Far Tails of the Positional Distribution in random walk: A Saddle-Point Approach
Abstract
Rare events and large-deviation theory have attracted considerable interest in the study of random walks. Here, we investigate the rare events of the positional distribution of continuous-time random walks under ordinary and equilibrium initial conditions using the saddle point method. We show that the far-tail behavior is sensitive to the expansion of the waiting-time distribution near its minimum value. For a general power-series expansion near the minimum waiting time, the positional distribution exhibits an approximately exponential decay in its far tails. In contrast, when the waiting-time distribution has a cutoff at its minimum value, the positional distribution exhibits an unexpected Gaussian decay, in contrast to the far-tail behavior of the renewal-count distribution. We further find that the most likely number of renewals required to reach a given position, as well as the associated rate functions, is insensitive to the initial conditions, whereas the far tails of the positional distribution retain a dependence on the initial conditions. Our results provide new insights into the rare-event statistics and large deviations of random walks.
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BibTeX
@misc{zhou2026tails,
title = {{Far Tails of the Positional Distribution in random walk: A Saddle-Point Approach}},
author = {Chen Zhou and YuDa Cheng and Yangji Li and Jing Xu and Wanli Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38492},
eprint = {2610.38492},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38492},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Far Tails of the Positional Distribution in random walk: A Saddle-Point Approach
AU - Chen Zhou
AU - YuDa Cheng
AU - Yangji Li
AU - Jing Xu
AU - Wanli Wang
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38492
UR - https://arxiv.org/abs/2610.38492
AB - Rare events and large-deviation theory have attracted considerable interest in the study of random walks. Here, we investigate the rare events of the positional distribution of continuous-time random walks under ordinary and equilibrium initial conditions using the saddle point method. We show that the far-tail behavior is sensitive to the expansion of the waiting-time distribution near its minimum value. For a general power-series expansion near the minimum waiting time, the positional distribution exhibits an approximately exponential decay in its far tails. In contrast, when the waiting-time distribution has a cutoff at its minimum value, the positional distribution exhibits an unexpected Gaussian decay, in contrast to the far-tail behavior of the renewal-count distribution. We further find that the most likely number of renewals required to reach a given position, as well as the associated rate functions, is insensitive to the initial conditions, whereas the far tails of the positional distribution retain a dependence on the initial conditions. Our results provide new insights into the rare-event statistics and large deviations of random walks.
ER -