Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble

The paper studies the spectral properties of large Wigner, band and sample covariance random matrices with heavy tails of the marginal distributions of matrix entries.Comment: This is an extended version of my talk at the QMath 9 conference at Giens, France on September 13-17, 200

Random matrices, log-gases and Holder regularity

The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner's original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices.Comment: Proceedings of ICM 201

Universality of Wigner Random Matrices

We consider $N\times N$ symmetric or hermitian random matrices with independent, identically distributed entries where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove that the local eigenvalue statistics in the bulk of the spectrum for these matrices coincide with those of the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE), respectively, in the limit $N\to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. We also show that the Wigner semicircle law holds locally on the smallest possible scales and we prove that eigenvectors are fully delocalized and eigenvalues repel each other on arbitrarily small scales.Comment: Submitted to the Proceedings of ICMP, Prague, 200

Identities and exponential bounds for transfer matrices

This paper is about analytic properties of single transfer matrices originating from general block-tridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity, If the block tridiagonal matrix is invertible, it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain, and the other half have an upper bound that is exponentially small. This is a consequence of a theorem by Demko, Moss and Smith on the decay of matrix elements of inverse of banded matrices.Comment: To appear in J. Phys. A: Math. and Theor. (Special issue on Lyapunov Exponents, edited by F. Ginelli and M. Cencini). 16 page

Winding Numbers, Complex Currents, and Non-Hermitian Localization

The nature of extended states in disordered tight binding models with a constant imaginary vector potential is explored. Such models, relevant to vortex physics in superconductors and to population biology, exhibit a delocalization transition and a band of extended states even for a one dimensional ring. Using an analysis of eigenvalue trajectories in the complex plane, we demonstrate that each delocalized state is characterized by an (integer) winding number, and evaluate the associated complex current. Winding numbers in higher dimensions are also discussed.Comment: 4 pages, 2 figure

Non-Hermitian Localization in Biological Networks

We explore the spectra and localization properties of the N-site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions, and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N, the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes, but also with respect to 90 degree rotations, with an unusual anisotropic divergence in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. All states are extended on the rim of this hole, while the localized eigenvalues outside the hole are unchanged. The bias dependent shape of this hole tracks the bias independent contours of constant localization length. We treat the large-N limit by a combination of direct numerical diagonalization and using transfer matrices, an approach that allows us to exploit an electrostatic analogy connecting the "charges" embodied in the eigenvalue distribution with the contours of constant localization length. We show that similar results are obtained for more realistic neural networks that obey "Dale's Law" (each site is purely excitatory or inhibitory), and conclude with perturbation theory results that describe the limit of large bias g, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns