Appears in collection : Non-local branching processes / Processus de branchement non local

Start with a rooted tree, and to each vertex attach a label consisting of an i.i.d. uniform(0,1) random variable plus a parameter theta times its distance from the root. Are there paths from the root of the tree to infinity along which the labels are increasing? Clearly this depends on the tree and the value of theta. The aim is to specify a critical theta, depending on the tree, above which the answer is yes with positive probability, and below which the answer is no. We also consider the same question on the integer lattice, where the bounds are worse but the pictures are better. This is a work in progress with Diana De Armas Bellon.