Log-gases on a quadratic lattice via discrete loop equations and q-boxed plane partitions

Evgeni Dimitrov; Alisa Knizel

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arXiv · 1710.01709

Log-gases on a quadratic lattice via discrete loop equations and q-boxed plane partitions

Evgeni Dimitrov·Alisa Knizel

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Abstract

We study a general class of log-gas ensembles on (shifted) quadratic lattices. We prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance. We apply our general results to analyze the asymptotic behavior of a $q$-boxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations of the height function on a fixed slice are described by a one-dimensional section of a pullback of the two-dimensional Gaussian free field. Our approach is based on a $q$-analogue of the Schwinger-Dyson (or loop) equations, which originate in the work of Nekrasov and his collaborators, and extends the methods developed by Borodin, Gorin and Guionnet to quadratic lattices.

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Evgeni Dimitrov, Alisa Knizel. 2017-10-04. Log-gases on a quadratic lattice via discrete loop equations and q-boxed plane partitions. https://arxiv.org/abs/1710.01709

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