111 research outputs found
An estimate for the average spectral measure of random band matrices
For a class of random band matrices of band width , we prove regularity of
the average spectral measure at scales , and find its
asymptotics at these scales.Comment: 19 pp., revised versio
Delocalization and Diffusion Profile for Random Band Matrices
We consider Hermitian and symmetric random band matrices in dimensions. The matrix entries , indexed by x,y \in
(\bZ/L\bZ)^d, are independent, centred random variables with variances s_{xy}
= \E |h_{xy}|^2. We assume that is negligible if exceeds the
band width . In one dimension we prove that the eigenvectors of are
delocalized if . We also show that the magnitude of the matrix
entries \abs{G_{xy}}^2 of the resolvent is self-averaging
and we compute \E \abs{G_{xy}}^2. We show that, as and , the behaviour of \E |G_{xy}|^2 is governed by a diffusion operator
whose diffusion constant we compute. Similar results are obtained in higher
dimensions
Power substitution in quasianalytic Carleman classes
Consider an equation of the form f(x)=g(xk), where k>1 and f(x) is a function in a given Carleman class of smooth functions. For each k, we construct a Carleman-type class which contains all the smooth solutions g(x) to such equations. We prove, under regularity assumptions, that if the original Carleman class is quasianalytic, then so is the new class. The results admit an extension to multivariate functions
Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution
We study zeroes of Gaussian analytic functions in a strip in the complex
plane, with translation-invariant distribution. We prove that the a limiting
horizontal mean counting-measure of the zeroes exists almost surely, and that
it is non-random if and only if the spectral measure is continuous (or
degenerate). In this case, the mean zero-counting measure is computed in terms
of the spectral measure. We compare the behavior with Gaussian analytic
function with symmetry around the real axis. These results extend a work by
Norbert Wiener.Comment: 24 pages, 1 figure. Some corrections were made and presentation was
improve
Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States
We prove general comparison theorems for eigenvalues of perturbed Schrodinger
operators that allow proof of Lieb--Thirring bounds for suitable non-free
Schrodinger operators and Jacobi matrices.Comment: 11 page
Spectral Statistics of Erd{\H o}s-R\'enyi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random
graphs, i.e.\ graphs on vertices where every edge is chosen independently
and with probability . We rescale the matrix so that its bulk
eigenvalues are of order one. Under the assumption , we prove
the universality of eigenvalue distributions both in the bulk and at the edge
of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of
the Erd{\H o}s-R\'enyi graph in the bulk of the spectrum has the same
distribution as that of the Gaussian orthogonal ensemble; and (2) that the
second largest eigenvalue of the Erd{\H o}s-R\'enyi graph has the same
distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As
an application of our method, we prove the bulk universality of generalized
Wigner matrices under the assumption that the matrix entries have at least moments
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)
Trace formulae for d-regular graphs are derived and used to express the
spectral density in terms of the periodic walks on the graphs under
consideration. The trace formulae depend on a parameter w which can be tuned
continuously to assign different weights to different periodic orbit
contributions. At the special value w=1, the only periodic orbits which
contribute are the non back- scattering orbits, and the smooth part in the
trace formula coincides with the Kesten-McKay expression. As w deviates from
unity, non vanishing weights are assigned to the periodic walks with
back-scatter, and the smooth part is modified in a consistent way. The trace
formulae presented here are the tools to be used in the second paper in this
sequence, for showing the connection between the spectral properties of
d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure
On a Watson-like Uniqueness Theorem and Gevrey Expansions
We present a maximal class of analytic functions, elements of which are in
one-to-one correspondence with their asymptotic expansions. In recent decades
it has been realized (B. Malgrange, J. Ecalle, J.-P. Ramis, Y. Sibuya et al.),
that the formal power series solutions of a wide range of systems of ordinary
(even non-linear) analytic differential equations are in fact the Gevrey
expansions for the regular solutions. Watson's uniqueness theorem belongs to
the foundations of this new theory. This paper contains a discussion of an
extension of Watson's uniqueness theorem for classes of functions which admit a
Gevrey expansion in angular regions of the complex plane with opening less than
or equal to (\frac \pi k,) where (k) is the order of the Gevrey expansion. We
present conditions which ensure uniqueness and which suggest an extension of
Watson's representation theorem. These results may be applied for solutions of
certain classes of differential equations to obtain the best accuracy estimate
for the deviation of a solution from a finite sum of the corresponding Gevrey
expansion.Comment: 18 pages, 4 figure
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