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Presented By: Integrable Systems and Random Matrix Theory Seminar - Department of Mathematics

ISRMT Seminar: Non-Hermitian orthogonality in the q^(Volume) tiling model

Ahmad Barhoumi (KTH)

Sample lozenge tiling of the hexagon. Generated using code kindly provided by Christophe Charlier. Sample lozenge tiling of the hexagon. Generated using code kindly provided by Christophe Charlier.
Sample lozenge tiling of the hexagon. Generated using code kindly provided by Christophe Charlier.
Consider a hexagon overlayed on a regular triangular grid, where the equilateral triangles have sidelength 1. A "lozenge" is a pair of adjacent triangles, and there are three types of lozenges that can be built on this grid. Using these lozenges, one can tile the entire hexagon. Lozenge tilings of a regular hexagon are in bijection with boxed plane partitions (i.e. stacks of boxes in the back of a cubic room) and can therefore be assigned a volume; a fact that is best illustrated by staring at a picture of one such tiling. The q^(Volume) tiling model is a measure on the space of tilings of the hexagon which assigns to each tiling a probability proportional to q^(Volume), where q is a real parameter. In this talk, I will recall the model and basic result about it and propose an approach to studying its statistical properties as the size of the hexagon grows by analyzing a related family of non-Hermitian orthogonal polynomials. The talk is based on ongoing joint work with Maurice Duits.
Sample lozenge tiling of the hexagon. Generated using code kindly provided by Christophe Charlier. Sample lozenge tiling of the hexagon. Generated using code kindly provided by Christophe Charlier.
Sample lozenge tiling of the hexagon. Generated using code kindly provided by Christophe Charlier.

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