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Edge-Reinforced Random Walk on a Ladder

Franz Merkl and Silke W. W. Rolles
The Annals of Probability
Vol. 33, No. 6 (Nov., 2005), pp. 2051-2093
Stable URL: http://www.jstor.org/stable/3481777
Page Count: 43
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Edge-Reinforced Random Walk on a Ladder
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Abstract

We prove that the edge-reinforced random walk on the ladder Z× {1,2} with initial weights a > 3/4 is recurrent. The proof uses a known representation of the edge-reinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal of a multicomponent Gibbsian process. A transfer operator technique and entropy estimates from statistical mechanics are used to analyze this Gibbsian process. Furthermore, we prove spatially exponentially fast decreasing bounds for normalized local times of the edge-reinforced random walk on a finite piece of the ladder, uniformly in the size of the finite piece.

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