Double interlacing in random tiling models

Abstract

Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid–liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain boundary, for large-sized domains, the tiles of a certain shape form a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a Gaussian unitary ensemble-matrix. Introducing non-convexities in large domains may lead to the appearance of several interacting liquid regions: They can merely touch, leading to either a split tacnode (hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the non-convex domains and probing about such split tacnodes, filaments, evolving in a bricklike sea of dimers of another type, will connect the liquid patches. Nearby, the tiling fluctuations are governed by a discrete tacnode kernel—i.e., a determinantal point process on a doubly interlacing set of dots belonging to a discrete array of parallel lines. This kernel enables us to compute the joint distribution of the dots along those lines. This kernel appears in two very different models: (i) domino tilings of skew-Aztec rectangles and (ii) lozenge tilings of hexagons with cuts along opposite edges. Soft tacnodes appear when two arctic curves gently touch each other amid a bricklike sea of dimers of one type, unlike the split tacnode. We hope that this largely expository paper will provide a view on the subject and be accessible to a wider audience.