tailieunhanh - Báo cáo toán học: " Random walks on generating sets for finite groups"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Random walks on generating sets for finite groups. | Random walks on generating sets for finite groups F. R. K. Chung 1 University of Pennsylvania Philadelphia PA 19104 R. L. Graham AT T Research Murray Hill NJ 07974 Submitted August 31 1996 Accepted November 12 1996 Dedicated to Herb Wilf on the occasion of his sixty-fifth birthday Abstract We analyze a certain random walk on the cartesian product Gn of a finite group G which is often used for generating random elements from G. In particular we show that the mixing time of the walk is at most crn2 log n where the constant cr depends only on the order r of G. 1. Introduction One method often used in computational group theory for generating random elements from a given non-trivial finite group G proceeds as follows . see 2 . A fixed integer n 2 is initially specified. Denote by Gn the set x1 . Xn Xj 2 G 1 i ng. If X x1 . xn 2 Gn we denote by x the subgroup of G generated by Xj 1 i ng. Let G c Gn denote the set of all X 2 Gn such that x G. We execute a random walk on G by taking the following general step. Suppose we are at a point p p1 . pn 2 G . Choose a random pair of indices i j with i j. Thus each such pair is chosen with probability n n1_1 . We then move to one of p0 pg . P0n where PiPj or PiPf1 if k i each with probability 1 2 Pk if k i . This rule determines the corresponding transition matrix Q of the walk. We note that with this rule we always have p0 2 G . It is also easy to check that for n n0 G this walk is irreducible and aperiodic see Section 5 for more quantitative remarks and has a stationary distribution ft which is uniform since G is a multigraph in which every vertex has degree 2n n 1 . 1 Research supported in part by NSF Grant No. DMS 95-04834 THE ELECTRONIC JOURNAL OF COMBINATORICS 4 NO. 2 1997 R7 2 Starting from some fixed initial distribution f0 on G we apply this procedure some number of times say t to reach a distribution f0Qt on G which we hope will be close to random when t is large. A crucial question which must be faced in this .