论文标题
经验措施，测地长度和第一组渗透中的变异公式
Empirical Measures, Geodesic Lengths, and a Variational Formula in First-Passage Percolation
论文作者
论文摘要
该专着的决心 - 在密集的案例中 - 关于I.I.D. $ \ mathbb {z}^d $上的第一笔percolation。我们的主要兴趣是从$ 0 $到$nξ$沿着测量学观察到的边缘权重的经验度量，其中$ξ$是固定的单位向量。对于各种稠密的边缘分布的家族，我们证明这些经验措施薄弱地汇总到确定性的限制为$ n \ to \ infty $，回答了霍夫曼的问题。这些家族包括任何给定分布的任意小$ l^\ ind $ bertertations，几乎所有有限支持的分布，无数的连续分布集合以及某些原子可以具有任何规定的概率序列的离散分布。此外，这些构造足以保证具有某些功能的示例，例如：连续和离散的分布，其支持的全部为$ [0，\ infty）$，以及由$ k $ times-times差异的密度函数给出的分布。所有结果也适用于$ξ$导向的无限大地测量学。相比之下，我们表明，如果$ \ mathbb {z}^d $被无限的$ d $ - ary树代替，那么权重的任何分布都可以沿着测量学沿着地球学的独特限制经验度量。在晶格和树案例中，我们的方法论是由时间常数的新变异公式驱动的，这不需要对边缘重量分布的假设。顺便说一句，这种变异方法还使我们能够为地球长度获得新的收敛结果，这是自1965年Hammersley和Welsh的开创性手稿以来的亚临界制度中未经改善的。
This monograph resolves - in a dense class of cases - several open problems concerning geodesics in i.i.d. first-passage percolation on $\mathbb{Z}^d$. Our primary interest is in the empirical measures of edge-weights observed along geodesics from $0$ to $nξ$, where $ξ$ is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these empirical measures converge weakly to a deterministic limit as $n\to\infty$, answering a question of Hoffman. These families include arbitrarily small $L^\infty$-perturbations of any given distribution, almost every finitely supported distribution, uncountable collections of continuous distributions, and certain discrete distributions whose atoms can have any prescribed sequence of probabilities. Moreover, the constructions are explicit enough to guarantee examples possessing certain features, for instance: both continuous and discrete distributions whose support is all of $[0,\infty)$, and distributions given by a density function that is $k$-times differentiable. All results also hold for $ξ$-directed infinite geodesics. In comparison, we show that if $\mathbb{Z}^d$ is replaced by the infinite $d$-ary tree, then any distribution for the weights admits a unique limiting empirical measure along geodesics. In both the lattice and tree cases, our methodology is driven by a new variational formula for the time constant, which requires no assumptions on the edge-weight distribution. Incidentally, this variational approach also allows us to obtain new convergence results for geodesic lengths, which have been unimproved in the subcritical regime since the seminal 1965 manuscript of Hammersley and Welsh.
