Related papers: Simple CLE in Doubly Connected Domains
We derive the boundary four-point Green's functions for conformal loop ensembles (CLE) with $\kappa\in(4,8)$. Specializing to $\kappa=6$ and $\kappa=16/3$, we establish the exact formulas for the boundary four-point connectivities in…
We show that the sum of the squares of the diameters of the complementary connected components of the CLE$_\kappa$ carpet/gasket is almost surely finite for $\kappa \in (8/3, 4) \cup (4, 8)$. This is a prerequisite for the application of a…
We show that when observing the range of a chordal SLE$_\kappa$ curve for $\kappa \in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related results about conformal loop ensembles…
The scaling limit of the probability that $n$ points are on the same cluster for 2D critical percolation is believed to be governed by a conformal field theory (CFT). Although this is not fully understood, Delfino and Viti (2010) made a…
Consider CLE$_4$ in the unit disk and let $\ell$ be the loop of the CLE$_4$ surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by $\ell$. We…
A natural class of conformally invariant ways for discovering the loops of a conformal loop ensemble $\text{CLE}_4$ is given by a certain family of $\text{SLE}_4^{\langle\mu\rangle}(-2)$ exploration processes for real $\mu$. Such an…
We construct the canonical geodesic metric on the gasket of conformal loop ensembles (CLE$_\kappa$) in the regime $\kappa \in (4,8)$ where the loops intersect themselves, each other, and the domain boundary. Previous work of the authors and…
We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $l$ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $l$…
We prove that the SLE$_\kappa$ loop measure arises naturally from the conformal welding of two $\gamma$-Liouville quantum gravity (LQG) disks for $\gamma^2 = \kappa \in (0,4)$. The proof relies on our companion work on conformal welding of…
We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity $\theta \in (0,1/2]$. We obtain an exact expression for the asymptotic probability of the existence of a cluster…
We define the Schramm-Loewner evolution (SLE) in multiply connected domains for kappa \leq 4 using the Brownian loop measure. We show that in the case of the annulus, this is the same measure obtained recently by Dapeng Zhan. We use the…
We study a class of approximation schemes aimed at constructing conformally covariant metrics defined in the gasket of a conformal loop ensemble (CLE$_\kappa$) for $\kappa \in (4,8)$. This is the range of parameter values so that the loops…
Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $\Omega$ in the complex plane ${{\mathbb{C}}}$ (assuming $\Omega\ne {{\mathbb{C}}}$) can be mapped bijectively to the unit disk by an…
A Sierpi\'nski packing in the $2$-sphere is a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We show that any Sierpi\'nski packing by continua whose diameters are square-summable can be…
We initiate a study of the quasisymmetric uniformization of naturally arising random fractals and show that many of them fall outside the realm of quasisymmetric uniformization to simple canonical spaces. We begin with the trace, the graph…
It is shown that there is a computable conformal map of the unit disk onto a domain $D$ that has a computable extension to the closure of the unit disk even though the boundary of $D$ is not effectively locally connected. The proof encodes…
We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE(6) and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We…
For a Brownian loop soup with intensity $c\in(0,1]$ in the unit disk, we show that almost surely, the set of simple (resp. double) points on any portion of boundary of any of its clusters has Hausdorff dimension $2-\xi_c(2)$ (resp.…
This paper initiates the study of the conformal field theory of the SLE$_\kappa$ loop measure $\nu$ for $\kappa\in(0,4]$, the range where the loop is almost surely simple. First, we construct two commuting representations…
Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE6 (the Stochastic Loewner Evolution with parameter k=6) was, in the work of…