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The Signed Loop Approach to the Ising Model: Foundations and Critical Point

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Abstract

The signed loop approach is a beautiful way to rigorously study the two-dimensional Ising model with no external field. In this paper, we explore the foundations of the method, including details that have so far been neglected or overlooked in the literature. We demonstrate how the method can be applied to the Ising model on the square lattice to derive explicit formal expressions for the free energy density and two-point functions in terms of sums over loops, valid all the way up to the self-dual point. As a corollary, it follows that the self-dual point is critical both for the behaviour of the free energy density, and for the decay of the two-point functions.

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Notes

  1. A picture of the same configurations appears in [7] to point out the error in Vdovichenko’s paper; it is crucial here to take the multiplicities of the loops into account.

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Authors and Affiliations

  1. Department of Mathematics, VU University, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    Wouter Kager, Marcin Lis & Ronald Meester

Authors
  1. Wouter Kager
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  2. Marcin Lis
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  3. Ronald Meester
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Correspondence to Wouter Kager.

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The second author was financially supported by Vidi grant 639.032.916 of the Netherlands Organisation for Scientific Research (NWO).

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Kager, W., Lis, M. & Meester, R. The Signed Loop Approach to the Ising Model: Foundations and Critical Point. J Stat Phys 152, 353–387 (2013). https://doi.org/10.1007/s10955-013-0767-z

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