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.
-
(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.
-
(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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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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DOI: https://doi.org/10.1007/BF01192961
Mathematics Subject Classification 1991
- 82B20
- 60K35
- 82B43