Short Talk: Diana de Armas Bellon

We consider a rooted tree where each vertex is labelled by an independent and identically distributed (i.i.d.) uniform(0,1) random variable, plus a parameter theta times its distance from the root. We study paths from the root to infinity along which the vertex labels are increasing. The existence of such increasing paths depends on both the structure of the tree and the value of theta. The goal is to determine the critical value of theta such that, above this value, increasing paths occur with positive probability, while below it, no such paths exist. Additionally, we extend this problem to consider the case of the integer lattice.