Capacity of the range of random walk: Moderate deviations in dimensions 4 and 5

• 成果类型: Article; Early Access
• 署名作者: Adhikari, Arka; Park, Jiyun
• 署名单位: University System of Maryland; University of Maryland College Park; Stanford University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01513-w
• 发表日期: 2026-07-03
• 关键词: Moderate deviation; random walk; Brownian Motion; capacity; lace expansion

摘要

We prove a moderate deviation principle for the capacity of the range of random walk in Z5 . Depending on the scale of deviation, we get two different regimes. We observe Gaussian tails when the deviation scale is smaller than n1/2(logn)3/4 . Otherwise, we get non-Gaussian tails with a constant arising from a generalized Gagliardo-Nirenberg inequality. This is analogous to the behavior of the volume of the random walk range in Z3 . Our methods can also be applied to the d=4 case to prove the moderate deviation principle in almost the full range of interest. This extends the work of Okada and the first author [], where they showed moderate deviations up to a deviation scale of loglogn times the standard deviation.