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Probabilistic models of interfaces and their scaling limit

  • 00-Xx General
  • Mathematics


Hokudai10.dvi PROBABILISTIC MODELS OF INTERFACES AND THEIR SCALING LIMIT TAKAO NISHIKAWA 1. INTRODUCTION We discuss the large scale hydrodynamic behavior of interfaces separating two distinct phases. The problem of the phase separation was first investigated from microscopic aspect in mathe- matically rigorous way by [2] for the Ising model in the low temperature regime. An interface at macroscopic level called the Wulff shape was derived from ˙ spin systems described by the finite volume canonical Gibbs measure with � spins’ boundary condition given the number of sites occupied by C spins inside the domain. The Wulff shape is characterized by a variational problem minimizing the total surface tension. The results of [2] have been generalized into sev- eral directions afterward. However, there exist very few dynamic results, for instance, for the Glauber or the Kawasaki dynamics in the low temperature regime, because of serious analyti- cal difficulties. In order to explain microscopic motions of interfaces, [10] introduced several simplified models including the Ginzburg-Landau r� interface model which is main object in this talk. We can also refer [3], which gives a detailed survey around the Ginzburg-Landau r� interface model and related models. 2. GINZBURG-LANDAU r� INTERFACE MODEL The Ginzburg-Landau r� interface model determines the stochastic dynamics of a dis- cretized hypersurface separating two phases. The position of the hypersurface is described by height variables � D f�.x/I x 2 � g measured from a fixed d -dimensional discrete hyperplane � . We then admit an energy (Hamiltonian) to the interface � by: (2.1) H.�/ D 1 2 X x;y2�; jx�yjD1 V.�.x/ � �.y//: The potential V in the Hamiltonian H is assumed to satisfy the conditions as follows: (i) V 2 C 2.R/. (ii) V is symmetric, i.e. V.�/ D V.��/ for all � 2 R. (iii) There exist constants cC; c� > 0 such that c� � V 00.�/ � cC; � 2 R holds. Once the energy H is introduced, the dynamics of the interfaces can be de

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