573 research outputs found
Limits for circular Jacobi beta-ensembles
Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the
circular Jacobi -ensemble, which is a generalization of the Dyson
circular -ensemble but equipped with an additional parameter , and
further studied its limiting spectral measure. We calculate the scaling limits
for expected products of characteristic polynomials of circular Jacobi
-ensembles. For the fixed constant , the resulting limit near the
spectrum singularity is proven to be a new multivariate function. When , the scaling limits in the bulk and at the soft edge agree with those of
the Hermite (Gaussian), Laguerre (Chiral) and Jacobi -ensembles proved
in the joint work with P Desrosiers "Asymptotics for products of characteristic
polynomials in classical beta-ensembles", Constr. Approx. 39 (2014),
arXiv:1112.1119v3. As corollaries, for even the scaling limits of point
correlation functions for the ensemble are given. Besides, a transition from
the spectrum singularity to the soft edge limit is observed as goes to
infinity. The positivity of two special multivariate hypergeometric functions,
which appear as one factor of the joint eigenvalue densities for spiked
Jacobi/Wishart -ensembles and Gaussian -ensembles with source,
will also be shown.Comment: 26 page
On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words
We consider the distributions of the lengths of the longest weakly increasing
and strongly decreasing subsequences in words of length N from an alphabet of k
letters. We find Toeplitz determinant representations for the exponential
generating functions (on N) of these distribution functions and show that they
are expressible in terms of solutions of Painlev\'e V equations. We show
further that in the weakly increasing case the generating function gives the
distribution of the smallest eigenvalue in the k x k Laguerre random matrix
ensemble and that the distribution itself has, after centering and normalizing,
an N -> infinity limit which is equal to the distribution function for the
largest eigenvalue in the Gaussian Unitary Ensemble of k x k hermitian matrices
of trace zero.Comment: 30 pages, revised version corrects an error in the statement of
Theorem
Asymptotics of Plancherel-type random partitions
We present a solution to a problem suggested by Philippe Biane: We prove that
a certain Plancherel-type probability distribution on partitions converges, as
partitions get large, to a new determinantal random point process on the set
{0,1,2,...} of nonnegative integers. This can be viewed as an edge limit
ransition. The limit process is determined by a correlation kernel on
{0,1,2,...} which is expressed through the Hermite polynomials, we call it the
discrete Hermite kernel. The proof is based on a simple argument which derives
convergence of correlation kernels from convergence of unbounded self-adjoint
difference operators.
Our approach can also be applied to a number of other probabilistic models.
As an example, we discuss a bulk limit for one more Plancherel-type model of
random partitions.Comment: AMS TeX, 19 pages. Version 2: minor typos fixe
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
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