Upper tail large deviation for the first passage time in the frog model on Zd with d≥ 2
Abstract
We study upper tail large deviations for the first passage time in the frog model on ℤd, d≥ 2. Let T(0,x) denote the first time at which the infection, started from the origin, reaches x, and let μ be the associated time constant. We prove that, for any Euclidean unit vector v∈ℝd and ξ>0, limn→∞ −1/sd(n) log ℙ(T(0,⌊ nv⌋) ≥ (μ(v)+ξ)n) =rdξ, where s2(n)=n/log n and r2=π for d=2; sd(n)=n and rd is the escape probability of simple random walk for d≥3. Moreover, on the upper tail large deviation event, we show that a certain slowdown mechanism occurs near the endpoint. To rule out other mechanisms, we analyze the growth of the infection near the starting point using a modified exploration process, and investigate its propagation through the intermediate region using the renormalization and covering arguments.
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BibTeX
@misc{can2026upper,
title = {{Upper tail large deviation for the first passage time in the frog model on $Z^d$ with $d\geq 2$}},
author = {Van Hao Can and Naoki Kubota and Shuta Nakajima},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14678},
eprint = {2610.14678},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14678},
}
RIS
TY - UNPB
TI - Upper tail large deviation for the first passage time in the frog model on Z^d with d≥ 2
AU - Van Hao Can
AU - Naoki Kubota
AU - Shuta Nakajima
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14678
UR - https://arxiv.org/abs/2610.14678
AB - We study upper tail large deviations for the first passage time in the frog model on ℤ^d, d≥ 2. Let T(0,x) denote the first time at which the infection, started from the origin, reaches x, and let μ be the associated time constant. We prove that, for any Euclidean unit vector v∈ℝ^d and ξ>0, lim_n→∞ −1/s_d(n) log ℙ(T(0,⌊ nv⌋) ≥ (μ(v)+ξ)n) =r_dξ, where s_2(n)=n/log n and r_2=π for d=2; s_d(n)=n and r_d is the escape probability of simple random walk for d≥3. Moreover, on the upper tail large deviation event, we show that a certain slowdown mechanism occurs near the endpoint. To rule out other mechanisms, we analyze the growth of the infection near the starting point using a modified exploration process, and investigate its propagation through the intermediate region using the renormalization and covering arguments.
ER -