Orateur
Andrea Sportiello
(CNRS and LIPN, Université Paris 13)
Description
Some models in two-dimensional statistical mechanics, in suitable
domains, show the emergence of phase-separation phenomena, and in
particular of an "arctic curve" separating a frozen and a liquid
region. Models in this class are characterised by the presence of a
conserved quantity, whose flow is directed (these 2D models are, in a
hidden way, 1+1-dimensional).
Some of these models are either "fully" solvable (they are
free-fermions, and local observables of certain fields form a
determinantal processes), or are nonetheless solvable "up to a certain
extent", because they are Yang-Baxter integrable. In the first case,
there are nowadays powerful techniques to determine the associated
Arctic curves, mostly due to Kenyon and Okounkov, we will not discuss
this at depth here. In the second case, a theory is moving its first
steps. In some lucky cases, a simple strategy, that we (F. Colomo and
myself) call "the Tangent Method", allows to determine these curves.
It is based on the study of modified domains, in which one unit of
flow of the conserved quantity has its endpoint on the boundary pinned
away from its usual location. The resulting directed path has a simple
behaviour in the frozen region, and the associated modified partition
functions (called "boundary observables") are often calculable. These
ingredients can be mixed in several ways, to produce algebraic systems
for the Arctic curve, in terms of the boundary observables. The
various incarnations of this strategies are the main topic of the
lectures, and are illustrated through a variety of examples.
Auteur principal
Andrea Sportiello
(CNRS and LIPN, Université Paris 13, Villetaneuse)
