Recurrence for vertex-reinforced random walks on Z with weak   reinforcements
  We prove that any vertex-reinforced random walk on the integer lattice with non-decreasing reinforcement sequence $w$ satisfying $w(k) = o(k^{\alpha})$ for some $\alpha < 1/2$ is recurrent. This improves on previous results of Volkov (2006) and Schapira (2012). 
Arvind Singh
math.PR