Related papers: Strong approximation of Gaussian $\beta$-ensemble …
We have discussed earlier the correlation functions of the random variables $\det(\la-X)$ in which $X$ is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the…
It has been shown recently [10] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random NxN Hermitian matrices. Our main goal is to…
We introduce a framework to study the random entire function $\zeta_\beta$ whose zeros are given by the Sine$_\beta$ process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power…
We develop a theory of multilevel distributions of eigenvalues which complements the Dyson's threefold $\beta=1,2,4$ approach corresponding to real/complex/quaternion matrices by $\beta=\infty$ point. Our central objects are G$\infty$E…
Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the circular Jacobi $\beta$-ensemble, which is a generalization of the Dyson circular $\beta$-ensemble but equipped with an additional parameter $b$, and further studied…
It is well known that the edge limit of Gaussian/Laguerre Beta-ensembles, as well as a large class of $\beta$-ensembles is given by the $\mathrm{Airy}(\beta)$ point process. We extend this universality result to a general class of additions…
The Airy$_{\beta }$ random point fields ($ \beta = 1,2,4$) are random point fields emerging as the soft-edge scaling limits of eigenvalues of Gaussian random matrices. We construct the unlabeled diffusion reversible with respect to the…
For general $\beta \geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \to \infty$. For each fixed time, this ensemble is…
We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a $N\times N$ random matrix taken from a $L$-deformed Chiral Gaussian Unitary Ensemble with an…
We point out a simple criterion for convergence of polynomials to a concrete entire function in the Laguerre-P\'{o}lya ($\mathcal{LP}$) class (of all functions arising as uniform limits of polynomials with only real roots). We then use this…
The Airy$_\beta$ point process, originally introduced by Ram\'irez, Rider, and Vir\'ag, is defined as the spectrum of the stochastic Airy operator $\mathcal{H}_\beta$ acting on a subspace of $L^2[0,\infty)$ with Dirichlet boundary…
We show in this paper that, when properly rescaled in time and in space, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a…
A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre $\beta$ ensemble, characterised by the Dyson parameter $\beta$, and the Laguerre…
We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $\beta$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are…
In the present context, superintegrability is a property of certain probability density functions coming from matrix models, which relates to the average over a distinguished basis of symmetric functions, typically the Jack or Macdonald…
The $\beta$ ensembles are a class of eigenvalue probability densities which generalise the invariant ensembles of classical random matrix theory. In the case of the Gaussian and Laguerre weights, the corresponding eigenvalue densities are…
We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation…
Ram\'irez and Rider (2009) established that the hard edge of the spectrum of the $\beta$-Laguerre ensemble converges, in the high-dimensional limit, to the bottom of the spectrum of the stochastic Bessel operator. Using stochastic analysis…
The stochastic Airy and sine operators, which are respectively a random Sturm-Liouville operator and a random Dirac operator, characterize the soft edge and bulk scaling limits of $\beta$-ensembles. Dirac and Sturm-Liouville operators are…
We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the Circular Unitary Ensemble (CUE) of Random Matrix Theory. In…