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;
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
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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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DOI: https://doi.org/10.1007/BF00324855
Keywords
- Stochastic Process
- Probability Theory
- Exponential Decay
- Random Field
- Mathematical Biology