Invariant sums of random matrices and the onset of level repulsion

Zdzisław Burda 1 Giacomo Livan 2 Pierpaolo Vivo 3, 4

Journal of Statistical Mechanics: Theory and Experiment, IOP Science, 2015, pp.P06024

Using a simple rotate-and-sum procedure, we construct and solve exactly a random matrix model with peculiar 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 $\mathbf{S}$ of the form $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{O}_i \mathbf{D}_i\mathbf{O}_i^{\mathrm{T}}\rangle$ or $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{U}_i \mathbf{D}_i\mathbf{U}_i^\dagger\rangle$ respectively. The diagonal matrices $\mathbf{D}_i=\mathrm{diag}\{\lambda_1^{(i)},\ldots,\lambda_N^{(i)}\}$ are constructed from real eigenvalues drawn independently from distributions $p^{(i)}(x)$, while the matrices $\mathbf{O}_i$ and $\mathbf{U}_i$ are all orthogonal or unitary and the average $\langle\cdot\rangle$ is performed over the respective group. While the original matrices $\mathbf{D}_i$ do not exhibit level repulsion, the resulting sum $\mathbf{S}$ develops it upon averaging over multiple $(M\geq 2)$ uncorrelated rotations. We focus on the cases where $p^{(i)}(x)$ is A.) a semicircle law, or B.) a Gaussian law for all $i=1,\ldots,M$. For the choice A, in the limit $N\to\infty$ this ensemble appears spectrally indistinguishable from the standard GOE or GUE, having same spectral density, two-point correlation function, and nearest-neighbor spacing distribution $p(s)$ after unfolding. However, working out the case $N=2$ in detail, we uncover a universal (independent of the $p^{(i)}(x)$) but different from Wigner-Dyson behavior as $s\to 0^+$. The generic interaction between eigenvalues of $\mathbf{S}$ is indeed not precisely Vandermondian, despite the rotationally invariant nature of the ensemble, and classical RMT universality is restored only asymptotically. (continue...)

  • 1. Faculty of Physics and Applied Computer Science, University of Science and Technology AGH
  • 2. Department of Computer Science
  • 3. King's College London
  • 4. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques