Groupe de Physique Statistique

Discrete analysis is an analog of complex analysis, for functions defined on a regular lattice. In the context of 2D statistical mechanics, some lattice operators have discretely holomorphic correlation functions, which become holomorphic fields of the Conformal Field Theory in the scaling limit. These discretely holomorphic operators were introduced as the first step for a rigorous proof of conformal invariance in the scaling limit of the Ising model, by S. Smirnov and collaborators.
The construction of discretely holomorphic operators was originally proposed for the Potts and O(n) lattice models, and we have extended it to several other exactly solved models. Moreover, I shall explain how we were able to relate discrete holomorphicity and integrability in all these models.