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Approximating Pairwise Correlations in the Ising Model

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Published:23 July 2019Publication History
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Abstract

In the Ising model, we consider the problem of estimating the covariance of the spins at two specified vertices. In the ferromagnetic case, it is easy to obtain an additive approximation to this covariance by repeatedly sampling from the relevant Gibbs distribution. However, we desire a multiplicative approximation, and it is not clear how to achieve this by sampling, given that the covariance can be exponentially small. Our main contribution is a fully polynomial time randomised approximation scheme (FPRAS) for the covariance in the ferromagnetic case. We also show that the restriction to the ferromagnetic case is essential—there is no FPRAS for multiplicatively estimating the covariance of an antiferromagnetic Ising model unless RP = #P. In fact, we show that even determining the sign of the covariance is #P-hard in the antiferromagnetic case.

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            • Published in

              cover image ACM Transactions on Computation Theory
              ACM Transactions on Computation Theory  Volume 11, Issue 4
              December 2019
              252 pages
              ISSN:1942-3454
              EISSN:1942-3462
              DOI:10.1145/3331049
              Issue’s Table of Contents

              Copyright © 2019 ACM

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              Association for Computing Machinery

              New York, NY, United States

              Publication History

              • Published: 23 July 2019
              • Revised: 1 April 2019
              • Accepted: 1 April 2019
              • Received: 1 October 2018
              Published in toct Volume 11, Issue 4

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