Tacnode GUE-minor Processes and Double Aztec Diamonds

TL;DR

This paper investigates the limiting behavior of L-particles in random domino tilings of a Double Aztec diamond, revealing a new tacnode GUE-minor process when the arctic circles are tangent.

Contribution

It introduces the limiting point process of L-particles in the overlap, called the tacnode GUE-minor process, and derives the inverse Kasteleyn matrix for the model.

Findings

L-particles near tangency point tend to a new tacnode GUE-minor process.

The inverse Kasteleyn matrix for the Double Aztec diamond is explicitly derived.

The behavior of L-particles in the tangent case is analogous to colliding GUE-minor processes.

Abstract

We study random domino tilings of a Double Aztec diamond, a region consisting of two overlapping Aztec diamonds. The random tilings give rise to two discrete determinantal point processes called the K-and L-particle processes. The correlation kernel of the K-particles was derived in Adler, Johansson and van Moerbeke (2011), who used it to study the limit process of the K-particles with different weights for horizontal and vertical dominos. Let the size of both, the Double Aztec diamond and the overlap, tend to infinity such that the two arctic ellipses just touch; then they show that the fluctuations of the K-particles near the tangency point tend to the tacnode process. In this paper, we find the limiting point process of the L-particles in the overlap when the weights of the horizontal and vertical dominos are equal, or asymptotically equal, as the Double Aztec diamond grows, while keeping the overlap finite. In this case the two limiting arctic circles are tangent in the overlap and the behavior of the L-particles in the vicinity of the point of tangency can then be viewed as two colliding GUE-minor process, which we call the tacnode GUE minor process. As part of the derivation of the kernel for the L-particles we find the inverse Kasteleyn matrix for the dimer model version of Double Aztec diamond.