We consider in this article the favorite sites of a random walk on a Galton–Watson tree with random environment. The set of favorite sites of a random walk... Read more
at a given time is the set of vertices with maximal local time – that is the vertices which have been most visited by the walk at that time. We prove that the evolution of this set may present different kind of behaviours, depending on the law of the environment.
The first one corresponds to the case where the environment satisfies certain hypotheses of finiteness of a second moment. In this case, we prove that there exists a finite set of vertices such that almost surely after a given time the set of favorite sites is included in this set. We also prove that in this case, almost surely, after a given time, the cardinal of the set of favorite sites is smaller than or equal to 3, and that this bound is optimal.
The second behaviour is encountered when the hypotheses of finiteness of the second moment are not satisfied. We prove that in that case, the set of favorite sites will oscillate between a neighbourhood of the root and vertices which may be taken as high as desired. However, the set of vertices which will be favorite sites infinitely many times is a finite set (non-empty) close to the root. The proof of this second result relies on the study of tail distributions of a certain Markov chain associated with the walk.
We consider here a nearest-neighbour random walk on a Galton–Watson tree in random environment. Under certain conditions on the law of ... Read more
the environment, this walk is subdiffusive.
Using the strategy developed in the previous article, we show that the height function of the walk converges towards the continuous-time height process of a spectrally positive strictly stable Lévy process. We also establish the joint convergence of the range towards the Lévy forest coded by the latter process.
The main contribution of this article is the determining of the tail distribution of a key random variable, that we show to follow a power law. To do this, we are led to study the process of the local times of a random walk on a biased Galton–Watson tree in random environment. Then, we link this process to the multiplicative cascade induced by the environment, on which we use in the end Kesten's renewal theorem.
We consider a Galton–Watson tree, and a nearest-neighbour random walk on this tree, biased towards the parent. We prove in this article that the range ... Read more
of the walk, that is the sub-tree made up of the vertices visited by the walk, converges towards the Brownian forest. To this end, we establish an explicit link between the aforesaid range and a certain leafed Galton–Watson forest with edge lengths (a notion defined in the article below).
Using the branching property, we also get the convergence in law of the height function of the random walk (that is the sequence of heights taken by the walk in the tree).
Finally, we extend our result to the random environment case, under finite-variance conditions.
We introduce in this article a new class of Galton–Watson trees, namely leafed Galton–Watson trees with edge lengths. This class of trees differs from simple Galton–Watson trees ... Read more
on two points: first, because edge lengths are not fixed, and second because these trees are made up of two types of vertices, one of which not being able to give progeny. Considering the weighted height process of a forest made up of such trees (that is the sequence of heights of the vertices taken in the lexicographical order), we prove that the latter converges in law towards the reflected Brownian motion, after re-scaling.
Then, using this first result, we establish the convergence in law of the height function of a multitype Galton–Watson forest with infinitely many types towards the reflected Brownian motion.