Aldous-type Spectral Gaps in Unitary Groups

TL;DR

This paper explores an analog of Aldous' spectral gap conjecture within the unitary group, demonstrating that for certain distributions, the spectral gap matches that of a simpler discrete process.

Contribution

It introduces a new Aldous-type conjecture for unitary groups and proves it in several non-trivial cases, linking continuous and discrete spectral gaps.

Findings

Spectral gap in certain unitary group random walks equals that of a discrete process.

Established cases where the conjecture holds for specific distributions.

Identified a connection between continuous unitary processes and discrete Markov chains.

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.