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Abstract
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We study a model of fully packed dimer configurations (or perfect matchings) on a
bipartite periodic graph that is two-dimensional but not planar. The graph is obtained
from
via the addition of an extensive number of extra edges that
break planarity (but not bipartiteness). We prove that, if the
weight
of the nonplanar edges is small enough, a suitably defined height
function scales on large distances to the Gaussian free field with a
-dependent
amplitude, that coincides with the anomalous exponent of dimer-dimer correlations.
Because of nonplanarity, Kasteleyn’s theory does not apply: the model is not
integrable. Rather, we map the model to a system of interacting lattice fermions in
the Luttinger universality class, which we then analyze via fermionic renormalization
group methods.
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Keywords
dimer model, nonplanar dimers, height function, GFF scaling
limit, renormalization group
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Mathematical Subject Classification
Primary: 82B20, 82B28, 82B41
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Milestones
Received: 10 August 2022
Revised: 9 July 2023
Accepted: 14 August 2023
Published: 29 November 2023
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© 2023 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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