# Spatial mixing and the connective constant: optimal bounds

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## Abstract

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded *connective constant*. More generally, we consider the problem of evaluating the partition functions of the *monomer-dimer model* (which is defined as a weighted sum over all matchings where each matching is given a weight \(\gamma ^{|V| - 2 |M|}\) in terms of a fixed parameter \(\gamma \) called the *monomer activity*) and the *hard core model* (which is defined as a weighted sum over all independent sets where an independent set *I* is given a weight \(\lambda ^{|I|}\) in terms of a fixed parameter \(\lambda \) called the *vertex activity*). The *connective constant* is a natural measure of the average degree of a graph which has been studied extensively in combinatorics and mathematical physics, and can be bounded by a constant even for certain unbounded degree graphs such as those sampled from the sparse Erdős–Rényi model \(\mathcal {G}(n, d/n)\). Our main technical contribution is to prove the best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant. These results on decay of correlations are obtained using a new framework based on the so-called *message* approach that has been extensively used recently to prove such results for bounded degree graphs. We then use these optimal decay of correlations results to obtain fully polynomial time approximation schemes (FPTASs) for the two problems on graphs of bounded connective constant. In particular, for the monomer-dimer model, we give a deterministic FPTAS for the partition function on all graphs of bounded connective constant for any given value of the monomer activity. The best previously known deterministic algorithm was due to Bayati et al. (Proc. 39th ACM Symp. Theory Comput., pp. 122–127, 2007), and gave the same runtime guarantees as our results but only for the case of bounded degree graphs. For the hard core model, we give an FPTAS for graphs of connective constant \(\varDelta \) whenever the vertex activity \(\lambda < \lambda _c(\varDelta )\), where \(\lambda _c(\varDelta ) :=\frac{\varDelta ^\varDelta }{(\varDelta - 1)^{\varDelta + 1}}\); this result is optimal in the sense that an FPTAS for any \(\lambda > \lambda _c(\varDelta )\) would imply that NP=RP (Sly and Sun, Ann. Probab. 42(6):2383–2416, 2014). The previous best known result in this direction was in a recent manuscript by a subset of the current authors (Proc. 54th IEEE Symp. Found. Comput. Sci., pp 300–309, 2013), where the result was established under the sub-optimal condition \(\lambda < \lambda _c(\varDelta + 1)\). Our techniques also allow us to improve upon known bounds for decay of correlations for the hard core model on various regular lattices, including those obtained by Restrepo et al. (Probab Theory Relat Fields 156(1–2):75–99, 2013) for the special case of \(\mathbb {Z}^2\) using sophisticated numerically intensive methods tailored to that special case.

## Mathematics Subject Classification

82B20 60J10 68W25 68W40## Notes

### Acknowledgments

We thank Elchanan Mossel, Allan Sly, Eric Vigoda and Dror Weitz for helpful discussions. We also thank the anonymous referees for detailed helpful comments.

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