ASYMPTOTIC EXPANSIONS FOR A CLASS OF FREDHOLM PFAFFIANS AND INTERACTING PARTICLE SYSTEMS

• 成果类型: Article
• 署名作者: Fitzgerald, Will; Tribe, Roger; Zaboronski, Oleg
• 署名单位: University of Manchester; University of Warwick
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1586
• 发表日期: 2022
• 页码: 2409-2474
• 关键词: largest real eigenvalue; DYNAMICS

摘要

Motivated by the phenomenon of duality for interacting particle systems, we introduce two classes of Pfaffian kernels describing a number of Pfaf-fian point processes in the bulk and at the edge. Using the probabilistic method due to Mark Kac, we prove two Szego-type asymptotic expansion theorems for the corresponding Fredholm Pfaffians. The idea of the proof is to introduce an effective random walk with transition density determined by the Pfaffian kernel, express the logarithm of the Fredholm Pfaffian through expectations with respect to the random walk, and analyse the expectations using general results on random walks. We demonstrate the utility of the theorems by calculating asymptotics for the empty interval and noncrossing probabilities for a number of examples of Pfaffian point processes: coalesc-ing/annihilating Brownian motions, massive coalescing Brownian motions, real zeros of Gaussian power series and Kac polynomials, and real eigenval-ues for the real Ginibre ensemble.