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

Christian Borgs, Jennifer Chayes, Stephan Mertens and Chandra Nair

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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 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 $\kappa >0$.

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BiBTeX Entry

@article{bcmn:rem2,
  author =	 {C. Borgs and J. Chayes and S. Mertens and C. Nair},
  title =	 {Proof of the local {REM} conjecture for number partitioning {II}: Growing energy scales},
  journal = {Rand.\ Struct.\ Alg.},
  year = {2009},
  volume = {34},
  pages = {241--284}
}

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npp-rem3.pdf (pdf, 260 kB)