The Tangent Method for the determination of Arctic Curves in Statistical Mechanics models

In the paper `Arctic curves of the six-vertex model on generic domains: the Tangent Method' [J. Stat. Phys. 164 (2016) 1488, arXiv:1605.01388], of Filippo Colomo and myself, we pose the basis for a method aimed at the determination of the `arctic curve' of large random combinatorial structures, i.e. the boundary between regions with zero and non-zero local entropy, in the scaling limit. This basic version of the Tangent Method (TM) is strikingly simple, but unfortunately it is not completely rigorous. Two other versions of the method exist, let us call them the `entropic' Tangent Method (E-TM) and the `double-refinement' Tangent Method (2R-TM). In this talk we shall first briefly review the basic TM, then we will introduce the two other methods and explain how the 2R-TM is completely rigorous, but it involves more complex quantities, while the E-TM has essentially the same technical difficulties of the TM, but it is even more heuristic. Finally, we close the circle, by showing how the Desnanot-Jacobi identity applied to the Izergin determinant implies the equivalence between the E-TM and the 2R-TM in the case of the six-vertex model with domain-wall boundary conditions.