Related papers: Generalizations of some integrals over the unitary…
Properties of universality have essential relevance for the theory of random matrices usually called the Wigner ensemble. The issue was analysed up to recent years with detailed and relevant results. We present a slightly different view and…
We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$, depending on a matrix $a\in M_{p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary…
Group algebras of permutations have proved highly useful in solving a number of problems in large N gauge theories. I review the use of permutations in classifying gauge invariants in one-matrix and multi-matrix models and computing their…
There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold…
We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the…
This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series…
Mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions. It has recently been applied to the Nekrasov instanton partition function of $\mathcal{N}=2$ 4d gauge theories. The…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…
This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses…
In the last few years several new Random Matrix Models have been proposed and studied. They have found application in various different contexts, ranging from the physics of mesoscopic systems to the chiral transition in lattice gauge…
We use character theory of finite groups of Lie type to establish new results on representation varieties of Fuchsian groups, and also on probabilistic generation of groups of Lie type.
We present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models, where vertices come in different types, and the probability of realizing an edge depends on the types of its…
Gaussian elimination is used in special linear groups to solve the word problem. In this paper, we extend Gaussian elimination to unitary groups. These algorithms have an application in building a public-key cryptosystem, we demonstrate…
We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and…
We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an…
We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show…
We consider perturbative quantum field theory in the causal framework. Gauge invariance is, in this framework, an identity involving chronological products of the interaction Lagrangian; it express the fact that the scattering matrix must…
We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove…
Generalization techniques have many applications, including template construction, argument generalization, and indexing. Modern interactive provers can exploit advancement in generalization methods over expressive type theories to further…
This paper considers generalized linear models using rule-based features, also referred to as rule ensembles, for regression and probabilistic classification. Rules facilitate model interpretation while also capturing nonlinear dependences…