In a classical work of 1952, Lee and Yang proved that zeros of certain polynomials (the "partition functions" of Ising models) always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. However, it is not an easy task to describe this distribution, so a little is known about it beyond the one-dimensional model.
We have studied distribution of the Lee-Yang zeros for a special "Diamond Hierarchical Lattice" which can be viewed as an approximation to the regular 2D lattice. In this case, it can be described in terms of dynamics of the "renorm-group transformation" which is given explicitly as a rational map in two variables. This map turns out to be partially hyperbolic on an invariant cylinder, which implies that the Lee-Yang zeros are organized in a transverse measure for its central foliation. From the global complex point of view, these measures get interpreted as slices of the Green (1,1)-current on the projective space.
In the mini-course we will first lay down a background on the Ising models, the Lee-Yang zeros, and renormalization. We will then discuss basics of the dynamical pluri-potential theory and explain its relation to the Lee-Yang zeros. We will complete with a discussion of the notion of partial hyperbolicity giving a proof of the above stated results. It is a mixture of methods from real and complex dynamics that exploits ideas of Kobayashi hyperbolicity and geometry of algebraic curves.