Ising and Potts models on quenched random gravity graphs

We report on single-cluster Monte Carlo simulations of the Ising, 4-state P orts and 10-state Potts models on quenched ensembles of planar, tri-valent (Phi(3)) random graphs. We confirm that the first-order phase transition of the 10-state Ports model on regular 2D lattices is softened by the quenche d connectivity disorder represented by the random graphs and that the expon ents of the Ising and 4-state Potts models are altered from their regular l attice counterparts. The behaviour of spin models on such graphs is thus mo re analogous to models with quenched bond disorder than to Poisonnian rando m lattices, where regular lattice critical behaviour persists. Using a wide variety of estimators we measure the critical exponents for al l three models, and compare the exponents with predictions derived from tak ing a quenched limit in the KPZ formula for the Ising and 4-state Potts mod els, Earlier simulations suggested that the measured values for the 10-stat e Potts model were quite close to the predicted quenched exponents of the f our-state Potts model. The analysis here, which employs a much greater rang e of estimators and also benefits from greatly improved statistics, still s upports these numerical values. (C) 2000 Elsevier Science B.V. All rights r eserved.