Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

For a fixed quadratic polynomial p in n non-commuting variables, and n independent N × N complex Ginibre matrices XN1, ⋯, XNn, we establish the convergence of the empirical measure of the eigenvalues of PN = p(XN1, ⋯, XNn) to the Brown measure of p evaluated at n freely independent circular elements c1, ⋯, cn in a non-commutative probability space. As in previous works on non-normal random matrices, a key step is to obtain quantitative control on the pseudospectrum of PN. Via a linearization trick of Haagerup-Thorbjørnsen for lifting non-commutative polynomials to tensors, we obtain this as a consequence of a lower tail estimate for the smallest singular value of patterned block matrices with strongly dependent entries. This reduces to establishing anticoncentration for determinants of random walks in a matrix space of bounded dimension, for which we encounter novel structural obstacles of an algebro-geometric nature.