How edge-reinforced random walk arises naturally (Q1404210)

The paper concerns the random walk on locally finite graph. It is introduced a notion of partial exchangeability and it is given a characterization of the random walk in terms of this notion, i.e., it is proved a theorem relating the fact that the random walk is a unique mixture of reversible Markov chains with the partial exchangeability condition. The second result concerns the case in which the graph is \(2\)-edge-connected. Under some assumptions on the nearest neighbor random walk on \(G\) it is proved that the random walk is an edge-reinfoced random walk or a non-reinfoced random walk. This result is obtained by introducing the definition of modified edge-reinfoced random walk and proving his characterization in terms of certain partially exchangeable sequences.