SÚminaire du 12 octobre 2009,
14h00: Filippo Colomo, UniversitÓ
di Firenze, Italy.
The limit shape of large Alternating Sign Matrices.
The problem of the limit shape of q-enumerated (0<q<5)
large Alternating Sign Matrices (ASMs) - which can also be rephrased
as the problem of the Arctic curve for interacting-domino tilings
of the Atzec Diamond - is addressed by exploiting the well-known
correspondence with the domain-wall six vertex model.
In particular we analyse the scaling limit behaviour of a multiple
representation of the Emptiness Formation Probability in the
We conjecture that the limit shape can be characterized by the
condition of condensation of almost all roots of the corresponding
saddle-point equations at the same, known, value.
Under this assumption, we are able to derive the limit shape of
q-enumerated (0<q<5) large ASMs. It is expressed
in parametric form, and it appears to be a
non-algebraic curve in general; it turns into an algebraic one
in the so-called root-of-unity cases (which includes, among others,
the famous particular cases: q = 1, 2, 3).
Last modified: Mon Oct 12 14:27:37 CEST 2009