论文标题
加权随机图中最后一段渗透常数的概率和分析特性
Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph
论文作者
论文摘要
到每个边缘（i，j），整数上完整的定向图的i <j我们分配了单位重量，概率为p或权重x，概率为1-p，从边缘到边缘独立于边缘，并给出每个路径重量等于其边缘重量的总和。如果w^x_ {0，n}是从0到n的所有路径的最大重量，则w^x_ {0，n}/n \至c_p（x），如n \至\ infty，几乎可以肯定，其中c_p（x）是正的且确定性的。我们将c_p（x）作为x的函数研究，对于固定的0 <p <1，并表明它是一个严格增加的凸函数，并且仅当x是非阳性有理或一个正整数时，除了1或倒数外，它是无可分析的。我们允许x是任何实际数字，甚至是负数，或者可能是 - \ infty。情况x = - \ infty对应于erd“ OS-r'enyi随机图（称为barak-erd” OS图）的有向有向版本，该图（称为barak-erd'OS图），该c_p（ - \ infty）= lim_ {x \ to- \ to- \ to- \ to- \ to- \ to- \ infty} c_p（x）在Paper的数字中作为P的函数研究了P。
To each edge (i,j), i<j of the complete directed graph on the integers we assign unit weight with probability p or weight x with probability 1-p, independently from edge to edge, and give to each path weight equal to the sum of its edge weights. If W^x_{0,n} is the maximum weight of all paths from 0 to n then W^x_{0,n}/n \to C_p(x), as n\to\infty, almost surely, where C_p(x) is positive and deterministic. We study C_p(x) as a function of x, for fixed 0<p<1 and show that it is a strictly increasing convex function that is not differentiable if and only if x is a nonpositive rational or a positive integer except 1 or the reciprocal of it. We allow x to be any real number, even negative, or, possibly, -\infty. The case x=-\infty corresponds to the well-studied directed version of the Erd"os-R'enyi random graph (known as Barak-Erd"os graph) for which C_p(-\infty) = lim_{x\to -\infty} C_p(x) has been studied as a function of p in a number of papers.
