Random even graphs

TL;DR

This paper investigates the properties of random even subgraphs in finite graphs, introduces a sampling algorithm, and explores phase transitions and connections to Schramm–L"owner evolutions.

Contribution

It presents a novel method to generate random even subgraphs from a random-cluster measure and analyzes their phase transition behavior.

Findings

Phase transition at parameter 1/2 times the critical point of the random-cluster model

Connection between random even subgraphs and Schramm–L"owner evolutions

Sampling algorithm based on coupling from the past

Abstract

We study a random even subgraph of a finite graph $G$ with a general edge-weight $p\in(0,1)$. We demonstrate how it may be obtained from a certain random-cluster measure on $G$, and we propose a sampling algorithm based on coupling from the past. A random even subgraph of a planar lattice undergoes a phase transition at the parameter-value $\frac 12 \pc$, where $\pc$ is the critical point of the $q=2$ random-cluster model on the dual lattice. The properties of such a graph are discussed, and are related to Schramm--L\"owner evolutions (SLE).