Potts glass on random graphsF. Krzakala1 and L. Zdeborová2, 3
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 in final form 4 January 2008; published March 2008
published online 4 February 2008
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.
75.10.Nr - Spin-glass and other random models.
89.20.Ff - Computer science and technology.
© EPLA 2008