相关论文: The Weingarten Calculus

It is given an efficient complete parametrization of wavelet matrices of rank $m$, genus $g+1$, and degree $g$, which are naturally identified with corresponding polynomial paraunitary matrix-functions. The parametrization depends on…

This is an introductory note concerning the distribution vectors in a unitary representation of a Lie group. We discuss the definition of matrix coefficients associated with a pair of distributions and how one can compute them. Most of the…

We present a compact formulation of the orthogonal Weingarten formula, with the traditional quantity $I(i_1,...,i_{2k}:j_1,...,j_{2k}) = \int_{O_n}u_{i_1j_1} ... u_{i_{2k}j_{2k}} du$ replaced by the more advanced quantity…

Wishart ensemble is a useful and important random matrix model used in diverse fields. By realizing induced random mixed quantum states as Wishart ensemble with the fixed-trace one, using matrix integral technique we give a fast track to…

Weingarten functions provide a tool for computing Haar measure matrix integrals of polynomials in the matrix entries. An important property of Weingarten functions, is their particularly simple large $N$ limits. In 2017 Benoit Collins and…

In this article, we shall derive by elementary calculations the Gauss map, spherical image, Weingarten map and the curvatures at identity of the special linear group, that is, the matrices of determinant 1. We could not find any reference…

A new parametrization (one-to-one onto map) of compact wavelet matrices of rank $m$ and of order and degree $N$ is proposed in terms of coordinates in the Euclidian space $R^{(m-1)N}$. The developed method depends on Wiener-Hopf…

We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically…

In this paper, we establish an explicit isomorphism between the symmetric group algebra and the path algebra of the Young graph. Specifically, we construct a family of matrix units in the group algebra. As a main application of this…

We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field.…

By using a symbolic method, known in the literature as the classical umbral calculus, the trace of a non-central Wishart random matrix is represented as the convolution of the trace of its central component and of a formal variable…

We study the ensemble of a product of n complex Gaussian i.i.d. matrices. We find this ensemble is Gaussian with a variance matrix which is averaged over a multi-Wishart ensemble. We compute the mixed moments and find that at large $N$,…

Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class…

The Wiener index of a graph $G$, denoted $W(G)$, is the sum of the distances between all pairs of vertices in $G$. \'E. Czabarka, et al. conjectured that for an $n$-vertex, $n\geq 4$, simple quadrangulation graph $G$,…

We show that the mixing times of random walks on compact groups can be used to obtain concentration inequalities for the respective Haar measures. As an application, we derive a concentration inequality for the empirical distribution of…

The RV coefficient measures the similarity between two multivariate configurations, and its significance testing has attracted various proposals in the last decades. We present a new approach, the invariant orthogonal integration,…

We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures.

This article provides a concise introduction to the theory of Haar measures on locally compact Hausdorff groups. We cover the necessary preliminaries on topological groups and measure theory, the Haar correspondence, unimodularity and Haar…

Integration of polynomials over the classical groups of unitary, orthogonal and symplectic matrices can be reduced to basic building blocks known as Weingarten functions. We present an elementary derivation of these functions.

This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed…