论文标题
关于不断增长的第一音阶渗透球的孔的数量和大小
On the number and size of holes in the growing ball of first-passage percolation
论文作者
论文摘要
首先使用i.i.d.在$ \ mathbb {z}^d $上定义的随机增长模型。边缘上的非负重$（τ_e）$。让$ t（x，y）$为重量引起的顶点$ x $和$ y $之间的距离，我们研究以原点为中心的半径$ t $的随机球，$ b（t）= \ {x \ in \ sathbb {z}^d：t（z}^d：t（0，x）众所周知，对于所有此类$τ_e$，$ b（t）$的顶点（体积）的数量至少为$ t^d $，在$τ_e$的轻度条件下，此卷的增长像确定性的常数时间$ t^d $。定义$ b（t）$的一个漏洞，成为补充$ b（t）^c $的有限组成部分，我们证明，如果$τ_e$不是确定性的，那么A.S.，对于所有$ t $，$ t $，$ b（t）$至少具有$ ct^{d-1} $的最大孔，并且至少是$ c \ c \ fog c \ fog c \ c。有条件地，在（未证实的）统一曲率假设上，我们证明，对于所有大$ t $，A.S.的孔数最多是$（\ log t）^c t^{d-1} $，对于$ d = 2 $，$ b（t）$中的$ d = 2 $的数量大于$（\ log log t）^c $。没有弯曲，我们表明没有孔的音量大于$ CT \ log t $。
First-passage percolation is a random growth model defined on $\mathbb{Z}^d$ using i.i.d. nonnegative weights $(τ_e)$ on the edges. Letting $T(x,y)$ be the distance between vertices $x$ and $y$ induced by the weights, we study the random ball of radius $t$ centered at the origin, $B(t) = \{x \in \mathbb{Z}^d : T(0,x) \leq t\}$. It is known that for all such $τ_e$, the number of vertices (volume) of $B(t)$ is at least order $t^d$, and under mild conditions on $τ_e$, this volume grows like a deterministic constant times $t^d$. Defining a hole in $B(t)$ to be a bounded component of the complement $B(t)^c$, we prove that if $τ_e$ is not deterministic, then a.s., for all large $t$, $B(t)$ has at least $ct^{d-1}$ many holes, and the maximal volume of any hole is at least $c\log t$. Conditionally on the (unproved) uniform curvature assumption, we prove that a.s., for all large $t$, the number of holes is at most $(\log t)^C t^{d-1}$, and for $d=2$, no hole in $B(t)$ has volume larger than $(\log t)^C$. Without curvature, we show that no hole has volume larger than $Ct \log t$.
