Vertex-reinforced random walk on Z with sub-square-root weights is   recurrent
  We prove that vertex-reinforced random walk on the integers with weight of order k to the power alpha, for alpha in [0, 1/2), is recurrent. This confirms a conjecture of Volkov for alpha<1/2. The conjecture for alpha in [1/2, 1) remains open. 
Jun Chen and Gady Kozma
math.PR