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On forest expansions for two-body partition functions on tree-like interaction graphs

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  • Caravelli, F.

Abstract

We study tree approximations to classical two-body partition functions on sparse and loopy graphs via the Brydges–Kennedy–Abdessalam–Rivasseau forest expansion. We show that for sparse graphs (with large cycles), the partition function above a certain temperature T∗ can be approximated by a graph polynomial expansion over forests of the interaction graph. Within this region, we show that the approximation can be written in terms of a reference tree T on the interaction graph, with corrections due to cycles. From this point of view, this implies that high-temperature models are easy to solve on sparse graphs, as one can evaluate the partition function using belief propagation. We also show that there exist a high- and low-temperature regime, in which T can be obtained via a maximal spanning tree algorithm on a (given) weighted graph. We study the algebra of these corrections and provide first- and second-order approximation to the tree Ansatz, and give explicit examples for the first-order approximation.

Suggested Citation

  • Caravelli, F., 2023. "On forest expansions for two-body partition functions on tree-like interaction graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    • Handle: RePEc:eee:phsmap:v:609:y:2023:i:c:s0378437122009037
    DOI: 10.1016/j.physa.2022.128345
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    References listed on IDEAS

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    1. M. Mézard & G. Parisi, 2001. "The Bethe lattice spin glass revisited," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 20(2), pages 217-233, March.
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