Accessibility percolation on N-ary trees

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\rightarrow \infty , we obtain the limit distribution of C_N,\, \beta N as \beta varies from 0 to +\infty and the joint limiting distribution of (C_N, C_N,\,\alpha N+t \sqrt\alpha N) for 0 < \alpha\leqslant 1 and t\in \mathbbR . 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 .