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

Abstract

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 $n \to \infty$, the suitably rescaled energy spectrum above some {\it 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 $α_n$ that grow with $n$, and show that the local REM conjecture holds as long as $n^{-1/4}α_n \to 0$, and fails if $α_n$ grows like $κn^{1/4}$ 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 $κ>0$.