35 research outputs found
From Random Matrices to Quasiperiodic Jacobi Matrices via Orthogonal Polynomials
We present an informal review of results on asymptotics of orthogonal
polynomials, stressing their spectral aspects and similarity in two cases
considered. They are polynomials orthonormal on a finite union of disjoint
intervals with respect to the Szego weight and polynomials orthonormal on R
with respect to varying weights and having the same union of intervals as the
set of oscillations of asymptotics. In both cases we construct double infinite
Jacobi matrices with generically quasiperiodic coefficients and show that each
of them is an isospectral deformation of another. Related results on asymptotic
eigenvalue distribution of a class of random matrices of large size are also
shortly discussed
On a Limiting Distribution of Singular Values of Random Band Matrices
An equation is obtained for the Stieltjes transform of the normalized
distribution of singular values of non-symmetric band random matrices in the
limit when the band width and rank of the matrix simultaneously tend to
infinity. Conditions under which this limit agrees with the quarter-circle law
are found. An interesting particular case of lower triangular random matrices
is also considered and certain properties of the corresponding limiting
singular value distribution are given
Eigenvalue Distribution of Large Random Matrices Arising in Deep Neural Networks: Orthogonal Case
The paper deals with the distribution of singular values of the input-output
Jacobian of deep untrained neural networks in the limit of their infinite
width. The Jacobian is the product of random matrices where the independent
rectangular weight matrices alternate with diagonal matrices whose entries
depend on the corresponding column of the nearest neighbor weight matrix. The
problem was considered in \cite{Pe-Co:18} for the Gaussian weights and biases
and also for the weights that are Haar distributed orthogonal matrices and
Gaussian biases. Basing on a free probability argument, it was claimed that in
these cases the singular value distribution of the Jacobian in the limit of
infinite width (matrix size) coincides with that of the analog of the Jacobian
with special random but weight independent diagonal matrices, the case well
known in random matrix theory. The claim was rigorously proved in
\cite{Pa-Sl:21} for a quite general class of weights and biases with i.i.d.
(including Gaussian) entries by using a version of the techniques of random
matrix theory. In this paper we use another version of the techniques to
justify the claim for random Haar distributed weight matrices and Gaussian
biases.Comment: arXiv admin note: text overlap with arXiv:2011.1143
Thomas precession, persistent spin currents and quantum forces
We consider T-invariant spin currents induced by spin-orbit interactions
which originate from the confined motion of spin carriers in nanostructures.
The resulting Thomas spin precession is a fundamental and purely kinematic
relativistic effect occurring when the acceleration of carriers is not parallel
to their velocity. In the case, where the carriers (e.g. electrons) have
magnetic moment the forces due to the electric field of the spin current can,
in certain conditions, exceed the van der Waals-Casimir forces by several
orders of magnitude. We also discuss a possible experimental set-up tailored to
use these forces for checking the existence of a nonzero anomalous magnetic
moment of the photon