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Seminario del 2021
2021
04 novembre
Andrea Sportiello, CNRS, and LIPN, Université Sorbonne Paris Nord, Villetaneuse, France
Seminario di algebra e geometria, fisica matematica, probabilità
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.