Double Interlacing in Random Tiling Models

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, the tiles of a certain shape form for large-size domains a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a GUE-matrix (Gaussian unitary ensemble). 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 (also called hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the nonconvex domains and probing about such split tacnodes, filaments of tiles of a certain type will connect the liquid patches: they evolve in a bricklike-sea of dimers of another type. 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 one 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, opposed to hard, tacnodes appear when two arctic curves gently touch each other amidst 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.