Simple Permutations Mix Even Better
Combinatorics
2007-05-23 v2
Abstract
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.
Cite
@article{arxiv.math/0411098,
title = {Simple Permutations Mix Even Better},
author = {Shlomo Hoory and Alex Brodsky},
journal= {arXiv preprint arXiv:math/0411098},
year = {2007}
}
Comments
Better statement and proof of Theorem 9