Strong Spatial Mixing with Fewer Colours for Lattice Graphs∗

Recursively-constructed couplings have been used in the past for mixing on trees. We show how to extend this technique to non-tree-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 Δ ≥ 3, the maximum degree of G is at most Δ. We show that the spin system consisting of q-colourings of G has strong spatial mixing, provided q > αΔ − γ, where α ≈ 1.76322 is the solution to α = e, and γ = 4α3−6α2−3α+4 2(α2−1) ≈ 0.47031. Note that we have no additional lower bound on q or Δ. 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 Z and the triangular lattice. For these graphs (in fact, for any graph in which the distance-k neighbourhood of a vertex grows sub-exponentially 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-colourings of triangle-free 4regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal (for the rectangular lattice) and by Bubley, Dyer, Greenhill and Jerrum.) It also gives a hand proof of strong spatial mixing for 5-colourings 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.) Towards 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 Z . 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-colourings of the triangular lattice. (Such a proof for 11 colours was given by Salas and Sokal.) For completeness, we also show that our strong spatial mixing proof implies rapid mixing of Glauber dynamics for sampling proper colourings of neighbourhood-amenable graphs. (It is known that strong spatial mixing often implies rapid mixing, but existing proofs seem to be written for Z.) Thus our strong spatial mixing results give rapid mixing corollaries for neighbourhood-amenable graphs such as lattices. ∗This work was partially supported by the EPSRC grant “Discontinuous Behaviour in the Complexity of Randomized Algorithms”. A preliminary version of this paper has appeared in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004), pp. 562–571.