Slow Mixing of Glauber Dynamics for the Hard-Core Model on Regular Bipartite Graphs

Let  = (V,E) be a finite, d-regular bipartite graph. For any  > 0 let be the probability measure on the independent sets of  in which the set I is chosen with probability proportional to |I| ( is the hard-core measure with activity  on ). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is . We show that when  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  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.