Walks on graphs and lattices – effective bounds and applications

Abstract

We continue the investigations started in [Rivin, Technical Report math., 1999, Rivin, Duke Math. J., 2006]. We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite group Γ. We consider all walks of length N on G , starting from ν i and ending at ν j . To each such walk w we assign the element of Γ equal to the product of the elements along the walk. The set of all walks of length N from ν i to ν j thus induces a probability distribution F N,i,j on Γ. In [Rivin, Technical Report math., 1999] we give necessary and sufficient conditions for the limit as N goes to infinity of F N,i,j to exist and to be the uniform density on Γ (a detailed argument is presented in [Rivin, Duke Math. J., 2006]). The convergence speed is then exponential in N . In this paper we consider ( G , Γ), where Γ is a group possessing Kazhdan's property T (or, less restrictively, property τ with respect to representations with finite image), and a family of homomorphisms ψ k : Γ → Γ k with finite image. Each F N,i,j induces a distribution on Γ k (by push-forward under ψ k ). Our main result is that, under mild technical assumptions, the exponential rate of convergence of to the uniform distribution on Γ k does not depend on k . As an application, we prove effective versions of the results of [Rivin, Duke Math. J., 2006] on the probability that a random (in a suitable sense) element of SL( n , ℤ) or Sp( n , ℤ) has irreducible characteristic polynomial, generic Galois group, etc.