Geodesic length and shifted weights in first-passage percolation
Geodesic length and shifted weights in first-passage percolation
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DOI：
10.1090/cams/18
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发表时间：
2021-01
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通讯作者：
Arjun Krishnan;F. Rassoul-Agha;T. Seppalainen
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作者：
Arjun Krishnan;F. Rassoul-Agha;T. Seppalainen
我们通过有限长度路径上的相关优化问题来研究首道渗流。路径长度变量与权值的移位呈对偶关系。这将Hammersley和Welsh， Smythe和Wierman以及Kesten关于测地线标准化欧几里德长度收敛的旧观察纳入凸对偶框架，并导致了关于测地线长度和形状函数作为权移函数的规则性的新结果。对于离原点足够远的点，测地线长度和到端点的1 \ell ^1距离的比值均匀地有界于1。形状函数是重量位移的严格凹函数。在这个函数中，权分布的原子产生奇点，即不可微点。我们将Steele和Zhang在平面伯努利情况下的旧奇异结果推广到所有分布、方向和维度。当重量分布有两个或多个原子时，密集的位移集产生奇点。结果来自于凸对偶性、不同首道优化问题的形状定理和修正论证的组合。
We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the ℓ 1 \ell ^1 distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produces singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments.