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 Xv 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 v1,v2,v3… such that Xv1<Xv2<Xv3<⋯ 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 αc=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 Fα record model. We also discuss a number of open problems concerning the accessibility percolation model for further consideration in future research.