The linearly edge reinforced random walk (ERRW) was
introduced in 1986 by Coppersmith and Diaconis and is one of the first
example of reinforced random walks. Recently a link has been found between
this model, the vertex reinforced jump process and a random spin model.
Because of these links it was possible to show that in dimension 3 and
above, the ERRW is recurrent for large reinforcement and transient for
small ones and thus exhibits a phase transition. We will present the links
between those models and show that the model has some monotonicity (the
larger the reinforcements the more recurrent it is) and that its phase
transition is unique.