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) ... such that X-v1, < X-v2 < X-v3 < ... 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) = f(i + 1)(alpha), 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 <= 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 Ft record model. We also discuss a number of open problems concerning the accessibility percolation model for further consideration in future research. (C) 2017 Elsevier B.V. All rights reserved. (AU)