Breadcrumb
How many eigenvalues of a truncated orthogonal matrix are real?
Fri 22 March 2013, 14:00
Boris Khoruzhenko
Queen Mary, University of London
Organiser: Nina Snaith
ABSTRACT
This talk is about eigenvalue distributions of truncated random orthogonal matrices. Consider a matrix chosen at random from the orthogonal group O(N). Truncate it by removing L columns rows. The resulting matrix is a random contraction with eigenvalues inside the unit disk in the complex plane. It turns out that it is possible to obtain the joint probability distribution of these eigenvalues in closed form and then study eigenvalue densities and correlations in the limit of large matrix dimensions. There are (at least) two interesting regimes in this limit: (i) weak non-orthogonality when L = O(1), e.g. L=1, and (ii) strong non-orthogonality when L is proportional to N, e.g. L=N/2. In the latter regime one find the same law of eigenvalue correlations as in the real Ginibre ensemble while in the former, away from the real axis, one recovers statistics of resonance widths. One of the striking features of large real random matrices is the accumulation of eigenvalues on the real line. For truncated orthogonals in the regime of weak non-orthogonality, on average, the total number of real eigenvalues grows logarithmically with N and their density is given by the Arctanh Law. This behaviour is similar to that of real roots of random Kac polynomials.
This talk is based on a joint work with Hans-Juergen Sommers and Karol Zyczkowski