Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

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.