Anh-Thai is a junior interested in statistical mechanics and probability. This summer, he looks forward to continuing some stat mech work with Professor Kenyon and exploring more of the New Haven area/East Coast.
Research proposal summary
Lattice models in statistical mechanics (e.g. dimer, six-vertex, Ising) are interesting not only as mathematical models of phase transitions, but also for the rich interplay between different areas of math, such as representation theory, combinatorics, and probability. Perhaps the models most interesting to mathematicians are “exactly solvable,” meaning one can compute closed-form expressions for relevant physical quantities (in the thermodynamic limit), such as the model’s free energy. The most ubiquitous tool for computing the free energy is the transfer matrix method, which has a storied history — for example, Onsager (a Yale professor!) used it to solve the two-dimensional Ising model without a magnetic field. This project explores the rich connection between the transfer matrix method and a well-known extension, the algebraic Bethe ansatz. While the transfer matrix encodes information about a model, the Bethe ansatz provides an educated guess for constructing the transfer matrix’s eigenvectors. Our starting point is the dimer model on the honeycomb lattice (equivalently tiling the plane with 60-degree rhombi). In this setting, the Bethe ansatz eigenvectors have a beautiful form, namely a Vandermonde determinant related to the Schur polynomials. From these eigenvectors, we are currently examining determinantal measures, where relevant probabilities are expressed as determinants of matrix minors. Along the way, there are other phenomena we hope to explore, such as limit shapes (that appear as one looks at random model configurations on a very large lattice) and emptiness formation probabilities (that allow for computations on more general domains).