Random Matrices: Theory and Applications

We review an elementary derivation of the Borodin–Sinclair–Forrester–Nagao Pfaffian point process, which characterizes the law of real eigenvalues for the real Ginibre ensemble in the large matrix size limit, in terms of averages of products of characteristic polynomials. This derivation relies on a number of interesting structures associated with the real Ginibre ensemble such as the hidden symplectic symmetry of the statistics of real eigenvalues. It leads to a representation for the $K$-point correlation function for any $K\in \mathbb{N}$ in terms of an integral over the symmetric space $U(2K)/USp(2K)$ and this paper gives the proof of asymptotic exactness for this integral.

The focus of this paper is on the distribution function of the rightmost eigenvalue for the complex elliptic Ginibre ensemble in the limit of weak non-Hermiticity. We show how the limiting distribution function can be expressed in terms of an integro-differential Painlevé-II function and how the same captures the nontrivial transition between Poisson and Airy point process extreme value statistics as the degree of non-Hermiticity decreases. Our most explicit new results concern the tail asymptotics of the limiting distribution function. For the right tail we compute the leading order asymptotics uniformly in the degree of non-Hermiticity, for the left tail we compute it close to Hermiticity.

This article develops a rectangular version of Male’s theory of traffic probability, in which an algebra is equipped with a trace evaluated on arbitrary graphs whose edges are labeled by elements and whose vertices are subspaces. Using the language of traffic distributions, we characterize the asymptotic behavior of independent families of rectangular random matrices which are bi-permutation invariant. In the process, we take a tour of non-commutative probabilities and their random matrix models. Special attention is paid to rectangular random matrices with independent or exchangeable entries, including the existence and description of limiting ∗-distributions for a broad range of models.

In this paper, we introduce a new class of large structured random matrices characterized by four fundamental properties which we discuss. We prove that this class is stable under matrix-valued and pointwise nonlinear operations. We then formulate an efficient method, based on an extremization problem, for computing the spectrum of subblocks of such large structured random matrices. We present different proofs — combinatorial or algebraic — of the validity of this method, which all have some connection with free probability. We illustrate this method with well-known examples of unstructured matrices, including Haar randomly rotated matrices, as well as with the example of structured random matrices arising in the quantum symmetric simple exclusion process.