28,372 research outputs found
Correlation kernels for sums and products of random matrices
Let be a random matrix whose squared singular value density is a
polynomial ensemble. We derive double contour integral formulas for the
correlation kernels of the squared singular values of and , where
is a complex Ginibre matrix and is a truncated unitary matrix. We also
consider the product of and several complex Ginibre/truncated unitary
matrices. As an application, we derive the precise condition for the squared
singular values of the product of several truncated unitary matrices to follow
a polynomial ensemble. We also consider the sum where is a GUE
matrix and is a random matrix whose eigenvalue density is a polynomial
ensemble. We show that the eigenvalues of follow a polynomial ensemble
whose correlation kernel can be expressed as a double contour integral. As an
application, we point out a connection to the two-matrix model.Comment: 33 pages, some changes suggested by the referee is made and some
references are adde
Logics of Finite Hankel Rank
We discuss the Feferman-Vaught Theorem in the setting of abstract model
theory for finite structures. We look at sum-like and product-like binary
operations on finite structures and their Hankel matrices. We show the
connection between Hankel matrices and the Feferman-Vaught Theorem. The largest
logic known to satisfy a Feferman-Vaught Theorem for product-like operations is
CFOL, first order logic with modular counting quantifiers. For sum-like
operations it is CMSOL, the corresponding monadic second order logic. We
discuss whether there are maximal logics satisfying Feferman-Vaught Theorems
for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th
birthday. The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-23534-9_1
M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities
We generalize the character formulas for multiplicities of irreducible
constituents from group theory to semigroup theory using Rota's theory of
M\"obius inversion. The technique works for a large class of semigroups
including: inverse semigroups, semigroups with commuting idempotents,
idempotent semigroups and semigroups with basic algebras. Using these tools we
are able to give a complete description of the spectra of random walks on
finite semigroups admitting a faithful representation by upper triangular
matrices over the complex numbers. These include the random walks on chambers
of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and
Diaconis. Applications are also given to decomposing tensor powers and exterior
products of rook matrix representations of inverse semigroups, generalizing and
simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update
Finite Dimensional Statistical Inference
In this paper, we derive the explicit series expansion of the eigenvalue
distribution of various models, namely the case of non-central Wishart
distributions, as well as correlated zero mean Wishart distributions. The tools
used extend those of the free probability framework, which have been quite
successful for high dimensional statistical inference (when the size of the
matrices tends to infinity), also known as free deconvolution. This
contribution focuses on the finite Gaussian case and proposes algorithmic
methods to compute the moments. Cases where asymptotic results fail to apply
are also discussed.Comment: 14 pages, 13 figures. Submitted to IEEE Transactions on Information
Theor
Approximating a Wavefunction as an Unconstrained Sum of Slater Determinants
The wavefunction for the multiparticle Schr\"odinger equation is a function
of many variables and satisfies an antisymmetry condition, so it is natural to
approximate it as a sum of Slater determinants. Many current methods do so, but
they impose additional structural constraints on the determinants, such as
orthogonality between orbitals or an excitation pattern. We present a method
without any such constraints, by which we hope to obtain much more efficient
expansions, and insight into the inherent structure of the wavefunction. We use
an integral formulation of the problem, a Green's function iteration, and a
fitting procedure based on the computational paradigm of separated
representations. The core procedure is the construction and solution of a
matrix-integral system derived from antisymmetric inner products involving the
potential operators. We show how to construct and solve this system with
computational complexity competitive with current methods.Comment: 30 page
- …