, The low temperature phase of
Kac-Ising models, J. Statist. Phys. 87, (1997),
pp. 311-332.
|
|
|
[Contents] | [Index] |
Collaborator: A. Bovier , C. Külske
Cooperation with: M. Zahradník (Charles University, Prague, Czech Republic)
Supported by: DFG Priority Program ``Interagierende Stochastische Systeme von hoher Komplexität'' (Interacting stochastic systems of high complexity)
Description: Models of statistical mechanics with weak long range interactions have been introduced by V. Kac half a century ago to obtain a rigorous version of the van der Waals mean field theory. Today we see these models anew as interesting candidates to reach a better understanding of disordered spin systems and in particular the relation between mean field theory and lattice models in the context of disordered systems. In this project we are undertaking a long-term effort to investigate such models and to develop appropriate mathematical tools for their analysis.
In the past year one of the main goals of this project could be achieved. In [1] we have developed a contour model representation for a large class of spin systems with interactions of Kac type that allows to analyze these systems in a domain of parameters that is uniform in the range of the interaction. The difficulties in achieving this arise from the fact that to gain uniform control on the energy of the so-called ``contours'', the definition of these objects must be quite non-local. As a consequence, the so-called ``restricted ensembles'', i.e. the regions in space that are free of contours, must satisfy non-local constraints. The crucial step, namely the cluster-expansion of the restricted ensembles is therefore quite non-standard, and a careful adjustment between the requirements for the convergence of this expansion and the necessity to obtain uniform estimates on the energies of contours necessitates a delicate balancing. The final formulation of the contour model with non-local interactions between contours that is provided in [1] allows then to use the Pirogov-Sinai theory to give a complete description of the phase diagram of the model ([3]).
In a separate development, an important step was taken to prepare the analysis of the random field Kac-Ising model. One of the challenges here is to control the low-temperature phase asymptotically up to the critical temperature of the mean field model, as was done in the ferromagnetic case in [2]. To do so requires a block transformation of the spin variables. While in the ferromagnetic case, one can work with suitable bounds on the resulting distribution of the block variables, in the random case one needs complete control of the effective action of the blocked model. This is achieved in [4] for suitably chosen block sizes as functions of the range of the interaction. The effective Hamiltonian of the blocked model is then represented as the desired ``mean-field'' type term plus a well-controlled ``small'' correction that arises from a high-temperature expansion. A conceptually interesting side-result is that the measure on the blocked variables is indeed a Gibbsian measure.
References:
, The low temperature phase of
Kac-Ising models, J. Statist. Phys. 87, (1997),
pp. 311-332.
|
|
|
[Contents] | [Index] |