相关论文: Products of Random Matrices

Products of random transfer matrices are applied to low dimensional disordered systems to evaluate numerically extensive quantities such as entropy and overlap probability distribution. The main advantage is the possibility to avoid…

In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian…

The singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a…

The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramer on…

We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $\ell_2$ norm of such a product applied to any fixed…

We present a simple, yet useful result about the expected value of the determinant of random sum of rank-one matrices. Computing such expectations in general may involve a sum over exponentially many terms. Nevertheless, we show that an…

Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a…

Consider $n$ complex random matrices $X_1,\ldots,X_n$ of size $d\times d$ sampled i.i.d. from a distribution with mean $E[X]=\mu$. While the concentration of averages of these matrices is well-studied, the concentration of other functions…

We derive a novel deterministic equivalence for the two-point function of a random matrix resolvent. Using this result, we give a unified derivation of the performance of a wide variety of high-dimensional linear models trained with…

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…

Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…

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 derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that…

We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other…

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of…

For fixed $m > 1$, we study the product of $m$ independent $N \times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the…

Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present…

The spectra of random feature matrices provide essential information on the conditioning of the linear system used in random feature regression problems and are thus connected to the consistency and generalization of random feature models.…

We introduce the notion of a random matrix-valued multiplicative function, generalizing Rademacher random multiplicative functions to matrices. We provide an asymptotic for the second moment based on a linear recurrence property for…

We formulate conjectures regarding the maximum value and maximizing matrices of the permanent and of diagonal products on the set of stochastic matrices with bounded rank. We formulate equivalent conjectures on upper bounds for these…