Double Aztec Diamonds and the Tacnode Process

TL;DR

This paper studies the domino tiling of overlapping Aztec diamonds, revealing a new universal critical process called the tacnode process that appears at the touching point of two arctic ellipses, extending known fluctuation phenomena.

Contribution

It introduces the tacnode process as a new universal limit in domino tilings of overlapping Aztec diamonds and connects it to non-intersecting random walks and Brownian motions.

Findings

The process is determinantal.

The tacnode process appears at the touching point of two arctic ellipses.

It is universal across different non-intersecting particle systems.

Abstract

Discrete and continuous non-intersecting random processes have given rise to critical "infinite dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the "tacnode process". It is also shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily.