Some remarks on commutation relations for SLE

Schramm-Loewner Evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e. several random curves). In particular, one derives infinitesimal commutation conditions, discuss some solutions, and show how to lift these infinitesimal relations to global relations in simple cases. For plane critical models of statistical physics, such as percolation or the Ising model, the general line of thinking of Conformal Field Theory leads to expect the existence of a non-degenerate scaling limit that satisfies conformal invariance properties. Though, it is not quite clear how to define this scaling limit and what conformal invariance exactly means. One way to proceed is to consider a model in a, say, bounded (plane) simply connected domain with Jordan boundary, and to set boundary conditions so as to force the existence of a macroscopic interface connecting two marked points on the boundary. In this set-up, Schramm has shown that the possible scaling limits satifying conformal invariance along with a “domain Markov” property are classified by a single positive parameter κ > 0, in the seminal article [16]. This defines the family of Schramm-Loewner Evolutions (SLEs), that are probability measures supported on non-self-traversing curves connecting two marked boundary points in a simply-connected domain. Consider the following situation for critical site percolation on the triangular lattice: a portion of the triangular lattice with mesh ε approximates a fixed simply connected domain D with two points x and y marked on the boundary. The boundary arc (xy) is set to blue and (yx) is set to yellow; sites are blue or yellow with probability 1/2. Then the interface between blue sites connected to (xy) and yellow sites connected to (yx) is a non-self traversing curve from x to y. In this set-up, Smirnov has proved that the interface converges to SLE6, as conjectured earlier by Schramm ([18, 4]). For discrete models such as percolation or the Ising model, the full information can be encoded as a collection of contours (interfaces between blue any yellow, + and − spins, . . . ). Hence it is quite natural to consider scaling limits as collection of countours, as in [1, 4]. Comparing the ideas of isolating one macroscopic interface by setting appropriate boundary conditions (following Schramm), and considering the scaling limit as a collection of contours, one is led to the problem of describing the joint law of a finite number of macroscopic interfaces created by appropriate boundary conditions. For instance, for percolation, consider a simply connected domain with 2nmarked points on the boundary, the 2n boundary arcs being alternatively blue and yellow. This gives n interfaces pairing the 2n points. One can then consider the joint scaling limits of these interfaces (either unconditionally or conditionally on a given pairing). Each of these interfaces close to its starting point is absolutely continuous w.r. t. SLE6. So we are defining n “non-crossing” SLE6’s; the problem is then to precisely quantify their interaction. Courant Institute