skip to content
 

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

Date: 
Thursday 27th March 2008 - 10:05 to 10:35
Venue: 
INI Seminar Room 1
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.

The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons