Glauber Dynamics on Trees and Hyperbolic Graphs

We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with $n$ vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap $|\lambda_1-\lambda_2|$) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in $n$. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time $\tau_2$ satisfies $\tau_2=O(1)$, then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp.Comment: To appear in Probability Theory and Related Field

Robust reconstruction on trees is determined by the second eigenvalue

Consider a Markov chain on an infinite tree T=(V,E) rooted at \rho. In such a chain, once the initial root state \sigma(\rho) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let \mu_j denote the resulting measure for \sigma(\rho)=j. The resulting measure \mu_j is defined on configurations \sigma=(\sigma(x))_{x\in V}\in A^V, where A is some finite set. Let \mu_j^n denote the restriction of \mu to the sigma-algebra generated by the variables \sigma(x), where x is at distance exactly n from \rho. Letting \alpha_n=max_{i,j\in A}d_{TV}(\mu_i^n,\mu_j^n), where d_{TV} denotes total variation distance, we say that the reconstruction problem is solvable if lim inf_{n\to\infty}\alpha_n>0. Reconstruction solvability roughly means that the nth level of the tree contains a nonvanishing amount of information on the root of the tree as n\to\infty. In this paper we study the problem of robust reconstruction. Let \nu be a nondegenerate distribution on A and \epsilon >0. Let \sigma be chosen according to \mu_j^n and \sigma' be obtained from \sigma by letting for each node independently, \sigma(v)=\sigma'(v) with probability 1-\epsilon and \sigma'(v) be an independent sample from \nu otherwise. We denote by \mu_j^n[\nu,\epsilon ] the resulting measure on \sigma'. The measure \mu_j^n[\nu,\epsilon ] is a perturbation of the measure \mu_j^n.Comment: Published at http://dx.doi.org/10.1214/009117904000000153 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average

In this work we show that for every $d < \infty$ and the Ising model defined on $G(n,d/n)$, there exists a $\beta_d > 0$, such that for all $\beta < \beta_d$ with probability going to 1 as $n \to \infty$, the mixing time of the dynamics on $G(n,d/n)$ is polynomial in $n$. Our results are the first polynomial time mixing results proven for a natural model on $G(n,d/n)$ for $d > 1$ where the parameters of the model do not depend on $n$. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than n \polylog(n). Our proof exploits in novel ways the local treelike structure of Erd\H{o}s-R\'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex $v$ of the graph has a neighborhood $N(v)$ of radius $O(\log n)$ in which the induced sub-graph is a tree union at most $O(\log n)$ edges and where for each simple path in $N(v)$ the sum of the vertex degrees along the path is $O(\log n)$. Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for $G(n,d/n)$ it applies for all external fields and $\beta < \beta_d$, where $d \tanh(\beta_d) = 1$ is the critical point for decay of correlation for the Ising model on $G(n,d/n)$.Comment: Corrected proof of Lemma 2.

Reconstruction thresholds on regular trees

We consider a branching random walk with binary state space and index set $T^k$, the infinite rooted tree in which each node has k children (also known as the model of "broadcasting on a tree"). The root of the tree takes a random value 0 or 1, and then each node passes a value independently to each of its children according to a 2x2 transition matrix P. We say that "reconstruction is possible" if the values at the d'th level of the tree contain non-vanishing information about the value at the root as $d\to\infty$. Adapting a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case $p_{11}=0$. The latter case is closely related to the "hard-core model" from statistical physics; a corollary of our results is that, for the hard-core model on the (k+1)-regular tree with activity $\lambda=1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any k.Comment: 12 page

Reconstruction Threshold for the Hardcore Model

In this paper we consider the reconstruction problem on the tree for the hardcore model. We determine new bounds for the non-reconstruction regime on the k-regular tree showing non-reconstruction when lambda < (ln 2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1. This is almost tight as reconstruction is known to hold when lambda> (e+o(1))ln^2(k). We discuss the relationship for finding large independent sets in sparse random graphs and to the mixing time of Markov chains for sampling independent sets on trees.Comment: 14 pages, 2 figure

Glauber dynamics on nonamenable graphs: Boundary conditions and mixing time

We study the stochastic Ising model on finite graphs with n vertices and bounded degree and analyze the effect of boundary conditions on the mixing time. We show that for all low enough temperatures, the spectral gap of the dynamics with (+)-boundary condition on a class of nonamenable graphs, is strictly positive uniformly in n. This implies that the mixing time grows at most linearly in n. The class of graphs we consider includes hyperbolic graphs with sufficiently high degree, where the best upper bound on the mixing time of the free boundary dynamics is polynomial in n, with exponent growing with the inverse temperature. In addition, we construct a graph in this class, for which the mixing time in the free boundary case is exponentially large in n. This provides a first example where the mixing time jumps from exponential to linear in n while passing from free to (+)-boundary condition. These results extend the analysis of Martinelli, Sinclair and Weitz to a wider class of nonamenable graphs.Comment: 31 pages, 4 figures; added reference; corrected typo