Strong spatial mixing for lattice graphs with fewer colours
Markov Chains in Algorithms and Statistical Physics
February 01, 2005 03:05 PM to 03:50 PM
Speakers:
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Abstract: |
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
Delta >= 3, the maximum degree of G is at most Delta. We show that the spin
system consisting of q-colourings of G has strong spatial mixing, provided q >
alpha*Delta-gamma, where alpha (approximately 1.76322) is the solution to
alpha^alpha=e, and gamma is approximately 0.47031. Note that we require no
additional lower bound on q or Delta. 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 such graphs strong spatial mixing implies that there is a unique
infinite-volume Gibbs measure. That is, there is one macroscopic equilibrium
rather than many. Furthermore, for a wide class of graphs, strong spatial
mixing implies rapid mixing of the natural single-site recolouring Markov chain
(Glauber dynamics) for sampling proper colourings.
Our general result gives, 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). It 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 Z3. 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-colourings of the
triangular lattice. (Such a proof for 11 colours was given by Salas and Sokal.)
Joint work with Leslie Ann Goldberg and Mike Paterson. |
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