  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 $\alpha\approx
1.763...$ 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.
