Abstract
We present a quenched weak large deviations principle for the Gibbs measures of a Random Field Kac Model (RFKM) in one dimension. The external random magnetic field is given by symmetrically distributed Bernouilli random variables. The results are valid for values of the temperature and magnitude of the field in the region where the free energy of the corresponding random Curie Weiss model has only two absolute minimizers. We give an explicit representation of the large deviation rate function and characterize its minimizers. We show that they are step functions taking two values, the two absolute minimizers of the free energy of the random Curie Weiss model. The points of discontinuity are described by a stationary renewal process related to the $h$-extrema of a bilateral Brownian motion studied by Neveu and Pitman, where $h$ depends on the temperature and magnitude of the random field. Our result is a complete characterization of the typical profiles of RFKM (the ground states) which was initiated in [2] and extended in [4].
Highlights
We consider a one-dimensional spin system interacting via a ferromagnetic two-body Kac potential and external random magnetic field given by symmetrically distributed Bernoulli random variables
We will comment later about this, but one should bear in mind that the results proven in [14] hold for almost all realizations of the random magnetic fields, the ones proven in [16] hold for a set of realizations of the random magnetic fields of probability that goes to one when γ ↓ 0, while the ones in the present paper hold merely in law
The macroscopic state of the system is determined by an order parameter which specifies the phase of the system. It has been proven in [14] that for almost all realizations of the random magnetic fields, for intervals whose length in macroscopic scale is of order (γ log log(1/γ))−1 the typical block spin profile is either rigid, taking one of the two values corresponding to the minima of the canonical free energy of the random field Curie Weiss model, or makes at most one transition from one of the minima to the other
Summary
We consider a one-dimensional spin system interacting via a ferromagnetic two-body Kac potential and external random magnetic field given by symmetrically distributed Bernoulli random variables. The macroscopic state of the system is determined by an order parameter which specifies the phase of the system It has been proven in [14] that for almost all realizations of the random magnetic fields, for intervals whose length in macroscopic scale is of order (γ log log(1/γ))−1 the typical block spin profile is either rigid, taking one of the two values (mβ or T mβ) corresponding to the minima of the canonical free energy of the random field Curie Weiss model, or makes at most one transition from one of the minima to the other. It could happen that even though at large the random fields undergoe to a positive (for example) fluctuation, locally there are negative fluctuations which make not convenient (in terms of the cost of the total free energy) for the system to have a magnetization profile close to the + phase in that interval Another problem in the previous analysis is due to the fact that the measure induced by the block-spin transformation contains multibody interaction of arbitrary order.
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