This paper introduces the annulus SLE$_κ$ processes in doubly connected domains. Annulus SLE$_6$ has the same law as stopped radial SLE$_6$, up to a time-change. For $κ\not=6$, some weak equivalence relation exists between annulus SLE$_κ$ and radial SLE$_κ$. Annulus SLE$_2$ is the scaling limit of the corresponding loop-erased conditional random walk, which implies that a certain form of SLE$_2$ satisfies the reversibility property. We also consider the disc SLE$_κ$ process defined as a limiting case of the annulus SLE's. Disc SLE$_6$ has the same law as stopped full plane SLE$_6$, up to a time-change. Disc SLE$_2$ is the scaling limit of loop-erased random walk, and is the reversal of radial SLE$_2$.
Excursion reflected Brownian motion (ERBM) is a strong Markov process defined in a finitely connected domain $D \subset \C$ that behaves like a Brownian motion away from the boundary of $D$ and picks a point according to harmonic measure from infinity to reflect from every time it hits a boundary component. We give a new construction of ERBM using its conformal invariance and discuss the relationship between the Poisson kernel and Green's function for ERBM and conformal maps into certain classes of finitely connected domains. One important reason for studying ERBM is the hope that it will be a…
We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE$_κ$ as $κ$ tends to 0 and the known reversibility of the SLE$_κ$ curves for small $κ$, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from B to A in D.
We investigate the relation between the Laplacian-$b$ motion and stochastic Komatu-Loewner evolution (SKLE) on multiply connected subdomains of the upper half-plane, both of which are analogues to SLE. In particular, we show that, if the driving function of an SKLE is given by a certain stochastic differential equation, then this SKLE is the same as a time-changed Laplacian-$b$ motion. As an application, we prove the finite time explosion of SKLE corresponding to Laplacian-$0$ motion, or $\mathrm{SLE_6}$, in the sense that the solution to the Komatu-Loewner equation for the slits blows up.
We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let ${\mathcal K}$ be the set of $κ\in (4,8)$ for which the adjacency graph of connected components of the complement of an SLE$_κ$ is a.s. connected, meaning that for every pair of complementary components $U, V$ there exist complementary components $U_1,\ldots,U_n$ with $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. We show that the range of an SLE$_κ$ for $κ\in {\mathcal K}$ is a.s…
We consider the Peano curve separating a spanning tree from its dual spanning tree on an embedded planar graph, where the tree and dual tree are weighted by $y$ to the number of active edges, and "active" is in the sense of the Tutte polynomial. When the graph is a portion of the square grid approximating a simply connected domain, it is known ($y=1$ and $y=1+\sqrt{2}$) or believed ($1<y<3$) that the Peano curve converges to a space-filling SLE$_κ$ loop, where $y=1-2\cos(4π/κ)$, corresponding to $4<κ\leq 8$. We argue that the same should hold for $0\le y<1$, which corresponds to $8<κ\leq 12$.
We define multiple-paths Schramm-Loewner evolution ($SLE_κ$) in multiply connected domains when $κ\leq 4$ and prove that in annuli, the partition function is smooth. Moreover, we give up-to-constant estimates for the partition function of bi-chordal annulus $SLE_κ$ measure and we establish a connection between this measure and two-sided $SLE_κ$ in the unit disk.
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 loop formulation to give a different derivation of the partial differential equation for the partition function for the annulus.
Let $D={\mathbb H}\setminus \bigcup_{j=1}^N C_j$ be a standard slit domain, where ${\mathbb H}$ is the upper half plane and $C_j,1\le j\le N,$ are mutually disjoint horizontal line segments in ${\mathbb H}$. A stochastic Komatu-Loewner evolution denoted by ${\rm SKLE}_{α,b}$ has been introduced in \cite{CF} as a family $\{F_t\}$ of random growing hulls with $F_t\subset D$ driven by a diffusion process $ξ(t)$ on $\partial {\mathbb H}$ that is determined by certain continuous homogeneous functions $α$ and $b$ defined on the space ${\cal S}$ of all labelled standard slit domains. We aim at identi…
A quantum surface (QS) is an equivalence class of pairs $(D,H)$ of simply connected domains $D\subsetneq\mathbb{C}$ and random distributions $H$ on $D$ induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with $N+1$ marked boundary points (MBPs) with $N\in\mathbb{Z}_{\ge 0}$. We propose the conformal welding problem for it in the case of $N\in\mathbb{Z}_{\ge 1}$. If $N=1$, it is reduced to the problem introduced by Sheffield, who solved it by coupling the QS with the Schramm--Loewner evolution (SLE). When $N \ge 3$, there natu…
We use the whole-plane Loewner equation to define a family of continuous LERW in finitely connected domains that are started from interior points. These continuous LERW satisfy conformal invariance, preserve some continuous local martingales, and are the scaling limits of the corresponding discrete LERW on the discrete approximation of the domains.
We study Conformal Loop Ensemble (CLE$_κ$) in doubly connected domains: annuli, the punctured disc, and the punctured plane. We restrict attention to CLE$_κ$ for which the loops are simple, i.e. $κ\in (8/3,4]$. In the paper "Conformal Loop Ensemble" by Sheffield and Werner, simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that…
Scharmm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice models and is suitable for strict proofs, it lacks computational and predictive power of conformal field theory. To provide a way for the concurrent use of SLE and CFT we consider CFT correlation functions which are martingales with respect to SLE. We establish connection between parameters of Schramm-Loewner evolution on coset space and algebraic data of coset…
The Brownian loop measure is a conformally invariant measure on loops in the plane that arises when studying the Schramm-Loewner evolution (SLE). When an SLE curve in a domain evolves from an interior point, it is natural to consider the loops that hit the curve and leave the domain, but their measure is infinite. We show that there is a related normalized quantity that is finite and invariant under Möbius transformations of the plane. We estimate this quantity when the curve is small and the domain simply connected. We then use this estimate to prove a formula for the Radon-Nikodym derivative…
Suppose that $η$ is a Schramm-Loewner evolution (SLE$_κ$) in a smoothly bounded simply connected domain $D \subset {\mathbb C}$ and that $ϕ$ is a conformal map from ${\mathbb D}$ to a connected component of $D \setminus η([0,t])$ for some $t>0$. The multifractal spectrum of $η$ is the function $(-1,1) \to [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial {\mathbb D}$ such that $|ϕ'( (1-ε) x)| = ε^{-s+o(1)}$ as $ε\to 0$. We rigorously compute the a.s. multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollar…
We consider collections of $N$ chordal random curves obtained from a critical lattice model on a planar graph, in the limit when a fine-mesh graph approximates a simply-connected domain. We define and study candidates for such limits in terms of conformally invariant collections of random curves, generated via iterated Loewner equations. These curves are a natural ``domain Markov extension'' of the earlier introduced local multiple SLE initial segments to global multiple SLE curves. For realizing them as scaling limits, we provide two a priori results to guarantee the precompactness of the dis…
We prove convergence of renormalized correlations of primary fields, i. e., spins, disorders, fermions and energy densities, in the scaling limit of the critical Ising model in arbitrary finitely connected domains, with fixed (plus or minus) or free boundary conditions, or mixture thereof. We describe the limits of correlations in terms of solutions of Riemann boundary value problems, and prove their conformal covariance. Moreover, we prove fusion rules, or operator product expansions, which describe asymptotics of the scaling limits of the correlations as some of the points collide together. …
