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

@article{Birkner2019CoalescingDR,
  title={Coalescing directed random walks on the backbone of a 1+1-dimensional oriented percolation cluster converge to the Brownian web},
  author={Matthias C. F. Birkner and Nina Gantert and Sebastian Steiber},
  journal={Latin American Journal of Probability and Mathematical Statistics},
  year={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 an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in [BCDG13] where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We… 

Figures from this paper

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

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

The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

  • R. Forien
  • Mathematics
    Electronic Journal of Probability
  • 2019
We study a one-dimensional spatial population model where the population sizes at each site are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away

Ancestral lineages in spatial population models with local regulation

TLDR
It is explained how an ancestral lineage can be interpreted as a random walk in a dynamic random environment and defined regeneration times allows to prove central limit theorems for such walks.

References

SHOWING 1-10 OF 38 REFERENCES

Directed random walk on the backbone of an oriented percolation cluster

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

The Brownian Web: Characterization and Convergence

The Brownian Web (BW) is the random network formally consist- ing of the paths of coalescing one-dimensional Brownian motions start- ing from every space-time point in R×R. We extend the earlier work

The Brownian web, the Brownian net, and their universality

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They

Convergence of Coalescing Nonsimple Random Walks to the Brownian Web

TLDR
This is the first time that convergence to the BW has been proved for models with crossing paths, and several corollaries are presented, including an analysis of the scaling limit of voter model interfaces that extends a result of Cox and Durrett.

Stochastic Flows in the Brownian Web and Net

It is known that certain one-dimensional nearest-neighbour random walks in i.i.d. random space-time environments have diffusive scaling limits. Here, in the continuum limit, the random environment is

Directed percolation and random walk

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

Brownian web in the scaling limit of supercritical oriented percolation in dimension 1 + 1

We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time

Consistent families of Brownian motions and stochastic flows of kernels

Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice Z. At unit rate, independently for each site, the mass at the site is split into two parts

Random walk on random walks

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

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

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