2019/06/30 by Sunil Chhita, Fabio Lucio Toninelli
Mathematics · Physics and Astronomy · #Algorithm #Anisotropy #Combinatorics #Computation #Domino #Geometry #Limit (mathematics) #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Physics #Quantum mechanics #Shuffling #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR
paper · pdf · doi:10.5802/ahl.95
published as Annales Henri Lebesgue vol. 4 (2021), 1005-1034 · 30 pages 9 figures; v3: minor changes
arxiv created 2020/10/30 · openalex publication_date 2021/08/26 · arxiv updated 2021/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
The domino-shuffling algorithm [EKLP92a, EKLP92b, Pro03] can be seen as a stochastic process describing the irreversible growth of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional discrete interface [CT19, Zha18]. Its stationary speed of growth <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mi mathvariant="monospace">w</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> depends on the average interface slope <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> , as well as on the edge weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="monospace">w</mml:mi> </mml:math> , that are assumed to be periodic in space. We show that this growth model belongs to the Anisotropic KPZ class [Ton18, Wol91]: one has <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix" movablelimits="true">det</mml:mo> <mml:mo>[</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi>v</mml:mi> <mml:mi mathvariant="monospace">w</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>]</mml:mo> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and the height fluctuations grow at most logarithmically in time. Moreover, we prove that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:msub> <mml:mi>v</mml:mi> <mml:mi mathvariant="monospace">w</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is discontinuous at each of the (finitely many) smooth (or “gaseous”) slopes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> ; at these slopes, fluctuations do not diverge as time grows. For a special case of spatially <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> periodic weights, analogous results have been recently proven [CT19] via an explicit computation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mi mathvariant="monospace">w</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . In the general case, such a computation is out of reach; instead, our proof goes through a relation between the speed of growth and the limit shape of domino tilings of the Aztec diamond.