论文标题
正方形，管和圆环上的第一组渗透的波动范围
Fluctuation bounds for first-passage percolation on the square, tube, and torus
论文作者
论文摘要
在第一阶段的渗透中，一个分配了I.I.D.非负权重$（t_e）$ to $ \ mathbb {z}^d $的边缘，并研究诱导距离（通过时间）$ t（x，y）$之间的$ x $ x $和$ y $。众所周知，对于$ d = 2 $，$ t（x，y）$的波动至少是$ \ sqrt {\ log | x-y |} $，在$ t_e $的温和假设下。我们研究了$ t_n $的波动下限的问题，这是$ n $的两个相对方面之间的最小通道时间（$ n $ square）。主要的结果是，在曲率假设下，此数量至少在任何$ε> 0 $的$ n^{1/8-ε} $中的波动，当$ t_e $指数分配时。由于以前的争论限制了$ t（x，y）$的波动仅给出$ t_n $（甚至假设曲率）的恒定下限，这是一个不同的论点，代表$ t_n $作为圆柱传导时间的最小限制，并得出了有关使用Markov属性的圆柱分布的更详细信息，并开发了更详细的信息。作为推论，在相同的曲率假设下，我们在离散圆环时间的较高中心矩上获得了第一个多项式下限。
In first-passage percolation, one assigns i.i.d. nonnegative weights $(t_e)$ to the edges of $\mathbb{Z}^d$ and studies the induced distance (passage time) $T(x,y)$ between vertices $x$ and $y$. It is known that for $d=2$, the fluctuations of $T(x,y)$ are at least order $\sqrt{\log |x-y|}$ under mild assumptions on $t_e$. We study the question of fluctuation lower bounds for $T_n$, the minimal passage time between two opposite sides of an $n$ by $n$ square. The main result is that, under a curvature assumption, this quantity has fluctuations at least of order $n^{1/8-ε}$ for any $ε>0$ when the $t_e$ are exponentially distributed. As previous arguments to bound the fluctuations of $T(x,y)$ only give a constant lower bound for those of $T_n$ (even assuming curvature), a different argument, representing $T_n$ as a minimum of cylinder passage times, and deriving more detailed information about the distribution of cylinder times using the Markov property, is developed. As a corollary, we obtain the first polynomial lower bounds on higher central moments of the discrete torus passage time, under the same curvature assumption.
