arXiv subjects
Publications and source records attributed to Dapeng Zhan.
We derive boundary arm exponents for SLE. Combining with the convergence of critical lattice models to SLE, these exponents would give the alternating half-plane arm exponents for the corresponding lattice models.
We first prove that, for $κ\in(0,4)$, a whole-plane SLE$(κ;κ+2)$ trace stopped at a fixed capacity time satisfies reversibility. We then use this reversibility result to prove that, for $κ\in(0,4)$, a chordal SLE$_κ$ curve stopped at a fixed capacity time can be mapped conformally to the initial segment of a whole-plane SLE$(κ;κ+2)$ trace. A similar but weaker result holds for radial SLE$_κ$. These results are then used to study the ergodic behavior of an SLE curve near its tip point at a fixed capacity time. The proofs rely on the symmetry of backward SLE laminations and conformal removability of SLE$_κ$ curves for $κ\in(0,4)$.
The backward chordal Schramm-Loewner Evolution naturally defines a conformal welding homeomorphism of the real line. We show that this homeomorphism is invariant under the automorphism $x\mapsto -1/x$, and conclude that the associated solution to the welding problem (which is a natural renormalized limit of the finite time Loewner traces) is reversible. The proofs rely on an analysis of the action of analytic circle diffeomorphisms on the space of hulls, and on the coupling techniques of the second author.
The main result of this paper is that, for $κ\in(0,4]$, whole-plane SLE$_κ$ satisfies reversibility, which means that the time-reversal of a whole-plane SLE$_κ$ trace is still a whole-plane SLE$_κ$ trace. In addition, we find that the time-reversal of a radial SLE$_κ$ trace for $κ\in(0,4]$ is a disc SLE$_κ$ trace with a marked boundary point. The main tool used in this paper is a stochastic coupling technique, which is used to couple two whole-plane SLE$_κ$ traces so that they overlap. Another tool used is the Feynman-Kac formula, which is used to solve a PDE. The solution of this PDE is then used to construct the above coupling.
For $κ\in(0,4]$, a family of annulus SLE$(κ;Λ)$ processes were introduced in [14] to prove the reversibility of whole-plane SLE$(κ)$. In this paper we prove that those annulus SLE$(κ;Λ)$ processes satisfy a restriction property, which is similar to that for chordal SLE$(κ)$. Using this property, we construct $n\ge 2$ curves crossing an annulus such that, when any $n-1$ curves are given, the last curve is a chordal SLE$(κ)$ trace.
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 use the coupling technique to prove that there exists a loop-erasure of a plane Brownian motion stopped on exiting a simply connected domain, and the loop-erased curve is the reversal of a radial SLE$_2$ curve.
We prove that the chordal SLE$_κ$ trace is reversible for $κ\in(0,4]$.
We prove that, for $κ\in(0,4)$ and $ρ\ge (κ-4)/2$, the chordal SLE$(κ;ρ)$ trace started from $(0;0^+)$ or $(0;0^-)$ satisfies the reversibility property. And we obtain the equation for the reversal of the chordal SLE$(κ;ρ)$ trace started from $(0;b_0)$, where $b_0>0$.
We define a family of stochastic Loewner evolution-type processes in finitely connected domains, which are called continuous LERW (loop-erased random walk). A continuous LERW describes a random curve in a finitely connected domain that starts from a prime end and ends at a certain target set, which could be an interior point, or a prime end, or a side arc. It is defined using the usual chordal Loewner equation with the driving function being $\sqrt{2}B(t)$ plus a drift term. The distributions of continuous LERW are conformally invariant. A continuous LERW preserves a family of local martingales, which are composed of generalized Poisson kernels, normalized by their behaviors near the target set. These local martingales resemble the discrete martingales preserved by the corresponding LERW on the discrete approximation of the domain. For all kinds of targets, if the domain satisfies certain boundary conditions, we use these martingales to prove that when the mesh of the discrete approximation is small enough, the continuous LERW and the corresponding discrete LERW can be coupled together, such that after suitable reparametrization, with probability close to 1, the two curves are uniformly close to each other.
We improve the geometric properties of SLE$(κ;\vecρ)$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $κ\in (4,8)$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from a random point. Using this fact together with a similar proposition in the case that $κ\ge 8$, we obtain a description of the boundary of a standard chordal SLE$(κ)$ hull for $κ>4$, at a finite stopping time. Finally, we prove that for $κ>4$, in many cases, the limit of a chordal or strip SLE$(κ;\vecρ)$ trace exists.
We derive some geometric properties of chordal SLE$(κ;\vecρ)$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(κ;\vecρ)$ process has the same distribution as the image of a chordal SLE$(κ';\vec{ρ'})$ trace, where $κ>4$, $κ'=16/κ$, and the forces $\vecρ$ and $\vec{ρ'}$ are suitably chosen. We find that for $κ\ge 8$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from $x$. Then we obtain a new proof of the fact that chordal SLE$(κ)$ trace is not reversible for $κ>8$. We also prove that the reversal of SLE$(4;\vecρ)$ trace has the same distribution as the time-change of some SLE$(4;\vec{ρ'})$ trace for certain values of $\vecρ$ and $\vec{ρ'}$.
We prove that the chordal SLE$_κ$ trace is reversible for $κ\in(0,4]$.
An annulus SLE$_κ$ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus SLE$_4$, SLE$_8$ and SLE$_{8/3}$. From the local martingale for annulus SLE$_4$ we find a candidate of discrete lattice model that may have annulus SLE$_4$ as its scaling limit. The local martingale for annulus SLE$_{8/3}$ is similar to those for chordal and radial SLE$_{8/3}$. But it seems that annulus SLE$_{8/3}$ does not satisfy the restriction property.
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$_2$.