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We study the accessibility percolation model on infinite trees. The model isdefined by associating an absolute continuous random variable $X_v$ to eachvertex $v$ of the tree. The main question to be considered is the existence ornot 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 eventdefined by the existence of such path is called {\it{percolation}}.  We consider the case of the accessibility percolation model on a sphericallysymmetric 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 apercolation 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 thecontinuity of the percolation probability function. Finally, we provide acomparison between this model with the well known $F^{\alpha}$ record model. Wealso discuss a number of open problems concerning the accessibility percolationmodel for further consideration in future research.

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