Improved Strong Spatial Mixing for Colorings on Trees
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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>α d where α≈ 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.
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