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.
Keywords
free convolution, random matrices, Airy point process,
quantized free convolution