Probability and Mathematical Physics Seminar

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Eigenvectors of non-hermitian random matrices

Speaker: Guillaume Dubach, Courant

Location: Warren Weaver Hall 512

Date: Friday, April 12, 2019, 10:30 a.m.

Synopsis:

Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system, to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps is Hermitian and positive-definite; it quantifies the stability of the spectrum, and characterizes the joint eigenvalues increments under Dyson-type dynamics. These variables first appeared in the physics literature, when Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results are expected to hold in other integrable models, and some have been established for quaternionic Gaussian matrices.