相关论文: Multiplicative free Convolution and Information-Pl…
We define a class of "algebraic" random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner…
Let M be a B-probability space. Assume that B itself is a D-probability space; then M can be viewed as a D-probability space as well. Let X be in M. We characterize freeness of X from B with amalgamation over D in terms of a certain…
Let k be a positive integer and let D_k denote the space of joint distributions for k-tuples of selfadjoint elements in C*-probability space. The paper studies the concept of "subordination distribution of \mu \boxplus \nu with respect to…
Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative…
The probability distribution for the free energy of directed polymers in random media (DPRM) with uncorrelated noise in $d=1+1$ dimensions satisfies the Tracy-Widom distribution. We inquire if and how this universal distribution is modified…
The goal of these expository notes is to give an introduction to random matrices for non-specialist of this topic focusing on the link between random matrices and systems of particles in interaction. We first recall some general results…
We study matricial approximations of master fields we constructed in a previous work. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in $\mathbb{R},…
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they…
Deep neural networks have been successfully applied to a broad range of problems where overparametrization yields weight matrices which are partially random. A comparison of weight matrix singular vectors to the Porter-Thomas distribution…
In random matrices with independent and continuous matrix entries, the degeneracy probability of the eigenvalues is known to be zero. In this paper, random matrices including discontinuous matrix entries are analyzed in order to observe how…
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can…
We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of…
Financial stock returns correlations have been studied in the prism of random matrix theory, to distinguish the signal from the "noise". Eigenvalues of the matrix that are above the rescaled Marchenko Pastur distribution can be interpreted…
The equivalence of the characteristic function approach and the probabilistic approach to monotone and boolean convolutions is proven for non-compactly supported probability measures. A probabilistically motivated definition of the…
It has been observed that the statistical distribution of the eigenvalues of random matrices possesses universal properties, independent of the probability law of the stochastic matrix. In this article we find the correlation functions of…
The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in…
In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible…
We suggest that Free Random Variables, represented here by large random matrices with spectral Levy disorder, may be relevant for several problems related to the modeling of financial systems. In particular, we consider a financial…
We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically…
In this mini-review, dedicated to the Jubilee of Professor Tadeusz Marek, we highlight in a popular way the power of so-called free random variables (hereafter FRV) calculus, viewed as a potential probability calculus for the XXI century,…