Random Matrices: Theory and Applications

We study the probability density function (PDF) of the smallest eigenvalue of Laguerre–Wishart matrices $W=X^{†}X$ where $X$ is a random $M\times N$ ($M\ge N$) matrix, with complex Gaussian independent entries. We compute this PDF in terms of semi-classical orthogonal polynomials, which are deformations of Laguerre polynomials. By analyzing these polynomials, and their associated recurrence relations, in the limit of large $N$, large $M$ with $M/N\to 1$ — i.e. for quasi-square large matrices $X$ — we show that this PDF, in the hard edge limit, can be expressed in terms of the solution of a Painlevé III equation, as found by Tracy and Widom, using Fredholm operator techniques. Furthermore, our method allows us to compute explicitly the first $1/N$ corrections to this limiting distribution at the hard edge. Our computations confirm a recent conjecture by Edelman, Guionnet and Péché. We also study the soft edge limit, when $M-N\sim {𝒪}(N)$, for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.

We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. Moreover, no moment assumptions are made about the distributions governing the entries. Our proofs are based on a concentration function inequality due to Kolmogorov, Rogozin and Kesten, which allows us to improve universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.

For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists and we describe it via an equation satisfied by its Stieltjes transform. No rate of convergence to zero of correlations is imposed, therefore the process is allowed to have long memory. In particular, if the symmetrized matrices are constructed from stationary Gaussian random fields which have spectral density, the result of this paper gives a complete solution to the limiting eigenvalue distribution. More generally, for matrices whose entries are functions of independent and identically distributed random variables the result also holds.