The voter model chordal interface in two dimensions

TL;DR

This paper investigates the geometric properties of the voter model's interface and coalescing classes in a large two-dimensional triangular lattice, revealing insights into their asymptotic behavior as the system size grows.

Contribution

It introduces a detailed study of the stationary interface and coalescing classes in the 2D voter model, inspired by percolation theory, and analyzes their large-scale geometric features.

Findings

Characterization of the voter model interface in large systems

Analysis of coalescing classes and their geometric structure

Insights into the asymptotic behavior of these objects as system size increases

Abstract

Consider the voter model on a box of side length $L$ (in the triangular lattice) with boundary votes fixed forever as type 0 or type 1 on two different halves of the boundary. Motivated by analogous questions in percolation, we study several geometric objects at stationarity, as $L\rightarrow \infty$. One is the interface between the (large -- i.e., boundary connected) 0-cluster and 1-cluster. Another is the set of large "coalescing classes" determined by the coalescing walk process dual to the voter model.