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Low temperature phases in models with long-range interactions

Collaborator: A. Bovier , C. Külske

Cooperation with: A. Le Ny (EURANDOM, Eindhoven, The Netherlands), F. Redig (Technical University, Eindhoven, The Netherlands), M. Zahradník (Charles University, Prague, Czech Republic)

Supported by: DFG Priority Program ``Interagierende Stochastische Systeme von hoher Komplexität'' (Interacting stochastic systems of high complexity)


In this year the study of statistical mechanics models with long-range interactions (Kac models) was continued. The challenge of these models is their description not only on the level of thermodynamical potentials, but also on the level of their Gibbsian distributions. A crucial tool needed for the understanding of these models is a reformulation in terms of suitable contour models. Since uniformity of the estimates in the range of the interaction is desired, standard tools for short-range models can not be used. Contours need to be defined carefully. Due to the long-range nature of the model, interactions between those contours cannot be avoided and need to be treated appropriately by expansion methods.

In the past year the final formulation of a large class of Kac spin systems as contour models could be achieved for translation-invariant interactions in [1]. Once such a representation has been obtained, statements for the random Gibbs measures in the spirit of the Pirogov-Sinai theory for short-range models can be derived (see [2]), and control of the original Gibbs measures of the spin system is obtained.

The present aim is to go beyond translation invariant models and investigate the influence of frozen impurities. Here we focus on the specific example of the Kac-Ising model in a random magnetic field. Using the contour representation developed in [1] we could provide a proof [3] showing ferromagnetic order for sufficiently small randomness, in three dimensions. However, while it is hoped that there should be ferromagnetic order when the inverse temperature is large but uniform in the range of the interactions, we still need, at present, that the inverse temperature is large compared to the logarithm of the range of the interaction. In this regime the contour representation of [1] can be treated by renormalization group techniques developed by Bricmont and Kupiainen [4] for the nearest neighbor Ising model in a random magnetic field.

We are hoping for uniform estimates but these pose additional difficulties and we will need further refinements of the expansion techniques.

In a related but different line of research we investigated the mathematical foundations of the so-called grand-ensemble approach to equilibrium statistical mechanics of disordered systems with frozen disorder. Here the joint measures on the product of disorder space and spin space are interpreted in a formal way as equilibrium measures for the joint variables (the pair of spin variables and disorder variables), for a suitable non-trivial interaction. What are general properties of the joint measures in infinite volume and their corresponding interaction? We were able to show in earlier research [5] that these measures typically leave the realm of classical Gibbs measures with uniformly summable interactions, but are only Gibbs measures in a weak sense. A fundamental question arises, motivated also by various other earlier examples of generalized Gibbs measures: To what extent can the classical Gibbsian formalism be restored for generalized Gibbs measures? Does the variational characterization of Gibbs measures as minimizers of the free energy functional still hold? In this year we made some significant progress in that direction [6], [7]: On the positive side, we were able to extend the variational principle to a certain class of generalized Gibbsian measures. On the other hand, it turned out that the joint measures of the random field Ising model (with short-range interaction) provide a concrete example for which the variational principle fails. This shows that the previously proposed class of so-called weakly Gibbsian measures is too broad for a variational principle to hold.


  1. A. BOVIER, M. ZAHRADNÍK, Cluster expansions and Pirogov-Sinai theory for long range spin systems, Markov Proc. Rel. Fields, 8 (2002), pp. 443-478.
  2. M. ZAHRADNÍK, Cluster expansions of small contours in abstract Pirogov-Sinai models, Markov Proc. Rel. Fields, 8 (2002), pp. 383-441.
  3. A. BOVIER, C. KÜLSKE, M. ZAHRADNÍK, Phase transitions in the Kac-random field Ising model, in preparation.
  4. J. BRICMONT, A. KUPIAINEN, Phase transition in the 3d random field Ising model, Comm. Math. Phys., 116 (1988), pp. 539-572.
  5. C. KÜLSKE, Weakly Gibbsian representations for joint measures of quenched lattice spin models, Probab. Theory Related Fields, 119 (2001), pp. 1-30.
  6. \dito 
, Regularity properties of potentials for joint measures of random spin systems, WIAS Preprint no. 781, 2002, to appear in: Markov Proc. Rel. Fields.
  7. C. KÜLSKE, A. LE NY, F. REDIG, Variational principle for generalized Gibbs measures, EURANDOM Preprint no. 035, 2002.

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