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Subset Glauber dynamics on graphs, hypergraphs and matroids of bounded tree-width.

Bordewich, M. and Kang, R. (2014) 'Subset Glauber dynamics on graphs, hypergraphs and matroids of bounded tree-width.', The electronic journal of combinatorics., 21 (4). P4.19.

Abstract

Motivated by the `subgraphs world' view of the ferromagnetic Ising model, we analyse the mixing times of Glauber dynamics based on subset expansion expressions for classes of graph, hypergraph and matroid polynomials. With a canonical paths argument, we demonstrate that the chains dened within this framework mix rapidly upon graphs, hypergraphs and matroids of bounded tree-width. This extends known results on rapid mixing for the Tutte polynomial, adjacency-rank (R2-)polynomial and interlace polynomial. In particular Glauber dynamics for the R2-polynomial was known to mix rapidly on trees, which led to hope of rapid mixing on a wider class of graphs. We show that Glauber dynamics for a very wide class of polynomials mixes rapidly on graphs of bounded tree-width, including many cases in which the Glauber dynamics does not mix rapidly for all graphs. This demonstrates that rapid mixing on trees or bounded tree-width graphs does not oer strong evidence towards rapid mixing on all graphs.

Item Type:Article
Keywords:Markov chain Monte Carlo, Graph polynomials, Tree-width, Canonical paths, Approximate counting.
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Status:Peer-reviewed
Publisher Web site:http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p19/pdf
Date accepted:No date available
Date deposited:23 July 2015
Date of first online publication:October 2014
Date first made open access:No date available

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