  We show transience of the edge-reinforced random walk (ERRW) for small
reinforcement in dimension d greater than 2. This proves the existence of a
phase transition between recurrent and transient behavior, thus solving an open
problem stated by Diaconis in 1986. The argument adapts the proof of
quasi-diffusive behavior of the SuSy hyperbolic model for fixed conductances by
Disertori, Spencer and Zirnbauer [CMP 2010], using the representation of ERRW
as a mixture of vertex-reinforced jump processes (VRJP) with independent gamma
conductances, and the interpretation of the limit law of VRJP as a
supersymmetric (SuSy) hyperbolic sigma model developed by Sabot and Tarr\`es in
[JEMS 2014].
