arXiv subjects

Leslie Molag

Publications and source records attributed to Leslie Molag.

6 recordsLinked to original sources

arXiv2026

Conditional thinning and multiplicative statistics of Laguerre-type orthogonal polynomial ensembles

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 particles. We prove that, under critical hard edge scaling and for a large class of potentials and deformation symbols, the correlation kernel of the conditional ensemble converges to a universal limit, which we identify as the conditional thinned Bessel point process. We derive an explicit expression for this limiting kernel in terms of the solution to a nonlocal integrable system depending on a parameter. For a special choice of the parameter, this system was recently identified in the study of multiplicative statistics of the Bessel point process. Our results establish that this system governs the full correlation structure of the conditional Bessel point process, extending the classical connection between the standard Bessel kernel and the Painlevé V equation.

math-ph↗

arXiv2025

Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices

We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} dμ(z) = (1-|z|^{2})^{α-1}|z-x|^γ\mathbf{1}_{|z| < 1}d^{2}z, \end{equation*} where $d^{2}z$ is the two dimensional area measure, $α$ is a parameter that can grow with $n$, while $γ>-2$ and $x>0$ are fixed. This measure arises naturally in the study of characteristic polynomials of non-Hermitian ensembles and generalises the example of a Gaussian weight that was recently studied by several authors. We obtain asymptotics in all regions of the complex plane and via an appropriate differential identity, we obtain the asymptotic expansion of the partition function. The main approach is to convert the planar orthogonality to one defined on suitable contours in the complex plane. Then the asymptotic analysis is performed using the Deift-Zhou steepest descent method for the associated Riemann-Hilbert problem.

math-ph↗

arXiv2023

Large deviations and fluctuations of real eigenvalues of elliptic random matrices

We study real eigenvalues of $N\times N$ real elliptic Ginibre matrices indexed by a non-Hermiticity parameter $0\leq τ<1$, in both the strong and weak non-Hermiticity regime. Here $N$ is assumed to be an even number. In both regimes, we prove a central limit theorem for the number of real eigenvalues. We also find the asymptotic behaviour of the probability $p_{N,k}^{(τ)}$ that exactly $k$ eigenvalues are real. In the strong non-Hermiticity regime, where $τ$ is fixed, we find \begin{align*} \lim_{N\to\infty} \frac{1}{\sqrt{N}} \log p_{N,k_N}^{(τ)} = -\sqrt\frac{1+τ}{1-τ} \frac{ζ(3/2)}{\sqrt{2π}} \end{align*} for any sequence $(k_N)_N$ of even numbers such that $k_N = o(\frac{\sqrt N}{\log N})$ as $N\to\infty$, where $ζ$ is the Riemann zeta function. In the weak non-Hermiticity regime, where $τ=1-\frac{α^2}{N}$, we obtain \begin{align*} \lim_{N\to\infty} \frac{1}{N} \log p_{N,k_N}^{(τ)} \leq \frac{2}π \int_0^1 \log\left(1-e^{-α^2 s^2}\right) \sqrt{1-s^2} \, ds \end{align*} for any sequence $(k_N)_N$ of even numbers such that $k_N=o(\frac{N}{\log N})$ as $n\to\infty$. This inequality is expected to be an equality.

math-ph↗

arXiv2019

The matching condition for larger size Riemann-Hilbert problems

In a larger size Riemann-Hilbert problem matching the local parametrices with the global parametrix is often a major issue. In this article we present a result that should tackle this problem in natural situations. We prove that, in a general setting, it is possible to obtain a double matching, that is, a matching condition on two circles instead of one circle. We discuss how this matching approach can be used to obtain local scaling limits of correlation kernels and apply our result to several examples from the existing literature.

math.CA↗

arXiv2019

Universality for conditional measures of the Bessel point process

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval $[0,R]$ is almost surely an orthogonal polynomial ensemble. In this article, we show that if $R$ tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure of the Bessel point process almost surely belongs.

math.PR↗

arXiv2017

Monodromy of the generalized hypergeometric equation in the Frobenius basis

We consider monodromy groups of the generalized hypergeometric equation \begin{equation*} \big[z(θ+α_{1})\cdots (θ+α_{n})-(θ+β_{1}-1)\cdots (θ+β_{n}-1)\big]f(z) = 0\text{, where }θ= z d/dz, \end{equation*} in a suitable basis, closely related to the Frobenius basis. We pay particular attention to the maximally unipotent case, where $β_{1}=\ldots=β_{n}=1$, and present a theorem that enables us to determine the form of the corresponding monodromy matrices in the case where $(X-e^{-2πiα_{1}})\cdots (X-e^{-2πiα_{n}})$ is a product of cyclotomic polynomials.

math.AG↗