# CSM

## Seminar

### Rapidly mixing Markov chains and the sharp transition in 2D ising percolation

van den Berg, J (CWI)
Thursday 27 March 2008, 10:05-10:35

Seminar Room 1, Newton Institute

#### Abstract

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.

#### Video

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