Bernard Derrida

Renormalization and disorder: a simple toy model

The problem of the depinning transition of a line from a random substrate is one of the simplest problems in the the theory of disordered systems. It has a long history among physicists and mathematicians. Still there are some open questions about the nature of this transition.

After a brief review of our present understanding of the problem, I will discuss a simple tree-like toy model which indicates that, when disorder is relevant, the depinning transition becomes an infinite order transition of the Berezinski-Kosterlitz-Thouless type. I will also try to present some recent developments allowing to understand how the precise nature of the singularity at the transition depends on the distribution of disorder.