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Sampling 3-colourings of regular bipartite graphs

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Sampling 3-colourings of regular bipartite graphs
2.	Creator	Author's name, affiliation, country	David J Galvin; University of Pennsylvania
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Mixing time, 3-colouring, Potts model, conductance, Glauber dynamics, discrete hypercube
3.	Subject	Subject classification	05C15, 82B20
4.	Description	Abstract	We show that if $G=(V,E)$ is a regular bipartite graph for which the expansion of subsets of a single parity of $V$ is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise non-adjacent vertices), and if $M$ is a Markov chain on the set of proper 3-colourings of $G$ which updates the colour of at most $c|V|$ vertices at each step and whose stationary distribution is uniform, then for $c < .22$ and $d$ sufficiently large the convergence to stationarity of $M$ is (essentially) exponential in $|V|$. In particular, if $G$ is the $d$-dimensional hypercube $Q_d$ (the graph on vertex set $\{0,1\}^d$ in which two strings are adjacent if they differ on exactly one coordinate) then the convergence to stationarity of the well-known Glauber (single-site update) dynamics is exponentially slow in $2^d/(\sqrt{d} \log d )$. A combinatorial corollary of our main result is that in a uniform 3-colouring of $Q_d$ there is an exponentially small probability (in $2^d$) that there is a colour $i$ such the proportion of vertices of the even subcube coloured $i$ differs from the proportion of the odd subcube coloured $i$ by at most .22. Our proof combines a conductance argument with combinatorial enumeration methods.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSF (grant DMS-0111298)
7.	Date	(YYYY-MM-DD)	2007-04-18
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/403
10.	Identifier	Digital Object Identifier	10.1214/EJP.v12-403
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 12
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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