Accessibility Percolation On Random Rooted Labeled Trees

The accessibility percolation models were inspired by some recent works in evolu-tionary biology.The numbers of the increasing paths in different deterministic graphs have been studied by several authors.The accessibility percolation model is investi-gated on random rooted labeled trees.We summarize the achievements in different deterministic graph and caculate the probability generating function of branching Pois-son tree.Since the random rooted labeled tree Tn on n vertices converges weakly to the PGW?(1)tree,as n??,we transfer the accessibility percolation problem on random rooted labeled trees to the problem on the PGW?(1)tree.It is shown that in a random rooted labeled tree of size n as n??,both the asymptotic distributions of the num-ber of increasing paths Zn and the number of accessible vertices Cn are the geometric distributions with parameter e/(1+e)and 1/e,respectively.