Analysis of the Hard-core Model on Square Lattices beyond the Tree Uniqueness Threshold

This research project studies the hard-core model (also know as the weighted independent set model) on the square lattice Z^2. For a (?nite) graph G = (V, E), the hard-core model is defined on the set ? of independent sets of G. There is an activity ? (corresponding to the fugacity of the gas in the statistical physics model), and independent set S has a weight w(S) = ?^{|S|}. The (?nite) Gibbs distribution µ corresponding to the equilibrium state of the system is then defined on ? as µ(S) = w(S)/Z where Z is known as the partition function. The fundamental notion studied in Statistical Physics is the phase transition between uniqueness and non-uniqueness of the Gibbs measure on the in?nite lattice. It is believed (based on simulations) that the critical ?_c(Z^2) for this phase transition from a unique to multiple Gibbs measures on Z^2 occurs at ? 3.79. Currently, the best rigorous lower bound follows from work of Weitz showing ?_c(Z^2) > 27/16, and the best rigorous upper bound was shown by Randall that ?_c (Z^2) < 8.066. In our research (joint with Ricardo Restrepo, Jinwoo Shin, Prasad Tetali, Eric Vigoda), we propose a new spatial mixing condition for spin systems based on Weitz's method. In particular, when it is applied to the hard-core model on Z^2, we are able to show that ?_c(Z^2) > 2.23.