In this paper we study the typical profiles of a random field Kac model. We give upper and lower bounds of the space scale where the profiles are constant. The results hold almost surely with respect to the realizations of the random field. The analysis is based on a block-spin construction, deviation techniques for the local empirical order parameters and concentration inequalities for the realizations of the random magnetic field. For the upper bound, we exhibit a scale related to the law of the iterated logarithm, where the random field makes an almost sure fluctuation that obliges the system to break its rigidity. For the lower bound, we prove that on a smaller scale the fluctuations are not strong enough to allow this transition.
Marzio Cassandro. Enza Orlandi. Pierre Picco. "Typical Configurations for One-Dimensional Random Field Kac Model." Ann. Probab. 27 (3) 1414 - 1467, July 1999. https://doi.org/10.1214/aop/1022677454