The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees. In this paper, we continue the investigation of the link between the branchingruin number and the criticality of random processes on trees. First, we study random walks on random conductances on trees, when the conductances have an heavy tail at 0, parametrized by some p > 1, where 1/p is the exponent of the tail. We prove a phase transition recurrence/transience with respect to p and identify the critical parameter to be equal to the branching-ruin number of the tree. Second, we study a multi-excited random walk on trees where each vertex has M cookies and each cookie has an infinite strength towards the root. Here again, we prove a phase transition recurrence/transience and identify the critical number of cookies to be equal to the branching-ruin number of the tree, minus 1. This result extends a conjecture of Volkov (2003). Besides, we study a generalized version of this process and generalize results of Basdevant and Singh (2009).
We consider a modified version of the biased random walk on a tree constructed from the set of finite self-avoiding walks on the hexagonal lattice, and use it to construct probability measures on infinite self-avoiding walk. Under theses probability measures, we prove that the infinite self-avoiding walks have the Russo-Seymour-Welsh property of the exploration curve of the critical Bernoulli percolation.
We study the scaling limit of essentially simple triangulations on the torus. We consider, for every n ≥ 1, a uniformly random triangulation G n over the set of (appropriately rooted) essentially simple triangulations on the torus with n vertices. We view G n as a metric space by endowing its set of vertices with the graph distance denoted by d Gn and show that the random metric space (V (G n ), n −1/4 d Gn ) converges in distribution in the Gromov-Hausdorff sense when n goes to infinity, at least along subsequences, toward a random metric space. One of the crucial steps in the argument is to construct a simple labeling on the map and show its convergence to an explicit scaling limit. We moreover show that this labeling approximates the distance to the root up to a uniform correction of order o(n 1/4 ).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.