Speaker
Sigurdur Stefansson
(Nordita)
Description
I will introduce an equilibrium statistical mechanical model of
trees with a local action which depends only on the degrees
of vertices in
the tree. Related models have been studied extensively, in
different
forms, by mathematicians with a history dating back to
Galton and Watson
in the 19th century who were interested in calculating the
probability of
the extinction of family names. Physicist became interested
in the model
in connection with simplicial gravity where they observed
that a certain
phase of the gravity model had tree-like features. The model
has two
phases which are called generic (fluid, elongated) and
non-generic
(condensed, crumpled) phase. I will review recent results on
the generic
phase and present new results on the non-generic phase which
I worked on
in my Ph.D. studies with Thordur Jonsson. We proved the
existence of a
Gibbs measure on infinite trees obtained as a weak limit of
the finite
volume measures and showed that in the infinite volume limit
there arises
exactly one vertex of infinite degree and the rest of the
tree is
distributed like a sub-critical Galton-Watson tree.
