1 research outputs found
Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses
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.Comment: 21 pages, 7 figs, to appear in Phys. Rev.