2,446 research outputs found
A random matrix decimation procedure relating to
Classical random matrix ensembles with orthogonal symmetry have the property
that the joint distribution of every second eigenvalue is equal to that of a
classical random matrix ensemble with symplectic symmetry. These results are
shown to be the case of a family of inter-relations between eigenvalue
probability density functions for generalizations of the classical random
matrix ensembles referred to as -ensembles. The inter-relations give
that the joint distribution of every -st eigenvalue in certain
-ensembles with is equal to that of another
-ensemble with . The proof requires generalizing a
conditional probability density function due to Dixon and Anderson.Comment: 19 pages, 1 figur
A Combinatorial Interpretation of the Free Fermion Condition of the Six-Vertex Model
The free fermion condition of the six-vertex model provides a 5 parameter
sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter
into the eigenfunctions of the transfer matrices of the model decouple, hence
allowing explicit solutions. Such conditions arose originally in early
field-theoretic S-matrix approaches. Here we provide a combinatorial
explanation for the condition in terms of a generalised Gessel-Viennot
involution. By doing so we extend the use of the Gessel-Viennot theorem,
originally devised for non-intersecting walks only, to a special weighted type
of \emph{intersecting} walk, and hence express the partition function of
such walks starting and finishing at fixed endpoints in terms of the single
walk partition functions
Difference system for Selberg correlation integrals
The Selberg correlation integrals are averages of the products
with respect to the Selberg
density. Our interest is in the case , , when this
corresponds to the -th moment of the corresponding characteristic
polynomial. We give the explicit form of a matrix linear
difference system in the variable which determines the average, and we
give the Gauss decomposition of the corresponding matrix.
For a positive integer the difference system can be used to efficiently
compute the power series defined by this average.Comment: 21 page
Growth models, random matrices and Painleve transcendents
The Hammersley process relates to the statistical properties of the maximum
length of all up/right paths connecting random points of a given density in the
unit square from (0,0) to (1,1). This process can also be interpreted in terms
of the height of the polynuclear growth model, or the length of the longest
increasing subsequence in a random permutation. The cumulative distribution of
the longest path length can be written in terms of an average over the unitary
group. Versions of the Hammersley process in which the points are constrained
to have certain symmetries of the square allow similar formulas. The derivation
of these formulas is reviewed. Generalizing the original model to have point
sources along two boundaries of the square, and appropriately scaling the
parameters gives a model in the KPZ universality class. Following works of Baik
and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled
cumulative distribution, in which a particular Painlev\'e II transcendent plays
a prominent role.Comment: 27 pages, 5 figure
Increasing subsequences and the hard-to-soft edge transition in matrix ensembles
Our interest is in the cumulative probabilities Pr(L(t) \le l) for the
maximum length of increasing subsequences in Poissonized ensembles of random
permutations, random fixed point free involutions and reversed random fixed
point free involutions. It is shown that these probabilities are equal to the
hard edge gap probability for matrix ensembles with unitary, orthogonal and
symplectic symmetry respectively. The gap probabilities can be written as a sum
over correlations for certain determinantal point processes. From these
expressions a proof can be given that the limiting form of Pr(L(t) \le l) in
the three cases is equal to the soft edge gap probability for matrix ensembles
with unitary, orthogonal and symplectic symmetry respectively, thereby
reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 page
{\bf -Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}
It has recently been emphasized that all known exact evaluations of gap
probabilities for classical unitary matrix ensembles are in fact
-functions for certain Painlev\'e systems. We show that all exact
evaluations of gap probabilities for classical orthogonal matrix ensembles,
either known or derivable from the existing literature, are likewise
-functions for certain Painlev\'e systems. In the case of symplectic
matrix ensembles all exact evaluations, either known or derivable from the
existing literature, are identified as the mean of two -functions, both
of which correspond to Hamiltonians satisfying the same differential equation,
differing only in the boundary condition. Furthermore the product of these two
-functions gives the gap probability in the corresponding unitary
symmetry case, while one of those -functions is the gap probability in
the corresponding orthogonal symmetry case.Comment: AMS-Late
Eigenvalue distributions for some correlated complex sample covariance matrices
The distributions of the smallest and largest eigenvalues for the matrix
product , where is an complex Gaussian matrix
with correlations both along rows and down columns, are expressed as determinants. In the case of correlation along rows, these expressions are
computationally more efficient than those involving sums over partitions and
Schur polynomials reported recently for the same distributions.Comment: 11 page
Scaling limit of vicious walks and two-matrix model
We consider the diffusion scaling limit of the one-dimensional vicious walker
model of Fisher and derive a system of nonintersecting Brownian motions. The
spatial distribution of particles is studied and it is described by use of
the probability density function of eigenvalues of Gaussian random
matrices. The particle distribution depends on the ratio of the observation
time and the time interval in which the nonintersecting condition is
imposed. As is going on from 0 to 1, there occurs a transition of
distribution, which is identified with the transition observed in the
two-matrix model of Pandey and Mehta. Despite of the absence of matrix
structure in the original vicious walker model, in the diffusion scaling limit,
accumulation of contact repulsive interactions realizes the correlated
distribution of eigenvalues in the multimatrix model as the particle
distribution.Comment: REVTeX4, 12 pages, no figure, minor corrections made for publicatio
From Random Matrices to Stochastic Operators
We propose that classical random matrix models are properly viewed as finite
difference schemes for stochastic differential operators. Three particular
stochastic operators commonly arise, each associated with a familiar class of
local eigenvalue behavior. The stochastic Airy operator displays soft edge
behavior, associated with the Airy kernel. The stochastic Bessel operator
displays hard edge behavior, associated with the Bessel kernel. The article
concludes with suggestions for a stochastic sine operator, which would display
bulk behavior, associated with the sine kernel.Comment: 41 pages, 5 figures. Submitted to Journal of Statistical Physics.
Changes in this revision: recomputed Monte Carlo simulations, added reference
[19], fit into margins, performed minor editin
Robust multi-fidelity design of a micro re-entry unmanned space vehicle
This article addresses the preliminary robust design of a small-scale re-entry unmanned space vehicle by means of a hybrid optimization technique. The approach, developed in this article, closely couples an evolutionary multi-objective algorithm with a direct transcription method for optimal control problems. The evolutionary part handles the shape parameters of the vehicle and the uncertain objective functions, while the direct transcription method generates an optimal control profile for the re-entry trajectory. Uncertainties on the aerodynamic forces and characteristics of the thermal protection material are incorporated into the vehicle model, and a Monte-Carlo sampling procedure is used to compute relevant statistical characteristics of the maximum heat flux and internal temperature. Then, the hybrid algorithm searches for geometries that minimize the mean value of the maximum heat flux, the mean value of the maximum internal temperature, and the weighted sum of their variance: the evolutionary part handles the shape parameters of the vehicle and the uncertain functions, while the direct transcription method generates the optimal control profile for the re-entry trajectory of each individual of the population. During the optimization process, artificial neural networks are utilized to approximate the aerodynamic forces required by the optimal control solver. The artificial neural networks are trained and updated by means of a multi-fidelity approach: initially a low-fidelity analytical model, fitted on a waverider type of vehicle, is used to train the neural networks, and through the evolution a mix of analytical and computational fluid dynamic, high-fidelity computations are used to update it. The data obtained by the high-fidelity model progressively become the main source of updates for the neural networks till, near the end of the optimization process, the influence of the data obtained by the analytical model is practically nullified. On the basis of preliminary results, the adopted technique is able to predict achievable performance of the small spacecraft and the requirements in terms of thermal protection materials
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