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Simple Permutations Mix Even Better

Shlomo Hoory, Alex Brodsky

TL;DR

The paper analyzes how many random width-2 simple permutations from a small O(n^3) family are needed so that their composition approximates k-wise independence on {0,1}^n. It advances the state of the art by establishing sharp mixing-time and spectral-gap bounds through a novel combination of the comparison technique and a randomized 3-cycle implementation, enabling efficient multicommodity flows on the associated Schreier graphs. In particular, it proves that up to polylog factors, O(n^2 k^2) random compositions suffice for k up to about 2^n/50, and provides matching asymptotics for broader regimes via refined comparison arguments. Additionally, it gives an explicit construction of a degree-O(n^3) Cayley graph of the alternating group on 2^n points with spectral gap Omega(2^{-n}/n^2), improving prior explicit-expander results and informing cryptographic and expander-graph applications.

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.

Simple Permutations Mix Even Better

TL;DR

The paper analyzes how many random width-2 simple permutations from a small O(n^3) family are needed so that their composition approximates k-wise independence on {0,1}^n. It advances the state of the art by establishing sharp mixing-time and spectral-gap bounds through a novel combination of the comparison technique and a randomized 3-cycle implementation, enabling efficient multicommodity flows on the associated Schreier graphs. In particular, it proves that up to polylog factors, O(n^2 k^2) random compositions suffice for k up to about 2^n/50, and provides matching asymptotics for broader regimes via refined comparison arguments. Additionally, it gives an explicit construction of a degree-O(n^3) Cayley graph of the alternating group on 2^n points with spectral gap Omega(2^{-n}/n^2), improving prior explicit-expander results and informing cryptographic and expander-graph applications.

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.

Paper Structure

This paper contains 7 sections, 19 theorems, 12 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Composing simple permutations
  3. Proof of Theorem \ref{['k2n3:theorem']}
  4. Proof of Theorem \ref{['k2n2:theorem']}
  5. Proof of Theorem \ref{['3-cycle:theorem']}
  6. Odds and Ends
  7. Some concluding remarks

Key Result

Theorem 1

$\tau({\epsilon}) = \tilde{O}( n^2 k^2 \cdot \log(1/{\epsilon}) )$, as long as $k \leq 2^{n/50}$.

Figures (1)

  • Figure 1: The evolution of $v_1$.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Lemma 10
  • ...and 16 more