Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices. (English. French summary) Zbl 1507.46045

Summary: For a fixed quadratic polynomial \(\mathfrak{p}\) in \(n\) non-commuting variables, and \(n\) independent \(N\times N\) complex Ginibre matrices \(X_1^N,\dots, X_n^N\), we establish the convergence of the empirical measure of the eigenvalues 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. As in previous works on non-normal random matrices, a key step is to obtain quantitative control on the pseudospectrum of \(P^N\). Via a linearization trick of Haagerup-Thorbjørnsen for lifting non-commutative polynomials to tensors [U. Haagerup and S. Thorbjørnsen, Ann. Math. (2) 162, No. 2, 711–775 (2005; Zbl 1103.46032)], 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 anti-concentration for determinants of random walks in a matrix space of bounded dimension, for which we encounter novel structural obstacles of an algebro-geometric nature.


46L54 Free probability and free operator algebras
60B20 Random matrices (probabilistic aspects)


Zbl 1103.46032


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