Partition function of periodic isoradial dimer models

Isoradial dimer models were introduced in Kenyon (Invent Math 150(2):409–439, 2002)— they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of (Kenyon in Invent Math 150(2):409–439, 2002), namely that for periodic isoradial dimer models, the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only. This is a surprising feature of periodic isoradial dimer models, which does not hold in the general periodic dimer case (Kenyon et al. in Ann Math, 2006). DOI: https://doi.org/10.1007/s00440-006-0041-2 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-21580 Accepted Version Originally published at: de Tilière, B (2007). Partition function of periodic isoradial dimer models. Probability Theory and Related Fields, 138(3-4):451-462. DOI: https://doi.org/10.1007/s00440-006-0041-2 ar X iv :m at h/ 06 05 58 3v 1 [ m at h. PR ] 2 2 M ay 2 00 6 Partition function of periodic isoradial dimer models Béatrice de Tilière ∗ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich. [email protected] Abstract Isoradial dimer models were introduced in [12] they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of [12], namely that for periodic isoradial dimer models, the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only. This is a surprising feature of periodic isoradial dimer models, which does not hold in the general periodic dimer case [14].Isoradial dimer models were introduced in [12] they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of [12], namely that for periodic isoradial dimer models, the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only. This is a surprising feature of periodic isoradial dimer models, which does not hold in the general periodic dimer case [14].