Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Singular values of convex functions of matrices

‎Let $A_{i},B_{i},X_{i},i=1,dots,m,$ be $n$-by-$n$ matrices such that $‎sum_{i=1}^{m}leftvert A_{i}rightvert ^{2}$ and $‎sum_{i=1}^{m}leftvert B_{i}rightvert ^{2}$  are nonzero matrices and each $X_{i}$ is‎ ‎positive semidefinite‎. ‎It is shown that if $f$ is a nonnegative increasing ‎convex function on $left[ 0,infty right) $ satisfying $fleft( 0right)‎ ‎=0 $‎, ‎then  $$‎2s_{j}left( fleft( fra...

Bulk universality of sparse random matrices

The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. We consider the adjacency matrix of the ensemble of Erd˝ os-Rényi random graphs which consists of graphs on N vertices in which each edge occurs independently with probability p. We prove that in the regime pN ≫ 1, these matrices exhibit bulk universality in the sense tha...

On the singular values of random matrices

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on R that is supported in a relatively small ball and linear functionals are uniformly bounded in Lp for some p > 8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1±c √ n/N ...

More about measures and Jacobians of singular random matrices

In this work are studied the Jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.

Universality in the bulk for random matrices Notes for Lectures