Proof of the local REM conjecture for number partitioning II: growing energy scales

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We continue our analysis of the number partitioning problem with n weights chosen i.i.d. from some fixed probability distribution with density ρ. In Part I of this work, we established the so-called local REM conjecture of Bauke, Franz and Mertens. Namely, we showed that, as n → ∞, the suitably rescaled energy spectrum above some fixed scale α tends to a Poisson process with density one, and the partitions corresponding to these energies become asymptotically uncorrelated. In this part, we analyze the number partitioning problem for energy scales αₙ that grow with n, and show that the local REM conjecture holds as long as n⁻¹/⁴αₙ → 0, and fails if αₙ grows like κn¹/⁴ with κ>0. We also consider the SK-spin glass model, and show that it has an analogous threshold: the local REM conjecture holds for energies of order o(n), and fails if the energies grow like κn with κ>0.