Propagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regime

TL;DR

This paper investigates the behavior of weakly interacting diffusions in a sparse regime, showing that only a few diffusions escape a trap and form a forest of independent excursions as the system size grows.

Contribution

It extends propagation of chaos results to a sparse regime with non-exchangeable initial conditions, revealing a forest structure of excursions in the limit.

Findings

System converges to a forest of excursions from the trap.

Initial independence propagates in the limit.

Sparse regime leads to a non-exchangeable, forest-like structure.

Abstract

Propagation of chaos is a well-studied phenomenon and shows that weakly interacting diffusions may become independent as the system size converges to infinity. Most of the literature focuses on the case of exchangeable systems where all involved diffusions have the same distribution and are "of the same size". In this paper, we analyze the case where only a few diffusions start outside of an accessible trap. Our main result shows that in this "sparse regime" the system of weakly interacting diffusions converges in distribution to a forest of excursions from the trap. In particular, initial independence propagates in the limit and results in a forest of independent trees.