arXiv subjects

Thomas Wolfs

Publications and source records attributed to Thomas Wolfs.

8 recordsLinked to original sources

arXiv2026

Invariant pseudo-unitary matrix ensembles and local statistics and conductance fluctuations of the DMPK ensemble

We consider the DMPK ensemble, which models quantum transport in a disordered wire, and describe its limiting local statistics as the number of particles tends to infinity. In the metallic regime, as well as at crossover I, we obtain sine statistics in the bulk, Bessel statistics at the hard edge and Airy statistics at the soft edge. In the insulating regime, we prove localization around certain explicit deterministic positions with Gaussian fluctuations. This confirms and quantifies the predicted universal and localized behavior in the physics literature. At crossover II, we obtain explicit limiting kernels in the bulk and at both edges, which interpolate between these regimes. The hard edge kernel involves Legendre functions and, to the best of our knowledge, is new, while the bulk and soft edge kernels agree with those derived for the squared singular values of critically scaled products of independent Ginibre matrices. To explain these similarities, we construct a class of invariant pseudo-unitary matrices that is closed under taking products and use it to interpret the DMPK ensemble as the joint density of the spectral variables of a product of such random matrices. More generally, the latter takes the form of a polynomial ensemble of derivative type involving a second order differential operator compatible with the Legendre functions. Finally, we obtain uniform asymptotic expansions of the mean and variance of the conductance and prove that the DMPK ensemble admits universal conductance fluctuations, i.e. the variance of the conductance tends to $1/15$ throughout the whole metallic regime. This again confirms the predictions in the physics literature.

math.PR↗

arXiv2026

Limit theorems for Coulomb gases on a Jordan curve in an external potential

We consider a Coulomb gas on a Jordan curve $γ$ in an external potential $V$ at inverse temperature $β>0$ and obtain an asymptotic expansion of the free energy up to $o(1)$ and a central limit theorem for linear statistics. We focus on the one-cut regime, where the density of the weighted equilibrium measure of $γ$ in $V$ is strictly positive on $γ$. The constant term in the (normalized) expansion consists of two parts: the Fredholm determinant of a generalized Grunsky operator and the Dirichlet energy of the logarithm of the density of the weighted equilibrium measure of $γ$. The coefficient of the latter vanishes for $β=2$. The variance of the fluctuations of the linear statistics only depends on the Dirichlet energy of the test function and is therefore independent of $V$. Essential in our approach is that the generalized Grunsky operator and the accompanying equilibrium parametrization allow us to transport the particles on the curve in the external potential to a reference object in a way that preserves the equilibrium measure. In our setting, the unit circle is the natural reference object.

math.CV↗

arXiv2026

Multiple orthogonal polynomial ensembles of derivative type

We characterize the biorthogonal ensembles that are both a multiple orthogonal polynomial ensemble and a polynomial ensemble of derivative type (also called a Pólya ensemble). We focus on the notions of multiplicative and additive derivative type that typically appear in connection with products and sums of random matrices respectively. Essential in the characterization is the use of the Mellin and Laplace transform: we show that the derivative type structure, which is a priori analytic in nature, becomes algebraic after applying the appropriate transform. Afterwards, we use the characterization to show that the eigenvalue densities of products of JUE and LUE matrices are essentially the only multiple orthogonal polynomial ensembles of multiplicative derivative type. We also show that the eigenvalue densities of sums of dilated LUE and GUE matrices are examples of multiple orthogonal polynomial ensemble of additive derivative type, but provide other examples as well. Finally, we explain how these notions of derivative type can be used to provide a partial solution to an open problem related to orthogonality of the finite finite free multiplicative and additive convolution of polynomials from finite free probability. In particular, we obtain families of multiple orthogonal polynomials that (de)compose naturally using these convolutions.

math.PR↗

arXiv2025

Rational approximation of Euler's constant using multiple orthogonal polynomials

We construct new rational approximants of Euler's constant that improve those of Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in terms of certain (mixed type) multiple orthogonal polynomials associated with the exponential integral. The dual family of multiple orthogonal polynomials leads to new rational approximants of the Gompertz constant that improve those of Aptekarev et al. (2007). Our approach is motivated by the fact that we can reformulate Rivoal's construction in terms of type I multiple Laguerre polynomials of the first kind by making use of the underlying Riemann-Hilbert problem. As a consequence, we can drastically simplify Rivoal's approach, which allows us to study the Diophantine and asymptotic properties of the approximants more easily.

math.NT↗

arXiv2024

Classical discrete multiple orthogonal polynomials: hypergeometric and integral representations

This work explores classical discrete multiple orthogonal polynomials, including Hahn, Meixner of the first and second kinds, Kravchuk, and Charlier polynomials, with an arbitrary number of weights. Explicit expressions for the recursion coefficients of Hahn multiple orthogonal polynomials are derived. By leveraging the multiple Askey scheme and the recently discovered explicit hypergeometric representation of type I multiple Hahn polynomials, corresponding explicit hypergeometric representations are provided for the type I polynomials and recursion coefficients of all the aforementioned descendants within the Askey scheme. Additionally, integral representations for these families within the Hahn class in the Askey scheme are presented. The multiple Askey scheme is further completed by providing the corresponding limits for the weights, polynomials, and recurrence coefficients.

math.CA↗

arXiv2024

Integral and hypergeometric representations for multiple orthogonal polynomials

This paper addresses two primary objectives in the realm of classical multiple orthogonal polynomials with an arbitrary number of weights. Firstly, it establishes new and explicit hypergeometric expressions for type I Hahn multiple orthogonal polynomials. Secondly, applying the residue theorem and the Mellin transform, the paper derives contour integral representations for several families of orthogonal polynomials. Specifically, it presents contour integral formulas for both type I and type II multiple orthogonal polynomials in the Laguerre of the first kind, Jacobi-Piñeiro, and Hahn families. The evaluation of these integrals leads to explicit hypergeometric representations.

math.CA↗

arXiv2024

Applications of multiple orthogonal polynomials with hypergeometric moment generating functions

We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval, the positive real line, or the unit circle and the multiple orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel orthogonal polynomials. We give explicit formulas for the type I and type II multiple orthogonal polynomials and study some of their properties. In particular, we describe the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials via the free convolution. Essential to our overall approach is the use of the Mellin transform. Finally, we discuss two applications. First, we show that the multiple orthogonal polynomials appear naturally in the study of the squared singular values of (mixed) products of truncated unitary random matrices and Ginibre matrices. Secondly, we use the multiple orthogonal polynomials to simultaneously approximate certain hypergeometric series and to provide an explicit proof of their $\mathbb{Q}$-linear independence.

math.CA↗

arXiv2023

Multiple orthogonal polynomials associated with the exponential integral

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^αe^{-x}$ the gamma density and $w_2(x) = x^αE_{ν+1}(x)$ a density related to the exponential integral $E_{ν+1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider a related family of mixed type multiple orthogonal polynomials.

math.CA↗