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arXiv · 1906.01666

Counting independent sets in unbalanced bipartite graphs

Sarah Cannon·Will Perkins
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Abstract

We give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides $(L,R)$ of the bipartition. This includes, among others, the biregular case when $λ=1$ (approximating the number of independent sets of $G$) and $Δ_R \geq 7Δ_L \log(Δ_L)$. Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. As a consequence of the method, we also prove that the hard-core model on such graphs exhibits exponential decay of correlations by utilizing connections between the cluster expansion and joint cumulants.

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Sarah Cannon, Will Perkins. 2019-06-04. Counting independent sets in unbalanced bipartite graphs. https://arxiv.org/abs/1906.01666

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