论文标题
径向化学距离为$ 2D $的估算值
An estimate for the radial chemical distance in $2d$ critical percolation clusters
论文作者
论文摘要
我们得出了以晶格间距测量的距离的估计值，在二维临界渗透簇内部，从侧面长度$ 2N $的边界的边界，以开放连接的存在为条件。我们获得的估计是在达姆隆，汉森和索西工作中发现的径向类似物。但是，在当前情况下，盒子中没有最低的交叉点可以进行比较，因此我们构建了一个从原点到距离$ n $的路径$γ$，该路径由“三臂”点组成，因此可以通过$ O（n^2π_3（n））估算其音量。在这里，$π_3（n）$是该来源连接到距离$ n $的“三臂概率”，三臂，两个打开和一个双关闭。然后，我们对框中的边缘$ e $存在快捷方式进行估计，以$ \ {e \ inγ\} $为条件，以获取$ o（n^{2-Δ}π_3（n））$的键，以$ u（n^{2-δ}π_3（n））$。
We derive an estimate for the distance, measured in lattice spacings, inside two-dimensional critical percolation clusters from the origin to the boundary of the box of side length $2n$, conditioned on the existence of an open connection. The estimate we obtain is the radial analogue of the one found in the work of Damron, Hanson, and Sosoe. In the present case, however, there is no lowest crossing in the box to compare to, so we construct a path $γ$ from the origin to distance $n$ that consists of "three-arm" points, and whose volume can thus be estimated by $O(n^2π_3(n))$. Here, $π_3(n)$ is the "three-arm probability" that the origin is connected to distance $n$ by three arms, two open and one dual-closed. We then develop estimates for the existence of shortcuts around an edge $e$ in the box, conditional on $\{e\in γ\}$, to obtain a bound of the form $O(n^{2-δ}π_3(n))$ for some $δ>0$.
