Linearly edge-reinforced random walks

Merkl, Franz, University of Munich; Rolles, Silke W. W., University of Bielefeld

doi:10.1214/074921706000000103

VOL. 48 | 2006 Linearly edge-reinforced random walks

Franz Merkl, Silke W. W. Rolles

Editor(s) Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy

IMS Lecture Notes Monogr. Ser., 2006: 66-77 (2006) DOI: 10.1214/074921706000000103

Abstract

We review results on linearly edge-reinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a representation as a random walk in an independent random environment. We review recent results for the random walk on ladders: recurrence, a representation as a random walk in a random environment, and estimates for the position of the random walker.