GranMa 2011
Oct 3 - 6, 2011
Institut Henri Poincaré, Paris
Anthony Metcalfe, KTH Royal Institute of Technology:
Universality properties of Gelfand-Tsetlin patterns
Gelfand-Tsetlin patterns are interlaced configurations of particles on the real line. Such patterns arise when considering the eigenvalues of the minors of Hermitian matrices. Therefore, choosing an Hermitian matrix randomly induces a probability distribution on the set of such patterns. We consider a related measure where the particles on the top line of the Gelfand-Tsetlin patterns are distinct and fixed, choosing uniformly from the set of patterns that result. Letting the size of the patterns increase, under the assumption that the empirical measure of the particles on the top line converges weakly to a bounded measure, we consider the asymptotic behaviour of particles on the same level in the bulk of the patterns. The behaviour is universal in nature in that, under some broad assumptions, the particles behave asymptotically like a determinantal random point field with the Sine kernel. A natural interpretation, arising from Free probability, of the conditions under which the Sine kernel is observed will also be discussed.
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