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Dapeng Zhan

Publications and source records attributed to Dapeng Zhan.

At least 19 recordsLinked to original sources
arXiv2026

Trivariate Hypergeometric Series Formulas for Pure Partition Functions of Multiple $3$-SLE$_\kappa$

Pure partition functions of multiple SLE are characterized by null-state partial differential equations, M\"obius covariance, and boundary asymptotics. After quotienting by M\"obius covariance, the case of three curves is the first genuinely multivariable one: the moduli space has three independent variables, naturally represented by the three unoriented cross-ratios of the three pairs of links. We solve this M\"obius-normalized three-variable problem for the two basic link-pattern types of multiple $3$-SLE$_\kappa$, namely the rainbow and neighbor patterns. Writing $\beta=4/\kappa$, we construct explicit trivariate hypergeometric-series normal forms and identify them with the corresponding pure partition functions {for all $\beta>1/2$ in both cases, equivalently for all $\kappa\in(0,8)$}. The proof is analytic. The null-state PDEs and M\"obius covariance yield recursion relations for the trivariate coefficient arrays. In the rainbow case, coefficient estimates give convergence and boundary regularity on the closed cube. In the neighbor case, Pfaff systems continue the local power series to a neighborhood of $[0,1)^3$, while {boundary analysis and corner propagation} give continuity on $[0,1]^3$ {throughout the full range $\beta>1/2$}. The two-dimensional boundary degenerations are classical Appell $F_1$ and Horn $G_2$ functions. The probabilistic identification uses SLE martingale arguments and It\^o calculus, together with positivity and boundary regularity. We also discuss boundary degenerations, including heuristic connections with boundary Green's functions.

math.PR↗
arXiv2023

Existence and uniqueness of nonsimple multiple SLE

We prove the existence and uniqueness of multiple SLE$_κ$ associated with any given link pattern for $κ\in (4,6]$. We also have the uniqueness for $κ\in (6,8)$. The multiple SLE$_κ$ law is constructed by first inductively constructing a $σ$-finite multiple SLE$_κ$ measure and then normalizing the measure whenever it is finite. The total mass of the measure satisfies the conformal covariance, asymptotics and PDE for multiple SLE$_κ$ partition functions in the literature subject to the assumption that it is smooth.

math.PR↗
arXiv2023

SLE$_κ(ρ)$ bubble measures

For $κ>0$ and $ρ>-2$, we construct a $σ$-finite measure, called a rooted SLE$_κ(ρ)$ bubble measure, on the space of curves in the upper half plane $\mathbb H$ started and ended at the same boundary point, which satisfies some SLE$_κ(ρ)$-related domain Markov property, and is the weak limit of SLE$_κ(ρ)$ curves in $\mathbb H$ with the two endpoints both tending to the root. For $κ\in(0,8)$ and $ρ\in ((-2)\vee(\fracκ2-4),\fracκ2-2)$, we derive decomposition theorems for the rooted SLE$_κ(ρ)$ bubble with respect to the Minkowski content measure of the intersection of the rooted SLE$_κ(ρ)$ bubble with $\mathbb R$, and construct unrooted SLE$_κ(ρ)$ bubble measures.

math.PR↗
arXiv2022

Green's function for cut points of chordal SLE attached with boundary arcs

Let $κ\in(4,8)$. Let $γ$ be an SLE$_κ$ curve in a Jordan domain $D$ connecting $a_1\ne a_2\in\partial D$. We attach $γ$ with two open boundary arcs $A_1,A_2$ of $D$, which share end points $b_1\ne b_2\in\partial D\setminus\{a_1,a_2\}$, and consider for each $z_0\in D$ the limit $$ \lim_{r \downarrow 0}r^{1-\frac 38κ} \mathbb{P}[γ\cup A_1\cup A_2 \mbox{ has a cut point in }\{|z-z_0|<r\}].$$ We prove that the limit converges, derive a rate of convergence, and obtain the exact formula of the limit up to a multiplicative constant depending only on $κ$.

math.PR↗
arXiv2022

Boundary Green's functions and Minkowski content measure of multi-force-point SLE$_κ(\underlineρ)$

We consider a transient chordal SLE$_κ(ρ_1,\dots,ρ_m)$ curve $η$ in $\mathbb{H}$ from $w$ to $\infty$ with force points $ v_1> \cdots >v_m$ in $(-\infty,w^-]$, which intersects and is not boundary-filling on $(-\infty,v_m)$. The main result is that there is an atomless locally finite Borel measure $μ_η$ on $η\cap (-\infty,v_m]$ such that for any $v<v_m$, the $d$-dimensional Minkowski content of $η\cap [v,v_m]$ exists and equals $μ_η[v,v_m]$, where $d=\frac{(\sum ρ_j+4)(κ-4-2\sum ρ_j)}{2κ}$ is the Hausdorff dimension of $η\cap [v,v_m]$. In the case that all $ρ_j=0$, this measure agrees with the covariant measure derived in [Alberts-Sheffield, 2011] for chordal SLE$_κ$ up to a multiplicative constant. %Such measure, called Minkowski content measure, satisfies conformal covariance properties. We call such measure a Minkowski content measure, extend it to a class of subsets of ${\mathbb{R}}^n$, and prove that they satisfy conformal covariance. To construct the Minkowski content measure on $η\cap [v,v_m]$, we follow the standard approach to derive the existence and estimates of the one- and two-point boundary Green's functions of $η$ on $(-\infty,v_m)$, which are the limits of the rescaled probability that $η$ passes through small discs or open real intervals centered at points on $(-\infty,v_m)$.

math.PR↗
arXiv2021

Time-reversal of multiple-force-point SLE$_κ(\underlineρ)$ with all force points lying on the same side

We define intermediate SLE$_κ(\underlineρ)$ and reversed intermediate SLE$_κ(\underlineρ)$ processes using Appell-Lauricella multiple hypergeometric functions, and use them to describe the time-reversal of multiple-force-point chordal SLE$_κ(\underlineρ)$ curves in the case that all force points are on the boundary and lie on the same side of the initial point, and $κ$ and $\underlineρ=(ρ_1,\dots,ρ_m)$ satisfy that either $κ\in(0,4]$ and $\sum_{j=1}^k ρ_j>-2$ for all $1\le k\le m$, or $κ\in(4,8)$ and $\sum_{j=1}^k ρ_j\ge \fracκ{2}-2$ for all $1\le k\le m$.

math.PR↗
arXiv2020

Two-curve Green's function for $2$-SLE: the boundary case

We prove that for $κ\in(0,8)$, if $(η_1,η_2)$ is a $2$-SLE$_κ$ pair in a simply connected domain $D$ with an analytic boundary point $z_0$, then $\lim_{r\to 0^+}r^{-α} \mathbb{P}[\mbox{dist}(z_0,η_j) 0$, which is called the two-curve Green's function. The exponent $α$ equals $\frac{12}κ-1$ or $2(\frac{12}κ-1)$ depending on whether $z_0$ is one of the endpoints of $η_1$ and $η_2$. We also find the convergence rate and the exact formula of the Green's function up to a multiplicative constant. To derive these results, we construct two-dimensional diffusion processes and use orthogonal polynomials to obtain their transition density.

math.PR↗
arXiv2020

Two-curve Green's function for $2$-SLE: the interior case

A $2$-SLE$_κ$ ($κ\in(0,8)$) is a pair of random curves $(η_1,η_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_κ$ curve in a complement domain. In this paper we prove that for any $z_0\in D$, the limit $\lim_{r\to 0^+}r^{-α_0} \mathbb{P}[\mbox{dist}(z_0,η_j)<r,j=1,2]$, where $α_0=\frac{(12-κ)(κ+4)}{8κ}$, exists. Such limit is called a two-curve Green's function. We find the convergence rate and the exact formula of the Green's function in terms of a hypergeometric function up to a multiplicative constant. For $κ\in(4,8)$, we also prove the convergence of $\lim_{r\to 0^+}r^{-α_0} \mathbb{P}[\mbox{dist}(z_0,η_1\cap η_2)<r]$, whose limit is a constant times the previous Green's function. To derive these results, we work on two-time-parameter stochastic processes, and use orthogonal polynomials to derive the transition density of a two-dimensional diffusion process that satisfies some system of SDE.

math.PR↗
arXiv2018

Optimal Hölder Continuity and Dimension Properties for SLE with Minkowski Content Parametrization

We make use of the fact that a two-sided whole-plane Schramm-Loewner evolution (SLE$_κ$) curve $γ$ for $κ\in(0,8)$ from $\infty$ to $\infty$ through $0$ may be parametrized by its $d$-dimensional Minkowski content, where $d=1+\fracκ8$, and become a self-similar process of index $\frac 1d$ with stationary increments. We prove that such $γ$ is locally $α$-Hölder continuous for any $α<\frac 1d$. In the case $κ\in(0,4]$, we show that $γ$ is not locally $\frac 1d$-Hölder continuous. We also prove that, for any deterministic closed set $A\subset \mathbb{R}$, the Hausdorff dimension of $γ(A)$ almost surely equals $d$ times the Hausdorff dimension of $A$.

math.PR↗
arXiv2017

Multipoint Estimates for Radial and Whole-plane SLE

We prove upper bounds for the probability that a radial SLE$_κ$ curve, $κ\in(0,8)$, comes within specified radii of $n$ different points in the unit disc. Using this estimate, we then prove a similar upper bound for a whole-plane SLE$_κ$ curve. We then use these estimates to show that the lower Minkowski content of both the radial and whole-plane SLE$_κ$ traces restricted in a bounded region have finite moments of any order.

math.PR↗
arXiv2017

Decomposition of backward SLE in the capacity parameterization

We prove that, for $κ\le 4$, backward chordal SLE$_κ$ admits backward chordal SLE$_κ(-4,-4)$ decomposition for the capacity parametrization. This means that, for any bounded measurable subset $U\subset Q_4:={\mathbb R}_+\times{\mathbb R}_-$, if we integrate the laws of extended backward chordal SLE$_κ(-4,-4)$ with different pairs of force points $(x,y)$ against some suitable density function $G(x,y)$ restricted to $U$, then we get a measure, which is absolutely continuous with respect to the law of backward chordal SLE$_κ$, and the Radon-Nikodym derivative is a constant depending on $κ$ times the capacity time that the generated welding curve $t\mapsto (d_t,c_t)$ spends in $U$, where $d_t>0>c_t$ are the pair of points that are swallowed by the process at time $t$. For the forward SLE curve, a similar analysis has been done for SLE in the natural parametrization ([1] $κ\leq 4$, [10] $κ<8$), and for the capacity parametrization ([10] $κ< \infty$).

math.PR↗
arXiv2017

SLE Loop Measures

We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_κ$ loop measures for $κ\in(0,8)$. First, we construct rooted SLE$_κ$ loop measures in the Riemann sphere $\widehat{\mathbb C}$, which satisfy Möbius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its $(1+\frac κ8)$-dimensional Minkowski content. Second, by integrating rooted SLE$_κ$ loop measures, we construct the unrooted SLE$_κ$ loop measure in $\widehat{\mathbb C}$, which satisfies Möbius invariance and reversibility. Third, we extend the SLE$_κ$ loop measures from $\widehat{\mathbb C}$ to subdomains of $\widehat{\mathbb C}$ and to two types of Riemann surfaces using Brownian loop measures, and obtain conformal invariance or covariance of these measures. Finally, using a similar approach, we construct SLE$_κ$ bubble measures in simply/multiply connected domains rooted at a boundary point. The SLE$_κ$ loop measures for $κ\in(0,4]$ give examples of Malliavin-Kontsevich-Suhov loop measures for all $c\le 1$. The space-time homogeneity of rooted SLE$_κ$ loop measures in $\widehat{\mathbb C}$ answers a question raised by Greg Lawler.

math.PR↗
arXiv2017

Green's function for chordal SLE curves

For a chordal SLE$_κ$ ($κ\in(0,8)$) curve in a domain $D$, the $n$-point Green's function valued at distinct points $z_1,\dots,z_n\in D$ is defined to be $$G(z_1,\dots,z_n)=\lim_{r_1,\dots,r_n\downarrow 0} \prod_{k=1}^n r_k^{d-2} \mathbb{P}[\mbox{dist}(γ,z_k)<r_k,1\le k\le n],$$ where $d=1+\fracκ{8}$ is the Hausdorff dimension of SLE$_κ$, provided that the limit converges. In this paper, we will show that such Green's functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green's functions as well. Finally, we give up-to-constant bounds for them.

math.PR↗
arXiv2017

On the reversal of radial SLE, I: Commutation Relations in Annuli

We aim at finding the reversal of radial SLE and proving the reversibility of whole-plane SLE. For this purpose, we define annulus SLE$(κ,Λ)$ processes in doubly connected domains with one marked boundary point. We derive some partial differential equation for $Λ$, which is sufficient for the annulus SLE$(κ,Λ)$ process to satisfy commutation relation. If $Λ$ satisfies this PDE, then using a coupling technique, we are able to construct a global commutation coupling of two annulus SLE$(κ,Λ)$ processes. If more conditions are satisfied, the coupling exists in the degenerate case, which becomes a coupling of two whole-plane SLE$_κ$ processes. The reversibility of whole-plane SLE$_κ$ follows from this coupling together with the assumption that such annulus SLE$(κ,Λ)$ trace ends at the marked point. We then conclude that the limit of such annulus SLE$(κ,Λ)$ trace is the reversal of radial SLE$_κ$ trace. In the end, we derive some particular solutions to the PDE for $Λ$.

math.PR↗
arXiv2017

Higher moments of the natural parameterization for SLE curves

In this paper, we will show that the higher moments of the natural parametrization of SLE curves in any bounded domain in the upper half plane is finite. We prove this by estimating the probability that an SLE curve gets near n given points.

math.PR↗
arXiv2017

Decomposition of Schramm-Loewner evolution along its curve

We show that, for $κ\in(0,8)$, the integral of the laws of two-sided radial SLE$_κ$ curves through different interior points against a measure with SLE$_κ$ Green function density is the law of a chordal SLE$_κ$ curve, biased by the path's natural length. We also show that, for $κ>0$, the integral of the laws of extended SLE$_κ(-8)$ curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE$_κ$ curve, biased by the path's capacity length restricted in that set. Another result is that, for $κ\in(4,8)$, if one integrates the laws of two-sided chordal SLE$_κ$ curves through different force points on $\mathbb R$ against a measure with density on $\mathbb R$, then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLE$_κ$ curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov's Theorem.

math.PR↗
arXiv2016

Boundary Arm Exponents for SLE$(κ)$

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.

math.PR↗