Arctic curves in path models from the tangent method

Philippe Di Francesco, Matthew F. Lapa

Research output: Contribution to journalArticle

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 languageEnglish (US)
Article number155202
JournalJournal of Physics A: Mathematical and Theoretical
Volume51
Issue number15
DOIs
StatePublished - Mar 19 2018

Fingerprint

tangents
Tangent line
Alternating Sign Matrices
Rhombus
Lattice Paths
Path
Curve
curves
Tiling
Asymptotic analysis
matrices
Domino Tilings
Point contacts
Dyck Paths
LU decomposition
parabolas
Parabola
hexagons
Diamonds
Ellipse

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

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

In: Journal of Physics A: Mathematical and Theoretical, Vol. 51, No. 15, 155202, 19.03.2018.

Research output: Contribution to journalArticle

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