Related papers: Hard-edge asymptotics of the Jacobi growth process
The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a…
We consider the Gibbs measure of a general interacting particle system for a certain class of ``weakly interacting" kernels. In particular, we show that the local point process converges to a Poisson point process as long as the inverse…
A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of…
In this paper, we are concerned with higher-order analogues of the Tracy-Widom distribution, which describe the eigenvalue distributions in unitary random matrix models near critical edge points. The associated kernels are constructed by…
The (BC type) z-measures are a family of four parameter $z, z', a, b$ probability measures on the path space of the nonnegative Gelfand-Tsetlin graph with Jacobi-edge multiplicities. We can interpret the $z$-measures as random point…
Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials…
$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $\beta$-Hermite, $\beta$-Laguerre, and $\beta$-Jacobi ensembles. For fixed $N$ there exist associated weak limit…
In this work type II Hermite-Pad\'e approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex…
Information about the behavior of zeros of classical families of multiple or Hermite-Pad\'e orthogonal polynomials as functions of the intrinsic parameters of the family is scarce. We establish the interlacing properties of the zeros of…
The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy's largest root test in…
The research about Harmonic Analysis associated with Jacobi expansions carried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper. Given the operator $\mathcal{J}^{(\alpha,\beta)}=J^{(\alpha,\beta)}-I$, where…
We obtain "large gap" asymptotics for a Fredholm determinant with a confluent hypergeometric kernel. We also obtain asymptotics for determinants with two types of Bessel kernels which appeared in random matrix theory.
We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to…
We introduce and study a class of discrete particle ensembles that naturally arise in connection with classical random matrix ensembles, log-gases and Jack polynomials. Under technical assumptions on a general analytic potential we prove…
We establish the Plancherel-Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases,…
For random matrix ensembles with unitary symmetry, there is interest in the large $N$ form of the moments of the absolute value of the characteristic polynomial for their relevance to the Riemann zeta function on the critical line, and to…
Several families of one-point interactions are derived from the system consisting of two and three $\delta$-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or…
We study the local statistics of orthogonal polynomial ensembles near a hard edge, subject to a multiplicative deformation of the measure. Probabilistically, this deformation corresponds to a position-dependent conditional thinning of the…
In this paper we return to the study of the Watson kernel for the Abel summabilty of Jacobi polynomial series. These estimates have been studied for over more than 30 years. The main innovations are in the techniques used to get the…
E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight $\left[x(x-\alpha)(x-\beta)\right]^{-\frac{1}{2}}$, $x\in[0,\alpha]$, $0<\alpha<\beta$. A related system was studied by C. J. Rees in 1945,…