8,750 research outputs found
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 symmetric or hermitian random matrices with
independent, identically distributed entries where the probability distribution
for each matrix element is given by a measure 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 . 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
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