4,185 research outputs found
Derivation of an eigenvalue probability density function relating to the Poincare disk
A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives
the eigenvalue probability density function for the top N x N sub-block of a
Haar distributed matrix from U(N+n). In the case n \ge N, we rederive this
result, starting from knowledge of the distribution of the sub-blocks,
introducing the Schur decomposition, and integrating over all variables except
the eigenvalues. The integration is done by identifying a recursive structure
which reduces the dimension. This approach is inspired by an analogous approach
which has been recently applied to determine the eigenvalue probability density
function for random matrices A^{-1} B, where A and B are random matrices with
entries standard complex normals. We relate the eigenvalue distribution of the
sub-blocks to a many body quantum state, and to the one-component plasma, on
the pseudosphere.Comment: 11 pages; To appear in J.Phys
Interlaced particle systems and tilings of the Aztec diamond
Motivated by the problem of domino tilings of the Aztec diamond, a weighted
particle system is defined on lines, with line containing
particles. The particles are restricted to lattice points from 0 to , and
particles on successive lines are subject to an interlacing constraint. It is
shown that marginal distributions for this particle system can be computed
exactly. This in turn is used to give unified derivations of a number of
fundamental properties of the tiling problem, for example the evaluation of the
number of distinct configurations and the relation to the GUE minor process. An
interlaced particle system associated with the domino tiling of a certain half
Aztec diamond is similarly defined and analyzed.Comment: 17 pages, 4 figure
Hypergeometric solutions to the q-Painlev\'e equation of type
We consider the q-Painlev\'e equation of type (a version of
q-Painlev\'e V equation) and construct a family of solutions expressible in
terms of certain basic hypergeometric series. We also present the determinant
formula for the solutions.Comment: 16 pages, IOP styl
{\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
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
Symmetrized models of last passage percolation and non-intersecting lattice paths
It has been shown that the last passage time in certain symmetrized models of
directed percolation can be written in terms of averages over random matrices
from the classical groups , and . We present a theory of
such results based on non-intersecting lattice paths, and integration
techniques familiar from the theory of random matrices. Detailed derivations of
probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure
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
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