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

TL;DR

This paper extends the analysis of the number partitioning problem, showing the local REM conjecture holds for growing energy scales under specific conditions, and explores similar phenomena in the SK-spin glass model.

Contribution

It demonstrates the validity of the local REM conjecture for increasing energy scales in number partitioning and SK-spin glass models, identifying precise thresholds for its applicability.

Findings

The local REM conjecture holds if $n^{-1/4}\alpha_n o 0$.

The conjecture fails if $\alpha_n$ grows like $\kappa n^{1/4}$ with $\kappa>0$.

In the SK-spin glass model, the conjecture holds for energies of order $o(n)$ and fails for energies proportional to $n$.

Abstract

We continue our analysis of the number partitioning problem with $n$ weights chosen i.i.d. from some fixed probability distribution with density $\rho$. 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 $\alpha$ 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 $\alpha_n$ that grow with $n$, and show that the local REM conjecture holds as long as $n^{-1/4}\alpha_n \to 0$, and fails if $\alpha_n$ grows like $\kappa n^{1/4}$ with $\kappa>0$. 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 $\kappa n$ with $\kappa >0$.