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The paper describes a system of rays declining at small angles in lasers. The correlation between a group of rays and binomial coefficients is shown. The correlation of distribution of rays in the system of numbers placed in a…
We analyze correlation functions in a toy model of a random geometry interacting with matter. We show that in general the connected correlator will contain a long-range scaling part which is in some sense a remnant of the disconnected part.…
This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and…
What is the connection of random matrices with integrable systems? Is this connection really useful? The answer to these questions leads to a new and unifying approach to the theory of random matrices. Introducing an appropriate time…
We study a model of colored multiwebs, which generalizes the dimer model to allow each vertex to be adjacent to \(n_v\) edges. These objects can be formulated as a random tiling of a graph with partial dimer covers. We examine the case of a…
We address the problem of filling missing entries in a kernel Gram matrix, given a related full Gram matrix. We attack this problem from the viewpoint of regression, assuming that the two kernel matrices can be considered as explanatory…
Most machine learning algorithms, such as classification or regression, treat the individual data point as the object of interest. Here we consider extending machine learning algorithms to operate on groups of data points. We suggest…
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 place ourselves in the setting of high-dimensional statistical inference where the number of variables $p$ in a dataset of interest is of the same order of magnitude as the number of observations $n$. We consider the spectrum of certain…
In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing…
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…
The double-direction orthogonalization algorithm is applied to construct sequences of polynomials, which are orthogonal over the interval [0,1]with the weighting function 1. Functional and recurrent relations are derived for the sequences…
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This…
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…
A classic result of Brooks, Smith, Stone and Tutte associates to any finite planar network with distinguished source and sink vertices, a tiling of a rectangle by smaller subrectangles whose aspect ratios are given by the conductances of…
This paper studies random operator-valued positive definite (p.d.) kernels and their connection to moment dilations. A class of random p.d. kernels is introduced in which the positivity requirement is imposed only in expectation, extending…
For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…
The kernel of a pair of linear systems is studied in the framework of commutative ring theory with applications to behavioral perspective of linear systems
The purpose of this article is to study operators whose kernel share some key features of Bergman kernels from complex analysis, and are approximate projectors. It turns out that they must be associated with a rich set of geometric data, on…
This paper presents a study of the properties of a matrix model that was introduced to describe transitions between all Wigner surmises of Random Matrix theory. New results include closed-form exact analytical expressions for the…