3,064 research outputs found
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
Recent developments in quantum mechanics with magnetic fields
We present a review on the recent developments concerning rigorous
mathematical results on Schr\"odinger operators with magnetic fields. This
paper is dedicated to the sixtieth birthday of Barry Simon.Comment: Update of the previous versions; some more references added and typos
and some minor errors correcte
The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case
We consider the spectral statistics of large random band matrices on
mesoscopic energy scales. We show that the two-point correlation function of
the local eigenvalue density exhibits a universal power law behaviour that
differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in
the physics literature by Altshuler and Shklovskii [4]; it describes the
correlations of the eigenvalue density in general metallic samples with weak
disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas
for band matrices. In two dimensions, where the leading term vanishes owing to
an algebraic cancellation, we identify the first non-vanishing term and show
that it differs substantially from the prediction of Kravtsov and Lerner [33]
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