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LA Probability Forum

Thursday, February 6, 2025
3:00pm to 4:00pm
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The Doubly Periodic Aztec Diamond Dimer Model: Gaussian Free Field and Discrete Gaussians
Matthew Nicoletti, Department of Mathematics, Stanford University,

USC Kaprielian (KAP) 414

The dimer model provides a large family of random surface models whose scaling limits exhibit spatial phase separation, and have universal properties such as conformal invariance. The Aztec diamond dimer model in particular has been extensively researched due to its integrability, in order to more precisely understand the various universal behaviors of dimer models. While dimer model height fluctuations have been computed in many cases, until now the exact characterization of height fluctuations of a dimer model with gaseous facets appearing in the bulk has remained open.

We analyze height fluctuations in Aztec diamond dimer models with nearly arbitrary periodic edge weights, which lead to the formation of gaseous facets in the limit shape. We show that the centered height function approximates the sum of two independent components: a Gaussian free field on the multiply connected rough region and a harmonic function with random, lattice-valued rough-gas boundary values. The boundary values are jointly distributed as a discrete Gaussian random vector. This discrete Gaussian distribution maintains a quasi-periodic dependence on~$N$, a phenomenon also observed in multi-cut random matrix models.

This is joint work with Tomas Berggren.

For more information, please contact Math Dept by phone at 626-395-4335 or by email at mathinfo@caltech.edu.