Related papers: Infinite Products of Large Random Matrices and Mat…
We investigate the product of $n$ complex non-Hermitian, independent random matrices, each of size $N\times N$ in the class of elliptic matrices, with independent identically distributed entries. The joint probability distribution of the…
In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a…
Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it…
Statistical properties of infinite products of random isotropically distributed matrices are investigated. Both for continuous processes with finite correlation time and discrete sequences of independent matrices, a formalism that allows to…
We briefly review the random matrix theory for large N by N matrices viewed as free random variables in a context of stochastic diffusion. We establish a surprising link between the spectral properties of matrix-valued multiplicative…
Products of $M$ i.i.d. non-Hermitian random matrices of size $N \times N$ relate Gaussian fluctuation of Lyapunov and stability exponents in dynamical systems (finite $N$ and large $M$) to local eigenvalue universality in random matrix…
In this paper, we consider $m$ independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the $m$ rectangular matrices is an $n$ by…
In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from…
In this short note, we revisit the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed entries with mean zero and unit variance. We prove under weaker assumptions that the limit…
One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic…
Phase transitions generically occur in random matrix models as the parameters in the joint probability distribution of the random variables are varied. They affect all main features of the theory and the interpretation of statistical models…
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…
We continue investigating spectral properties of a Hermitised random matrix product, which, contrary to previous product ensembles, allows for eigenvalues on the full real line. When a GUE matrix with an external source is involved, we…
Consider two types of products of independent random matrices, including products of Ginibre matrices and inverse Ginibre matrices and products of truncated Haar unitary matrices and inverse truncated Haar matrices. Each product matrix has…
We derive analytic expressions for infinite products of random 2x2 matrices. The determinant of the target matrix is log-normally distributed, whereas the remainder is a surprisingly complicated function of a parameter characterizing the…
The open problem of calculating the limiting spectrum (or its Shannon transform) of increasingly large random Hermitian finite-band matrices is described. In general, these matrices include a finite number of non-zero diagonals around their…
By using the independence structure of points following a determinantal point process, we study the radii of the spherical ensemble, the truncation of the circular unitary ensemble and the product ensemble with parameter n and k. The…
In this work, we introduce matrix-valued diffusion processes which describe the non-equilibrium situation of the matrix models for the beta-Hermite and the beta-Laguerre ensembles. We also study the corresponding spectral measure process…
The notion of (auto) catalytic networks has become a cornerstone in understanding the possibility of a sudden dramatic increase of diversity in biological evolution as well as in the evolution of social and economical systems. Here we study…
We consider products of independent large random rectangular matrices with independent entries. The limit distribution of the expected empirical distribution of singular values of such products is computed. The distribution function is…