A more rapidly mixing Markov chain for graph colourings

Very rapidly mixing Markov Chains for 2Δ-colourings and for independent sets in a graph with maximum degree 4

Sampling Grid Colourings with Fewer Colours

We provide an optimally mixing Markov chain for 6-colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for k-colourings of the square grid.

On Systematic Scan

In this thesis we study the mixing time of systematic scan Markov chains on finite spin systems. A systematic scan Markov chain is a Markov chain which updates the sites in a deterministic order and this type of Markov chain is often seen as intuitively appealing in terms of implementation to scientists conducting experimental work. Until recently systematic scan Markov chains have largely resi...

Random Sampling of 3-colourings in Z *

We consider the problem of uniformly sampling proper 3-colourings of an m × n rectangular region of Z. We show that the single-site “Glauber-dynamics” Markov chain is rapidly mixing. Our result complements an earlier result of Luby, Randall and Sinclair, which demonstrates rapid mixing when there is a fixed boundary (whose colour cannot be changed). c © John Wiley & Sons, Inc.

Counting and Sampling H-Colourings

For counting problems in #P which are “essentially self-reducible”, it is known that sampling and approximate counting are equivalent. However, many problems of interest do not have such a structure and there is already some evidence that this equivalence does not hold for the whole of #P. An intriguing example is the class ofH-colouring problems, which have recently been the subject of much st...