Results and conjectures on a toy model of depinning

We review recent results and conjectures for a simplified version of the depinning problem in presence of disorder, which was introduced by Derrida and Retaux in 2014. For this toy model, the depinning transition has been predicted to be of the Berezinskii–Kosterlitz–Thouless type. Here we discuss under which conditions this prediction can be proved and how it is modified otherwise.