Harmonic analysis for rank-1 Randomised Horn Problems

Abstract

The randomised Horn problem, in both its additive and multiplicative version, has recently drawn increasing interest. Especially, closed analytical results have been found for the rank-1 perturbation of sums of Hermitian matrices and products of unitary matrices. We will generalise these results to rank-1 perturbations for products of positive definite Hermitian matrices and prove the other results in a new unified way. Our ideas work along harmonic analysis for matrix groups via spherical transforms that have been successfully applied in products of random matrices in the past years. In order to achieve the unified derivation of all three cases, we define the spherical transform on the unitary group and prove its invertibility.

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

“…This operation is equivalent to put an additional β 2 term in the exponent of each determinant in (15), except for the last one.…”

Stability of large complex systems with heterogeneous relaxation dynamics

“…(103). However, the case with the exponential function in the second Vandermonde, appeared in the randomized multiplicative Horn problem [51], in the DPMK equation for transport in semiconductors [52] and in the multiplicative analogue of Dyson Brownian Motion [53].…”

Derivative principles for invariant ensembles

“…the eigenvalue distributions are Dirac deltas, Horn's problem generalises to a sum of randomized adjoint orbits of O(n) and U(n), see [13,34,7,11]. We would like to draw also attention to some recent works discussing a multiplicative version of Horn's problem [11,33]. For general eigenvalues, a general unitarily invariant ensemble, the Pólya ensemble, on Herm(n) have been identified in [28,12].…”

“…For general eigenvalues, a general unitarily invariant ensemble, the Pólya ensemble, on Herm(n) have been identified in [28,12]. For this particular problem it has been proven, analogous to elementary probability theory, that harmonic analysis on matrix spaces is an essential tool for studies of sums of random matrices, e.g., see [28,12,33].…”

“…Indeed, one only needs to replace f diag by a product of weight functions. In recent works [12,22,23,24,25,33], harmonic analysis for random matrices has been extended to sums of other matrix spaces as well as to products of random matrices with the help of other Harish-Chandra-Itzykson-Zuber-type integrals [17,21] like the Berezin-Karpelevich integral [2,16] or the Gelfand-Naȋmark integral [15]. Interestingly, the corresponding Pólya ensembles that arise out of these computations have a distribution that have a similar structure as the right side of (1.2).…”