
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.
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- 4905 Statistics
- 0104 Statistics
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Published In
DOI
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- Statistics & Probability
- 4905 Statistics
- 0104 Statistics