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

Birkner, Matthias; Gantert, Nina; Steiber, Sebastian

doi:10.48550/arxiv.1812.03733

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“…For this we have to study the coalescence of pairs of random walks in a random environment. This was done in a particular setting in [BGS18], where the authors consider a family of random walks following the model in [BCDG13] which coalesce whenever they find themselves in the same site. They show that, adequately rescaled, the resulting process converges in distribution to the Brownian web (i.e.…”

Section: Introductionmentioning

confidence: 99%

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

Forien¹

2019

Electron. J. Probab.

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E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractWe study a one-dimensional spatial population model where the population sizes of the subpopulations are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the effective diffusion rate of genes within the population and the effective local population density. We find that, in our model, the local heterogeneity leads to a slower effective diffusion rate and a larger effective population density than in a uniform population. Our proof relies on duality techniques, an invariance principle for reversible random walks in a random environment and a convergence result for a system of coalescing random walks in a random environment.(0.1) where p t (x) is the proportion of individuals carrying the type in deme x at time t, m xy is the probability that an individual in deme x is issued from one living in deme y in the previous generation, N (x) is number of individuals in deme x and (B x t , t ≥ 0, x ∈ Z) is a family of independent Brownian motions [Eth11, Shi88]. Kimura and Weiss [KW64], followed by Sawyer [Saw76] showed how the correlations in the genetic composition of demes decrease with the distance between them. In particular, in a one-dimensional space, they showed that the correlation coefficient between the genetic composition of two demes separated by a distance r decreases like e −λr , where λ > 0 depends on the parameters of the model. This model can also be extended to a (one-dimensional) continuous geographical space in the following way [Shi88, DEF + 00]. For n ≥ 1 and x ∈ 1 √ n Z, set p n (t, x) = p nt ( √ nx), (0.2) and assume m xy = 1 2 m if |x − y| = 1, (1 − m) if x = y and 0 otherwise and that N (x) = √n/γ for all x ∈ Z for some γ > 0. Then, as n → ∞, the sequence of processes EJP 24 (2019), paper 57.

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Section: Introductionmentioning

confidence: 99%

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

Forien¹

2019

Electron. J. Probab.

View full text Add to dashboard Cite

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

Cited by 1 publication

References 0 publications

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

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

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