A Spectral Approach to Survival Bounds for the Elephant Random Walk
Abstract
We study a rate-one Poissonization of the one-dimensional elephant random walk killed upon reaching ± N. For every fixed memory parameter 0<p<1, we give a self-contained proof of two-sided exponential survival bounds on the rate scale N−2, uniformly for sufficiently large N and times t≥ TN3. Our approach combines the classical Dirichlet Poincaré inequality with an O(N/t) estimate on the memory-dependent perturbation and a quantitative initialization of the principal-eigenvector projection. The lower bound covers the critical and superdiffusive regimes 3/4≤ p<1 in this late-time range. The confined-path comparison used for initialization also gives a direct proof of the lower bound. The spectral argument controls the exponentially small forcing terms and jumps introduced by Poisson truncation.
Related
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BibTeX
@misc{coletti2026spectral,
title = {{A Spectral Approach to Survival Bounds for the Elephant Random Walk}},
author = {Cristian Favio Coletti and Rafael de Mattos Grisi and Ioannis Papageorgiou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.08884},
eprint = {2606.08884},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08884},
}
RIS
TY - UNPB
TI - A Spectral Approach to Survival Bounds for the Elephant Random Walk
AU - Cristian Favio Coletti
AU - Rafael de Mattos Grisi
AU - Ioannis Papageorgiou
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2606.08884
UR - https://arxiv.org/abs/2606.08884
AB - We study a rate-one Poissonization of the one-dimensional elephant random walk killed upon reaching ± N. For every fixed memory parameter 0<p<1, we give a self-contained proof of two-sided exponential survival bounds on the rate scale N^−2, uniformly for sufficiently large N and times t≥ TN^3. Our approach combines the classical Dirichlet Poincaré inequality with an O(N/t) estimate on the memory-dependent perturbation and a quantitative initialization of the principal-eigenvector projection. The lower bound covers the critical and superdiffusive regimes 3/4≤ p<1 in this late-time range. The confined-path comparison used for initialization also gives a direct proof of the lower bound. The spectral argument controls the exponentially small forcing terms and jumps introduced by Poisson truncation.
ER -