Arctic curves in path models from the tangent method

Philippe Di Francesco, Matthew F. Lapa

Research output: Contribution to journalArticlepeer-review


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
Issue number15
StatePublished - Mar 19 2018


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modeling and Simulation
  • Mathematical Physics
  • General Physics and Astronomy


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