Interacting diffusions on sparse graphs: hydrodynamics from local weak limits

2018/12/31 by Roberto I. Oliveira, Oliveira, Roberto I., Guilherme H. Reis +3 · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1812.11924

We did several improvements on the manuscript. We demoted the Propagation of Chaos Theorem to a Corollary. We improved the numerical simulations considering a larger time window

arxiv created 2019/12/30 · arxiv updated 2020/01/01

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 Erdős-Rényi 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.

Cited by

• Marginal dynamics of probabilistic cellular automata on trees

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• Hydrodynamic Limits of non-Markovian Interacting Particle Systems on Sparse Graphs
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