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Weakly interacting oscillators on dense random graphs

Part of: Graph theory Equations of mathematical physics and other areas of application Special processes Stochastic analysis Equilibrium statistical mechanics

Published online by Cambridge University Press:  30 June 2023

Gianmarco Bet Open the ORCID record for Gianmarco Bet [Opens in a new window] ,
Fabio Coppini  and
Francesca Romana Nardi
Gianmarco Bet*
Affiliation:
Università degli Studi di Firenze
Fabio Coppini*
Affiliation:
Università degli Studi di Firenze
Francesca Romana Nardi*
Affiliation:
Università degli Studi di Firenze and Eindhoven University of Technology
*
*Postal address: Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Università degli Studi di Firenze, Firenze, Italy.
*Postal address: Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Università degli Studi di Firenze, Firenze, Italy.
*Postal address: Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Università degli Studi di Firenze, Firenze, Italy.
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Abstract

We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e. two particles are interacting if and only if they are connected in the underlying graph. We establish a law of large numbers for the empirical measure of the system that holds whenever the graph sequence is convergent to a graphon. The limit is the solution of a non-linear Fokker–Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on both deterministic and random graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the classical McKean–Vlasov mean-field limit.

Keywords

Interacting oscillatorsrandom graphonsmean-field systemsFokker–Planck equationexchangeable graphsMcKean–Vlasov

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 05C80: Random graphs
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60H20: Stochastic integral equations 35Q84: Fokker-Planck equations

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 255 - 278
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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