abstract:
Maps are
combinatorial objects arising in physics as the natural discretization of
random surfaces used for instance in two-dimensional quantum gravity, as
well as in matrix models. After briefly reviewing these relations, I shall
present some elementary constructions, that establish correspondences
between maps and (embedded) trees. A first application is the enumeration of
maps: several results can be (re)derived, and lead to the evaluation of
critical exponents for various models of random surfaces. In a second time,
I shall show how the constructions can be used to access to a fine
information on the intrinsic geometry of maps. Analytical exact results can
be obtained thanks to an unexpected "integrability" property.
abstract:
Linearly interacting diffusions model the evolution of colonies of populations. When the interaction kernel is of an appropriate form and the diffusions are indexed by the hierarchical group, the large scale space-time behavior of the system exhibits universality which can be fully characterized using a renormalization analysis first developed by Dawson and Greven. Such renormalization analysis has previously been successfully carried out for interacting diffusions which are either one-dimensional or live on a compact state space. In most of these cases, there is a single type of large scale space-time behavior and the system exhibits so-called full universality.
In this talk, we give an overview of
the renormalization program and then discuss recent progress in the analysis
of hierarchically interacting two-type branching diffusions, where each
diffusion lives on [0,\infty)^2 and the structure of the system's large
scale space-time behavior is much richer. This is work in progress joint
with D. Dawson, A. Greven, F. den Hollander and J. M. Swart.