We consider a polymer measure based on random walks which are based on sums of iid stable random variables.
A Gibbs measure is defined which models an attraction to the origin for these walks. A phase transition occurs as the the strength of the attraction to the origin occurs.
We examine various "thermodynamic" quantities and show they are all related to each other in a simple way and exhibit universality.

I will present a criterion for the validity of the central limit theorem
for a class of dependent random variables and then I will discuss some applications of
it on random, boundary homogenization problems of nonlinear PDEs such nonlinear
parabolic ones and Navier walls.

Abstract: consider a finite graph, with an actor sitting at each node, and a
dollar on each edge. Negotiations will be conducted between pairs of
adjacent actors over splitting the dollar on the intervening edge.
At the end of negotiations, each actor may sign at most one contract with a
neighbour, agreeing on some possibly uneven split of the dollar.
How much money is each actor likely to receive? And which matchings of the
graph are likely to arise?
Kleinberg and Tardos analysed the limiting answer - a balanced solution -
that arises from assuming that actors iteratively revise current deals using
Nash bargaining, taking the best alternative deal currently available as a
backup.

Most of the talk will be expository, I'll explain the concepts of Nash bargaining and balanced solution. If there is time, I will discuss
the rate of
convergence to the balanced solution of this type of negotiation.