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.
Publié le :
DOI : 10.4171/aihpd/96
Mots-clés : Lattice models, Dimer model, height function, GFF
@article{AIHPD_2021__8_1_1_0,
author = {Russkikh, Marianna},
title = {Dominos in hedgehog domains},
journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
pages = {1--33},
volume = {8},
number = {1},
year = {2021},
doi = {10.4171/aihpd/96},
mrnumber = {4228618},
zbl = {1467.82025},
language = {en},
url = {http://www.numdam.org/articles/10.4171/aihpd/96/}
}
Russkikh, Marianna. Dominos in hedgehog domains. Annales de l’Institut Henri Poincaré D, Tome 8 (2021) no. 1, pp. 1-33. doi : 10.4171/aihpd/96. http://www.numdam.org/articles/10.4171/aihpd/96/
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