Random Matrices: Theory and Applications

This work proposes algorithms for computing additive and multiplicative free convolutions of two given probability measures. We consider probability measures with compact support whose free convolution results in a probability measure with a density function that exhibits a square-root decay at the boundary (for example, the semicircle distribution or the Marchenko–Pastur distribution). A key ingredient of our method is rewriting the intermediate quantities of the free convolution using the Cauchy integral formula and then discretizing these integrals using the trapezoidal quadrature rule, which converges exponentially fast under suitable analyticity properties of the functions to be integrated.

Let $Z={\mathrm{\Sigma }}^{\frac{1}{2}}X\in {ℝ}^{p\times n}\mathrm{\text{ where }}{({\mathrm{\Sigma }}^{\frac{1}{2}})}^2=\mathrm{\Sigma }$ is a positive definite matrix and $X=\frac{1}{\sqrt{n}}{(X_{ij})}_{p\times n}$ with $X_{ij}$ being independent and identically distributed (i.i.d.) centered real random variables. Consider the matrix

In this paper, we analyze the asymptotic fluctuations of linear eigenvalue statistics of random centrosymmetric matrices with i.i.d. entries. We prove that for a complex analytic test function, the centered and normalized linear eigenvalue statistics of random centrosymmetric matrices converge to a normal distribution. We find the exact expression of the variance of the limiting normal distribution via combinatorial arguments. Moreover, we argue that the limiting spectral distribution of properly scaled centrosymmetric matrices follows the circular law.

We introduce the first random matrix model of a complex $\beta$-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite $\beta$-ensembles discovered by [I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002) 5830–5847]. The main feature of the model is that the exponent $\beta$ of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in ${ℝ}_{+}.$ However, when $\beta =2$, the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.

In this paper, we provide a rate of convergence for a version of the Carathéodory convergence for the multiple SLE model with a Dyson Brownian motion driver towards its hydrodynamic limit, for $\beta =1$ and $\beta =2$. The results are obtained by combining techniques from the field of Schramm–Loewner Evolutions with modern techniques from random matrices. Our approach shows how one can apply modern tools used in the proof of universality in random matrix theory to the field of Schramm–Loewner Evolutions.