Title:

Spatial and Rapid Mixing on Lattice Graphs

In this thesis we consider the antiferromagnetic Potts model on lattice graphs. A spin
system under this model is characterised by an underlying connected graph and two
parameters: q, the number of spins, and), = exp(f3), where f3 is the 'inverse temperature'.
1\vo questions of interest are to determine if a spin system has strong spatial
mixing and if the Glauber dynamics is rapidly mixing on the configuration space. These
two properties are closely related, in particular it is known that strong spatial mixing
often implies that the Glauber dynamics is rapidly mixing. Rapidly mixing Glauber
dynamics implies that there is a fullypolynomial randomised approximation scheme
for the partition function. For graphs in which the distanced neighbourhood of a vertex
grows subexponentially in d, strong spatial mixing implies that there is a unique
infinitevolume Gibbs distribution.
We use recursivelyconstructed couplings to derive mixing bounds for any graph
and any temperature. The result improves previously known general mixing bounds.
Our main objective is to have results which are applicable to the lattices studied in
statistical physics. In this thesis we focus on the square lattice (Z2), the triangular
lattice and the kagome lattice. By considering the geometry of the lattice we are able
to achieve better mixing bounds. Rather than constructing recursive couplings from a
single recurrence, we use a system of recurrences, which is highly dependent on the
geometry of the lattice. For the square lattice we give a proof of strong spatial mixing
and rapid mixing for q 2: 6 and any).. We also show that mixing occurs for a larger
range of ). than was previously known for q = 3, 4 and 5. Certain probabilities that
are used in the proof are obtained with computer assistance which makes the proof
machine assisted.
The antiferromagnetic Potts model corresponds to proper colourings when the
temperature is zero. By refining the technique of recursively constructing couplings,
we provide proofs of mixing for the triangular lattice with q = 9 and ). = 0, and the
kagome lattice with q = 5 and), = o. This improves previously known mixing bounds.
The systems of recurrences we use here are rather large and require a computer to be
constructed. This makes the proofs machine assisted. The Glauber dynamics is not
necessarily irreducible on the kagome lattice with q = 5 and), = 0 if we impose a
boundary condition. However, we show rapid mixing under the free boundary.
We also study the mixing time of a systematic scan Markov chain for sampling
from the uniform distribution on proper 7colourings of a finite rectangular subgrid of
the square lattice. Asystematic scan Markov chain updates finitesize subsets ofvertices
in a deterministic order. We use a heuristicbased computation in order to establish a
rigorous result about the mixing time. The proof is computer assisted and improves
previously known mixing bounds for systematic scan on the square lattice.
