• Corpus ID: 119158674

# Once reinforced random walk on $\mathbb{Z}\times \Gamma$

@article{Kious2018OnceRR,
title={Once reinforced random walk on \$\mathbb\{Z\}\times \Gamma\$},
author={Daniel Kious and Bruno Schapira and Arvind Singh},
journal={arXiv: Probability},
year={2018}
}
• Published 18 July 2018
• Mathematics
• arXiv: Probability
We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $\mathbb{Z}\times \Gamma$, with $\Gamma$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might…
1 Citations

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## References

SHOWING 1-10 OF 16 REFERENCES
Reinforced random walk
SummaryLetai,i≧1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion $$\overrightarrow X$$ =X0,X1, ... on the integers as follows. Initially the weight of each interval
The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees
• Mathematics
Communications on Pure and Applied Mathematics
• 2019
The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For
A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs
• Mathematics, Physics
Journal of the American Mathematical Society
• 2018
This paper concerns the vertex reinforced jump process (VRJP), the edge reinforced random walk (ERRW), and their relation to a random Schrödinger operator. On infinite graphs, we define a 1-dependent
Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees
• Mathematics
The Annals of Probability
• 2018
In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees,
Probability on Trees and Networks
• Mathematics
• 2017
Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together
A random Schr\"odinger operator associated with the Vertex Reinforced Jump Process and the Edge Reinforced Random Walk
• Mathematics
• 2015
This paper concerns the Vertex reinforced jump process (VRJP) and the Edge reinforced random walk (ERRW) and their link with a random Schr\"odinger operator. On infinite graphs, we define a
The Vertex Reinforced Jump Process and a Random Schr\"odinger operator on finite graphs
• Mathematics, Physics
• 2015
We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the Inverse Gaussian distribution. Considered as the potential of a random
Transience of Edge-Reinforced Random Walk
• Mathematics, Physics
• 2014
We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension $${d\ge3}$$d≥3. This proves the existence of a phase transition between recurrent and transient
Localization for Linearly Edge Reinforced Random Walks
• Mathematics, Physics
• 2012
We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs
Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model
• Mathematics, Physics
• 2011
Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process that takes values in the vertex set of a graph G, which is more likely to cross edges it has