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Aldous-type Spectral Gaps in Unitary Groups

Gil Alon, Doron Puder

Abstract

Aldous' spectral gap conjecture, proven by Caputo, Liggett and Richthammer, states the following: for any set of transpositions in the symmetric group $\mathrm{Sym}(n)$, the spectral gap of the corresponding random walk on the group -- an $n!$-state process -- coincides with that of the corresponding random walk of a single element -- an $n$-state process. This paper presents an analog of this conjecture in the unitary group $\mathrm{U}(n)$, and proves it in several non-trivial cases. The phenomenon we discover is that for some natural families of probability distributions on $\mathrm{U}(n)$, the spectral gap of the corresponding random walk, which has a continuous state space, is identical to that of a discrete KMP process (also known as the uniform reshuffling process) with two indistinguishable particles on a hypergraph on $n$ vertices -- a discrete Markov chain with $\binom{n+1}{2}$ states.

Aldous-type Spectral Gaps in Unitary Groups

Abstract

Aldous' spectral gap conjecture, proven by Caputo, Liggett and Richthammer, states the following: for any set of transpositions in the symmetric group , the spectral gap of the corresponding random walk on the group -- an -state process -- coincides with that of the corresponding random walk of a single element -- an -state process. This paper presents an analog of this conjecture in the unitary group , and proves it in several non-trivial cases. The phenomenon we discover is that for some natural families of probability distributions on , the spectral gap of the corresponding random walk, which has a continuous state space, is identical to that of a discrete KMP process (also known as the uniform reshuffling process) with two indistinguishable particles on a hypergraph on vertices -- a discrete Markov chain with states.
Paper Structure (39 sections, 51 theorems, 156 equations)

This paper contains 39 sections, 51 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. The Aldous phenomenon in $\mathrm{Sym}(n)$
  3. Hypergraph measures on $\mathrm{U}(n)$
  4. An Aldous phenomenon in $\mathrm{U}(n)$
  5. A discrete KMP (or multiset) process
  6. The $\mathrm{U}(n)$-spectrum of a hypergraph contains its $\mathrm{Sym}(n)$-spectrum
  7. Outline and notation
  8. Representations:
  9. Hypergraph measures and Laplacians:
  10. Torus-invariant subspace:
  11. KMP:
  12. Additional notation:
  13. Related works
  14. Aldous-type phenomena in groups.
  15. Random walks on $\mathrm{U}(n)$.
  16. ...and 24 more sections

Key Result

Theorem 1.1

caputo2010proof Let $Q\colon \mathrm{Sym}(n)\to \mathbb{R}_{\ge0}$ be supported on transpositions. Then for every $\mathrm{triv}\ne\rho\in\mathrm{Irr}(\mathrm{Sym}(n))$,

Theorems & Definitions (104)

  • Theorem 1.1
  • Conjecture 1.2: Caputo's hypergraph conjecture
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 94 more