Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses
Abstract
We develop a systematic cluster expansion for dilute systems in the highly dilute phase. We first apply it to the calculation of the entropy of the K-satisfiability problem in the satisfiable phase. We derive a series expansion in the control parameter, the average connectivity, that is identical to the one obtained by using the replica approach with a replica symmetric ({\sc rs}) {\it Ansatz}, when the order parameter is calculated via a perturbative expansion in the control parameter. As a second application we compute the free-energy of the Viana-Bray model in the paramagnetic phase. The cluster expansion allows one to compute finite-size corrections in a simple manner and these are particularly important in optimization problems. Importantly enough, these calculations prove the exactness of the {\sc rs} {\it Ansatz} below the percolation threshold and might require its revision between this and the easy-to-hard transition.
Cite
@article{arxiv.cond-mat/0102259,
title = {Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses},
author = {Guilhem Semerjian and Leticia F. Cugliandolo},
journal= {arXiv preprint arXiv:cond-mat/0102259},
year = {2009}
}
Comments
21 pages, 7 figs, to appear in Phys. Rev. E