Probability and Mathematical Physics Seminar
Probability and the City Seminar (at Columbia)
Speaker: Tomas Berggren (KTH) and Mateusz Piorkowski (KTH)
Location: Columbia University, Mathematics Hall, 2990 Broadway 417
Date: Friday, February 28, 2025, 10 a.m.
Synopsis:
Tomas Berggren (10-11):
Title: Gaussian free field and discrete Gaussians in periodic dimer models
Abstract: Random dimer models (or equivalently random tiling models) have been extensively studied in mathematics and physics for several decades. A fundamental result is that the associated height function converges, in the large-scale limit, to a deterministic surface known as the limit shape. In this talk, we discuss fluctuations of the height function around the limit shape in the presence of smooth (or gas) regions, specifically on the doubly periodic Aztec diamond dimer model. We show that the height function approximates the sum of two independent components: a Gaussian free field on the multiply connected rough region and a harmonic function with random rough-smooth boundary values. These boundary values are jointly distributed as a discrete Gaussian random vector which maintains a quasi-periodic dependence on $N$. Joint work with Matthew Nicoletti.
Mateusz Piorkowski (11:15-12:15):
Title: Arctic curves of periodic dimer models and generalized discriminants
Abstract: In this talk I will discuss arctic curves of the k x l-periodic Aztec diamond recently studied by Berggren & Borodin '23 (arXiv:2306.07482). I will show how polynomial equations for these arctic curves can be obtained using theta function of the associated spectral curve. As a corollary we obtain a simple formula for their degree in terms of the number of smooth (or gaseous) and frozen regions. Similar formulas also hold for certain tiling models of the hexagon studied by Bobenko & Bobenko '24 (arXiv:2407.19462).
The key to this result is a generalization of the classical discriminant of a polynomial to the setting of meromorphic sections on compact Riemann surfaces. This talk is based on arXiv:2410.17138.