abstract:
In the last three years there has been an intense activity on the mathematical aspects of spin-glasses, which lead to establish the rigorous proof of Parisi formula for the Sherrington-Kirkpatrick model, through works of F. Guerra (who proved an upper bound for the pressure) and M.Talagrand (who derived the corresponding lower bound). The upper bound has been put in a more general framework of "extended variational principle" by Aizenman, Sims and Star.
We will show how the Generalized Random Energy Model fits into this scheme.
The talk is based on a joint work with
Shannon Star.
abstract:
If we pair every face of the
triangular lattice with one of its neighbors, we gets a tiling of the plane
with rhombi. One can ask how to construct probability measures on all
possible tilings obtained that way.
Following the work of Kenyon, Okounkov and Sheffield, we will explain how to
construct such measures and describe their phase diagram.
We will then describe the behaviour of these measures in the neighborhood of
the phase transition, that is related to the random matrix theory.