The tangent method for the determination of Arctic Curves: the simplest rigorous application
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], by 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.
In this paper many things are claimed, and few are proven. In
particular a few questions remain only vaguely answered:
* how rigorous is this method?
* in which cases does it apply, rigorously or heuristically?
* in the cases where other methods exist, how does it compare?
We will try to answer to this partially, by giving a "top-to-bottom"
rigorous derivation for the simplest and oldest case: the arctic
circle phenomenon for "domino tilings of the aztec diamond", first
discovered by Jockusch, Propp and Shor [arXiv:math/9801068, but in
fact from 1995]. We suppose that, of the nowadays many possible
derivations of the arctic circle phenomenon, those coming from the
tangent method (and restricted to the rigorous versions of it) are the
fastest and cheapest ones. The audience will judge...