finite, we prove that the spectral norm of $\mathbf {M}_n$ scaled by $\sqrt{2n\log n}$ converges almost surely to 1.

math.PR↗

arXiv2004

Cugliandolo-Kurchan equations for dynamics of Spin-Glasses

We study the Langevin dynamics for the family of spherical $p$-spin disordered mean-field models and prove that in the limit of system size $N$ approaching infinity, the empirical state correlation and integrated response functions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations introduced by Cugliandolo and Kurchan.

math.PR↗

arXiv2003

Cover Times for Brownian Motion and Random Walks in Two Dimensions

Let T(x,r) denote the first hitting time of the disc of radius r centered at x for Brownian motion on the two dimensional torus. We prove that sup_{x} T(x,r)/|log r|^2 --> 2/pi as r --> 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z_n^2 is asymptotic to (2n log n)^2/pi. Determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied non-rigorously in the physics literature. We also establish a conjecture, due to Kesten and Revesz, that describes the asymptotics for the number of steps needed by simple random walk in Z^2 to cover the disc of radius n.

math.PR↗

arXiv2003

A large-deviation theorem for tree-indexed Markov chains

Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed by taking all descendants of a single vertex. We prove a large deviation principle in $n$, with explicit rate function, for the empirical subtree measures of multitype Galton-Watson trees conditioned to have exactly $n$ vertices. In the process, we extend the notions of shift-invariance and specific relative entropy--as typically understood for Markov fields on deterministic graphs such as $\mathbb Z^d$--to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with $n$ vertices.

math.PR↗

arXiv2001

Aging properties of Sinai's model of random walk in random environment

We study in this short note aging properties of Sinai's (nearest neighbour) random walk in random environment. With $\PP^o$ denoting the annealed law of the RWRE $X_n$, our main result is a full proof of the following statement due to P. Le Doussal, C. Monthus and D. S. Fisher: $$\lim_{η\to0} \lim_{n\to\infty} \PP^o (\frac{|X_{n^h} - X_n|}{(\log n)^2} < η) = \frac{1}{h^2} [ {5/3} - {2/3} e^{-(h-1)} ]. $$

math.PR↗