Proof of the local REM conjecture for number partitioning I: Constant energy scales

TL;DR

This paper proves the local REM conjecture for the number partitioning problem, showing that at constant energy scales, the energy levels behave like a Poisson process and are uncorrelated.

Contribution

It provides a rigorous proof of the local REM conjecture for number partitioning, establishing the uncorrelated nature of energy levels and spin configurations at constant energy scales.

Findings

Energy levels form a Poisson process at constant scales

Energy spectrum is locally like a random energy model

Spin configurations are uncorrelated, with vanishing overlaps

Abstract

The number partitioning problem is a classic problem of combinatorial optimization in which a set of $n$ numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the $n$ numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies -- corresponding to the costs of the partitions, and overlaps -- corresponding to the correlations between partitions. Although the energy levels of this model are {\em a priori} highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.