26 research outputs found
Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
We study a model of non-intersecting squared Bessel processes in the
confluent case: all paths start at time at the same positive value , remain positive, and are conditioned to end at time at . In
the limit , after appropriate rescaling, the paths fill out a
region in the -plane that we describe explicitly. In particular, the paths
initially stay away from the hard edge at , but at a certain critical
time the smallest paths hit the hard edge and from then on are stuck to
it. For we obtain the usual scaling limits from random matrix
theory, namely the sine, Airy, and Bessel kernels. A key fact is that the
positions of the paths at any time constitute a multiple orthogonal
polynomial ensemble, corresponding to a system of two modified Bessel-type
weights. As a consequence, there is a matrix valued
Riemann-Hilbert problem characterizing this model, that we analyze in the large
limit using the Deift-Zhou steepest descent method. There are some novel
ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure
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
Non-intersecting squared Bessel paths: critical time and double scaling limit
We consider the double scaling limit for a model of non-intersecting
squared Bessel processes in the confluent case: all paths start at time
at the same positive value , remain positive, and are conditioned to end
at time at . After appropriate rescaling, the paths fill a region in
the --plane as that intersects the hard edge at at a
critical time . In a previous paper (arXiv:0712.1333), the scaling
limits for the positions of the paths at time were shown to be
the usual scaling limits from random matrix theory. Here, we describe the limit
as of the correlation kernel at critical time and in the
double scaling regime. We derive an integral representation for the limit
kernel which bears some connections with the Pearcey kernel. The analysis is
based on the study of a matrix valued Riemann-Hilbert problem by
the Deift-Zhou steepest descent method. The main ingredient is the construction
of a local parametrix at the origin, out of the solutions of a particular
third-order linear differential equation, and its matching with a global
parametrix.Comment: 53 pages, 15 figure
Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes
Random Hermitian matrices with a source term arise, for instance, in the
study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and
sample covariance matrices \cite{Baik:2005}.
We consider the case when the external source matrix has two
distinct real eigenvalues: with multiplicity and zero with multiplicity
. The source is small in the sense that is finite or , for . For a Gaussian potential, P\'ech\'e
\cite{Peche:2006} showed that for sufficiently small (the subcritical
regime) the external source has no leading-order effect on the eigenvalues,
while for sufficiently large (the supercritical regime) eigenvalues
exit the bulk of the spectrum and behave as the eigenvalues of
Gaussian unitary ensemble (GUE). We establish the universality of these results
for a general class of analytic potentials in the supercritical and subcritical
regimes.Comment: 41 pages, 4 figure
Asymptotics for a special solution to the second member of the Painleve I hierarchy
We study the asymptotic behavior of a special smooth solution y(x,t) to the
second member of the Painleve I hierarchy. This solution arises in random
matrix theory and in the study of Hamiltonian perturbations of hyperbolic
equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is
known and relatively simple, but it turns out to be more subtle when x and t
tend to infinity simultaneously. We distinguish a region of algebraic
asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain
rigorous asymptotics in both regions. We also discuss two critical transitional
asymptotic regimes.Comment: 19 page
Historic buildings and the creation of experiencescapes: looking to the past for future success
Purpose: The purpose of this paper is to identify the role that the creative re-use of historic buildings can play in the future development of the experiences economy. The aesthetic attributes and the imbued historic connotation associated with the building help create unique and extraordinary âexperiencescapesâ within the contemporary tourism and hospitality industries. Design/methodology/approach: This paper provides a conceptual insight into the creative re-use of historic buildings in the tourism and hospitality sectors, the work draws on two examples of re-use in the UK. Findings: This work demonstrates how the creative re-use of historic buildings can help create experiences that are differentiated from the mainstream hospitality experiences. It also identifies that it adds an addition unquantifiable element that enables the shift to take place from servicescape to experiencescape. Originality/value: There has been an ongoing debate as to the significance of heritage in hospitality and tourism. However, this paper provides an insight into how the practical re-use of buildings can help companies both benefit from and contribute to the experiences economy
On bulk singularities in the random normal matrix model
We extend the method of rescaled Ward identities of Ameur-Kang-Makarov to
study the distribution of eigenvalues close to a bulk singularity, i.e. a point
in the interior of the droplet where the density of the classical equilibrium
measure vanishes. We prove results to the effect that a certain "dominant part"
of the Taylor expansion determines the microscopic properties near a bulk
singularity. A description of the distribution is given in terms of a special
entire function, which depends on the nature of the singularity (a
Mittag-Leffler function in the case of a rotationally symmetric singularity).Comment: This version clarifies on the proof of Theorem
Noncolliding Squared Bessel Processes
We consider a particle system of the squared Bessel processes with index conditioned never to collide with each other, in which if
the origin is assumed to be reflecting. When the number of particles is finite,
we prove for any fixed initial configuration that this noncolliding diffusion
process is determinantal in the sense that any multitime correlation function
is given by a determinant with a continuous kernel called the correlation
kernel. When the number of particles is infinite, we give sufficient conditions
for initial configurations so that the system is well defined. There the
process with an infinite number of particles is determinantal and the
correlation kernel is expressed using an entire function represented by the
Weierstrass canonical product, whose zeros on the positive part of the real
axis are given by the particle-positions in the initial configuration. From the
class of infinite-particle initial configurations satisfying our conditions, we
report one example in detail, which is a fixed configuration such that every
point of the square of positive zero of the Bessel function is
occupied by one particle. The process starting from this initial configuration
shows a relaxation phenomenon converging to the stationary process, which is
determinantal with the extended Bessel kernel, in the long-term limit.Comment: v3: LaTeX2e, 26 pages, no figure, corrections made for publication in
J. Stat. Phy
Universality in the two matrix model with a monomial quartic and a general even polynomial potential
In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix
model with a quartic monomial and a general even polynomial potential. We
studied the correlation kernel for the eigenvalues of one of the matrices in
asymptotic limit. We extended the results of Duits and Kuijlaars to the case
when the limiting eigenvalue density for one of the matrices is supported on
multiple intervals. The results are achieved by constructing the parametrix to
a Riemann-Hilbert problem obtained by Duits and Kuijlaars with theta functions
and then showing that this parametrix is well-defined by studying the theta
divisor.Comment: 35 pages, 8 figure
Topological expansion in the complex cubic log-gas model. One-cut case
We prove the topological expansion for the cubic log-gas partition function, with a complex parameter and defined on an unbounded contour on the complex plane. The complex cubic log-gas model exhibits two phase regions on the complex t-plane, with one cut and two cuts, separated by analytic critical arcs of the two types of phase transition: split of a cut and birth of a cut. The common point of the critical arcs is a tricritical point of the Painleve I type. In the present paper we prove the topological expansion for the partition function in the one-cut phase region. The proof is based on the Riemann-Hilbert approach to semiclassical asymptotic expansions for the associated orthogonal polynomials and the theory of S-curves and quadratic differentials