The Manhattan and Lorentz Mirror Models: A Result on the Cylinder with Low Density of Mirrors

Abstract

AbstractWe study the Manhattan and Lorentz mirror models on an infinite cylinder of finite even width n, with the mirror probability p satisfying $$p<Cn^{-1}$$ p < C n - 1 , C a constant. We show that the maximum height along the cylinder reached by a walker is order $$p^{-2}$$ p - 2 . We observe an algebraic structure, which helps organise our argument. The models on the cylinder can be thought of as Markov chains on the Brauer (in the Mirror case) or Walled Brauer (in the Manhattan case) algebra, with the transfer matrix given by multiplication by an element of the algebra.