Geometry of the doubly periodic Aztec dimer model

The purpose of the present work is to provide a detailed asymptotic analysis of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k times script l"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>×</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">k\times \ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> doubly periodic Aztec diamond dimer model of growing size for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and under mild conditions on the edge weights. We explicitly describe the limit shape and the ‘arctic’ curves that separate different phases, as well as prove the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures away from the arctic curves. We also obtain a homeomorphism between the rough region and the amoeba of an associated Harnack curve, and illustrate, using this homeomorphism, how the geometry of the amoeba offers insight into various aspects of the geometry of the arctic curves. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. Our framework essentially relies on three somewhat distinct areas: (1) Wiener-Hopf factorization approach to computing dimer correlations; (2) Algebraic geometric ‘spectral’ parameterization of periodic dimer models; and (3) Finite-gap theory of linearization of (non-linear) integrable partial differential and difference equations on the Jacobians of the associated algebraic curves. In addition, in order to access desired asymptotic results we develop a novel approach to steepest descent analysis on Riemann surfaces via their amoebas.