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 $Λ_{\mathrm{del}}=\sum_{u\in\mathcal P_{\mathrm{trap}}}ζ(u)π(u)$ 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 $Λ_{\mathrm{del}}$ 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.

翻译：自调节随机游走（SRRW）是一种在图上进行的去中心化令牌传递过程，允许节点仅基于返回时间的“年龄”统计量，在本地进行令牌的“分叉”、“终止”或“传递”。我们研究有限连通图上的懒惰可逆游走下的SRRW，其中外生的“陷阱”删除由吸收压力 $Λ_{\mathrm{del}}=\sum_{u\in\mathcal P_{\mathrm{trap}}}ζ(u)π(u)$ 和全局每次访问分叉上限 $q$ 来概括。利用返回时间尾部的指数包络，我们构建了图依赖的拉普拉斯包络，该包络普遍地限制了任何基于年龄策略的稳态分叉强度，从而导出了一个有效触发年龄 $A_{\mathrm{eff}}$。随后，基于混合性的块漂移分析得出了与控制器无关的稳定性极限：任何避免灭绝和爆炸的策略必须满足一个“生存性”不等式（在低种群水平下，新生令牌能够克服 $Λ_{\mathrm{del}}$）和一个“安全性”不等式（在高种群水平下，陷阱删除加上主动终止主导了新生令牌）。在这些条件的通道化版本下，我们得到了种群向有限通道的正则递归性。