Concentration of the complexity of spherical pure p-spin models at arbitrary energies

Abstract

We consider critical points of the spherical pure p-spin spin glass model with Hamiltonian HNσ=1Np−1/2∑i1,…,ip=1NJi1,…,ipσi1…σip, where σ=σ1,…,σN∈SN−1≔σ∈RN:σ2=N and Ji1,…,ip are i.i.d. standard normal variables. Using a second moment analysis, we prove that for p ≥ 32 and any E > −E⋆, where E⋆ is the (normalized) ground state, the ratio of the number of critical points σ with HN(σ) ≤ NE and its expectation asymptotically concentrate at 1. This extends to arbitrary E, a similar conclusion of Subag [Ann. Probab. 45, 3385–3450 (2017)].