Scaling Limits of Controlled Stochastic Systems Near Criticality

<div xmlns="http://www.tei-c.org/ns/1.0"> We investigate the asymptotic behaviour of controlled stochastic systems defined on large graphs or spatial structures as the system approaches criticality-namely, a phase transition point in the underlying geometry or dynamics. The models considered include locally interacting diffusions, Markov chains, or stochastic differential equations equipped with feedback control laws that act locally and preserve decentralized structure. We focus on settings where the underlying system undergoes a topological or dynamical phase transition, such as the emergence of a giant component in an Erdős-Rényi graph, a loss of ergodicity in sparse networks, or a change in controllability of the linearized dynamics. Our main results establish scaling limits for the controlled processes in various critical regimes and show that the limiting behaviour is universal and governed by geometric properties of the system at criticality. In particular, we characterize how local control affects the global system near the critical point and derive weak convergence results for rescaled versions of the state process, often toward diffusions or interacting particle systems on continuum limits (such as the continuum random tree or critical Galton-Watson limits). Our analysis combines probabilistic coupling, control-theoretic energy bounds, and scaling limit arguments from random graph theory and critical phenomena. The results reveal sharp tradeoffs between control strength, noise, and geometric connectivity near the threshold and provide insight into decentralized control's fundamental limitations and scaling laws in large, critical stochastic systems. </div>