Jeff Steif

Fortuin-Kastelyn representations for Threshold Gaussian and Stable Vectors: aka Divide and Color processes

We consider the following simple model: one starts with a finite (or countable) set V, a random partition of V and a parameter p in [0,1]. The "Generalized Divide and Color Model" is the {0,1}-valued process indexed by V obtained by independently, for each partition element in the random partition chosen, with probability p assigning all the elements of the partition element the value 1, and with probability 1−p, assigning all the elements of the partition element the value 0. Many models fall into this context: (1) the 0 external field Ising model (where the random partition is given by FK percolation), (2) the stationary distributions for the voter model (where the random partition is given by coalescing random walks), (3) random walk in random scenery and (4) the original "Divide and Color Model" introduced and studied by Olle Häggström.

In earlier work, together with Johan Tykesson, we studied what one could say about such processes.

In joint work with Malin Palö Forsström, we study the question of which threshold Gaussian and stable vectors have such a representation: (A threshold Gaussian (stable) vector is a vector obtained by taking a Gaussian (stable) vector and a threshold h and looking where the vector exceeds the threshold h). The answer turns out to be quite varied depending on properties of the vector and the threshold; it turns out that h=0 behaves quite
differently than h different from 0. Among other results, in the large h regime, we obtain a phase transition in the stability exponent alpha for stable vectors and the critical value is alpha=1/2.

I will also briefly describe some related results by Forsström concerning such questions for the Ising Model with a nonzero external field.

Joint work with Malin Palö Forsström

Note: this talk is part of the Bath--Bristol probability colloquium.