Columbia / Courant Joint Probability Seminar Series:

Random Tilings

Columbia University, February 27, 2015



Coffee, tea and light breakfast (room 508)
Vadim Gorin [Multilevel Dyson Brownian Motion and its edge limits]
Richard Kenyon[Fixed-energy harmonic functions]
Greta Panova [Statistical mechanics via asymptotics of symmetric functions]

Practical Information
The talks will take place at Columbia University Mathematics building in room 520, on February 27, 2015. No registration necessary. However, if you would like to attend the group lunch after the talks, please RSVP.

Directions to Columbia. Event poster.

For further information, please contact the organizers.

Titles and Abstracts

Vadim Gorin (MIT)
Multilevel Dyson Brownian Motion and its edge limits

Abstract: The GUE Tracy-Widom distribution is known to govern the large-time asymptotics for a variety of interacting particle systems on one side, and the asymptotic behavior for largest eigenvalues of random Hermitian matrices on the other side. In my talk I will explain some reasons for this connection between two seemingly unrelated classes of stochastic systems, and how this relation can be extended to general beta random matrices. A multilevel extension of the Dyson Brownian Motion will be the central object in the discussion. (Based on joint papers with Misha Shkolnikov).

Richard Kenyon (Brown)
Fixed-energy harmonic functions

Abstract: We solve the Dirichlet problem for networks, fixing energies rather than fixing conductances. More precisely, we show that for any given choice of edge energies there is a choice of conductances for which the resulting harmonic function realizes those energies. In fact the set of solutions is the number of compatible acyclic orientations of the graph. As an application we study fixed-area rectangulations of planar domains. This is joint work with Aaron Abrams.

Greta Panova (U. Penn)
Statistical mechanics via asymptotics of symmetric functions

Abstract: What do lozenge tilings (a.k.a. plane partitions, dimer covers of the hexagonal lattice), alternating sign matrices (or the 6-vertex model) and the dense loop model have in common? For one, their limiting behavior can be studied with the help of some "asymptotic" algebraic combinatorics.
We develop methods to analyze normalized symmetric functions (Schur functions and more general Lie group characters), as the indexing partition converges to a limiting profile. We apply this analysis together with some combinatorial interpretations to study the limiting behavior of the integrable models listed above. In particular, we show that the positions of horizontal lozenges near a vertical flat boundary are distributed like the eigenvalues of GUE matrices, and this holds for a wide class of domains (including such with free boundary). These methods can also be used to establish the existence of limit shapes also for free boundary domains. We discover Gaussian distribution for some observables of the Alternating Sign Matrices, leading again to GUE eigenvalues for the positions of 1s near the border (result of V. Gorin). We also find the asymptotics for the [conjectural] expected value of the mean total current between two adjacent points in the dense loop model.
Based on joint work with Vadim Gorin.