Accessiblility Percolation with Crossing Valleys on $n$-ary Trees

TL;DR

This paper investigates a variation of accessibility percolation on $n$-ary trees, analyzing the probability of monotone subsequences with crossing valleys, and establishes thresholds based on the growth rate of $n$ relative to tree height.

Contribution

The paper introduces a new model of accessibility percolation with crossing valleys on $n$-ary trees and derives asymptotic probability thresholds for the existence of certain monotone subsequences.

Findings

Probability tends to 1 if $n$ grows faster than $ oot{k}{h/(ek)}$

Probability tends to 0 if $n$ grows slower than $ oot{k}{h/(ek)}$

Threshold behavior depends on the growth rate of $n$ relative to tree height h

Abstract

In this paper we study a variation of the accessibility percolation model, this is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leaf, where any $k$ consecutive vertices in the path contain at least one vertex of the subsequence. An $n$-ary tree, with height $h$, is a tree whose vertices at distance at most $h-1$ to the root have $n$ children. For the case of $n$-ary trees, we prove that, as $h$ tends to infinity the probability of having such subsequence: tends to 1, if $n$ grows significantly faster than $\sqrt[k]{h/(ek)}$; and tends to 0, if $n$ grows significantly slower than $\sqrt[k]{h/(ek)}$.