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Counting critical points of random functions in many dimensions

Seminar | November 16 | 4:10-5 p.m. | 60 Evans Hall | Note change in date


Gérard Ben Arous, New York University, visiting MSRI

Statistics, Department of


I will survey the recent progress of an on-going joint work with Antonio Auffinger and Jiri Cerny. We compute the number of critical points of a general Gaussian random smooth function on the N-dimensional sphere. These corresponds to Hamiltonians of well-known models of statistical physics, i.e spherical spin glasses. Using the classical Kac-Rice formula, this counting boils down to a problem in Random Matrix Theory. This allows us to show an interesting picture for the complexity of these random Hamiltonians, for the bottom of the energy landscape, and in particular a strong correlation between the index and and the critical value. We also propose a new invariant for the possible transition between he so-called 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme.


510-642-2781