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Airy point process via supersymmetric lifts

Andrew Ahn
Vol. 3 (2022), No. 4, 869–938
DOI: 10.2140/pmp.2022.3.869
Abstract

We study the local asymptotics at the edge for particle systems arising either from eigenvalues of sums of unitarily invariant random Hermitian matrices or from signatures corresponding to decompositions of tensor products of representations of the unitary group. Our method treats these two models in parallel, and is based on new formulas for observables described in terms of a special family of lifts, which we call supersymmetric lifts, of Schur functions and multivariate Bessel functions. We obtain explicit expressions for a class of supersymmetric lifts inspired by determinantal formulas for supersymmetric Schur functions due to Moens and Van der Jeugt (J. Algebraic Combin. 17:3 (2003), 283–307). Asymptotic analysis of these lifts enable us to probe the edge. We focus on several settings where the Airy point process arises.

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Keywords
free convolution, random matrices, Airy point process, quantized free convolution
Mathematical Subject Classification
Primary: 60B15, 60B20
Secondary: 22E46
Milestones
Received: 12 October 2021
Accepted: 6 June 2022
Published: 21 February 2023
Authors
Andrew Ahn
Cornell University
Ithaca, NY
United States