Continuity and finiteness of the radius of convergence of a p-adic differential equation via potential theory

We study the radius of convergence of a differential equation on a smooth Berkovich curve over a non-archimedean complete valued field of characteristic O. Several properties of this function are known: F. Baldassarri proved that it is continuous (see [2]) and the authors showed that it factorizes by the retraction through a locally finite graph (see [12] and [10]). Here, assuming that the curve has no boundary or that the differential equation is overconvergent, we provide a shorter proof of both results by using potential theory on Berkovich curves.