Zero-temperature dynamics for the ferromagnetic Ising model on random graphs

We consider Glauber dynamics at zero temperature for the ferromagnetic Ising model on the usual random graph model on N vertices, with on average gamma edges incident to each vertex, in the limit as N --> infinity. Based on numerical simulations, Svenson (Phys. Rev. E 64 (2001) 036122) reported that the dynamics fails to reach a global energy minimum for a range of values of gamma. The present paper provides a mathematically rigorous proof that this failure to find the global minimum in fact happens for all gamma > 0. A lower bound on the residual energy is also given. (C) 2002 Elsevier Science B.V. All rights reserved.