Strong Spatial Mixing with Fewer Colors for Lattice Graphs

Abstract

Recursively-constructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontree-like graphs such as lattices. Using this method, we obtain the following general result. Suppose that $G$ is a triangle-free graph and that for some $\degree \geq 3$, the maximum degree of $G$ is at most $\degree$. We show that the spin system consisting of $q$-colorings of $G$ has strong spatial mixing, provided $q > \alpha \degree-\gamma$, where $\alpha\approx 1.76322$ is the solution to $\alpha^\alpha=e$, and $\gamma = \frac{4\alpha^3-6\alpha^2-3\alpha+4}{2(\alpha^2-1)}\approx 0.47031$. Note that we have no additional lower bound on $q$ or $\degree$. This is important for us because our main objective is to have results which are applicable to the lattices studied in statistical physics, such as the integer lattice $\zset^d$ and the triangular lattice. For these graphs (in fact, for any graph in which the distance-$k$ neighborhood of a vertex grows subexponentially in $k$), strong spatial mixing implies that there is a unique infinite-volume Gibbs measure. That is, there is one macroscopic equilibrium rather than many. Our general result gives, for example, a ``hand proof'' of strong spatial mixing for $7$-colorings of triangle-free $4$-regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp. 551--579] (for the rectangular lattice) and by Bubley, Dyer, Greenhill, and Jerrum [\textit{SIAM J. Comput.}, 29 (1999), pp. 387--400].) It also gives a hand proof of strong spatial mixing for $5$-colorings of triangle-free $3$-regular graphs. (A computer-assisted proof for the special case of the hexagonal lattice was provided earlier by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp. 551--579].) Toward the end of the paper we show how to improve our general technique by considering the geometry of the lattice. The idea is to construct the recursive coupling from a system of recurrences rather than from a single recurrence. We use the geometry of the lattice to derive the system of recurrences. This gives us an analysis with a horizon of more than one level of induction, which leads to improved results. We illustrate this idea by proving strong spatial mixing for $q=10$ on the lattice $\zset^3$. Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us a (machine-assisted) proof of strong spatial mixing for $10$-colorings of the triangular lattice. (Such a proof for $11$ colors was given by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp. 551--579].) For completeness, we also show that our strong spatial mixing proof implies rapid mixing of Glauber dynamics for sampling proper colorings of neighborhood-amenable graphs. (It is known that strong spatial mixing often implies rapid mixing, but existing proofs seem to be written for $\zset^d$.) Thus our strong spatial mixing results give rapid