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

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 $p^{-2}$.

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