Abstract
J. Math. Phys. 64 (2023) Special Collection in Honor of Freeman
Dyson 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.