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Annales de l’Institut Henri Poincaré D

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Volume 8, Issue 1, 2021, pp. 1–33
DOI: 10.4171/AIHPD/96

Published online: 2020-12-10

Dominos in hedgehog domains

Marianna Russkikh[1]

(1) Massachusetts Institute of Technology, Cambridge, USA

We introduce a new class of discrete approximations of planar domains that we call “hedgehog domains”. In particular, this class of approximations contains two-step Aztec diamonds and similar shapes. We show that fluctuations of the height function of a random dimer tiling on hedgehog discretizations of a planar domain converge in the scaling limit to the Gaussian Free Field with Dirichlet boundary conditions. Interestingly enough, in this case the dimer model coupling function satisfies the same Riemann-type boundary conditions as fermionic observables in the Ising model.

In addition, using the same factorization of the double-dimer model coupling function as in [18], we show that in the case of approximations by hedgehog domains the expectation of the double-dimer height function is harmonic in the scaling limit.

Keywords: Lattice models, Dimer model, height function, GFF

Russkikh Marianna: Dominos in hedgehog domains. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), 1-33. doi: 10.4171/AIHPD/96