Fundamental Limits of Decentralized Self-Regulating Random Walks

Self-regulating random walks (SRRWs) are decentralized token-passing processes on a graph allowing nodes to locally \emph{fork}, \emph{terminate}, or \emph{pass} tokens based only on a return-time \emph{age} statistic. We study SRRWs on a finite connected graph under a lazy reversible walk, with exogenous \emph{trap} deletions summarized by the absorption pressure $\Lambda_{\mathrm{del}}=\sum_{u\in\mathcal P_{\mathrm{trap}}}\zeta(u)\pi(u)$ and a global per-visit fork cap $q$. Using exponential envelopes for return-time tails, we build graph-dependent Laplace envelopes that universally bound the stationary fork intensity of any age-based policy, leading to an effective triggering age $A_{\mathrm{eff}}$. A mixing-based block drift analysis then yields controller-agnostic stability limits: any policy that avoids extinction and explosion must satisfy a \emph{viability} inequality (births can overcome $\Lambda_{\mathrm{del}}$ at low population) and a \emph{safety} inequality (trap deletions plus deliberate terminations dominate births at high population). Under corridor-wise versions of these conditions, we obtain positive recurrence of the population to a finite corridor.