Eur. Phys. J. B 20, 217-233 (2001)
https://doi.org/10.1007/PL00011099

The Bethe lattice spin glass revisited

¹ Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris Sud, Bâtiment 100, 91405 Orsay Cedex, France
² Dipartimento di Fisica, Sezione INFN and Unità INFM, Università di Roma "La Sapienza", Piazzale Aldo Moro 2, 00185 Rome, Italy

Corresponding author: ^a mezard@ipno.in2p3.fr

Received: 27 September 2000
Published online: 15 March 2001

Abstract

So far the problem of a spin glass on a Bethe lattice has been solved only at the replica symmetric level, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a replica symmetry breaking solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non perturbative solution of the Bethe lattice spin glass problem at a level of approximation which is equivalent to a one step replica symmetry breaking solution. The results compare well with numerical simulations. The method can be used for many finite connectivity problems appearing in combinatorial optimization.