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A percolation problem for {±1}-valued strongly mixing random fields on ℤd ⋆
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  • Published: November 1986

A percolation problem for {±1}-valued strongly mixing random fields on ℤd ⋆

  • Yasunari Higuchi1 

Probability Theory and Related Fields volume 73, pages 597–611 (1986)Cite this article

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Summary

We present here an upper estimate for the probability that the origin is + connected to the boundary of the cube {|x i|≦k, 1≦i≦d{, under the condition that the expectation E(r(W +0 )d-1) is finite, where W +0 is the+cluster of the origin, and r(W +0 ) is its radius;

$$r(W_0^ + ) = \max \left\{ {\left| {x^i } \right|;1 \leqq i \leqq d,x \in W_0^ + } \right\}$$

The upper estimate is given in terms of the mixing coefficient. In particular, if the mixing coefficient decays exponentially, then our upper estimate supply an “almost exponential” decay of the probability in question; it decays faster than exp (-Ck/log k) as k→∞, for some positive constant C.

As an example we discuss the two-dimensional Ising model except at the critical point. By using our result, we show the above almost exponential decay for parameters (β, h) satisfying

$$h < - 4\beta ^{ - 1} (\beta _c - \beta )v0$$

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Authors and Affiliations

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

    Yasunari Higuchi

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  1. Yasunari Higuchi
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Additional information

Part of this work was done while the author was visiting Sonderforschungsbereich 123, Universität Heidelberg, from August to November 1984, supported by Deutsche Forschungsgemeinschaft

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Higuchi, Y. A percolation problem for {±1}-valued strongly mixing random fields on ℤd ⋆ . Probab. Th. Rel. Fields 73, 597–611 (1986). https://doi.org/10.1007/BF00324855

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  • Received: 11 January 1985

  • Revised: 20 April 1986

  • Issue Date: November 1986

  • DOI: https://doi.org/10.1007/BF00324855

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Keywords

  • Stochastic Process
  • Probability Theory
  • Exponential Decay
  • Random Field
  • Mathematical Biology
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