Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ 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 $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ 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 $Z^\dagger Z$, where $Z$ is an $n \times m$ complex Gaussian matrix with correlations both along rows and down columns, are expressed as $m \times m$ 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 $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t=0$ at the same positive value $x=a$, remain positive, and are conditioned to end at time $t=1$ at $x=0$. After appropriate rescaling, the paths fill a region in the $tx$--plane as $n\to \infty$ that intersects the hard edge at $x=0$ at a critical time $t=t^{*}$. In a previous paper (arXiv:0712.1333), the scaling limits for the positions of the paths at time $t\neq t^{*}$ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as $n\to \infty$ of the correlation kernel at critical time $t^{*}$ 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 $3\times 3$ 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 $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. The source is small in the sense that $r$ is finite or $r=\mathcal O(n^\gamma)$, for $0< \gamma<1$. For a Gaussian potential, P\'ech\'e \cite{Peche:2006} showed that for $|a|$ sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for $|a|$ sufficiently large (the supercritical regime) $r$ eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of $r\times r$ 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 “experiencescapes” 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 $\nu > -1$ conditioned never to collide with each other, in which if $-1 < \nu < 0$ 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 $J_{\nu}$ 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