The number of saddles of the spherical $p$-spin model

Abstract

We show that the quenched complexity of saddles of the spherical pure $p$-spin model agrees with the annealed complexity when both are positive. Precisely, we show that the second moment of the number of critical values of a given finite index in a given interval has twice the growth rate of the first moment.

arXiv comment52 pages, 1 figure

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Cited by in corpus (4)

• Exponential growth of random determinants beyond invariance
• Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates
• Concentration of the complexity of spherical pure $p$-spin models at arbitrary energies
• Glass--like transition described by toppling of stability hierarchy