Self-avoiding walks on hyperbolic graphs

The self-avoiding walk is a model of statistical physics which has been studied extensively on the hypercubic lattice ${\mathbb{Z}}^d$. Over the last few decades, the study of self-avoiding walk beyond ${\mathbb{Z}}^d$ has received increasing attention. In this talk, we will consider the case of regular tessellations of the hyperbolic plane. We will show that there are exponentially fewer self-avoiding polygons than self-avoiding walks, and we will deduce that the self-avoiding walk is ballistic.