Probability and Mathematical Physics Seminar

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Concentration of Equilibria and Relative Instability in the Asymmetric p-Spin Model

Speaker: Pax Kivimae, Courant

Location: Warren Weaver Hall 1302

Date: Friday, December 2, 2022, 11:10 a.m.

Synopsis:

We study the number of equilibria (stationary points) in a non-relaxational analog of the gradient flow for the spherical p-spin model. The asymptotics for the expected number of equilibria in this model were recently computed by Fyodorov, followed by a similar computation for the expected number of stable equilibria by Garcia. These results suggest a critical threshold in terms of the strength of the non-relaxational term, above which the system would transition from having an abundance of stable equilibria, to having none at all. This mirrors the transition from relative to absolute instability found in the generalized May-Wigner model.

We confirm this picture by showing that for $p>9$ the number of equilibria, as well as the number of stable equilibria, concentrate around their respective averages, generalizing recent results of Subag and Zeitouni in the relaxational case. This confirms the existence of the above transition, as well as shows that the fraction of equilibria which are stable coincides with its annealed variant.