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.Comment: Published at http://dx.doi.org/10.1214/074921706000000103 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org