Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

Abstract

We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poissonlike statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour.

Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles

“…It is quite likely that other emergent spectral statistics are possible that can be approached that are not covered when taking first m → ∞ (as we have done in the present work) and, then, N → ∞. Especially the phenomena of an emerging non-trivial mesoscopic statistics observed in [56] seems to corroborate this guess.…”

“…The point is that one needs almost surely asymptotically free stable random matrices to find these kinds of level densities, see [65]. For instance some elliptical stable invariant random matrix ensembles do not belong to this class [56]. Then, one obtains deviations from the free probability results.…”

“…Very recently, it was shown [56] that also unitarily invariant Hermitian random matrices can exhibit Poisson statistics or a mixture of Poisson and sine-kernel statistics in the tail of the level density. This was corroborated by Monte-Carlo simulations and supersymmetry computations.…”

“…Thus, the Poisson statistics is not bounded to the localisation of the eigenvectors. Even non-trivial mesoscopic spectral statistics, meaning on a scale between the mean level spacing of consecutive eigenvalues and the scale of the macroscopic level density, like the Wigner semicircle [81] or the Marčenko-Pastur distribution [60], have been observed [56]. Born out from these findings, new question have arisen.…”

A rate of convergence when generating stable invariant Hermitian random matrix ensembles

“…Going away from Wigner matrices and introducing group invariance such as unitarily invariant random matrices [48,2,30,15,14,3,1,18,25,26,32], meaning matrices that are invariant under the conjugate action of the unitary group (see (1.6)), may change the picture. While also Poisson statistics has been observed [32] for the largest eigenvalues in the limit of large matrix dimensions, the eigenvectors remain always delocalised. This is due to unitarily invariance as then the eigenvectors are distributed with respect to the Haar measure of the unitary group.…”