Improved Strong Spatial Mixing for Colorings on Trees

Abstract

Strong spatial mixing (SSM) is a form of correlation decay that has played an essential role in the design of approximate counting algorithms for spin systems. A notable example is the algorithm of Weitz (2006) for the hard-core model on weighted independent sets. We study SSM for the $q$-colorings problem on the infinite $(d+1)$-regular tree. Weak spatial mixing (WSM) captures whether the influence of the leaves on the root vanishes as the height of the tree grows. Jonasson (2002) established WSM when $q>d+1$. In contrast, in SSM, we first fix a coloring on a subset of internal vertices, and we again ask if the influence of the leaves on the root is vanishing. It was known that SSM holds on the $(d+1)$-regular tree when $q>\alpha d$ where $\alpha \approx \mathrm{1.763...}$ is a constant that has arisen in a variety of results concerning random colorings. Here we improve on this bound by showing SSM for $q>1.59d$. Our proof establishes an $L^2$ contraction for the BP operator. For the contraction we bound the norm of the BP Jacobian by exploiting combinatorial properties of the coloring of the tree.