Liouville量子引力度量的Minkowski内容测度

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…

随机自相似码树分形的平均 Minkowski 含量与平均分形曲率

Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals

本文研究基于码树的随机自相似分形，在一致强开集条件下证明其平行集 Lipschitz-Killing 曲率平均值的重标度极限存在，并给出积分表示，推广了确定性与部分随机情形下的已知结果，同时涵盖平均 Minkowski 含量。

Natural parametrization of percolation interface and pivotal points

We prove that the interface of critical site percolation on the triangular lattice converges to SLE$_6$ in its natural parametrization, where the discrete interface is parametrized such that each edge is crossed in one unit of time, while the limiting curve is parametrized by the $7/4$-dimensional Minkowski content. We also prove that the scaling limit of counting measure on the pivotal points, which was proved to exist by Garban, Pete, and Schramm (2013), is the $3/4$-dimensional Minkowski content up to a deterministic multiplicative constant.

Heat content asymptotics of some random Koch type snowflakes

We consider the short time asymptotics of the heat content $E$ of a domain $D$ of $\mathbb{R}^d$. The novelty of this paper is that we consider the situation where $D$ is a domain whose boundary $\partial D$ is a random Koch type curve. When $\partial D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\partial D$ from the short time behaviour of $E(s)$. Furthermore, in some situations where the Minkowski dimension exists, finer geometric fluctuations can be recovered and the heat content is controlled by $s^αe^{f(\log(1/s))}$ for small $s$, fo…

SLE曲线量子长度的初等方法

An elementary approach to quantum length of SLE

本文通过初等方法证明了SLE_κ曲线的左右侧√κ-量子长度相等，利用了高斯乘法混沌与共形Minkowski内容的关系。

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, w…

随机游走割点的占据测度的标度极限

Scaling limit of the occupation measure of random walk cut points

本文研究二维和三维立方晶格上简单随机游走割点的占据测度，证明其弱拓扑下的标度极限为基于闵可夫斯基容度定义的布朗割点自然分形测度。

Random fields and the geometry of Wiener space

In this work we consider infinite dimensional extensions of some finite dimensional Gaussian geometric functionals called the Gaussian Minkowski functionals. These functionals appear as coefficients in the probability content of a tube around a convex set $D\subset\mathbb{R}^k$ under the standard Gaussian law $N(0,I_{k\times k})$. Using these infinite dimensional extensions, we consider geometric properties of some smooth random fields in the spirit of [Random Fields and Geometry (2007) Springer] that can be expressed in terms of reasonably smooth Wiener functionals.