Interfaces of the two-dimensional voter model in the context of SLE

TL;DR

This paper studies the geometrical properties of interfaces in the 2D voter model, revealing that while locally fractal, they become straight at large scales, contrasting with the critical Ising model and challenging SLE descriptions.

Contribution

It provides a detailed analysis of voter model interfaces, showing their local fractal nature and large-scale straightness, and compares these properties with those of the critical Ising model.

Findings

Voter model interfaces are locally fractal with dimension 3/2.

At large scales, voter model interfaces become straight, not describable by SLE.

Contrasts with the critical Ising model, which has SLE$_3$ interfaces.

Abstract

This paper investigates various geometrical properties of interfaces of the two-dimensional voter model. Despite its simplicity, the model exhibits dual characteristics, resembling both a critical system with long-range correlations, while also showing a tendency towards order similar to the Ising-Glauber model at zero temperature. This duality is reflected in the geometrical properties of its interfaces, which are examined here from the perspective of Schramm-Loewner evolution. Recent studies have delved into the geometrical properties of these interfaces within different lattice geometries and boundary conditions. We revisit these findings, focusing on a system within a box of linear size $L$ with Dobrushin boundary conditions, where values of the spins are fixed to either $+1$ or $-1$ on two distinct halves of the boundary, in order to enforce the presence of a pinned interface with fixed endpoints (or chordal interface). We also expand the study to compare the geometrical properties of the interfaces of the voter model with those of the critical Ising model and other related models. Scaling arguments and numerical studies suggest that, while locally the chordal interface of the voter model has fractal dimension $d_{\rm f}=3/2$, corresponding to a parameter $\kappa=4$, it becomes straight at large scales, confirming a conjecture made by Holmes et al \cite{holmes}, and ruling out the possibility of describing the chordal interface of the voter model by SLE$_{\kappa}$, for any non zero value of $\kappa$. This contrasts with the critical Ising model, which is described by SLE$_3$, and whose interface fluctuations remain of order $L$, and more generally with related critical models, which are in the same universality class.