{
  "id": "math/0703533",
  "version": "v1",
  "published": "2007-03-19T00:36:11.000Z",
  "updated": "2007-03-19T00:36:11.000Z",
  "title": "Walks on graphs and lattices -- effective bounds and applications",
  "authors": [
    "Igor Rivin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "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 homomorphisms\\psi_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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-19T00:36:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G99",
      "20E05",
      "57M60"
    ],
    "keywords": [
      "effective bounds",
      "application",
      "generic galois group",
      "finite group gamma",
      "finite directed graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3533R"
    }
  }
}