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For a fixed quadratic polynomial $mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $Ntimes N$ complex Ginibre matrices $X_1^N,dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =mathfrak{p}(X_1^N,dots, X_n^N)$ to the Brown measure of $mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.
Let $G_n$ be an $n times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n times n$ matrix with $Vert A Vert le 1$, and let $gamma in (0,1)$. We show that with probability $0.99$, $A + gamma G_n$ has all of its eigenvalue condition
In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence statistics. We discuss how these statistics should originate fr
Let $O(2n+ell)$ be the group of orthogonal matrices of size $left(2n+ellright)times left(2n+ellright)$ equipped with the probability distribution given by normalized Haar measure. We study the probability begin{equation*} p_{2n}^{left(ellright)} = ma
In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distr
We propose a boundary regularity condition for the $M_n(mathbb{C})$-valued subordination functions in free probability to prove the local limit theorem and delocalization of eigenvectors for polynomials in two random matrices. We prove this through e