For q-Volume lozenge tilings with q = e^{c/(2N)}, the liquid-region boundary converges to the Airy process away from the hexagon boundary, and at inflection points no parabola term appears over tangent distances o(N^{-2/9}).
Aggarwal, Universality for lozenge tiling local statistics, Ann
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The $q^{\mathrm{Volume}}$ lozenge tiling model via non-Hermitian orthogonal polynomials
For q-Volume lozenge tilings with q = e^{c/(2N)}, the liquid-region boundary converges to the Airy process away from the hexagon boundary, and at inflection points no parabola term appears over tangent distances o(N^{-2/9}).