Percolation and Ising Model on Graphs with Tree-like Structure

Abstract

In the main part of this dissertation we present a method for finding the critical probability for the Bernoulli bond percolation and the critical inverse temperature for Ising model on graphs with the so-called tree-like structure. Such graphs can be decomposed into a tree of pieces, which have finitely many isomorphism classes. This class of graphs includes Cayley graphs of amalgamated products, HNN extensions or general groups acting on trees. It also includes all transitive graphs with more than one end.

We show that any Cayley graph of a virtually free group (that is, a group acting on a tree with finite vertex stabilizers) with respect to any finite generating set has a tree-like structure with finite pieces. In particular we show how to compute the critical probability and the critical inverse temperature of the Cayley graph of a free group with respect to any finite generating set. The method is illustrated on several examples, including free products, the Cayley graph of the special linear group SL(2,Z), and grandparent tree.

Next we prove that with probability tending to 1, a 1-relator group with at least 3 generators and the relator of length n is residually finite, virtually residually (finite p-)group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions.