Related papers: Riemann-Hilbert approach to a generalised sine ker…

We develop a framework for function classes generated by parametric ridge kernels: one-dimensional kernels composed with affine projections and averaged over a parameter measure. The induced kernels are positive definite, and the resulting…

We characterize Fredholm determinants of a class of Hankel composition operators via matrix-valued Riemann-Hilbert problems, for additive and multiplicative compositions. The scalar-valued kernels of the underlying integral operators are…

We establish a link between Fourier optics and a recent construction from the machine learning community termed the kernel mean map. Using the Fraunhofer approximation, it identifies the kernel with the squared Fourier transform of the…

We consider the gap probability for the Pearcey and Airy processes; we set up a Riemann--Hilbert approach (different from the standard one) whereby the asymptotic analysis for large gap/large time of the Pearcey process is shown to…

The Bessel process models the local eigenvalue statistics near $0$ of certain large positive definite matrices. In this work, we consider the probability \begin{align*} \mathbb{P}\Big( \mbox{there are no points in the Bessel process on }…

We introduce two kernels that extend the mean map, which embeds probability measures in Hilbert spaces. The generative mean map kernel (GMMK) is a smooth similarity measure between probabilistic models. The latent mean map kernel (LMMK)…

For a locally finite point set $\Lambda \subset \mathbb{R}$, consider the collection of exponential functions given by $\mathcal{E}_{\Lambda}:= \{e^{i \lambda x} : \lambda \in L \}$. We examine the question whether $\mathcal{E}_{\Lambda}$…

We consider a family of linear operators, diagonalized by the Hankel transform. The Fredholm determinants of these operators, restricted to $L_2[0, R]$, are expressed in a convenient form for asymptotic analysis as $R\to\infty$. The result…

For many machine learning problem settings, particularly with structured inputs such as sequences or sets of objects, a distance measure between inputs can be specified more naturally than a feature representation. However, most standard…

A standard Gelfand-Tsetlin pattern of depth $n$ is a configuration of particles in $\{1,...,n\} \times \R$. For each $r \in \{1,...,n\}$, $\{r\} \times \R$ is referred to as the $r^\text{th}$ level of the pattern. A standard Gelfand-Tsetlin…

We study a 2-parametric family of probability measures on the space of countable point configurations on the punctured real line (the points of the random configuration are concentrated near zero). These measures (or, equivalently, point…

We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance $N^{-3/4+\beta}$ for some positive…

We study how iterated and composed completely positive maps act on operator-valued kernels. Each kernel is realized inside a single Hilbert space where composition corresponds to applying bounded creation operators to feature vectors. This…

This paper introduces a novel approach to probabilistic deep learning, kernel density matrices, which provide a simpler yet effective mechanism for representing joint probability distributions of both continuous and discrete random…

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their…

We use discrete analogs of Riemann-Hilbert problem's methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define discrete analogs of a Riemann-Hilbert problem…

We consider the kernel of a hypoelliptic diffusion beyond the case of sub-ellipticity or polynomial coefficients. We get a full asymptotic expansion for small times, based on a Duhamel-type comparison with an approximate polynomial kernel.…

We derive the asymptotic behavior of determinants of truncated Wiener-Hopf operators generated by symbols having Fisher-Hartwig singularities. This task is achieved thanks to an asymptotic resolution of the Riemann-Hilbert problem…

In this paper, we survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications,…

The purpose of this article is to develop a theory behind the occurrence of "path-integral" kernels in the study of extended determinantal point processes and non-intersecting line ensembles. Our first result shows how determinants…