Mihai Ciucu
and Christian Krattenthaler
Boundary dents, the arctic circle and the arctic ellipse
(36 pages)
Abstract.
The original motivation for this paper goes back to the mid-1990's,
when James Propp was interested in natural situations when the number
of domino tilings of a region increases if some of its unit squares
are deleted. Guided in part by the intuition one gets from earlier
work on parallels between the number of tilings of a region with holes
and the 2D Coulomb energy of the corresponding system of electric
charges, we consider Aztec diamond regions with unit square defects
along two adjacent sides. We show that for large regions, if these
defects are at fixed distances from a corner, the ratio between the
number of domino tilings of the Aztec diamond with defects and the
number of tilings of the entire Aztec diamond approaches a Delannoy
number.
When the locations of the defects are not fixed but instead approach
given points on the boundary of the scaling limit S (a square) of
the Aztec diamonds, we prove that, provided the line segment
connecting these points is outside the circle inscribed in S, this
ratio has the same asymptotics as the Delannoy number corresponding to
the locations of the defects; if the segment crosses the circle, the
asymptotics is radically different. We use this to deduce (under the
assumption that an arctic curve exists) that the arctic curve for
domino tilings of Aztec diamonds is the circle inscribed in S. We
also discuss counterparts of this phenomenon for lozenge tilings of
hexagons.
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