Algebraic Combinatorics Seminar - Terrence George

Title: Arctic curves for groves

Abstract:

The limit shape phenomenon is a "law of large numbers" for random surfaces: the random surface looks macroscopically like the "average surface". The first result of this kind was the celebrated arctic circle theorem for domino tilings of the aztec diamond. The limit shape has macroscopic regions with different qualitative behavior, and the arctic curve is the boundary separating these regions. The work of Kenyon, Okounkov, Sheffield and others has shown that periodic lattices with non-trivial Newton polygons lead to rich arctic curves with many frozen and gaseous regions. Groves are another model, closely related to spanning trees, that exhibits an arctic circle theorem, due to Petersen and Speyer. We compute arctic curves for groves with non-trivial Newton polygons using analytic combinatorics results of Baryshnikov, Pemantle and Wilson, and provide a geometric description of asymptotic edge probabilities.