Fractal characterisation of non-Gaussian critical Markov random fields

We characterize a class of self-similar non-Gaussian Markov random fields (MRFs) that we call critical MRFs (CMRFs). We show that since the partition function in a Gibbs distribution of a CMRF is necessarily scale invariant, all order statistics generated from the partition function are generalized homogenous functions. This implies that the correlation function has long-range memory since it decays as a power-law function, a very important characteristic of many textures. This characterization is potentially of great use in modeling a wide variety of multi-dimensional spatial phenomena.