AI 中文总结
研究高维临界伯努利键渗流，通过证明重标经验测度收敛及临界簇质量界，得出重标簇在豪斯多夫度量下收敛，进而获得尖锐单臂渐近性，改进了前人结果。
AI 中文摘要
对于高维情形下\(\mathbb{Z}^d\)上的临界伯努利键渗流，我们证明了原点处簇的重标经验测度在合适的\(\sigma -\)有限意义下收敛到超布朗运动的总占据测度。结合我们所证明的关于临界簇相对于外在（欧几里得）度量的一致低质量界（这本身也有独立意义），测度收敛进一步得出重标簇作为紧集在豪斯多夫度量下的收敛。结果，我们能够得到尖锐单臂渐近性\(r^2\,\mathbb{P}(0\leftrightarrow \partial B_r)\to \theta_1\in(0,\infty)\)，从而改进了Kozma和Nachmias的结果。
英文摘要
For critical Bernoulli bond percolation on $\Z^d$ in the high-dimensional regime, we prove that the rescaled empirical measure of the cluster of the origin converges, in a suitable $σ$-finite sense, to the total occupation measure of super-Brownian motion. Combined with a uniform lower mass bound for the critical cluster with respect to the extrinsic (Euclidean) metric, which we also prove and which is of independent interest, the measure convergence further yields the convergence of the rescaled cluster as a compact set, in the Hausdorff metric. As a consequence, we are able to obtain the sharp one-arm asymptotics $r^2\,\bP(0\leftrightarrow \partial B_r)\to θ_1\in(0,\infty)$, hence refining a result of Kozma and Nachmias.
Comments62 pages, 6 figures
