Critical behaviour of Dyson's hierarchical model with a random field

Abstract

The critical behaviour of Dyson's hierarchical Ising model is studied in the presence of a random field. Simple scaling relations determine the exponents eta , eta exactly for random fields with either short- or long-range correlations. The Schwartz-Soffer inequality eta <or=2 eta is satisfied as an equality provided the random-field correlations are not too long ranged. For short-range random fields, the exponent v is calculated to order in = lambda -4/3, where lambda is the 'range parameter' of the Dyson model. The exponent v is also computed to O( in 2) with in =2 lambda -3, for the hierarchical O(n) model in zero field: the results reveal striking similarities with the exponents of the corresponding one-dimensional model with long-range interactions.