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A sharp transition for the two-dimensional Ising percolation
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  • Published: December 1993

A sharp transition for the two-dimensional Ising percolation

  • Yasunari Higuchi1 

Probability Theory and Related Fields volume 97, pages 489–514 (1993)Cite this article

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Summary

We show that the percolation transition for the two-dimensional Ising model is sharp. Namely, we show that for every reciprocal temperature β>0, there exists a critical valueh c (β) of external magnetic fieldh such that the following two statements hold.

  1. (i)

    Ifh>h c (β), then the percolation probability (i.e., the probability that the origin is in the infinite cluster of + spins) with respect to the Gibbs state μβ,h for the parameter (β,h) is positive.

  2. (ii)

    Ifh<h c (β), then the connectivity function τ +β,h (0,x) (the probability that the origin is connected by + spins tox with respect to μβ,h ) decays exponentially as |x|→∞.

We also shows that the percolation probability is continuous in (β,h) except on the half line {(β, 0); β≧β c }.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kobe University, Rokko, 657, Kobe, Japan

    Yasunari Higuchi

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  1. Yasunari Higuchi
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Additional information

Work supported in part by Grant in Aid for Scientific Research no. 04640230

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Higuchi, Y. A sharp transition for the two-dimensional Ising percolation. Probab. Th. Rel. Fields 97, 489–514 (1993). https://doi.org/10.1007/BF01192961

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  • Received: 11 November 1992

  • Revised: 31 May 1993

  • Issue Date: December 1993

  • DOI: https://doi.org/10.1007/BF01192961

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Mathematics Subject Classification 1991

  • 82B20
  • 60K35
  • 82B43
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