论文标题
沿着大地测量的经验分布的估计值
Estimates for the empirical distribution along a geodesic in first-passage percolation
论文作者
论文摘要
在第一阶段的渗透中，我们分配了i.i.d.〜非负权重$（t_e）$ $ \ \ mathbb {z}^d $的最近的邻居边缘，并研究了诱导的伪计$ t = t = t（x，y）$。在本文中，我们专注于测量学或$ t $的最佳路径，并估算其沿$ T $的经验分布。我们证明，所有地理器的结合的重量$ \ geq m $的预期边缘数量从$ 0 $到$ q（m）\ mathbb {p}（t_e \ geq m）| x | $，其中$ q（m）$ q（m）\ leq e^{ - cm cm cmm} $。这表明，沿着测量的预期经验分布的尾巴比指数因子的原始重量分布的尾巴更轻。我们还为任何地理上的重量$ \ geq m $的预期最小边缘提供了下限例如，这两个意味着，如果$ t_e $具有$ \ mathbb {p}（t_e \ geq m）\ sim m^{ - α} $的功率定律的尾巴，则预期经验分布的尾巴在$ e^{ - cm cm \ log m} $和$ e^$和$ e^{ - cmm}之间。我们还提供了估计的估算值，重量为（a）任意$ a $，（b）$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ t_e $和（c）$ a = [0，a] $ for somer $ a \ egeq 0 $的支撑的估计值$ a $ a $ a $ a $ a $ a $ a $。
In first-passage percolation, we assign i.i.d.~nonnegative weights $(t_e)$ to the nearest-neighbor edges of $\mathbb{Z}^d$ and study the induced pseudometric $T = T(x,y)$. In this paper, we focus on geodesics, or optimal paths for $T$, and estimate the empirical distribution of weights along them. We prove an upper bound for the expected number of edges with weight $\geq M$ in the union of all geodesics from $0$ to $x$ of the form $q(M) \mathbb{P}(t_e \geq M)|x|$, where $q(M) \leq e^{-cM}$. This shows that the tail of the expected empirical distribution along a geodesic is lighter than that of the original weight distribution by an exponential factor. We also give a lower bound for the expected minimal number of edges with weight $\geq M$ in any geodesic from $0$ to $x$ in terms of $\mathbb{P}(t_e \geq M)$ and $\mathbb{P}(t_e \in [M,2M])$. For example, these two imply that if $t_e$ has a power law tail of the form $\mathbb{P}(t_e \geq M) \sim M^{-α}$, then the tail of the expected empirical distribution asymptotically lies between $e^{-CM \log M}$ and $e^{-cM}$. We also provide estimates for the expected number of edges in a geodesic with weight in a set $A$ for (a) arbitrary $A$, (b) $A$ an interval separated from the infimum of the support of $t_e$ and (c) $A=[0,a]$ for some $a \geq 0$.
