Use of statistically leinert sets to calculate return probabilities of random walks in F_s1 x F_s2

Abstract

Hastings first presented bounds on the second largest eigenvalue for matrices in a Hermitian complete positive map in 2007. In this work we extend his work to tighten these bounds. To do this, we introduce the idea of Statistically Leinert Sets to modify the generating functions presented in Woess in 1986 and recompute the radii of convergence in his paper in 1986. We primarily use techniques from combinatorics and calculate norms using the ideas presented by Akemann and Ostrang in their paper in 1976.