Stochastic Löwner Evolution and the scaling limit of critical models

Great progress in the understanding of conformally invariant scaling limits of stochastic models, has been given by the Stochastic Löwner Evolutions (SLE). This approach has been pioneered by Schramm [46] and by Lawler, Schramm and Werner [31]. It describes a one-parameter family of conformally invariant measures of curves in the plane or a two-dimensional domain. This family is commonly re-ferred to as SLEκ, where κ parametrizes the family. It has been shown to be the scaling limit of many well-known and less well-known statistical lattice models. These models are typically members of the families of critical and tricritical [40] q-state Potts models [61] and of O(n) models [17], or believed to be in the corresponding universality class.
SLE describes the scaling limit of various open, non-crossing, stochastic paths on the lattice, which are, at least on one side, attached to the boundary. Therefore its application to polygons is restricted in various ways. In the first place it describes only the scaling limit. In many studies of lattice polygons, of course, the scaling limit is considered the most interesting aspect. The restriction to open paths attached to the boundary is more severe. This restriction has been lifted to some extent by recursively considering domains bounded by closed paths resulting from a previous SLE process. This approach applies only to paths that have a tendency to touch themselves (without, of course crossing), and this generalization is not the subject of this chapter. In most cases the paths under consideration by their nature occur in extensive numbers. However, one may concentrate on one of them, and treat the interaction with the others only as an ingredient that defines the stochastic measure of the path under consideration. This, in fact is precisely what SLE does.