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.