×

Coarse-graining techniques for (random) Kac models. (English) Zbl 1081.82008

Deuschel, Jean-Dominique (ed.) et al., Interacting stochastic systems. Berlin: Springer (ISBN 3-540-23033-5/hbk). 11-28 (2005).
In this paper the authors review a number of their results on Kac models, mostly obtained by combining coarse-graining and Pirogov-Sinai methods. Both deterministic and disordered models are treated. They discuss the scaling of the coarse-graining with the Kac parameter in some detail. In particular, as a new result, they announce, and indicate the arguments how to prove, a version of the well-known Bricmont-Kupiainen result that there exists a phase transition in the three- or higher-dimensional random-field Ising model for the Kac version, for a temperature interval which is uniform in the Kac parameter. I found it a clear and accessible review.
For the entire collection see [Zbl 1053.82002].

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics
PDFBibTeX XMLCite