Ordered random walks and the Airy line ensemble

• 成果类型: Article; Early Access
• 署名作者: Denisov, Denis; Fitzgerald, Will; Wachtel, Vitali
• 署名单位: University of Manchester; University of Bielefeld
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01481-1
• 发表日期: 2026-03-18
• 关键词: Ordered random walks; Airy line ensemble; Dyson Brownian motion; Doob h-transforms; KPZ universality class; LIMIT-THEOREMS; UNIVERSALITY; fluctuations; eigenvalues; Invariance; particles

摘要

The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang universality class. The aim of this paper is to prove a universality property of the Airy line ensemble. We study growing numbers of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. We consider a general class of increment distributions; a sufficient condition is the existence of an exponential moment and a log-concave density. We prove that the top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power (with a non-optimal exponent 3/50) of the expected number of random walk steps. Furthermore, in a similar regime we prove that the law of large numbers and fluctuations of linear statistics agree with non-intersecting Brownian motions.