On the existence of accessibility in a tree-indexed percolation model

We study the accessibility percolation model on infinite trees. The model is defined by associating an absolute continuous random variable $X_v$ to each vertex $v$ of the tree. The main question to be considered is the existence or not of an infinite path of nearest neighbors $v_1,v_2,v_3\ldots$ such that $X_{v_1}<X_{v_2}<X_{v_3}<\cdots$ and which spans the entire graph. The event defined by the existence of such path is called {\it{percolation}}. We consider the case of the accessibility percolation model on a spherically symmetric tree with growth function given by $f(i)=\lceil (i+1)^ \alpha \rceil$, where $\alpha>0$ is a given constant. We show that there is a percolation threshold at $\alpha_c =1$ such that there is percolation if $\alpha> 1$ and there is absence of percolation if $\alpha \leq 1$. Moreover, we study the event of percolation starting at any vertex, as well as the continuity of the percolation probability function. Finally, we provide a comparison between this model with the well known $F^{\alpha}$ record model. We also discuss a number of open problems concerning the accessibility percolation model for further consideration in future research.