Information geometry of finite Ising models

A model in statistical mechanics, characterised by a Gibbs measure, inherits a natural parameter-space geometry through an embedding into the space of square-integrable functions. This geometric structure reflects the underlying physics of the model in various ways. Here, we study the associated geometry and curvature for finite one- and two-dimensional Ising models as the lattice size N is varied. We show that there are temperature T and magnetic field It dependent critical values for the system size N* (T, h) where the curvature varies rapidly and undergoes a change of sign. Such finite volume geometric transitions are necessarily continuous. By comparison with known indicators, we demonstrate that the criterion N much greater than N* provides a consistent constraint that lattice systems are qualitatively in their thermodynamic regime. (C) 2002 Elsevier Science B.V. All rights reserved.