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Description

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.

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 $lambda_c(d) < lambda < lambda_c(d) + varepsilon(d)$ where $$lambda_c = frac{(d-1)^{d-1}}{(d-2)^d}$$ is the uniqueness threshold on the $d$-regular tree and $varepsilon(d)>0$ is a positive constant. Weitz produced an FPTAS for approximating the partition function when $0<lambda < lambda_c(d)$ so this result demonstrates that the computation threshold exactly coincides with the statistical physics phase transition thus confirming the main conjecture of [MWW, '09]. We further analyze the special case of $lambda=1, d=6$ and show there is no polynomial time approximation scheme for approximately counting independent sets on graphs of maximum degree $d= 6$, which is optimal, improving the previous bound of $d= 24$.

Our proof is based on specially constructed random bi-partite graphs which act as gadgets in a reduction to MAX-CUT. Building on the involved second moment method analysis of cite{MWW:09} 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.

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