Slow Mixing for the Hard-Core Model on Z^2

The hard-core model has attracted much attention across several disciplines, representing lattice gases in statistical physics and independent sets in discrete mathematics and computer science. On finite graphs, we are given a parameter $\lambda$, and an independent set $I$ arises with probability proportional to $\lambda^{|I|}$. We are interested in determining the mixing time of local Markov chains that add or remove a small number of vertices in each step. On finite regions of $Z^2$ it is conjectured that there is a phase transition at some critical point $\lambda_c$ that is approximately $3.79$. It is known that local chains are rapidly mixing when $\lambda < 2.3882$. We give complementary results showing that local chains will mix slowly when $\lambda > 5.3646$ on regions with periodic (toroidal) boundary conditions and when $\lambda > 7.1031$ with non-periodic (free) boundary conditions. The proofs use a combinatorial characterization of configurations based on the presence or absence of fault lines and an enumeration of a new class of self-avoiding walks called taxi walks. This is work with Antonio Blanca, David Galvin and Prasad Tetali.