Title | ||
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Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs |
Abstract | ||
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In a seminal paper [12], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (equivalently, approximating the partition function of the hard-core model from statistical physics) on graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the infinite d-regular tree. More recently Sly [10] showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the infinite d-regular tree then NP = RP. In this paper, we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [9] to the case of the anti-ferromagnetic Ising model with arbitrary field. By a standard correspondence, these results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. |
Year | DOI | Venue |
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2012 | https://doi.org/10.1007/s10955-014-0947-5 | Journal of Statistical Physics |
Keywords | DocType | Volume |
Phase transitions,Complexity theory,Approximation algorithms,Decay of correlations,Two-spin systems on trees | Conference | 155 |
Issue | ISSN | Citations |
4 | 0022-4715 | 30 |
PageRank | References | Authors |
1.07 | 11 | 3 |
Name | Order | Citations | PageRank |
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Alistair Sinclair | 1 | 1506 | 308.40 |
Piyush Srivastava | 2 | 65 | 4.55 |
Marc Thurley | 3 | 99 | 6.02 |