论文标题
二维第一通道渗透中横向增量的波动
Fluctuations of Transverse Increments in Two-dimensional First Passage Percolation
论文作者
论文摘要
我们考虑了第一通道渗透模型（FPP），其中标准二维欧几里得晶格的最近邻居边缘配备了随机变量。这些变量是i.i.d. \，非负，连续的，并且在$ 0 $的附近具有有限的力矩生成功能。假设该模型满足某些属性，我们会导致有关通道时间横向增量的后果。大约是假定的属性如下：我们假设规模上的通道时间的标准偏差为某些订单$σ（r）$，而$ \ weft \ {σ（r），r> 0 \ right \ \} $将大约作为$ r $的功率增长。此外，距离$ r $的通道时间分布的尾巴满足了在$ r $上均匀的规模$σ（r）$的指数限制。此外，在某个固定方向$θ$的附近，极限形状的边界具有均匀的二次曲率。通过横向增量，我们指的是从原点到一对点的通行时间差，如下所示：它们大约朝着相同的方向（例如$θ$），与原点相同；其中一个从另一个方向是极限形状边界的切线，在极限形状在$θ$的方向上。得出的主要结果是以下内容。如果$σ（r）$变化为$ r^χ$，对于某些$χ> 0 $，而$χ= $χ=2ξ-1$，那么在距离$ r $ r $之间的横向通道时间的横向增量的波动是$ r^{χ/ξ} $。
We consider a model of first passage percolation (FPP) where the nearest-neighbor edges of the standard two-dimensional Euclidean lattice are equipped with random variables. These variables are i.i.d.\, nonnegative, continuous, and have a finite moment generating function in a neighborhood of $0$. We derive consequences about transverse increments of passage times, assuming the model satisfies certain properties. Approximately, the assumed properties are the following: We assume that the standard deviation of the passage time on scale $r$ is of some order $σ(r)$, and $\left\{σ(r), r > 0\right\}$ grows approximately as a power of $r$. Also, the tails of the passage time distributions for distance $r$ satisfy an exponential bound on a scale $σ(r)$ uniformly over $r$. In addition, the boundary of the limit shape in a neighborhood of some fixed direction $θ$ has a uniform quadratic curvature. By transverse increment we mean the difference of passage times from the origin to a pair of points which are located as follows: they are approximately in the same direction, say $θ$, from the origin; the direction of one of them from the other is the direction of the tangent of the boundary of the limit shape at the point on the limit shape in the direction $θ$. The main consequence derived is the following. If $σ(r)$ varies as $r^χ$ for some $χ>0$, and $ξ$ is such that $χ=2ξ-1$, then the fluctuation of the transverse increment of passage time between a pair of points situated at distance $r$ from each other is of the order of $r^{χ/ξ}$.
