Random tilings : gap probabilities, local and global asymptotics

In the thesis we explore and develop two different approaches to the study of random tiling models. First, we consider tilings of a hexagon by rombi, viewed as 3D random stepped surfaces with a measure proportional to q-volume. Such model is closely related to q-Hahn orthogonal polynomial ensembles, and we use this connection to obtain results about the local behavior of this model. In terms of the q-Hahn orthogonal polynomial ensemble, our goal is to show that the one-interval gap probability function can be expressed through a solution of the asymmetric q-Painleve V equation. The case of the q-Hahn ensemble we consider is the most general case of the orthogonal polynomial ensembles that have been studied in this context. Our approach is based on the analysis of q-connections on P1 with a particular singularity structure. It requires a new derivation of a q-difference equation of Sakai's hierarchy [75] of type A(1)/2. We also calculate its Lax pair. Following [7], we introduce the notion of the [tau]-function of a q-connection and its isomonodromy transformations. We show that the gap probability function of the q-Hahn ensemble can be viewed as the [tau]-function for an associated q-connection and its isomonodromy transformations. Second, in collaboration with Alexey Bufetov we consider asymptotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a special class of examples, an explicit parametrization of the frozen boundary is given. It turns out to be an algebraic curve with very special properties. Moreover, we establish the convergence of the fluctuations of the height functions to the Gaussian Free Field in appropriate coordinates. Our main tool is a recently developed moment method for discrete particle systems.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2017

Author dc:creator

• Knizel, Alisa

Advisor dc:contributor.advisor

• Alexei Borodin.

• Mathematics.

Statement dc:rights

• MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/112900

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/112900