The Cusp-Airy Process

TL;DR

This paper introduces the Cusp-Airy process, a new determinantal point process emerging at cusp points in random tiling models, extending the known Airy kernel process and providing insights into complex boundary behaviors.

Contribution

It defines and analyzes the Cusp-Airy process as a novel limiting point process at cusp points in random tiling models, expanding understanding of boundary phenomena.

Findings

Identification of the Cusp-Airy process as a new limiting process

Extension of the Airy kernel to a two-sided process

Application to interlacing particle systems in tiling models

Abstract

At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point.