Quenched complexity of marginal states in the Sherrington--Kirkpatrick spin glass

Abstract: In the Sherrington-Kirkpatrick model the exponentially many metastable states are marginal, so counting them requires breaking the BRST supersymmetry or a two-group replica Ansatz. For four decades this complexity was known only in the annealed approximation, which is unstable at low free energy and predicts states below the Parisi equilibrium free energy. We compute it quenched, with one step of replica symmetry breaking. It leaves the annealed curve where it becomes unstable and vanishes next to the full-RSB equilibrium free energy: the lowest marginal states are thus plausibly the equilibrium states themselves.