Group-invariant random processes

First we consider some isometry-invariant point process on Rd and examine what types of graphs can be defined on the points of the process in such a way that the point process and the graph have an equivariant distribution. We show that 1-ended trees and Zn can be achieved for invariant point processes in Rd that satisfy some nontriviality condition. The rest of the dissertation is about random subgraphs of infinite transitive graphs. Babson and Benjamini showed that for the Cayley graphs of finitely presented 1-ended groups there is a unique infinite cluster when the percolation parameter is in a small neighborhood of 1. We give a simpler proof for their result and solve two of their questions about how certain properties of cutsets are preserved under quasi-isometries. Babson and Benjamini did not know whether the assumption finitely presented" is necessary for their method in proving p_u<1 to work, or whether the key property that they actually need is satisfied by every 1-ended infinite Cayley graph. We show that the lamplighter group does not have this key property". In the next section we prove that if there are infinitely many infinite clusters at Bernoulli percolation on a transitive unimodular graph, then no two of these clusters can contain infinitely many vertices that are adjacent to the other cluster. Minimal spanning trees on infinite graphs are important because of their close connection to Bernoulli percolation. We show that all the trees in the free minimal spanning forest of an infinite transitive unimodular graph have the same number of ends almost always. Finally, we go beyond unimodular graphs, and generalize some results that have only been known for unimodular graphs to nonunimodular graphs. We also show some remarkable differences between these two classes from the point of view of the behavior of invariant random subgraphs. One result of the section is that critical percolation on a nonunimodular graph cannot have infinitely many infinite clusters (the case for so-called heavy clusters had been shown by Lyons, Peres and Schramm).

Grantor dc:publisher

[Bloomington, Ind.] : Indiana University

Year dc:date.issued

2010

Author dc:creator

• Timar, Adam

Advisor dc:contributor.advisor

• Lyons, Russell

• nonamenable
• percolation
• nonunimodular
• group-invariant

Language dc:language.iso

EN

Handle dc:identifier.uri

https://hdl.handle.net/2022/7030

OAI identifier oai:identifier

oai:scholarworks.iu.edu:2022/7030