Abstracts:

Dmitri Znamenski

Title: "On the variety of critical thresholds in the Bak-Sneppen evolution model."

Keywords:
Interacting Particle Systems, Self-Organized Criticality, differential equation.

In the presentation we will consider a collection of Bak-Sneppen like models and a number of recent results concerning the original model on the circle.   The last one can be defined as follows. Each vertex of a regular polygon accommodate a random variable `state' (fitness) with value in (0,1). At the initial moment all the states are independent and U[0,1] distributed. We update the states in a Markov way. Every time step we choose a vertex with minimal state and give to this vertex and to its neighbors three new independent states with distribution U[0,1].   The principle open problems related to the model are to determine the limit distribution F when the number of vertices tends to infinity and to prove the power low of some characteristics.   An important step to understanding of Bak-Sneppen model is to consider it as a sequence of avalanches from a given threshold in (0,1). We will introduce a number of critical thresholds and will show how their uniqueness determines the shape of the limit distribution F.

F.L.Toninelli

Mean field spin glasses: rigorous results from
interpolation methods

In the study of mean field spin glasses, for instance the
well known Sherrington-Kirkpatrick model, very simple
interpolation techniques have proven to be able to give
highly non trivial results. I will
explain the main idea and outline some applications:
existence of the thermodynamic limit, control of the high
temperature region and central limit theorem for the
fluctuations,
broken replica symmetry bounds for the free energy at low
temperature.
Recent extensions to finite connectivity mean field
models and random optimization problems will be also
considered.
This is based on joint work with Francesco Guerra.