Transience of Edge-Reinforced Random Walk

We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension . 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 supersymmetric (SuSy) hyperbolic model for fixed conductances by Disertori et al. (Commun Math Phys 300:435-486, 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 SuSy hyperbolic sigma model developed by Sabot and TarrSs (J Eur Math Soc, arXiv:1111.3991, 2015).