Aldous-type Spectral Gaps in Unitary Groups

Gil Alon (The Open University of Israel) https://simons.berkeley.edu/talks/gil-alon-open-university-israel-2026-09-22 Local to Global Principles Aldous’ spectral-gap conjecture, proved by Caputo, Liggett and Richthammer, states that the interchange process on the symmetric group and the underlying random walk have the same spectral gap. We formulate a unitary analogue: a weighted hypergraph generates a random walk on the unitary group U(n) by Haar-randomizing the coordinate subspaces associated with its hyperedges. The resulting spectrum is surprisingly rich: it contains the spectra of all the associated discrete KMP processes, with arbitrary numbers of particles, as well as the entire spectrum of the corresponding symmetric-group walk, and much more. We show that the spectral gap can nevertheless be sought entirely in the torus-invariant, or zero-weight, parts of mixed tensor powers having equally many standard and dual factors. We furthermore conjecture that the spectral gap of the unitary walk always equals that of the two-particle KMP process. We prove the conjecture in the mean-field and codimension-one cases. Joint work with Doron Puder.