Forest expansion of two-body partition functions for sparse interaction graphs
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917583582658560 |
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| author | Caravelli, Francesco |
| author_facet | Caravelli, Francesco |
| contents | 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 "forest phase", we show that the approximation can be written in terms of a reference tree $\mathcal T$ 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 exists a high- and low-temperature regime, in which $\mathcal T$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_00113 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Forest expansion of two-body partition functions for sparse interaction graphs Caravelli, Francesco Combinatorics Statistical Mechanics Mathematical Physics 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 "forest phase", we show that the approximation can be written in terms of a reference tree $\mathcal T$ 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 exists a high- and low-temperature regime, in which $\mathcal T$ 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. |
| title | Forest expansion of two-body partition functions for sparse interaction graphs |
| topic | Combinatorics Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2009.00113 |