Eur. Phys. J. B 15, 493-498 (2000)
https://doi.org/10.1007/s100510051151

Metastable states of spin glasses on random thin graphs

IRSAMC, Laboratoire de Physique Quantique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France

Received: 14 October 1999
Revised: 14 December 1999
Published online: 15 June 2000

Abstract

In this paper we calculate the mean number of metastable states for spin glasses on so called random thin graphs with couplings taken from a symmetric binary distribution . Thin graphs are graphs where the local connectivity of each site is fixed to some value c. As in totally connected mean field models we find that the number of metastable states increases exponentially with the system size. Furthermore we find that the average number of metastable states decreases as c in agreement with previous studies showing that finite connectivity corrections of order 1/c increase the number of metastable states with respect to the totally connected mean field limit. We also prove that the average number of metastable states in the limit is finite and converges to the average number of metastable states in the Sherrington-Kirkpatrick model. An annealed calculation for the number of metastable states of energy E is also carried out giving a lower bound on the ground state energy of these spin glasses. For small c one may obtain analytic expressions for .