AI 中文总结
研究证明存在正常数\(A\)和\(V\)，给出高维临界渗流\(k\)点函数表达式，验证了Aizenman和Newman的猜想。还证明原点簇定律重标度下收敛到积分超布朗游程规范测度等，解决了一些猜想，回答了相关问题。
AI 中文摘要
我们证明存在正常数A和V，使得当\(\min_{i\neq j}\|x_i - x_j\|\to \infty\)时，高维临界渗流k点函数由\[ T_{p_c}(x_1,x_2,\ldots,x_{k}) \sim V^{k - 2}A^{2k - 3} \sum_{T\in \mathsf{Tr}(k)} \sum_{\substack{\Phi:V(T)\to \mathbb{Z}^d \\ \Phi(i)=x_i \forall 1\leq i \leq k}} \prod_{\substack{u,v\in V(T)\u\sim v}}G(\Phi(u),\Phi(v)) \]给出，其中\(\mathsf{Tr}(k)\)是具有k个标记叶\(\{1,\ldots,k\}\)且所有未标记内顶点度数为3的树的同构类代表集，G是格点格林函数。这验证了Aizenman和Newman（1984）的一个猜想。由此定理可知，原点簇的定律在重标度下收敛到积分超布朗游程的规范测度。通过计算k点函数各种更复杂变体的渐近性，我们还证明了更强的结果，即簇作为嵌入的度量测度空间收敛到配备布朗嵌入到\(\mathbb{R}^d\)的连续随机树。这种收敛对于簇上的化学距离、枢轴距离和电阻距离同时成立，我们证明它们彼此渐近于常数倍数。这解决了Hara和Slade（1998）以及van der Hofstad和Slade（2003）的猜想。作为我们结果的推论，我们证明存在正常数C，使得\(\mathbb{P}_{p_c}(0\leftrightarrow \mathbb{Z}^d \setminus [-r,r]^d)\sim C r^{-2}\)，回答了Heydenreich和van der Hofstad（2017）的一个问题。
英文摘要
We prove that there exist positive constants $A$ and $V$ such that the high-dimensional critical percolation $k$-point function is given by \[ T_{p_c}(x_1,x_2,\ldots,x_{k}) \sim V^{k-2} A^{2k-3} \sum_{T\in \mathsf{Tr}(k)} \sum_{\substack{Φ:V(T)\to \mathbb{Z}^d \\ Φ(i)=x_i \forall 1\leq i \leq k}} \prod_{\substack{u,v\in V(T)\\u\sim v}}G(Φ(u),Φ(v)) \] as $\min_{i\neq j}\|x_i-x_j\|\to \infty$, where $\mathsf{Tr}(k)$ is a set of isomorphism class representatives of trees with $k$ labelled leaves $\{1,\ldots,k\}$ and unlabelled internal vertices all of which have degree $3$ and $G$ is the lattice Green's function. This verifies a conjecture of Aizenman and Newman (1984) subject to the usual perturbative conditions needed for convergence of the lace expansion. It follows from this theorem that the law of the cluster of the origin, considered as the counting measure on its range, converges under rescaling to the canonical measure of the integrated super-Brownian excursion. By computing the asymptotics of various more complicated variations on the $k$-point function, we also prove the stronger result that the cluster converges as an embedded metric-measure space to the continuum random tree equipped with its Brownian embedding into $\mathbb{R}^d$. This convergence holds simultaneously with respect to the chemical distance, pivotal distance, and resistance distance on the cluster, which we prove are asymptotic to constant multiples of each other. This resolves conjectures of Hara and Slade (1998) and van der Hofstad and Slade (2003). As a corollary of our results we prove that there exists a positive constant $C$ such that $\mathbb{P}_{p_c}(0\leftrightarrow \mathbb{Z}^d \setminus [-r,r]^d)\sim C r^{-2}$, answering a question of Heydenreich and van der Hofstad (2017).
Comments86 pages. Abstract shortened to meet arXiv requirements
