Abstract
We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there. For low temperatures, the disordered region consists of two disjoint ellipses. In the high temperature regime the two ellipses merge into a single simply connected region. At the transition from the low to the high temperature a tacnode appears. The key to our asymptotic study is a recent approach introduced by Duits and Kuijlaars providing a double integral representation for the correlation kernel. One of the factors in the integrand is the Christoffel–Darboux kernel associated to polynomials that satisfy non-Hermitian orthogonality relations with respect to a complex-valued weight on a contour in the complex plane. We compute the asymptotic behavior of these orthogonal polynomials and the Christoffel–Darboux kernel by means of a Riemann–Hilbert analysis. After substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments.
Highlights
We study random lozenge tilings of large regular hexagons
Note that if α = 1 all tilings occur with the same probability and the probability measure reduces to the uniform measure on all possible tilings
The main results in this paper concern the asymptotic behavior of the random tilings as the size of the hexagon grows large, i.e., as N → ∞, and how this asymptotic behavior depends on the parameter α
Summary
J. Kuijlaars: Supported by long term structural funding-Methusalem grant of the Flemish Government, and by FWO Flanders projects G.0864.16 and G.0910.20, and EOS 30889451.
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