Insertion-tolerance and repetitiveness of random graphs

TL;DR

This paper explores the properties of random graphs derived from bond percolation on Cayley graphs and from repetitive subgraphs, showing they have mutually singular probability laws through a general lemma on Borel equivalence relations.

Contribution

It introduces a new connection between bond percolation on Cayley graphs and repetitive subgraphs, demonstrating their mutually singular laws using a general Borel equivalence relation lemma.

Findings

Bond percolation on Cayley graphs yields specific random graph models.

Repetitive subgraphs of Cayley graphs form a distinct class with different probabilistic laws.

The two graph families are mutually singular in their probability measures.

Abstract

Bond percolation on Cayley graphs provides examples of random graphs. Other examples arise from the dynamical study of proper repetitive subgraphs of Cayley graphs. In this paper we demonstrate that these two families have mutually singular laws as a corollary of a general lemma about countable Borel equivalence relations on first countable Hausdorff spaces.