Federico Camia, Christophe Garban – Journée spéciale

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Date(s) - 19 juin 2009 20 juin 2009

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-*Federico Camia (Vrije Universitet Amsterdam) et Christophe Garban (ENS Paris). Programme: Ising Euclidean Fields and (Conformal) Measure Ensembles | 10h-11h | Federico Camia : A Brief and Gentle Introduction to the Ising Model and Scaling Limits | | 11h15-12h15 | Federico Camia : Conformal Measure Ensembles and Euclidean Fields | | 13h45-14h45 | Christophe Garban : The Scaling Limit of Area Measures | | 15h-16h | Christophe Garban : Conformal Covariance of the Magnetization Field | Abstract: The two-dimensional Ising model is one of the most studied models of statistical mechanics and has played a fundamental role in the theory of phase transitions. This series of talks will focus on the spin or magnetization field, which describes the spatial fluctuations of the local magnetic field generated by the spins and is one of the main objects in the Ising field theory. In the scaling limit, as the lattice spacing is sent to zero, the magnetization field can be desribed as a random generalized function. Above the critical temperature, for instance, the scaling limit of the lattice magnetization field is Gaussian white noise. The situation is more interesting at the critical point. In that case, thermal fluctuations extend over all scales, leading to scale invariance and a conformally covariant field. One also expects cluster boundaries to converge in the scaling limit to SLE-type curves, and a proof of this fact has been recently announced by S. Smirnov. The first two lectures will be based on (F. Camia and C.M. Newman, PNAS 106 (2009) 5457-5463) and will be devoted to introducing a representation of the magnetization field in the scaling limit based on its interpretation as the sum of signed areas of Fortuin-Kasteleyn (FK) clusters. This representation provides a connection between the field-theoretic and the SLE approach and, combined with Russo-Seymour-Welsh (RSW) type estimates, it leads to a simple proof of the existence of subsequential limits for the magnetization field. The same representation is potentially applicable to the three-dimensional Ising model and to the 3- and 4-state Potts models in two dimensions. In the third and fourth lectures, based on work in progress by F. Camia, C. Garban and C.M. Newman, it will be explained how the representation mentioned above can be used to obtain uniqueness and the conformal covariance properties of the magnetization field in the scaling limit. The key ingredient is an ensemble of measures of fractal support coming from the scaling limit of the rescaled areas of critical FK clusters. It will also be explained how the presentation can be useful in studying off-critical scaling limits, particularly in the case of an external magnetic field vanishing in the scaling limit.