For fixed-q q-Racah tilings of a large hexagon, the 2D Gibbs measure collapses to a random 1D barcode staircase; the paper proves exponential concentration to a deterministic waterfall profile and conjectures the limiting kernel.
Non-intersecting paths and Hahn orthogonal polynomial ensemble
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abstract
We compute the bulk limit of the correlation functions for the uniform measure on lozenge tilings of a hexagon. The limiting determinantal process is a translation invariant extension of the discrete sine process, which also describes the ergodic Gibbs measure of an appropriate slope.
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Random Lozenge Waterfall: Dimensional Collapse of Gibbs Measures
For fixed-q q-Racah tilings of a large hexagon, the 2D Gibbs measure collapses to a random 1D barcode staircase; the paper proves exponential concentration to a deterministic waterfall profile and conjectures the limiting kernel.