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Christophe Charlier (KTH Stockholm) : "Periodic weights for lozenge tilings of hexagons"
CYCL 05 (Marc de Hemptinne)
Marc de Hemptinne
The model of lozenge tilings of a hexagon is equivalent to a model of non-intersecting paths on a discrete lattice, i.e. the positions (or heights) of these paths, as well as their domains (times), are discrete.
We will show how this correspondance works, supported by pictures.
The discrete lattice can be viewed as a graph where we assign a weight on the arrows joining the vertices.
In the simplest case, the weight function is periodic of period 1 in both space and time directions.
We review some known results in this case.
Then, we introduce some larger periodicities in the weight.
We will focus on the case when the weight is periodic of period 2 in both space and time directions.
The heights of the paths at a given time in a determinantal point process.
In the second part of the talk, based on a recent preprint by M. Duits and A.B.J. Kuijlaars, we introduce matrix valued orthogonal polynomial of size 2x2 related to a 4x4 Riemann-Hilbert problem (RHP), and we show how the correlation kernel can be expressed in terms of this RHP.
We will show some steps in the Deift/Zhou steepest descent which does not appear in usual steepest descent for 2x2 RHPs.
We will finish by showing some pictures when we allow higher periodicities in the weight.
This talk is intended to be accessible to non-specialists, and is based on a work in progress with M. Duits, A.B.J. Kuijlaars and J. Lenells.