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Invariant sums of random matrices and the onset of level repulsion

Research output: Contribution to journalArticlepeer-review

Zdzislaw Burda, Giacomo Livan, Pierpaolo Vivo

Original languageEnglish
Article numberP06024
JournalJournal of Statistical Mechanics (JSTAT)
Volume2015
DOIs
Published12 Jun 2015

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  • BLVv10jstat

    BLVv10jstat.pdf, 493 KB, application/pdf

    Uploaded date:22 Jul 2015

    Version:Accepted author manuscript

King's Authors

Abstract

We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary) and yet the interaction between eigenvalues is not Vandermondian. The ensemble contains real symmetric or complex Hermitian matrices S of the form ${\bf S}=\sum_{i=1}^M \langle {\bf O}_i {\bf D}_i{\bf O}_i^{{\rm T}}\rangle$ or ${\bf S}=\sum_{i=1}^M \langle {\bf U}_i {\bf D}_i{\bf U}_i^\dagger\rangle$ respectively. The diagonal matrices ${\bf D}_i={\rm diag}\{\lambda_1^{(i)},\ldots,\lambda_N^{(i)}\}$ are constructed from real eigenvalues drawn independently from distributions p(i)(x), while the matrices Oi and Ui are all orthogonal or unitary. The average 〈centerdot〉 is simultaneously performed over the symmetry group and the joint distribution of $\{\lambda_j^{(i)}\}$ . We focus on the limits (i.) N → ∞ and (ii.) M → ∞, with N = 2. In the limit (i.), the resulting sum S develops level repulsion even though the original matrices do not feature it, and classical RMT universality is restored asymptotically. In the limit (ii.) the spacing distribution attains scaling forms that are computed exactly: for the orthogonal case, we recover the β = 1 Wigner's surmise, while for the unitary case an entirely new universal distribution is obtained. Our results allow to probe analytically the microscopic statistics of the sum of random matrices that become asymptotically free. We also give an interpretation of this model in terms of radial random walks in a matrix space. The analytical results are corroborated by numerical simulations.

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