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Exact results from random matrix theory are used to systematically analyse the relationship between microscopic Dirac spectra and finite-volume partition functions. Results are presented for the unitary ensemble, and the chiral analogs of…
This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries. The proof relies on the Erd{\H…
A set partition technique that is useful for identifying wires in cables can be recast in the language of 0--1 matrices, thereby resolving an open problem stated by R.~L. Graham in Volume 1 of this journal. The proof involves a construction…
This paper addresses the problem of computing valuations of the Dieudonn\'e determinants of matrices over discrete valuation skew fields (DVSFs). Under a reasonable computational model, we propose two algorithms for a class of DVSFs, called…
In this paper natural necessary and sufficient conditions for quantifier elimination of matrix rings $M_n(K)$ in the language of rings expanded by two unary functions, naming the trace and transposition, are identified. This is used…
Many problems in physics, chemistry and other fields are perturbative in nature, i.e. differ only slightly from related problems with known solutions. Prominent among these is the eigenvalue perturbation problem, wherein one seeks the…
By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface.…
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network…
Investigating the dynamics of learning in machine learning algorithms is of paramount importance for understanding how and why an approach may be successful. The tools of physics and statistics provide a robust setting for such…
In this paper, we present several estimators of the diagonal elements of the inverse of the covariance matrix, called precision matrix, of a sample of iid random vectors. The focus is on high dimensional vectors having a sparse precision…
In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of 5n+4 into five equal classes. This gave a combinatorial explanation of…
Matrix-valued stochastic processes have been of significant importance in areas such as physics, engineering and mathematical finance. One of the first models studied has been the so-called Wishart process, which is described as the…
Einsiedler, Mozes, Shah and Shapira [Compos. Math. 152 (2016), 667-692] prove an equidistribution theorem for rational points on expanding horospheres in the space of d-dimensional Euclidean lattices, with d>2. Their proof exploits measure…
This paper derives exponential tail bounds and polynomial moment inequalities for the spectral norm deviation of a random matrix from its mean value. The argument depends on a matrix extension of Stein's method of exchangeable pairs for…
Wishart random matrices are often used to model multivariate systems in physics, finance, biology and wireless communication. Extreme value statistics, such as those of the smallest eigenvalue, can be used to test the accuracy of the model.…
We develop a theory for the eigenvalue density of arbitrary non-Hermitian Euclidean matrices. Closed equations for the resolvent and the eigenvector correlator are derived. The theory is applied to the random Green's matrix relevant to wave…
In this work we have considered formal power series and partial differential equations, and their relationship with Coding Theory. We have obtained the nature of solutions for the partial differential equations for Cycle Poisson Case. The…
This PhD thesis is organized as follows. In the first two chapters I will review some basic notions of statistical physics of disordered systems, such as random graph theory, the mean-field approximation, spin glasses and combinatorial…
Every sufficiently big matrix with small spectral norm has a nearby low-rank matrix if the distance is measured in the maximum norm (Udell & Townsend, SIAM J Math Data Sci, 2019). We use the Hanson--Wright inequality to improve the estimate…
The fractional diffraction optics theory has been elaborated using the Green function technique. The optics-fractional equation describing the diffraction X-ray scattering by imperfect crystals has been derived as the fractional matrix…