| Abstract: |
| It is well-known that important statistical quantities, such as gap probabilities, in various discrete probabilistic models of random matrix type satisfy the so-called discrete Painlev\`e equations, which provides an effective way to computing gap probabilities. In this talk we discuss this correspondence for a particular class of models, known as boxed plane partitions (equivalently, lozenge tilings of a hexagon). For uniform probability distribution, this is one of the most studied models of random surfaces. Borodin, Gorin, and Rains showed that it is possible to assign a very general elliptic weight to the distribution, with various degenerations of this weight corresponding to the degeneration cascade of discrete polynomial ensembles, such as Racah and Hahn ensembles and their q-analogues. This also correspond to the degeneration scheme of discrete Painlev\`e equations, due to Sakai. The connection between the two is given by the isomonodromy theory. In this project we use geometry to study this correspondence, as well as the degeneration. In particular, we show that gap probabilities in the q-Racah tiling model can be computed in terms of discrete Painlev\`e equation of type E_{7}^{(1)}. This, in turn, gives us a new symmetric Lax pair for this equation. |
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