# Directed random walk on the backbone of an oriented percolation cluster

@article{Birkner2012DirectedRW, title={Directed random walk on the backbone of an oriented percolation cluster}, author={Matthias C. F. Birkner and Jiř{\'i} {\vC}ern{\'y} and Andrej Depperschmidt and Nina Gantert}, journal={Electronic Journal of Probability}, year={2012}, volume={18}, pages={1-35} }

We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the "ancestral lineage'' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e. for almost any…

## 17 Citations

### Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster

- Mathematics
- 2021

We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions d+ 1 with d ≥ 3 being the spatial dimension. For this random walk we prove an…

### Coalescing directed random walks on the backbone of a 1+1-dimensional oriented percolation cluster converge to the Brownian web

- MathematicsLatin American Journal of Probability and Mathematical Statistics
- 2019

We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 +1. A directed random walk on this backbone can be seen as an "ancestral line" of…

### Random walks on weighted, oriented percolation clusters

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- 2015

We consider a weighted random walk on the backbone of an oriented percolation cluster. We determine necessary conditions on the weights for Brownian scaling limits under the annealed and the quenched…

### Random walks on oriented percolation and in recurrent environments

- Mathematics
- 2017

We present a central limit theorem for random walks on oriented percolation clusters. These random walks are representations of ancestral lineages in a population model with local competition. We…

### Random walk on random walks

- Mathematics
- 2014

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson…

### Scaling of a random walk on a supercritical contact process

- Mathematics
- 2012

A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a…

### Random walks in dynamic random environments and ancestry under local population regulation

- Mathematics
- 2016

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of…

### Quenched invariance principle for random walks with time-dependent ergodic degenerate weights

- Mathematics
- 2016

We study a continuous-time random walk, $X$, on $\mathbb{Z}^d$ in an environment of dynamic random conductances taking values in $(0, \infty)$. We assume that the law of the conductances is ergodic…

### The number of open paths in oriented percolation

- Mathematics
- 2013

We study the number $N_n$ of open paths of length $n$ in supercritical oriented percolation on $\Zd \times \N$, with $d \ge 1$. We prove that on the percolation event $\{\inf N_n>0\}$, $N_n^{1/n}$…

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In [HN01], Howard and Newman gave an essentially complete picture about existence and uniqueness of asymptotically directed infinite geodesics on Poisson point process on $\mathbb{R}^d$. For $d=2$,…

## References

SHOWING 1-10 OF 26 REFERENCES

### Directed percolation and random walk

- Mathematics
- 2002

Techniques of ‘dynamic renormalization’, developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several…

### Limit theorems for random walks in dynamic random environment

- Mathematics
- 2011

We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties and concentration…

### Law of Large Numbers for a Class of Random Walks in Dynamic Random Environments

- Mathematics
- 2009

In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on…

### Slowdown estimates and central limit theorem for random walks in random environment

- Mathematics
- 2000

Abstract.This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in…

### Random walk in Markovian environment

- Mathematics
- 2008

We prove a quenched central limit theorem for random
walks with bounded increments in a randomly evolving environment on
$\Z^d$. We assume that the transition probabilities of the walk depend not…

### Ergodic theorems for the multitype contact process

- Mathematics
- 1992

SummaryThis paper studies a process involving competition of two types of particles (1 and 2) for the empty space (0). Each site of the latticeZd is therefore in one of three possible states: 0, 1,…

### On the Satic and Dynamic Points of View for Certain Random Walks in Random Environment

- Mathematics
- 2002

In this work we prove the equivalence between static and dynamic points of views for certain ballistic random walks in random environment on Zd, when d greater than or equal to 4 and the disorder is…

### Multitype contact process on Z: extinction and interface

- Mathematics
- 2010

We consider a two-type contact process on the integers. Both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only…

### The Central Limit Theorem for the Right Edge of Supercritical Oriented Percolation

- Mathematics
- 1989

On demontre un theoreme central limite pour la croissance des bords droits d'une percolation orientee supercritique. La technique de demonstration utilisee est de trouver des points avec des…