AIRY PROCESSES WITH WANDERERS AND NEW UNIVERSALITY CLASSES

• 成果类型: Article
• 署名作者: Adler, Mark; Ferrari, Patrik L.; van Moerbeke, Pierre
• 署名单位: Brandeis University; Universite Catholique Louvain; University of Bonn
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP493
• 发表日期: 2010
• 页码: 714-769
• 关键词: random matrices; LARGEST EIGENVALUE; brownian-motion; external source; distributions; GROWTH; dyson; pdes

摘要

Consider n + m nonintersecting Brownian bridges, with n of them leaving from 0 at time t = -1 and returning to 0 at time t = -1, while the m remaining ones (wanderers) go from m points a(i) to m points b(i). First, we keep m fixed and we scale a(i), b(i) appropriately with n. In the large-n limit, we obtain a new Airy process with wanderers, in the neighborhood of root 2n, the approximate location of the rightmost particle in the absence of wanderers. This new process is governed by an Airy-type kernel, with a rational perturbation. Letting the number m of wanderers tend to infinity as well, leads to two Pearcey processes about two cusps, a closing and an opening cusp, the location of the tips being related by an elliptic curve. Upon tuning the starting and target points, one can let the two tips of the cusps grow very close; this leads to a new process, which might be governed by a kernel, represented as a double integral involving the exponential of a quintic polynomial in the integration variables.