Percolation on Self-Dual Polygon Configurations

Опубликовано 17 августа 2016, 1:17

In this talk I shall sketch some results Oliver Riordan of Oxford and I have obtained on the critical probabilities in percolation. The story starts with Scullard and Zi, who recently pointed out that a broad class of planar percolation models are self-dual. They stated that in a variety of classes of models depending on a parameter, the parameter giving self-duality is the critical value for percolation. However, noticing self-duality is simply the starting point: the mathematical difficulty is precisely showing that self-duality implies criticality. Riordan and I have managed to overcome this difficulty: we have shown that, for a generalization of the models considered by Scullard and Zi, self-duality indeed implies criticality.