Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphs

  • Authors:

  • David Galvin;Prasad Tetali

  • Affiliations:

  • Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395;School of Mathematics & College of Computing, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2006

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Abstract

Let ∑ = (V,E) be a finite, d-regular bipartite graph. For any λ 0 let πλ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ|I| (πλ is the hard-core measure with activity λ on ∑). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is πλ. We show that when λ is large enough (as a function of d and the expansion of subsets of single-parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d-dimensional hypercube {0,1}d we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006