In this paper we study the time required for a @math -biased ( @math ) walk to visit all nodes of a supercritical Galton-Watson tree up to generation @math . Inspired by the extremal landscape approach in [ 2018] for simple random walk on binary trees, we establish the near-independent nature of extremal points for the @math -biased walk, and deduce the scaling limit of the cover time.
As for trees, the first order approximation for @math -ary trees was first obtained in Aldous @cite_3 using recursive equations, @math General walks on Galton-Watson trees were studied Andreoletti and Debs @cite_2 by a second moment method: on the recurrent case with some regularity assumptions, @math generations are covered in @math steps, where @math is an explicit constant (reciprocal to the constant of law of large number for branching random walk).
@cite_3 @cite_2
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In this paper we deal with a random walk in a random environment on a super-critical Galton–Watson tree. We focus on the recurrent cases already studied by Hu and Shi (Ann. Probab. 35:1978–1997, 2007; Probab. Theory Relat. Fields 138:521–549, 2007), (Probab. Theory Relat. Fields, 2011, in press), and Faraud (Electron. J. Probab. 16(6):174–215, 2011). We prove that the largest generation entirely visited by these walks behaves like logn, and that the constant of normalization, which differs from one case to another, is a function of the inverse of the constant of Biggins’ law of large numbers for branching random walks (Biggins in Adv. Appl. Probab. 8:446–459, 1976). Consider a uniform random rooted tree on vertices labelled by @math , with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number @math of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider @math and let @math denote the event that all @math cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if @math , we have @math , whereas if @math we have @math . We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson( @math ) number of cars arrive at each vertex. Let @math be the number of cars which visit the root of the tree. Then for @math , we have @math , whereas for @math , we have @math . This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution. Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability @math , independently of each other, and a deterministic spreading rule with a fixed parameter @math : if a vacant site has at least @math occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on @math ; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of @math for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than @math , then the critical probability is 1, while it is @math on the @math -ary tree. A related result is that in any rooted tree @math there is a way of erasing @math children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree @math has branching number @math . We also prove that on any @math -regular non-amenable graph, the critical probability for the @math -rule is strictly positive. These notes provide an elementary and self-contained introduction to branching ran- dom walks. Section 1 gives a brief overview of Galton-Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly in- dispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-gout to the main part, Section 3, where branching random walks are studied from a deeper point of view, and are connected to the model of directed polymers on a tree. Tree-related random processes form a rich and exciting research subject. These notes cover only special topics. For a general account, we refer to the St-Flour lecture notes of Peres (47) and to the forthcoming book of Lyons and Peres (42), as well as to Duquesne and Le Gall (23) and Le Gall (37) for continuous random trees.