Despite great progress in the study of critical percolation on $\mathbb{Z}^d$ for $d$ large, properties of critical clusters in high-dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high-dimensional critical exponents; see in particular the conjecture by Kozma-Nachmias that the probability that $0$ and $(n, n, n, \ldots)$ are connected within $[-n,n]^d$ scales as $n^{-2-2d}$. In this paper, we study the properties of critical clusters in high-dimensional half-spaces and boxes. In half-spaces, we show that the probability of an open connection ("arm") from $0$ to the boundary of a sidelength $n$ box scales as $n^{-3}$. We also find the scaling of the half-space two-point function (the probability of an open connection between two vertices) and the tail of the cluster size distribution. In boxes, we obtain the scaling of the two-point function between vertices which are any macroscopic distance away from the boundary.

中文翻译：

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高维受限渗透临界指数

尽管在 $\mathbb{Z}^d$ 上对 $d$ 大的临界渗流的研究取得了很大进展，但与二维情况不同，高维分数空间和盒子中临界簇的性质仍然知之甚少。密切相关的模型，如临界分支随机游走，对相关高维临界指数的值给出了自然的猜想；特别参见 Kozma-Nachmias 的猜想，即 $0$ 和 $(n, n, n, \ldots)$ 在 $[-n,n]^d$ 范围内连接的概率为 $n^{-2- 2d}$。在本文中，我们研究了高维半空间和盒子中临界簇的性质。在半空间中，我们展示了从 $0$ 到边长 $n$ 框边界的开放连接（“arm”）的概率为 $n^{-3}$。我们还发现了半空间两点函数的缩放（两个顶点之间开放连接的概率）和集群大小分布的尾部。在盒子中，我们获得了与边界相距任何宏观距离的顶点之间的两点函数的缩放比例。