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We introduce a toy model, which represents a simplified version of theproblem of the depinning transition in the limit of strong disorder. This toymodel can be formulated as a simple renormalization transformation for theprobability distribution of a single real variable. For this toy model, {thecritical line is known exactly in one particular case} and it can be calculatedperturbatively in the general case. One can also show that, at the transition,there is no strong disorder fixed distribution accessible by renormalization.Instead, both our numerical and analytic approaches indicate a transition ofinfinite order (of the Berezinskii-Kosterlitz-Thouless type). We give numericalevidence that this infinite order transition persists for the problem of thedepinning transition with disorder on the hierarchical lattice.

The depinning transition in presence of disorder: a toy model