Cover times and ranges of elephant random walks

Abstract

We study cover times on discrete tori and ranges on ℤ^d for elephant random walks with memory parameter p∈[0,1). In dimension one there is a phase transition at p=3/4. Away from criticality we determine the exact first-order asymptotics of the mean cover time, while at p=3/4 the mean is of order L²/√(log L), a factor √(log L) larger than the natural fluctuation scale L²/log L. We identify the limiting distribution under each of the three natural fluctuation normalizations. In dimensions d≥2, for every fixed p<1, the cover time has the same order as for simple random walk, in expectation and with high probability. The leading constants also agree when p<(2d+1)/(4d), and at equality when d≥3. For the range on ℤ^d, we prove L¹ convergence to the simple random walk asymptotics when d=2 and p<5/8, and for every p<1 when d≥3. Finally, we give a cover-time upper bound for generalized step-reinforced random walks on finite groups in terms of a conditional L^∞ mixing profile.

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Cite

BibTeX RIS

BibTeX

@misc{qin2026cover,
  title = {{Cover times and ranges of elephant random walks}},
  author = {Shuo Qin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17264},
  eprint = {2609.17264},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17264},
}

RIS

TY  - UNPB
TI  - Cover times and ranges of elephant random walks
AU  - Shuo Qin
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17264
UR  - https://arxiv.org/abs/2609.17264
AB  - We study cover times on discrete tori and ranges on ℤ^d for elephant random walks with memory parameter p∈[0,1). In dimension one there is a phase transition at p=3/4. Away from criticality we determine the exact first-order asymptotics of the mean cover time, while at p=3/4 the mean is of order L^2/√(log L), a factor √(log L) larger than the natural fluctuation scale L^2/log L. We identify the limiting distribution under each of the three natural fluctuation normalizations. In dimensions d≥2, for every fixed p<1, the cover time has the same order as for simple random walk, in expectation and with high probability. The leading constants also agree when p<(2d+1)/(4d), and at equality when d≥3. For the range on ℤ^d, we prove L^1 convergence to the simple random walk asymptotics when d=2 and p<5/8, and for every p<1 when d≥3. Finally, we give a cover-time upper bound for generalized step-reinforced random walks on finite groups in terms of a conditional L^∞ mixing profile.
ER  -

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