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We prove that any vertex-reinforced random walk on the integer lattice withnon-decreasing reinforcement sequence $w$ satisfying $w(k) = o(k^{\alpha})$ forsome $\alpha < 1/2$ is recurrent. This improves on previous results of Volkov(2006) and Schapira (2012).

Recurrence for vertex-reinforced random walks on Z with weak reinforcements