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A $0$-$1$ law for vertex-reinforced random walks on $\mathbb{Z}$ with weight of order $k^\alpha$, $\alpha\in[0,1/2)$

   
@article{ECP2084,
	author = {Bruno Schapira},
	title = {A $0$-$1$ law for vertex-reinforced random walks on $\mathbb{Z}$ with weight of order $k^\alpha$, $\alpha\in[0,1/2)$},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {17},
	year = {2012},
	keywords = {Self-interacting random walk; Reinforced random walk; \$0\$-\$1\$ law},
	abstract = {We prove that Vertex Reinforced Random Walk on $\mathbb{Z}$ with weight  of order $k^\alpha$, with $\alpha\in [0,1/2)$, is either almost surely recurrent or almost surely transient.  This improves a previous result of Volkov who showed that the set of sites which are visited infinitely often was a.s. either empty or infinite.},
	pages = {no. 22, 1-8},
	issn = {1083-589X},
	doi = {10.1214/ECP.v17-2084},    
        url = {http://ecp.ejpecp.org/article/view/2084}}