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2020 Interacting diffusions on sparse graphs: hydrodynamics from local weak limits
Roberto I. Oliveira, Guilherme H. Reis, Lucas M. Stolerman
Electron. J. Probab. 25: 1-35 (2020). DOI: 10.1214/20-EJP505

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 Erdös-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.

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Roberto I. Oliveira. Guilherme H. Reis. Lucas M. Stolerman. "Interacting diffusions on sparse graphs: hydrodynamics from local weak limits." Electron. J. Probab. 25 1 - 35, 2020. https://doi.org/10.1214/20-EJP505

Information

Received: 15 January 2019; Accepted: 3 August 2020; Published: 2020
First available in Project Euclid: 15 September 2020

zbMATH: 07252704
MathSciNet: MR4150522
Digital Object Identifier: 10.1214/20-EJP505

Subjects:
Primary: 05C80, 60F15, 60K35, 60K37

JOURNAL ARTICLE
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