Near-independence of permutations and an almost sure polynomial bound on the diameter of the symmetric group

Abstract

We address the long-standing conjecture that all permutations have polynomially bounded word length in terms of any set of generators of the symmetric group Sn This is equivalent to polynomial-time (O(nc)) mixing of the (lazy) random walk on Sn where one step is multiplication by a generator or its inverse.We prove that the conjecture is true for almost all pairs of generators. Specifically, our bound is O(n7). For almost all pairs of generators, words of this length representing any given permutation can be constructed in Las Vegas polynomial time. The best previous bound on the word length for a random pair of generators was nInn(1/2+o(1)) (Babai-Hetyei, 1992).We build on recent major progress by Babai-Beals-Seress (SODA, 2004), confirming the conjecture under the assumption that at least one of the generators has degree