Recurrence and transience of random walks in monotonically changing environments
Abstract
Let (ct)t≥ 0 be a deterministic family of edge conductances on a countable vertex set, monotone in t, and let (Xt) be the random walk that takes its t-th step using the conductances ct. We prove that if ct↑ c∞ and c∞ is recurrent (respectively, c0 is transient), then (Xt) 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 (ct) is monotone non-increasing: if ct↓ c∞ and c∞ is transient, (Xt) need not be transient, and similarly if c0 is recurrent, (Xt) need not be recurrent, even if c∞≥αc0 for some α>0.
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BibTeX
@misc{li2026recurrence,
title = {{Recurrence and transience of random walks in monotonically changing environments}},
author = {Rupert Li and Jiyun Park},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28848},
eprint = {2609.28848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28848},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Recurrence and transience of random walks in monotonically changing environments
AU - Rupert Li
AU - Jiyun Park
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.28848
UR - https://arxiv.org/abs/2609.28848
AB - Let (c_t)_t≥ 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↑ c_∞ and c_∞ 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↓ c_∞ and c_∞ is transient, (X_t) need not be transient, and similarly if c_0 is recurrent, (X_t) need not be recurrent, even if c_∞≥αc_0 for some α>0.
ER -