Glauber dynamics on trees and hyperbolic graphs

Abstract

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 |λ 1 − λ 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 for general graphs, if the relaxation time τ 2 satisfies τ 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.

The Ising model on trees: boundary conditions and mixing time

“…This is the content of our first main result; in fact, we are able to prove that the mixing time is Ç´Ò ÐÓ Òµ, which is optimal: This analysis has several advantages over previous ones [4,36]: it is more direct, applies also when there is an external field, gives a technically stronger result (as explained below), and applies to models more general than the Ising model. We now proceed to sketch some of our techniques and point out the main technical innovations.…”

“…The Glauber dynamics for the Ising model on trees has also been studied. In a recent paper [4], it is shown that the mixing time with a free boundary on a complete -ary tree with Ò vertices is Ç´Ò ÐÓ Òµ at high and intermediate temperatures (i.e., when ¬ ¬ ½ ) Ý . Moreover, as soon as ¬ ¬ ½ the mixing time becomes Ò ½·ª´½µ , and the exponent is unbounded as ¬ ½.…”

“…Thus in the Ý Actually [4] proves this only for sufficiently high temperatures, but the argument can be extended to all ¬ ¬ ½ [36].…”

“…The importance of (9) is that Ñ Ò Ü Ô´ Ü µ depends only on the size of Ü and ¬, but not on the size of Ì ; in fact, it is at least ª´ ´ ¬µ¡ µ [4]. Therefore, in order to show that Ô is bounded by a constant independent of the size of Ì , it is enough to show that, for some finite , Î Ö´ µ ÓÒ×Ø ¢ ´ µ for all functions .…”

“…We stress that, while the tree is simpler in some respects than due to the lack of cycles, in other respects it is more complex: e.g., it exhibits a "double phase transition" (see below). Moreover, the Ising model on trees has recently received a lot of attention as the canonical example of a statistical physics model on a "non-amenable" graph (i.e., one whose boundary is of comparable size to its volume) -see, e.g., [4,6,7,12,19,24,27,39]. In the next subsection, we briefly describe the Ising model on trees before stating our results in more detail.…”