Random lozenge tilings of polygons and their asymptotic behavior

Random lozenge tilings of polygons and their asymptotic behavior

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Leonid Petrov , Northeastern University
Fine Hall 601

I will discuss the model of uniformly random tilings of polygons drawn on the triangular lattice by lozenges of three types (equivalent formulations: dimer models on the honeycomb lattice, or random 3-dimensional stepped surfaces). Asymptotic questions about these tilings (when the polygon is fixed and the mesh of the lattice goes to zero) have received a significant attention over the past years. Using a new formula for the determinantal correlation kernel of the model, we manage to establish the conjectural local asymptotics of random tilings in the bulk (leading to ergodic translation invariant Gibbs measures on tilings of the whole plane), and the predicted behavior of interfaces between so-called liquid and frozen phases (leading to the Airy process). Bulk asymptotic behavior allows to reconstruct the limit shapes of random stepped surfaces obtained by Cohn, Propp, Kenyon, and Okounkov. We also prove a conjecture of Kenyon (2004) that the large-scale asymptotics of random tilings are asymptotically governed by the Gaussian Free Field.  As a particular case, all our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).