Computational Transition at the Uniqueness Threshold

The hardcore model is a model of lattice gas systems which has received much attention in statistical physics, probability theory and theoretical computer science. It is the probability distribution over independent sets $I$ of a graph weighted proportionally to $\lambda^{|I|}$ with fugacity parameter $\lambda$. We prove that at the uniqueness threshold of the hardcore model on the $d$-regular tree, approximating the partition function becomes computationally hard on graphs of maximum degree $d$. Specifically, we show that unless NP$=$RP there is no polynomial time approximation scheme for the partition function (the sum of such weighted independent sets) on graphs of maximum degree $d$ for fugacity $\lambda_c(d) < \lambda < \lambda_c(d) + \epsilon(d)$ where $\lambda_c = \frac{(d-1)^{d-1}}{(d-2)^d}$ is the uniqueness threshold on the $d$-regular tree and $\epsilon(d)>0$. Weitz produced an FPTAS for approximating the partition function when $0<\lambda < \lambda_c(d)$ so this result demonstrates that the computational threshold exactly coincides with the statistical physics phase transition thus confirming the main conjecture of [28]. We further analyze the special case of $\lambda=1, d=6$ and show there is no polynomial time algorithm for approximately counting independent sets on graphs of maximum degree $d= 6$ which is optimal. Our proof is based on specially constructed random bi-partite graphs which act as gadgets in a reduction to MAX-CUT. Building on the second moment method analysis of [28] and combined with an analysis of the reconstruction problem on the tree our proof establishes a strong version of 'replica' method heuristics developed by theoretical physicists. The result establishes the first rigorous correspondence between the hardness of approximate counting and sampling with statistical physics phase transitions.