相关论文: How to generate random matrices from the classical…

Markov-chain Monte Carlo algorithms rely on trial moves that are either rejected or accepted based on certain criteria. Here, we provide an efficient algorithm to generate random rotation matrices in four dimensions (4D) covering an…

In a recent paper the author proved a theorem to the effect that the matrix of normalized Euclidean distances on the set of specially distributed random points in the $n$-dimensional Euclidean space $\mathbb R^{n}$ with independent…

In these lectures we discuss some elementary concepts in connection with the theory of symmetric spaces applied to ensembles of random matrices. We review how the relationship between random matrix theory and symmetric spaces can be used in…

Correlation functions for matrix ensembles with orthogonal and unitarysymplectic rotation symmetry are more complicated to calculate than in the unitary case. The supersymmetry method and the orthogonal polynomials are two techniques to…

In [earlier work by the author], it was shown that if U is a random n x n unitary matrix, then for any p>=n, the eigenvalues of U^p are i.i.d. uniform; similar results were also shown for general compact Lie groups. We study what happens…

Brief lecture notes for a course about random matrices given at the University of Cambridge.

These lecture notes are a concise introduction of recent techniques to prove local spectral universality for a large class of random matrices. The general strategy is presented following the recent book with H.T. Yau. We extend the scope of…

We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.

We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such…

We classify in this paper Poisson structures on modules over semisimple Lie algebras arising from classical r-matrices. We then study their quantizations and the relation to classical invariant theory.

We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT) -- a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M…

We consider the characteristic polynomials of random unitary matrices $U$ drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The variances of these ``secular…

Directed acyclic graphs are the basic representation of the structure underlying Bayesian networks, which represent multivariate probability distributions. In many practical applications, such as the reverse engineering of gene regulatory…

Consider the collection of all binary matrices having a specific sequence of row and column sums and consider sampling binary matrices uniformly from this collection. Practical algorithms for exact uniform sampling are not known, but there…

We introduce a geometrically natural probability measure on the group of all M\"obius transformations of the circle. Our aim is to study "random" groups of M\"obius transformations, and in particular random two-generator groups. By this we…

Observables in random tensor theory are polynomials in the entries of a tensor of rank $d$ which are invariant under $U(N)^d$. It is notoriously difficult to evaluate the expectations of such polynomials, even in the Gaussian distribution.…

Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices…

Given a classical $r$-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian formulation of Lax equations. Examples for which our formalism applies include the Benny…

We demonstrate quantum algorithms to implement pseudo-random operators that closely reproduce statistical properties of random matrices from the three universal classes: unitary, symmetric, and symplectic. Modified versions of the…

We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the…