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

TL;DR

This work extends the conformal restriction approach to SLE in multiply connected domains, addressing whether the partition function Z_{D,κ}(x,y) is finite beyond annuli. It provides two explicit constructions: (i) SLE_4 as a mixture of GFF level-lines in multiply connected domains, including both non-crossing and crossing endpoint configurations, with a concrete expression and finiteness of the partition function; and (ii) SLE_κ for κ∈(8/3,4] in the non-crossing case as a mixture of restriction hulls and CLE with proven restriction properties and finite partition functions. The analysis leverages Gaussian free field couplings, local sets, and detailed Dirichlet-energy calculations, connecting to Brownian excursion measures and CLE via loop-soup frameworks. Together, these results give concrete, finite partition functions and rigorous constructions for SLE in complex topologies, strengthening links to conformal field theory and Liouville quantum gravity while offering tools for related lattice-model and CLE studies. The findings illuminate how restriction measures and CLE can encode SLE in moduli-rich domains, enabling new avenues for studying boundary perturbations, BPZ-type equations, and probabilistic representations of CFT data in non-simply connected geometries.

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 $σ$-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 $κ= 4$ 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 $\mathrm{SLE}_κ$ in multiply-connected domains for the whole range $κ\in (8/3,4]$; 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 $\mathrm{CLE}_κ$ 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 $\mathrm{SLE}_κ$.

Figures (4)

• Figure 1: An example of a domain with the definitions above, together with a test domain $D'$ shaded in light blue. Here the curve $\eta$ depicted in the middle is in $\mathcal{A}_\mathbf{b}$ for $\mathbf{b} = \left \{ -\lambda, + \lambda, + \lambda \right \}$. In particular, the boundary values of $g_\mathbf{b}$ are $-\lambda$ and $+\lambda$ exactly on the parts of $\partial D$ that lie on the left and on the right of $\eta$, respectively.
• Figure 2: An example of two non path-homotopic curves in the event $\mathcal{A}_{\left \{ -\lambda,+\lambda \right \}}$. In general, there are infinitely (countably) many homotopy classes of distinct curves corresponding to every event $\mathcal{A}_\mathbf{b}$.
• Figure 3: $D"$ corresponds to the darkest shade, $D'$ to the two darker shades (left of orange curve $\mathcal{B}_r'$) and $D$ to the three shades.
• Figure 4: A configuration with a curve in $\mathcal{A}_{0,+3\lambda,+\lambda}$.

Theorems & Definitions (37)

• Definition 1: Conformal restriction $\mathrm{SLE}_\kappa$ in multiply-connected domains
• Lemma 1.1: Some identities for loop measures
• proof
• Proposition 1.2: Proposition 2.2 in dubedat_sle_2009
• Definition 1.3: Conformal restriction SLE, lawler_defining_2011
• Definition 2.1: Local sets
• Lemma 2.2: Lemma 3.1 in aru_extremal_2022
• Lemma 2.3: Proposition 1.2.8 in da_prato_second_2002
• Theorem 2.4: $\mathrm{SLE}_4$ is a local set for the GFF
• Proposition 2.5: General level-lines, aru_bounded-type_2019