Potts glass on random graphs

¹  PCT, UMR 7083 CNRS-ESPCI - 10 rue Vauquelin, 75231 Paris, France
²  Université Paris-Sud, LPTMS, UMR8626 - Bât. 100, Université Paris-Sud, 91405 Orsay cedex, France
³  CNRS, LPTMS, UMR8626 - Bât. 100, Université Paris-Sud, 91405 Orsay cedex, France

Received: 20 October 2007
Accepted: 4 January 2008

Abstract

We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves as a mean-field spin glass. We use the cavity method to compute the temperature-coordination phase diagram and to determine the location of the dynamic and static glass transitions, and of the Gardner instability. We show that for q≥ 4 the model possesses a phenomenology similar to the one observed in structural glasses. We also illustrate the links between the positive- and the zero-temperature cavity approaches, and discuss the consequences for the coloring of random graphs. In particular, we argue that in the colorable region the one-step replica symmetry-breaking solution is stable towards more steps of replica symmetry breaking.