Does Eulerian percolation on $Z^2$ percolate ?

TL;DR

This paper investigates Eulerian percolation on the two-dimensional lattice, showing its connection to the Ising model and analyzing its percolation properties to understand phase transitions.

Contribution

It establishes the equivalence between Eulerian percolation and Ising model contours, and studies their percolation behavior on Z^2.

Findings

Eulerian percolation coincides with Ising model contours for a specific parameter.

Percolation properties depend on the parameter p, indicating phase transition behavior.

The study provides insights into the percolation threshold for Eulerian configurations.

Abstract

Eulerian percolation on Z 2 with parameter p is the classical Bernoulli bond percolation with parameter p conditioned on the fact that every site has an even degree. We first explain why Eulerian percolation with parameter p coincides with the contours of the Ising model for a well-chosen parameter $\beta$(p). Then we study the percolation properties of Eulerian percolation.