Random currents expansion of the Ising model

  • Hugo Duminil-Copin

    Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France and Université de Genève, Switzerland
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Critical behavior at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, Spectral Theory, etc) have contributed to a more and more elaborate description of the possible critical behaviors for a large variety of models. The Ising model is maybe one of the most striking success of this cross-fertilization, for this model of ferromagnetism is now very well understood both physically and mathematically. In this article, we review an approach, initiated in [7, 24] and based on the notion of random currents, enabling a deep study of the model.