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Lozenge Tilings with Gaps in a 90° Wedge Domain with Mixed Boundary Conditions

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Abstract

We consider a triangular gap of side two in a 90° angle on the triangular lattice with mixed boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line boundary along the other. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its three images in the sides of the angle. This provides evidence for a unified way of understanding the interaction of gaps with the boundary under mixed boundary conditions, which we present as a conjecture. Our conjecture is phrased in terms of the steady state heat flow problem in a uniform block of material in which there are a finite number of heat sources and sinks. This new physical analogy is equivalent in the bulk to the electrostatic analogy we developed in previous work, but arises as the correct one for the correlation with the boundary.

The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi-parameter generalization of a classical plane partition enumeration problem (that of enumerating symmetric, self-complementary plane partitions).

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Mihai Ciucu

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  1. Mihai Ciucu
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Correspondence to Mihai Ciucu.

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Communicated by H. Spohn

Research supported in part by NSF Grant DMS-1101670.

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Ciucu, M. Lozenge Tilings with Gaps in a 90° Wedge Domain with Mixed Boundary Conditions. Commun. Math. Phys. 334, 507–532 (2015). https://doi.org/10.1007/s00220-014-2138-2

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  • DOI: https://doi.org/10.1007/s00220-014-2138-2

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