TY - JOUR

T1 - Arctic curves in path models from the tangent method

AU - Di Francesco, Philippe

AU - Lapa, Matthew F.

N1 - Funding Information:
We are thankful to F Colomo and A Sportiello for extensive discussions on the tangent method. PDF thanks the organizers of the program ‘Combi17: Combinatorics and interactions’ held at the Institut Henri Poincaré, Paris where the present work originated. MFL would like to acknowledge the support of his advisor Taylor Hughes, and support from NSF grant DMR 1351895-CAR. MFL thanks the Galileo Galilei Institute in Florence for hospitality during the 2017 ‘Lectures on Statistical Field Theories’ winter school program where the first stages of the present project were carried out, and also the organizers of the 2017 ‘Exact methods in low-dimensional physics’ summer school at Institut d’Etudes Scientifiques in Cargèse. PDF is partially supported by the Morris and Gertrude Fine endowment.
Publisher Copyright:
© 2018 IOP Publishing Ltd.

PY - 2018/3/19

Y1 - 2018/3/19

N2 - Recently, Colomo and Sportiello introduced a powerful method, known as the tangent method, for computing the arctic curve in statistical models which have a (non- or weakly-) intersecting lattice path formulation. We apply the tangent method to compute arctic curves in various models: the domino tiling of the Aztec diamond for which we recover the celebrated arctic circle; a model of Dyck paths equivalent to the rhombus tiling of a half-hexagon for which we find an arctic half-ellipse; another rhombus tiling model with an arctic parabola; the vertically symmetric alternating sign matrices, where we find the same arctic curve as for unconstrained alternating sign matrices. The latter case involves lattice paths that are non-intersecting but that are allowed to have osculating contact points, for which the tangent method was argued to still apply. For each problem we estimate the large size asymptotics of a certain one-point function using LU decomposition of the corresponding Gessel-Viennot matrices, and a reformulation of the result amenable to asymptotic analysis.

AB - Recently, Colomo and Sportiello introduced a powerful method, known as the tangent method, for computing the arctic curve in statistical models which have a (non- or weakly-) intersecting lattice path formulation. We apply the tangent method to compute arctic curves in various models: the domino tiling of the Aztec diamond for which we recover the celebrated arctic circle; a model of Dyck paths equivalent to the rhombus tiling of a half-hexagon for which we find an arctic half-ellipse; another rhombus tiling model with an arctic parabola; the vertically symmetric alternating sign matrices, where we find the same arctic curve as for unconstrained alternating sign matrices. The latter case involves lattice paths that are non-intersecting but that are allowed to have osculating contact points, for which the tangent method was argued to still apply. For each problem we estimate the large size asymptotics of a certain one-point function using LU decomposition of the corresponding Gessel-Viennot matrices, and a reformulation of the result amenable to asymptotic analysis.

KW - arctic curve

KW - continuum limit

KW - non-intersecting lattice paths

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U2 - 10.1088/1751-8121/aab3c0

DO - 10.1088/1751-8121/aab3c0

M3 - Article

AN - SCOPUS:85046959990

VL - 51

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 15

M1 - 155202

ER -