PreprintarXiv licencearXiv
Recurrence, Transience, and the Rate of Escape for Elephant Random Walks with Two Memory Channels
Abstract
We study the one-dimensional elephant random walk with two memory channels introduced by Saha [Phys.\ Rev.\ E \textbf{106}, L062105 (2022)] with memory parameter p∈(0,1). Maulik, Roy and Sadhukhan [arXiv:2509.10225] proved recurrence for p≤11/16 and transience for p>7/8, leaving the range 11/16<p≤7/8 open. We close this gap by proving that p=11/16 is the exact recurrence--transience threshold and by determining how fast the walk escapes from the origin throughout the transient regime. For 11/16<p<7/8, we also show that the scaling limit obtained by Maulik, Roy and Sadhukhan is almost surely nonzero. At p=7/8, we prove that the walk is transient with zero asymptotic velocity and escapes at the scale n/√(log n).
Related
- Recurrence and transience of random walks with drift ρxα/tβ
- Recurrence and transience of random walks in monotonically changing environments
- Recurrence and transience of open Jackson networks with balanced nodes
- Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes
- Moments approach for the elephant random walk
BibTeX
@misc{ngoc2026recurrence,
title = {{Recurrence, Transience, and the Rate of Escape for Elephant Random Walks with Two Memory Channels}},
author = {Ngo Phuoc Nguyen Ngoc},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06641},
eprint = {2609.06641},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06641},
}
RIS
TY - UNPB
TI - Recurrence, Transience, and the Rate of Escape for Elephant Random Walks with Two Memory Channels
AU - Ngo Phuoc Nguyen Ngoc
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06641
UR - https://arxiv.org/abs/2609.06641
AB - We study the one-dimensional elephant random walk with two memory channels introduced by Saha [Phys.\ Rev.\ E \textbf{106}, L062105 (2022)] with memory parameter p∈(0,1). Maulik, Roy and Sadhukhan [arXiv:2509.10225] proved recurrence for p≤11/16 and transience for p>7/8, leaving the range 11/16<p≤7/8 open. We close this gap by proving that p=11/16 is the exact recurrence--transience threshold and by determining how fast the walk escapes from the origin throughout the transient regime. For 11/16<p<7/8, we also show that the scaling limit obtained by Maulik, Roy and Sadhukhan is almost surely nonzero. At p=7/8, we prove that the walk is transient with zero asymptotic velocity and escapes at the scale n/√(log n).
ER -