Increasing paths on N-ary trees

TL;DR

This paper analyzes the asymptotic behavior of accessible paths in N-ary trees with i.i.d. variables, focusing on population growth and critical survival thresholds as N becomes large.

Contribution

It provides new asymptotic results on the growth and survival probability of increasing paths in N-ary trees with random variables.

Findings

Population behavior at the αN-th generation characterized for large N

Critical survival probability analyzed at the (eN - 1.5 log N)-th generation

Asymptotic descriptions of accessible vertices in large N-ary trees

Abstract

Consider a rooted $N$-ary tree. To every vertex of this tree, we attach an i.i.d. continuous random variable. A vertex is called accessible if along its ancestral line, the attached random variables are increasing. We keep accessible vertices and kill all the others. For any positive constant $\alpha$, we describe the asymptotic behaviors of the population at the $\alpha N$-th generation as $N$ goes to infinity. We also study the criticality of the survival probability at the $(eN-\frac{3}{2}\log N)$-th generation in this paper.