Typical configurations for one-dimensional random field Kac model. (English) Zbl 0983.60091

Typical interfaces in one-dimensional Kac systems (at vanishing external field) are rather well understood, see the recent results by M. Cassandro, E. Orlandi and E. Presutti [Probab. Theory Relat. Fields 96, No. 1, 57-96 (1993; Zbl 0791.60096)] or T. Bodineau [Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 5, 559-590 (1997; Zbl 0893.60014)]. Here, the authors study new phenomena arising in such systems after adding a zero-mean external random field given by symmetric i.i.d. Bernoulli process.
From the authors’ summary: 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.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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