Walks on graphs and lattices -- effective bounds and applications
| dc.creator | Rivin, Igor | |
| dc.date | 2007-03-19 | |
| dc.date.accessioned | 2026-07-07T07:52:36Z | |
| dc.date.available | 2026-07-07T07:52:36Z | |
| dc.description | 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 Gamma. We consider all walks of length N on G, starting from v_i and ending at v_j To each such walk $w$ we assign the element of Gamma equal to the product of the elements along the walk. The set of all walks of length N from v_i to v_j thus induces a probability distribution $F_N on Gamma In previous work we have given necessary and sufficient conditions for the limit as N goes to infinity of F_N to exist and to be the uniform density on Gamma. The convergence speed is then exponential in N. In this paper we consider (G, Gamma) where Gamma is a group possessing Kazhdan's property T (or, less restrictively, property tau with respect to representations with finite image), and a family of homomorphismsψ_k: Gamma -> Gamma_k with finite image. Each F_N induces a distribution $F_{N, k} on Gamma_k (by push-forward). Our main result is that, under mild technical assumptions, the exponential rate of convergence of $F_{N, k} to the uniform distribution on Gamma_k does not depend on k. As an application, we prove effective versions of the results of the author on the probability that a random (in a suitable sence) element of SL(n, Z) or Sp(n, Z) has irreducible characteristic polynomial, generic Galois group, etc. | |
| dc.identifier | https://arxiv.org/abs/math/0703533 | |
| dc.identifier | http://arxiv.org/abs/math/0703533 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/125939 | |
| dc.subject | Number Theory | |
| dc.subject | 11G99; 20E05; 57M60 | |
| dc.title | Walks on graphs and lattices -- effective bounds and applications | |
| dc.type | text |