Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses

Guilhem Semerjian (1) , Leticia F. Cugliandolo (1, 2)

1 LPTENS - Laboratoire de Physique Théorique de l'ENS [École Normale Supérieure]
2 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

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.

Domaines: Combinatoire [math.CO]; Mécanique statistique [cond-mat.stat-mech]

Guilhem Semerjian, Leticia F. Cugliandolo. Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2001, 64, pp.036115. ⟨hal-00117084⟩