In this talk we introduce a new si mple but powerful general technique for the study of edge- and vertex-reinforced processes with supe r-linear reinforcement\, based on the use of order statistics for the number of edge\, respectively of vertex\, traversals. The technique relies on up per bound estimates for the number of edge travers als\, proved in a different context by Cotar and L imic [Ann. Appl. Probab. (2009)] for finite graphs with edge reinforcement. We apply our new method both to edge- and to vertex-reinforced random walk s with super-linear reinforcement on arbitrary inf inite connected graphs of bounded degree. We stres s that\, unlike all previous results for processes with super-linear reinforcement\, we make no othe r assumption on the graphs.

For edge-reinfo rced random walks\, we complete the results of Lim ic and Tarres [Ann. Probab. (2007)] and we settle a conjecture of Sellke [Technical Report 94-26\, P urdue University (1994)] by showing that for any r eciprocally summable reinforcement weight function w\, the walk traverses a random attracting edge a t all large times.

For vertex-reinforced ra ndom walks\, we extend results previously obtained on Z by Volkov [Ann. Probab. (2001)] and by Basde vant\, Schapira and Singh [Ann. Probab. (2014)]\, and on complete graphs by Benaim\, Raimond and Sch apira [ALEA (2013)]. We show that on any infinite connected graph of bounded degree\, with reinforce ment weight function w taken from a general class of reciprocally summable reinforcement weight func tions\, the walk traverses two random neighbouring attracting vertices at all large times.

Re lated Links

- https://arxiv.org/abs/1509.00807 \;- We bpage where the paper can be found