Uniqueness of the critical probability for percolation in the two dimensional Sierpinski carpet lattice

$\u211d^3$中无约束Sierpinski地毯上的Dirichlet型

Dirichlet forms on unconstrained Sierpinski carpets in $\mathbb{R}^3$

该研究证明三维无约束Sierpinski地毯上存在带次高斯热核估计的强局部正则自相似Dirichlet型，且其Hausdorff维数恒大于行走维数。

无约束Sierpinski地毯上阻力形式的唯一性与收敛性

Uniqueness and convergence of resistance forms on unconstrained Sierpinski carpets

本文证明了无约束Sierpinski地毯上自相似$D_4$对称阻力形式的唯一性，并揭示了地毯序列在Hausdorff度量下收敛时，关联扩散过程依分布收敛与测地度量等度连续的等价条件。

Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs

Consider the free field on a fractal graph based on a high-dimensional Sierpinski carpet (e.g. the Menger sponge), that is, a centered Gaussian field whose covariance is the Green's function for simple random walk on the graph. Moreover assume that a "hard wall" is placed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on $\mathbb{Z}^d$, $d \geq 3$, to the …

Fractional Gaussian fields on the Sierpinski gasket and related fractals

We define and study a fractional Gaussian field $X$ with Hurst parameter $H$ on the Sierpiński gasket $K$ equipped with its Hausdorff measure $μ$. It appears as a solution, in a weak sense, of the equation $(-Δ)^s X =W$ where $W$ is a Gaussian white noise on $L_0^2(K,μ)$, $Δ$ the Laplacian on $K$ and $s= \frac{d_h+2H}{2d_w}$, where $d_h$ is the Hausdorff dimension of $K$ and $d_w$ its walk dimension. The construction of those fields is then extended to other fractals including the Sierpiński carpet.

Densely defined non-closable curl on carpet-like metric measure spaces

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have positive two-dimensional Lebesgue measure and carry nontrivial $2$-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on $1$-forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abst…

低维随机空间的拟对称几何

Quasisymmetric geometry of low-dimensional random spaces

该研究针对随机分形的拟对称单值化问题，证明布朗运动迹、SLE_κ变体非拟弧，CLE_κ的环外点集与标准谢尔宾斯基地毯同胚但不与圆形地毯拟对称等价，推进了低维随机空间拟对称几何领域研究。

简单共形环集地毯上共形协变测地度量的存在性和唯一性

Existence and uniqueness of the conformally covariant geodesic metric on simple conformal loop ensemble carpets

本文证明了在κ∈(8/3,4)范围内，简单共形环集地毯上的共形协变测地度量存在且唯一，并推测其描述了离散环模型在尺度极限下的化学距离度量。

Dimensions of random statistically self-affine Sierpinski sponges in $\mathbb R^k$

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \mathbb{R}^k$ ($k\ge 2$) obtained by using some percolation process in $[0,1]^k$. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on $K$. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing $\dim_H K$ as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on $K$, and show th…

分形型空间中的边界 Harnack 原理与 3G 原理

The Boundary Harnack Principle and the 3G Principle in Fractal-Type Spaces

本文证明了一大类 Dirichlet 空间中有界内一致域上 Green 函数的广义 3G 原理，结果适用于高维分形及无明确 Hausdorff 维数的广义分形型空间，并探讨了其在 Schrödinger 算子上的应用。