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The Large Deviation Principle for Interacting Dynamical Systems on Random Graphs

Abstract

Using the weak convergence approach to large deviations, we formulate and prove the large deviation principle (LDP) for W-random graphs in the cut-norm topology. This generalizes the LDP for Erdős–Rényi random graphs by Chatterjee and Varadhan. Furthermore, we translate the LDP for random graphs to a class of interacting dynamical systems on such graphs. To this end, we demonstrate that the solutions of the dynamical models depend continuously on the underlying graphs with respect to the cut-norm and apply the contraction principle.

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. Several proofs are available, including one that extends Sanov’s theorem. However, the most direct argument is to note that Mogulskii’s theorem asserts that if \(Y^n(x)=\int _0^x F^n(t)dt\), then \(\{Y^n\}\) satisfies an LDP in C([0, 1]) with the rate function \(I(\phi )\) equal to \(\int _0^1 L(\dot{\phi }(t))dt\) if \(\phi \) is absolutely continuous with \(\phi (0)=0\) and \(\infty \) otherwise. Using \(F^n(x)=\dot{Y}^n(x)\) a.s. in x, we can find the result stated for the sequence \(\{F^n\}\) using integration by parts.

  2. Here and below, C stands for a generic constant.

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Correspondence to Georgi S. Medvedev.

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Communicated by S. Chatterjee.

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P. Dupuis: research supported in part by the NSF (DMS-1904992). G. S. Medvedev: research supported in part by the NSF (DMS-2009233).

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Dupuis, P., Medvedev, G.S. The Large Deviation Principle for Interacting Dynamical Systems on Random Graphs. Commun. Math. Phys. 390, 545–575 (2022). https://doi.org/10.1007/s00220-022-04312-1

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  • DOI: https://doi.org/10.1007/s00220-022-04312-1