In recent works, Christophe Sabot and Pierre Tarrès established a link between edge-reinforced random walks and a model from quantum field theory [ST15a]. Introduced by Coppersmith and Diaconis in 1986, reinforced random walks are random process in an environnement which is modified by their own behavior through reinforcement by the edges or the cells that it has already visited.
Thanks to this link with quantum fields theory, the authors, together with Margherita Disertori, have been able to prove a conjecture stated by Diaconis on the large time behavior of these walks and conjecturing a recurrence/transience phase transition [DST15]. For weak reinforcement, the process is transient, no cell is visited infinitely many. On the contrary, for sufficiently strong reinforcement, and if a cell is visited infinitely often, then the same will hold for the whole graph : the process is said to be recurrent. The vertex-reinforced jump process enjoy other interesting links with statistical physics and random Schrödinger equations, on which Christophe Sabot, Pierre Tarrès, and Xiaolin Zeng have obtained sophisticated new results his year [STZ15].
[DST15] M. Disertori, C. Sabot, et P. Tarrès. Transience of edge-reinforced random walk. Comm. Math. Phys., 339(1) : 121–148, 2015.
[RM06] Franz Merkl and Silke W. W. Rolles, Linearly edge-reinforced random walks. IMS Lecture Notes–Monograph Series Dynamics & Stochastics, 48, 66–77, 2006.
[ST15a] C. Sabot et P. Tarrès. Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. (JEMS), 17(9) : 2353–2378, 2015.
[STZ15] C. Sabot, P. Tarrès, et X. Zeng. The vertex-reinforced jump process and a random Schrödinger operator on finite graphs. Prépublication, disponible sur http://arxiv.org/abs/1507.04660, 2015.