论文标题
在第一次通行的连续模型中，时间常数的连续性
Continuity of the time constant in a continuous model of first passage percolation
论文作者
论文摘要
对于给定的尺寸d $ \ ge $ 2和有限的度量$ν$（0， +$ \ infty $），我们考虑$ξ$ r d x（0， +$ \ iffty $）上的Poisson点进程具有强度度量dc $ \ outimes $ \ otimes $ $ $ $ $ $ $ $ $ $ $ $ $ n dc表示dc表示dc the lebesgue对R d d d d d。我们考虑布尔型$σ$ = $ = $ \ cup $（c，r）$ \ in $$ξ$ b（c，r），其中b（c，r）表示以radius r为中心的开放球。对于每个x，y $ \ in $ r d，我们将t（x，y）定义为从x到y旅行所需的最短时间，该旅行者以$σ$的速度和$σ$内的无限速度行驶的旅行者以1速行走。通过Kingman次级定理的标准应用，人们很容易表明t（0，x）的行为像$ $ $ x时，当x进入无穷大时，$μ$是经典第一段段落中的时间常数。在本文中，我们调查了$μ$的规律性，这是与基础布尔模型相关的度量$ν$的函数。
For a given dimension d $\ge$ 2 and a finite measure $ν$ on (0, +$\infty$), we consider $ξ$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $ν$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $Σ$ = $\cup$ (c,r)$\in$$ξ$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $Σ$ and at infinite speed inside $Σ$. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like $μ$ x when x goes to infinity, where $μ$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of $μ$ as a function of the measure $ν$ associated with the underlying Boolean model.
