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Loïc de Raphélis

Publications and source records attributed to Loïc de Raphélis.

5 recordsLinked to original sources
arXiv2025

Branching random walk conditioned on large martingale limit

We consider a branching random walk in the non-boundary case where the additive martingale $W_n$ converges a.s. and in mean to some non-degenerate limit $W_\infty$. We first establish the joint tail distribution of $W_\infty$ and the global minimum of this branching random walk. Next, conditioned on the event that the minimum is atypically small or conditioned on very large $W_\infty$, we study the branching random walk viewed from the minimum and obtain the convergence in law in the vague sense. As a byproduct, we also get the right tail of the limit of derivative martingale.

math.PR↗
arXiv2020

Maximal local time of randomly biased random walks on a Galton-Watson tree

We consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson tree with countably infinitely many types. When the walk is the diffusive or sub-diffusive, by studying the maximal type of this Galton-Watson tree, we establish the asymptotic behaviour of the largest local times of this walk during n excursions, under the annealed law.

math.PR↗
arXiv2019

Scaling limit of the subdiffusive random walk on a Galton-Watson tree in random environment

We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable Lévy process, jointly with the convergence of the renormalised range of the walk towards the real tree coded by the latter continuous-time height process.

math.PR↗
arXiv2016

Favorite sites of randomly biased walks on a supercritical Galton-Watson tree

Erdős and Révész initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Gaton-Watson tree. We prove that there is some parameter $κ\in (1, \infty]$ such that the set of the favorite sites of the biased walk is almost surely bounded in the case $κ\in (2, \infty]$, tight in the case $κ=2$, and oscillates between a neighborhood of the root and the boundary of the range in the case $κ\in (1, 2)$. Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case $κ\in (2, \infty]$. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton-Watson trees.

math.PR↗