N叉树上的可达渗流

Accessibility percolation on N-ary trees

考虑一棵有根的N叉树,赋予此树的每个顶点一个独立同分布的连续随机变量。如果从根点到某个顶点路径上的随机变量是递增的,我们就称这个顶点是可到达的。考虑在N叉树中前k层可到达顶点的数目C_(N,k)和可到达顶点的总数目C_(N),当N→∞时,得到了C_(N,βN)不同的极限分布(随着β从0到+∞变化),建立了C_(N)的极限性质。当N固定时,也考虑了N叉树中前n层最长递增路径长度的大数定律。

Consider a rooted N-ary tree.To each of its vertices,we assign an independent and identically distributed continuous random variable.A vertex is called accessible if the assigned random variables along the path from the root to it are increasing.We study the number C_(N,k) of accessible vertices of the first k levels and the number C_(N) of accessible vertices in the N-ary tree.As N→∞,we obtain the limit distribution of C_(N,βN) asβvaries from 0 to+∞and the joint limiting distribution of(C_(N),C_(N,αN+t√(αN)))for 0<α≤1 and t∈R.In this work,we also obtain a weak law of large numbers for the longest increasing path in the first n levels of the N-ary tree for fixed N.