Simple permutations mix even better

We study the random composition of a small family of O(n3 ) 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 [12] and Hoory et al. [13] and show that up to a polylogarithmic factor, n2 k2 compositions of random permutations from this family suffice. In addition, our results give an explicit construction of a degree O(n3 ) Cayley graph of the alternating group of 2n objects with a spectral gap (2-n /n2 ), which is a substantial improvement over previous constructions.

• Type: Journal
• Year: 2008
• Where: RSA
• Authors: Alex Brodsky, Shlomo Hoory
• Keywords: Random Composition, Random Permutations, RSA 2008, Simple Permutations