Asymptotic analysis of random lozenge tilings on a vertical
Gaëtan Borot
Max Plank Bonn
Mon, Sep. 26th 2016, 11:00-12:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers

We consider the Jack measures on lozenge tilings of a polygonal 2d domain -- non-necessarily planar neither simply-connected. The induced distribution of lozenge on a vertical 1d section of this domain is described by a matrix model whose eigenvalues are restricted to a lattice, and repel each other with an index $\theta > 0$. We develop a rigorous asymptotic analysis of this system in the thermodynamic limit including finite size corrections, in particular obtain the asymptotic expansion of the partition function, and prove the analog of a central limit theorem for a position of the lozenge on the vertical. This analysis is made possible by the existence of a discrete analog of Schwinger-Dyson equations, found by Nekrasov in the more general context of supersymmetric gauge theories. \par This model has been studied in many previous works. In particular, Kenyon and Okounkov described the limit shape of the liquid region (in which tiling fluctuate) and conjectured Gaussian Free Field behavior -- which has been proved for tilings of the hexagon. Here we show for general domains (maybe non-planar) that the distribution on a vertical section is indeed the restriction of a suitable GFF on the spectral curve of the model, which we identify with Kenyon-Okounkov curve. Also for general regions, Eynard proposed that the asymptotic expansion of the partition function is described by the topological recursion. We show explicitly that this conjecture has to be amended by correction terms, which is analog to passing from a rational to a trigonometric expression although their structure to all orders is yet unclear. \\ \\ This is based on a work in progress with Gorin and Guionnet.

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