Quenched CLT for ancestral lineages of logistic branching random walks

TL;DR

This paper proves a quenched central limit theorem for ancestral lineages modeled as random walks in dynamic environments derived from logistic branching random walks, using coarse-graining and regeneration techniques.

Contribution

It establishes the quenched CLT for these lineages in high-density regimes and introduces auxiliary models based on oriented percolation to facilitate the proof.

Findings

Quenched CLT holds for high-density logistic branching random walks.

Weak influence of the environment is sufficient for the CLT.

Coupling and regeneration methods are effective in proving the CLT.

Abstract

We consider random walks in dynamic random environments which arise naturally as spatial embeddings of ancestral lineages in spatial locally regulated population models. In particular, as the main result, we prove the quenched central limit theorem for a random walk in dynamic random environment generated by time reversal of logistic branching random walks in a regime where the population density is sufficiently high. As an important tool we consider as auxiliary models random walks in dynamic random environments defined in terms of the time-reversal of oriented percolation. We show that the quenched central limit theorem holds if the influence of the random medium on the walks is suitably weak. The proofs of the quenched central limit theorems in these models rely on coarse-graining arguments and a construction of regeneration times for a pair of conditionally independent random walks in the same medium, combined with a coupling that relates them to a pair of independent random walks in two independent copies of the medium.