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Title: Spatial and rapid mixing on lattice graphs
Author: Jalsenius, Markus
ISNI:       0000 0001 3588 6399
Awarding Body: University of Liverpool
Current Institution: University of Liverpool
Date of Award: 2008
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In this thesis we consider the anti-ferromagnetic 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 fully-polynomial randomised approximation scheme for the partition function. For graphs in which the distance-d neighbourhood of a vertex grows sub-exponentially in d, strong spatial mixing implies that there is a unique infinite-volume Gibbs distribution. We use recursively-constructed 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 anti-ferromagnetic 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 7-colourings of a finite rectangular sub-grid of the square lattice. Asystematic scan Markov chain updates finite-size subsets ofvertices in a deterministic order. We use a heuristic-based 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral