A singular Toeplitz determinant and the discrete tacnode kernel for skew-Aztec Rectangles

TL;DR

This paper studies domino tilings of skew-Aztec rectangles with cuts, revealing that fluctuations near split tacnodes are governed by the discrete tacnode kernel, indicating its universality in double interlacing patterns.

Contribution

It introduces a novel analysis of the discrete tacnode kernel in skew-Aztec rectangles, involving the inversion of a singular Toeplitz matrix, and demonstrates its universality in this context.

Findings

Fluctuations near split tacnodes follow the discrete tacnode kernel.

The analysis involves inverting a singular Toeplitz matrix.

The discrete tacnode kernel appears as a universal limit in double interlacing patterns.

Abstract

Random tilings of geometrical shapes with dominos or lozenges have been a rich source of universal statistical distributions. This paper deals with domino tilings of checker board rectangular shapes such that the top two and bottom two adjacent squares have the same orientation and the two most left and two most right ones as well. It forces these so-called "skew-Aztec rectangles" to have cuts on either side. For large sizes of the domain and upon an appropriate scaling of the location of the cuts, one finds split tacnodes between liquid regions with two distinct adjacent frozen phases descending into the tacnode. Zooming about such split tacnodes, filaments appear between the liquid patches evolving in a bricklike sea of dimers of another type. This work shows that the random fluctuations in a neighborhood of the split tacnode are governed asymptotically by the discrete tacnode kernel, providing strong evidence that this kernel is a universal discrete-continuous limiting kernel occurring naturally whenever we have double interlacing patterns. The analysis involves the inversion of a singular Toeplitz matrix which leads to considerable difficulties.