Arctic curves in path models from the tangent method

作者：Di Francesco Philippe^*; Lapa Matthew F

来源：Journal of Physics A-Mathematical and Theoretical, 2018, 51(15): 155202.

DOI：10.1088/1751-8121/aab3c0

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

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.

• 出版日期2018-4-13
• 单位中国地震局