The Manhattan and Lorentz Mirror Models -- A result on the Cylinder with low density of mirrors

TL;DR

This paper investigates the behavior of Manhattan and Lorentz Mirror models on a finite-width cylinder, demonstrating that the maximum height reached by a walker scales as the inverse square of the mirror probability, using algebraic methods.

Contribution

It introduces a novel algebraic approach to analyze the maximum height in mirror models on a cylinder with low mirror density.

Findings

Maximum height scales as p^{-2} for low mirror probability p

Uses Brauer and Walled Brauer algebras for analysis

Results apply to models with mirror probability p<Cn^{-1}

Abstract

We study the Manhattan and Lorentz Mirror models on an infinite cylinder of finite even width $n$, with the mirror probability $p$ satisfying $p<Cn^{-1}$, $C$ a constant. We use the Brauer and Walled Brauer algebras to show that the maximum height along the cylinder reached by a walker is order $p^{-2}$.