  The property of spatial mixing and strong spatial mixing in spin systems has
been of interest because of its implications on uniqueness of Gibbs measures on
infinite graphs and efficient approximation of counting problems that are
otherwise known to be #P hard. In the context of coloring, strong spatial
mixing has been established for regular trees when $q \geq \alpha^{*} \Delta +
1$ where $q$ the number of colors, $\Delta$ is the degree and $\alpha^* =
1.763..$ is the unique solution to $xe^{-1/x} = 1$. It has also been
established for bounded degree lattice graphs whenever $q \geq \alpha^* \Delta
- \beta$ for some constant $\beta$, where $\Delta$ is the maximum vertex degree
of the graph. The latter uses a technique based on recursively constructed
coupling of Markov chains whereas the former is based on establishing decay of
correlations on the tree. We establish strong spatial mixing of list colorings
on arbitrary bounded degree triangle-free graphs whenever the size of the list
of each vertex $v$ is at least $\alpha \Delta(v) + \beta$ where $\Delta(v)$ is
the degree of vertex $v$ and $\alpha > \alpha ^*$ and $\beta$ is a constant
that only depends on $\alpha$. We do this by proving the decay of correlations
via recursive contraction of the distance between the marginals measured with
respect to a suitably chosen error function.
