SLE and its partition function in multiply connected domains via the Gaussian Free Field and restriction measures

Abstract

One way to uniquely define Schramm-Loewner Evolution (SLE) in multiply connected domains is to use the restriction property. This gives an implicit definition of a $\sigma$-finite measure on curves; yet it is in general not clear how to construct such measures nor whether the mass of these measures, called the partition function, is finite. We provide an explicit construction of the such conformal restriction SLEs in multiply connected domains when $\kappa = 4$ using the Gaussian Free Field (GFF). In particular, both when the target points of the curve are on the same or on distinct boundary components, we show that there is a mixture of laws of level lines of GFFs that satisfies the restriction property. This allows us to give an expression for the partition function of $\mathrm{SLE}_4$ on multiply connected domains and shows that the partition function is finite, answering the question raised in [Lawler, J. Stat. Phys. 2009]. In a second part, we provide a second construction of $\mathrm{SLE}_\kappa$ in multiply-connected domains for the whole range $\kappa \in (8/3,4]$; specific, however, to the case of the two target points belonging to the same boundary components. This is inspired by [Werner, Wu, Electron. J. Probab. 2013] and consists of a mixture of laws on curves obtained by following $\mathrm{CLE}_\kappa$ loops and restriction hulls attached to parts of the boundary of the domain. In this case as well, we obtain as a corollary the finiteness of the partition function for this type of $\mathrm{SLE}_\kappa$.