arXiv math/04110982004-11math.CO

Simple Permutations Mix Even Better

Shlomo Hoory, Alex Brodsky

备注：: Better statement and proof of Theorem 9

英文摘要：

We study the random composition of a small family of O(n^3) simple permutations on {0,1}^n. Specifically we ask how many randomly selected simple permutations need be composed to yield a permutation that is close to k-wise independent. We improve on the results of Gowers 1996 and Hoory, Magen, Myers and Rackoff 2004, and show that up to a polylogarithmic factor, n^2*k^2 compositions of random permutations from this family suffice. In addition, our results give an explicit construction of a degree O(n^3) Cayley graph of the alternating group of 2^n objects with a spectral gap Omega(2^{-n}/n^2), which is a substantial improvement over previous constructions.

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