2005/08/25 by Christian Borgs, Jennifer Chayes, Borgs, Christian +5 · 1 citation
Mathematics · Physics and Astronomy · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/0508600
42 pages
arxiv created 2005/08/25 · openalex publication_date 2005/08/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \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 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.