Range and critical generations of a random walk on Galton-Watson trees

@article{Andreoletti2015RangeAC,
  title={Range and critical generations of a random walk on Galton-Watson trees},
  author={Pierre Andreoletti and Xinxing Chen},
  journal={arXiv: Probability},
  year={2015}
}
In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range $R\_n$ of this walk up to time $n$ and obtain its correct asymptotic in probability which is of order $n/\log n$. Thisresult is a consequence of the asymptotical behavior of the number of visited sites at generations of order $(\log n)^2$,which turn out to be the most visited generations. Our proof which involves a quenched… 

Figures from this paper

The heavy range of randomly biased walks on trees
Favourite generations and range of a randomly biased random walk on Galton-Watson tree
We are interested in the boundary case where E[ ∑ν i=1Ai] = 1 and E[ ∑ν i=1Ai logAi] = 0. Hu and Shi [2] proved that the generation of Xn is of order (log n) 2 in distribution. We count, up to time
Dimension Drop for Transient Random Walks on Galton-Watson Trees in Random Environments
We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and
PR ] 2 M ay 2 01 9 Conductance of a subdiffusive random weighted tree
We work on a Galton–Watson tree with random weights, in the so-called “subdiffusive” regime. We study the rate of decay of the conductance between the root and the n-th level of the tree, as n goes
Conductance of a subdiffusive random weighted tree
We work on a Galton--Watson tree with random weights, in the so-called "subdiffusive" regime. We study the rate of decay of the conductance between the root and the $n$-th level of the tree, as $n$
Marches aleatoires sur les arbres aleatoires
Cette these a pour objet d’etude divers modeles de marches aleatoires sur les arbres aleatoires.Nous nous sommes consacres principalement aux aspects qui relevaient a la fois de la theorie des

References

SHOWING 1-10 OF 32 REFERENCES
The slow regime of randomly biased walks on trees
We are interested in the randomly biased random walk on the supercritical Galton--Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_n)$ is null recurrent, making
The potential energy of biased random walks on trees
Biased random walks on supercritical Galton--Watson trees are introduced and studied in depth by Lyons (1990) and Lyons, Pemantle and Peres (1996). We investigate the slow regime, in which case the
The Number of Generations Entirely Visited for Recurrent Random Walks in a Random Environment
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,
A subdiffusive behaviour of recurrent random walk in random environment on a regular tree
We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle (Ann. Probab. 20, 125–136, 1992) give a precise recurrence/transience criterion. Our paper focuses
Almost sure convergence for stochastically biased random walks on trees
We are interested in the biased random walk on a supercritical Galton–Watson tree in the sense of Lyons (Ann. Probab. 18:931–958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields
A Central Limit Theorem for Random Walk in a Random Environment on a Marked Galton-Watson Tree.
Models of random walks in a random environment were introduced at first by Chernoff in 1967 in order to study biological mechanisms. The original model has been intensively studied since then and is
On Random Walks in Random Environment on Trees and Their Relationship with Multiplicative Chaos
The purpose of this paper is to report on recent results concerning random walks in a random environment on monochromatic and coloured trees and their relationship with multiplicative chaos. The
Convergence in law of the minimum of a branching random walk
We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [Ann. Probab. 37 (2009) 1044–1079] proved the tightness of the minimum centered around its mean value. We
SENETA-HEYDE NORMING IN THE BRANCHING RANDOM WALK
!. !. !. malization of the general Crump! Mode! Jagers branching process is obtained; in this case the convergence holds almost surely. The results rely heavily on a detailed study of the functional
...
...