Strong spatial mixing for lattice graphs with fewer colours

Recursively-constructed couplings have been used in the past for mixing on trees. We show for the first time how to extend this technique to nontree-like graphs such as the integer lattice. 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, where α ≈ 1.76322 is the solution to α^α = e. 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^d 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. We extend our general result, obtaining, for example, the first "hand proof" of strong spatial mixing for 7-colourings of triangle-free 4-regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal (1997) for the rectangular lattice and by Bubley et al. (1999)). The extension also gives the first 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 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 Z³. Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us the first (machi- ne-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.).