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Towards conformal invariance of 2D lattice models. (English) Zbl 1112.82014

Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 1421-1451 (2006).
This paper deals with the mathematical treatment of an important conjecture in conformal field theory and other areas of theoretical physics, the conformal invariance conjecture, saying that 2D statistical mechanics lattice models at criticality have a continuum scaling limit that is conformally invariant and universal, in the sense that this limit is the same for the same model on different lattices. The author reviews known results and conjectures, describes briefly the characterization of the possible scaling limits in terms of the Schramm Lowener’s evolution (SLE) of a single parameter \(\kappa\), and thereafter focuses on the sketch of a new proof he gave in another paper for the existence and the conformal invariance of the scaling limit for a single interface in Ising and Ising random cluster models on the square lattice at critical temperature. It identifies the corresponding scaling limits as SLE(3) and SLE(16/3) respectively, and also uses the same method to get SLE(6) as the scaling limit of percolation. Other conjectures and new directions are also developped for two families of lattice models which have nice loop representations, so-called loop models on hexagonal lattices and Fortuyn-Kasteleyn random cluster models, for which martingale principles and conformal martingales can be used to identify the parameter \(\kappa\) of the SLE in the limit.
For the entire collection see [Zbl 1095.00005].

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
30C35 General theory of conformal mappings
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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