Phase Transition in Vertex-Reinforced Random Walks on {\mathbb{Z}} with Non-linear Reinforcement |
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Authors: | Stanislav Volkov |
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Institution: | (1) Department of Mathematics, University of Bristol, BS8 1TW Bristol, UK |
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Abstract: | Vertex-reinforced random walk is a random process which visits a site with probability proportional to the weight w
k
of the number k of previous visits. We show that if w
k
∼ k
α, then there is a large time T
0 such that after T
0 the walk visits 2, 5, or ∞ sites when α < 1, = 1, or > 1, respectively. More general results are also proven.
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Keywords: | Vertex-reinforced random walks urn models Rubin’ s construction |
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