On the convexity of interacting particle systems in the plane

Yaroslav Yudin
Theory of Stochastic Processes
Vol.30 (46), no.1, 2026, pp.104-110

In this article we consider a system of n interacting points in R² with dynamics defined by a linear term and bounded C¹ perturbations with bounded Jacobian matrix. We study the asymptotic behavior of the configuration after recentering and rescaling by the linear flow. Next, we present the results about the convergence of the rescaled system and discuss the long-term behavior of some geometric properties, e.g. the convexity of the polygon formed by the interacting points. We prove that for almost all initial conditions the time average of the convexity indicator of the evolving polygon equals the convexity indicator of the limiting configuration of the rescaled system.

DOI: https://doi.org/10.3842/tsp-0976043245-46
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