Monte Carlo study of the hull distribution for the q=1 Brauer model

TL;DR

This paper uses Monte Carlo simulations to analyze the hull distribution in a special q=1 Brauer model, revealing its similarity to reflected Brownian motion in the large system limit.

Contribution

It provides the first detailed Monte Carlo analysis of the hull distribution in the q=1 Brauer model, connecting it to Brownian motion with boundary reflection.

Findings

Hull distribution converges to that of reflected Brownian motion as system size increases.

Results support the hypothesis of universality with SLE(6) and critical percolation.

Monte Carlo data aligns with theoretical predictions for large systems.

Abstract

We study a special case of the Brauer model in which every path of the model has weight q=1. The model has been studied before as a solvable lattice model and can be viewed as a Lorentz lattice gas. The paths of the model are also called self-avoiding trails. We consider the model in a triangle with boundary conditions such that one of the trails must cross the triangle from a corner to the opposite side. Motivated by similarities between this model, SLE(6) and critical percolation, we investigate the distribution of the hull generated by this trail (the set of points on or surrounded by the trail) up to the hitting time of the side of the triangle opposite the starting point. Our Monte Carlo results are consistent with the hypothesis that for system size tending to infinity, the hull distribution is the same as that of a Brownian motion with perpendicular reflection on the boundary.