Asymptotic behavior of edge-reinforced random walks. (English) Zbl 1206.82082

Summary: We study linearly edge-reinforced random walk on general multi-level ladders for large initial edge weights. For infinite ladders, we show that the process can be represented as a random walk in a random environment, given by random weights on the edges. The edge weights decay exponentially in space. The process converges to a stationary process. We provide asymptotic bounds for the range of the random walker up to a given time, showing that it localizes much more than an ordinary random walker. The random environment is described in terms of an infinite-volume Gibbs measure.


82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
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