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

Abstract 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 … such that X v 1 X v 2 X v 3 ⋯ and which spans the entire graph. The event defined by the existence of such path is called percolation . We consider the case of the accessibility percolation model on a spherically symmetric tree with growth function given by f ( i ) = ⌈ ( i + 1 ) α ⌉ , where α > 0 is a given constant. We show that there is a percolation threshold at α c = 1 such that there is percolation if α > 1 and there is absence of percolation if α ≤ 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 α record model. We also discuss a number of open problems concerning the accessibility percolation model for further consideration in future research.