Rapidly mixing Markov chains and the sharp transition in 2D ising percolation
Seminar Room 1, Newton Institute
One of the most well-known classical results for site percolation on the square lattice is that, for all values of the parameter p (except its critical value) the following holds: Either a.s. there is an infinite open cluster or a.s. there is an infinite closed `star' cluster. This result is closely related to the percolation transition being sharp.
The analog of this result has been proved by Higuchi in 1993 for two-dimensional Ising percolation (at fixed temperature T > Tc) with external field h, the parameter of the model.
Using a Markov chain and rapid-mixing results of Martinelli and Olivieri, we show that the Ising model can be `decoded' in terms of i.i.d. random variables in such a way that certain general approximate zero-one laws (sharp-threshold results) can be applied. This leads to an alternative proof of Higuchi's result and puts it in a more general framework.
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