Simple Permutations Mix Even Better

TL;DR

This paper demonstrates that composing a small family of simple permutations can produce near k-wise independence efficiently, and provides an explicit construction of a Cayley graph with a significant spectral gap.

Contribution

It improves bounds on the number of compositions needed for near independence and offers an explicit Cayley graph construction with a better spectral gap.

Findings

Approximately n^2 * k^2 compositions suffice for near k-wise independence.

Constructs a degree O(n^3) Cayley graph with spectral gap Omega(2^{-n}/n^2).

Enhances previous results by Gowers (1996) and Hoory et al. (2004).

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.