发表机构
Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单调变化电导环境上的随机游走，证明电导单调递增时游走常返性/暂态性与极限环境一致，并给出单调递减时的反例，解决多个猜想。
AI 中文摘要
设$(c_t)_{t\geq 0}$是可数顶点集上依赖于$t$的确定性边电导族，且关于$t$单调，并设$(X_t)$是使用电导$c_t$迈出第$t$步的随机游走。我们证明，若$c_t\uparrow c_\infty$且$c_\infty$常返（相应地，$c_0$暂态），则$(X_t)$几乎必然常返（相应地，暂态），即几乎必然无限（相应地，有限）次访问每个顶点。我们还建立了连续时间下的类似结果。这证明了Amir、Benjamini、Gurel-Gurevich和Kozma的猜想，而取值为$\set{0,1}$的电导特例对应于增长图上的简单随机游走，在此特例中我们的结果证明了Dembo、Huang和Sidoravicius的一个猜想。此外，当$(c_t)$单调非增时，我们提供了相应猜想的反例：若$c_t\downarrow c_\infty$且$c_\infty$暂态，$(X_t)$未必暂态；类似地，若$c_0$常返，$(X_t)$未必常返，即使对某个$\alpha>0$有$c_\infty\geq\alpha c_0$。
英文摘要
Let $(c_t)_{t\geq 0}$ be a deterministic family of edge conductances on a countable vertex set, monotone in $t$, and let $(X_t)$ be the random walk that takes its $t$-th step using the conductances $c_t$. We prove that if $c_t\uparrow c_\infty$ and $c_\infty$ is recurrent (respectively, $c_0$ is transient), then $(X_t)$ is almost surely recurrent (respectively, transient), i.e., visits every vertex infinitely (respectively, finitely) often. We also establish the analogous results in continuous time. This proves conjectures of Amir, Benjamini, Gurel-Gurevich, and Kozma, and the special case of $\{ 0, 1 \}$-valued conductances corresponds to simple random walk on a growing graph, and in this special case our results prove a conjecture of Dembo, Huang, and Sidoravicius. In addition, we provide counterexamples to the corresponding conjectures when $(c_t)$ is monotone non-increasing: if $c_t\downarrow c_\infty$ and $c_\infty$ is transient, $(X_t)$ need not be transient, and similarly if $c_0$ is recurrent, $(X_t)$ need not be recurrent, even if $c_\infty\geqαc_0$ for some $α>0$.
Comments18 pages
