### Abstract

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.

Original language | English (US) |
---|---|

Article number | 155202 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 51 |

Issue number | 15 |

DOIs | |

State | Published - Mar 19 2018 |

### Fingerprint

### Keywords

- arctic curve
- continuum limit
- non-intersecting lattice paths

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*51*(15), [155202]. https://doi.org/10.1088/1751-8121/aab3c0

**Arctic curves in path models from the tangent method.** / Di Francesco, Philippe; Lapa, Matthew F.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 51, no. 15, 155202. https://doi.org/10.1088/1751-8121/aab3c0

}

TY - JOUR

T1 - Arctic curves in path models from the tangent method

AU - Di Francesco, Philippe

AU - Lapa, Matthew F.

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

UR - http://www.scopus.com/inward/record.url?scp=85046959990&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046959990&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aab3c0

DO - 10.1088/1751-8121/aab3c0

M3 - Article

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 -