Aldous-type Spectral Gaps in Unitary Groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917300567801856 |
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| author | Alon, Gil Puder, Doron |
| author_facet | Alon, Gil Puder, Doron |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_00353 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Aldous-type Spectral Gaps in Unitary Groups Alon, Gil Puder, Doron Probability Mathematical Physics Combinatorics Group Theory 20c15, 60K35 (Primary) 60B15, 60B20, 60J27, 05c50, 22c05, 05c81, 05c65 (Secondary) 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. |
| title | Aldous-type Spectral Gaps in Unitary Groups |
| topic | Probability Mathematical Physics Combinatorics Group Theory 20c15, 60K35 (Primary) 60B15, 60B20, 60J27, 05c50, 22c05, 05c81, 05c65 (Secondary) |
| url | https://arxiv.org/abs/2603.00353 |