# Interacting diffusions on sparse graphs: hydrodynamics from local weak limits

## TL;DR

This paper establishes limit theorems for interacting diffusion systems on sparse graphs, including hydrodynamic limits and chaos propagation, with applications to models like the stochastic Kuramoto model on Erdős-Rényi graphs.

## Contribution

It introduces a general framework for analyzing diffusions on sparse graphs via local weak limits, including a new locality estimate and applications to synchronization phenomena.

## Key findings

- Hydrodynamic limit for diffusions on sparse graphs
- Propagation of chaos in the stochastic Kuramoto model
- Numerical evidence of phase transitions in synchronization

## Abstract

We prove limit theorems for systems of interacting diffusions on sparse graphs. For example, we deduce a hydrodynamic limit and the propagation of chaos property for the stochastic Kuramoto model with interactions determined by Erd\H{o}s-R\'{e}nyi graphs with constant mean degree. The limiting object is related to a potentially infinite system of SDEs defined over a Galton-Watson tree. Our theorems apply more generally, when the sequence of graphs ("decorated" with edge and vertex parameters) converges in the local weak sense. Our main technical result is a locality estimate bounding the influence of far-away diffusions on one another. We also numerically explore the emergence of synchronization phenomena on Galton-Watson random trees, observing rich phase transitions from synchronized to desynchronized activity among nodes at different distances from the root.