图与格上的游动--有效界及其应用
Walks on graphs and lattices -- effective bounds and applications
摘要：
我们考虑以下情况：G是一个有限的有向图，其中G的每个顶点都被赋予一个有限群Gamma的元素.我们考虑G上从v_i开始到v_j结束的所有长度为N的行走。对于每一个这样的行走$w$，我们赋予Gamma的元素等于沿着行走的元素的乘积。因此，从v_i到v_j的长度为N的所有遍历的集合在Gamma_k上导出概率分布$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}（通过前推）。我们的主要结果是，在温和的技术假设下，$ F_{N, k}的指数收敛速率的均匀分布伽玛_k不依赖于k。作为应用，我们证明了作者关于SL（n，Z）或Sp（n，Z）的随机元（在适当意义下）有不可约特征多项式的概率、一般伽罗瓦群等结果的有效形式.
英文摘要：
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.
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