Coupling with the stationary distribution and improved sampling for colorings and independent sets
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
ACM Transactions on Algorithms (TALG)
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Randomly coloring sparse random graphs with fewer colors than the maximum degree
Random Structures & Algorithms
Rapid mixing of Gibbs sampling on graphs that are sparse on average
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
| Hi-index | 0.00 |
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 non-tree-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 驴 驴 3, the maximumdegree of G is at most 驴 We show that the spin system consisting of q-colourings of G has strong spatial mixing, provided q 驴驴, where 驴 驴 1.76322 is the solution to \alpha ^\alpha = e. Note that we have no additional lower bound on q or 驴. 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 \mathbb{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 (for the rectangular lattice) and by Bubley, Dyer, Greenhill and Jerrum). 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 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 \mathbb{Z}^3. Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us the first (machine-assisted) proof of strong spatial mixing for 10-colouringsof the triangular lattice. (Such a proof for 11 colours was given by Salas and Sokal.)