Edge-reinforced random walk on a ladder

Details Edge-reinforced random walk on a ladder Edge-reinforced random walk on a ladder

2005-01-10

arxiv.org/abs/math/0501137v2

Annals of Probability 2005, Vol. 33, No. 6, 2051-2093

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000396 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000396

We prove that the edge-reinforced random walk on the ladder ${\mathbb{Z}\times\{1,2\}}$ 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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