Arctic Curves In Path Models from The Tangent Method

Philippe Di Francesco (1, 2) , Matthew F. Lapa (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 Department of Mathematics [Urbana]

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.

Domaines: Physique [physics]

Philippe Di Francesco, Matthew F. Lapa. Arctic Curves In Path Models from The Tangent Method. Journal of Physics A: Mathematical and Theoretical, 2018, 51, pp.155202. ⟨10.1088/1751-8121/aab3c0⟩. ⟨cea-01692535⟩