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Article overview
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Proof of the local REM conjecture for number partitioning II: growing energy scales | Christian Borgs
; Jennifer Chayes
; Stephan Mertens
; Chandra Nair
; | Date: |
25 Aug 2005 | Subject: | Disordered Systems and Neural Networks; Statistical Mechanics | cond-mat.dis-nn cond-mat.stat-mech | Abstract: | We continue our analysis of the number partitioning problem with $n$ weights chosen i.i.d. from some fixed probability distribution with density $
ho$. In Part I of this work, we established the so-called local REM conjecture of Bauke, Franz and Mertens. Namely, we showed that, as $n o infty$, 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 $n^{-1/4}alpha_n o 0$, 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 $kappa >0$. | Source: | arXiv, cond-mat/0508600 | Services: | Forum | Review | PDF | Favorites |
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