11,712 research outputs found
On a symbolic representation of non-central Wishart random matrices with applications
By using a symbolic method, known in the literature as the classical umbral
calculus, the trace of a non-central Wishart random matrix is represented as
the convolution of the trace of its central component and of a formal variable
involving traces of its non-centrality matrix. Thanks to this representation,
the moments of this random matrix are proved to be a Sheffer polynomial
sequence, allowing us to recover several properties. The multivariate symbolic
method generalizes the employment of Sheffer representation and a closed form
formula for computing joint moments and cumulants (also normalized) is given.
By using this closed form formula and a combinatorial device, known in the
literature as necklace, an efficient algorithm for their computations is set
up. Applications are given to the computation of permanents as well as to the
characterization of inherited estimators of cumulants, which turn useful in
dealing with minors of non-central Wishart random matrices. An asymptotic
approximation of generalized moments involving free probability is proposed.Comment: Journal of Multivariate Analysis (2014
Quantum Hall Ground States, Binary Invariants, and Regular Graphs
Extracting meaningful physical information out of a many-body wavefunction is
often impractical. The polynomial nature of fractional quantum Hall (FQH)
wavefunctions, however, provides a rare opportunity for a study by virtue of
ground states alone. In this article, we investigate the general properties of
FQH ground state polynomials. It turns out that the data carried by an FQH
ground state can be essentially that of a (small) directed graph/matrix. We
establish a correspondence between FQH ground states, binary invariants and
regular graphs and briefly introduce all the necessary concepts. Utilizing
methods from invariant theory and graph theory, we will then take a fresh look
on physical properties of interest, e.g. squeezing properties, clustering
properties, etc. Our methodology allows us to `unify' almost all of the
previously constructed FQH ground states in the literature as special cases of
a graph-based class of model FQH ground states, which we call \emph{accordion}
model FQH states
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