Related papers: Critical edge behavior in the singularly perturbed…

We consider the planar orthogonal polynomial $p_{n}(z)$ with respect to the measure supported on the whole complex plane $${\rm e}^{-N|z|^2} \prod_{j=1}^\nu |z-a_j|^{2c_j}\,{\rm d} A(z)$$ where ${\rm d} A$ is the Lebesgue measure of the…

We study the asymptotic properties of a class of multiple orthogonal polynomials with respect to a Nikishin system generated by two measures $(\sigma_1, \sigma_2)$ with unbounded supports (${supp}(\sigma_1) \subset \mathbb{R}_+$,…

We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function…

The Jacobi polynomials $\hat{P}_n^{(\alpha,\beta)}(x)$ conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function $(1-x)^\alpha (1+x)^\beta, \alpha,\beta>-1,$ on the interval…

Any square complex matrix of size $n\times n$ can be partially characterized by its $n$ eigenvalues and/or $n$ singular values. While no one-to-one correspondence exists between those two kinds of values on a deterministic level, for random…

In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of $$w(x,t)=(x+t)^{\lambda}x^{\alpha}e^{-x}$$ with $ x\geq 0,\ t>0,\ \alpha>0,\ \alpha+\lambda+1 > 0.$ The Deift-Zhou nonlinear steepest…

We consider the orthogonal polynomials $\{P_{n}(z)\}$ with respect to the measure $|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex…

Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence…

We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$…

Let $w$ be a weight on the unit disk $\mathbb{D}$ having the form \[w(z)=|v(z)|^2\prod_{k=1}^s\left|\frac{z-a_k}{1-z\overline{a}_k}\right|^{m_k}\,,\quad m_k>-2,\ |a_k|<1,\] where $v$ is analytic and free of zeros in $\overline{\mathbb{D}}$,…

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\beta$-ensembles of $N\times N$ random matrices. More specifically, we calculate scaling limits of the expectation value of products of…

We discuss the asymptotic behavior (as $n\to \infty$) of the entropic integrals $$ E_n= - \int_{-1}^1 \log \big(p^2_n(x) \big) p^2_n(x) w(x) d x, $$ and $$ F_n = -\int_{-1}^1 \log (p_n^2(x)w(x)) p_n^2(x) w(x) dx, $$ when $w$ is the…

We consider Angelesco ensembles with respect to two modified Jacobi weights on touching intervals [a,0] and [0,1], for a < 0. As a \to -1 the particles around 0 experience a phase transition. This transition is studied in a double scaling…

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for $a_n\equiv 1$, $b_n =-C n^{-\beta}$ ($0<\beta< \frac23)$, one has $d\mu(x)= w(x) dx$ on $(-2,2)$, and near…

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function $e^{ni\lambda z}$ on $[-1,1]$, where $\lambda$ is a positive parameter. This family of polynomials has appeared in…

We obtain large $n$ asymptotics of $n \times n$ Hankel determinants whose weight has a one-cut regular potential and Fisher-Hartwig singularities. We restrict our attention to the case where the associated equilibrium measure possesses…

The investigation of universality questions for local eigenvalue statistics continues to be a driving force in the theory of Random Matrices. For Matrix Models [53] the method of orthogonal polynomials can be used and the asymptotics of the…

We investigate the large $n$ behavior of Jacobi polynomials with varying parameters $P_{n}^{(an+\alpha,\,bn+\beta)}(1-2\lambda^{2})$ for $a,b >-1$ and $\lambda\in(0,\,1)$. This is a well-studied topic in the literature but some of the…

Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their $\beta$ generalisations at the hard and soft edge. It has been…

We study the Jacobi unitary ensemble perturbed by an algebraic singularity at $t>1$. For fixed $t$, this is the modified Jacobi ensemble studied by Kuijlaars {\it{et al.}} The main focus here, however, is the case when the algebraic…