Information geometry of the ising model on planar random graphs

It has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterization of the phase structure, particularly in the case where there are two such parameters, such as the Ising model with inverse temperature ß and external field h. In various two-parameter calculable models, the scalar curvature R of the information metric has been found to diverge at the phase transition point ß_c and a plausible scaling relation postulated: R~\ß-ß_c^a-2. For spin models the necessity of calculating in nonzero field has limited analytic consideration to one-dimensional, mean-field and Bethe lattice Ising models. In this paper we use the solution in field of the Ising model on an ensemble of planar random graphs (where a= -1, ß=1/2, ?=2) to evaluate the scaling behavior of the scalar curvature, and find R~~\ß -ß_c|^-2. The apparent discrepancy is traced back to the effect of a negative a. ©2002 The American Physical Society.