Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

We prove that the mixing time of the Glauber dynamics for random k-colorings of the complete tree with branching factor b undergoes a phase transition at $k=b(1+o_b(1))/\ln{b}$. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for $k=Cb/\ln{b}$ colors with constant C. For $C\geq1$ we prove the mixing time is $O(n^{1+o_b(1)}\ln{n})$. On the other side, for $C<1$ the mixing time experiences a slowing down; in particular, we prove it is $O(n^{1/C+o_b(1)}\ln{n})$ and $\Omega(n^{1/C-o_b(1)})$. The critical point C=1 is interesting since it coincides (at least up to first order) with the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.Comment: Published in at http://dx.doi.org/10.1214/11-AAP833 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Matrix norms and rapid mixing for spin systems

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sampling Random Colorings of Sparse Random Graphs

We study the mixing properties of the single-site Markov chain known as the Glauber dynamics for sampling $k$-colorings of a sparse random graph $G(n,d/n)$ for constant $d$. The best known rapid mixing results for general graphs are in terms of the maximum degree $\Delta$ of the input graph $G$ and hold when $k>11\Delta/6$ for all $G$. Improved results hold when $k>\alpha\Delta$ for graphs with girth $\geq 5$ and $\Delta$ sufficiently large where $\alpha\approx 1.7632\ldots$ is the root of $\alpha=\exp(1/\alpha)$; further improvements on the constant $\alpha$ hold with stronger girth and maximum degree assumptions. For sparse random graphs the maximum degree is a function of $n$ and the goal is to obtain results in terms of the expected degree $d$. The following rapid mixing results for $G(n,d/n)$ hold with high probability over the choice of the random graph for sufficiently large constant~$d$. Mossel and Sly (2009) proved rapid mixing for constant $k$, and Efthymiou (2014) improved this to $k$ linear in~$d$. The condition was improved to $k>3d$ by Yin and Zhang (2016) using non-MCMC methods. Here we prove rapid mixing when $k>\alpha d$ where $\alpha\approx 1.7632\ldots$ is the same constant as above. Moreover we obtain $O(n^{3})$ mixing time of the Glauber dynamics, while in previous rapid mixing results the exponent was an increasing function in $d$. As in previous results for random graphs our proof analyzes an appropriately defined block dynamics to "hide" high-degree vertices. One new aspect in our improved approach is utilizing so-called local uniformity properties for the analysis of block dynamics. To analyze the "burn-in" phase we prove a concentration inequality for the number of disagreements propagating in large blocks

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.