The 2021 Bowen Lectures will be delivered by Richard Kenyon, Yale University.
Each lecture starts at 4:10 p.m. and ends at 5:00 p.m. in Lewis Hall Room 100.
Title of the Series: Multinomial random tilings and other models.
Series Overview:
This is joint work with Cosmin Pohoata. Random tiling models are an important subarea of combinatorics, probability and statistical mechanics. Typically, however, they are quite difficult to analyze. We define a variant of the random tiling model, the "multinomial random tiling model" which is much easier to study. We find explicit and diverse behavior including phase transitions, crystallization and quasicrystallization, and conformal invariance. We can also define multinomial versions of other stat mech models such as the Ising or Potts model; these can be effectively analyzed on essentially arbitrary graphs, in contrast to standard versions of these models which are NP hard on most graphs.
Tuesday, October 5, 4:10-5 PM Lewis Hall Room 100.
Lecture 1: Multinomial random tilings.
Abstract:
We give definitions and combinatorial results. We relate the model to a certain high-dimensional random walk. We study the special case of multinomial “domino tilings” and its limit shapes.
Wednesday, October 6, 4:10-5 PM Lewis Hall Room 100.
Lecture 2: Fluctuations and conformal invariance.
Abstract:
We study the fluctuations of the tile densities in the model, showing that they are Gaussian and giving an explicit description of their covariance function. We study multinomial tilings in the plane with various sets of polyominos such as the L polyomino. We describe the associated crystals, quasicrystals and Gaussian scaling limits.
Friday, October 8 4:10-5 PM Lewis Hall Room 100.
Lecture 3: The multinomial Ising and Potts model
Abstract:
We define the models, compute the various critical temperatures on an arbitrary graph, and describe the fluctuations. We also describe the limit shapes in 1d and of the “Ising droplet” on Z^2.
Biography of Speaker:
Richard Kenyon received his Ph.D. from Princeton University in 1990 under the direction of William Thurston. After a postdoc at IHES, he held positions at CNRS in Grenoble, Lyon, and Orsay and then became professor at UBC, Brown University and then Yale where he is currently Erastus L. Deforest Professor of Mathematics. He was awarded the CNRS bronze medal, the Rollo Davidson prize, the Loève prize, is a member of the American Academy of Arts and Sciences, and is a Simons Investigator.