How edge-reinforced random walk arises naturally

Abstract

We give a characterization of a modi ed edge-reinforced random walk in terms of certain partially exchangeable sequences. In particular, we obtain a characterization of edge-reinforced random walk (introduced by Coppersmith and Diaconis) on a 2-edge-connected graph. Modifying the notion of partial exchangeability i n troduced by D i a c onis and Freedman in 2], we c haracterize unique mixtures of reversible Markov c hains under a recurrence assumption.

A random environment for linearly edge-reinforced random walks on infinite graphs

“…In this case, it is known that the measure Q errw satisfies the assertions of Theorem 2.2 (see e.g. Theorem 3.1 of [Rol03]). …”

“…Theorem 3.1 of [Rol03]) that the edge-reinforced random walk on a finite graph G has the same distribution as a random walk in a random environment given by random weights on the edges. Let…”

“…Even more, on any finite graph, the reinforced random walk is a mixture of reversible Markov chains; this follows e.g. by a de Finetti theorem for reversible Markov chains (see [Rol03]). However, on many infinite, locally finite graphs with some initial weights, edge-reinforced random walk is not recurrent, and on some infinite, locally finite graphs, including Z d for d ≥ 2, it is not known whether edge-reinforced random walk is recurrent.…”

On the recurrence of edge-reinforced random walk on ℤ×G

“…The environment is given by random time-independent edge weights with a complicated dependence structure. This representation follows from a de Finetti theorem for reversible Markov chains proved in [Rol03], refining a de Finetti theorem for Markov chains of Diaconis and Freedman [DF80]. The distribution of the random environment was discovered by Coppersmith and Diaconis (see [Dia88]).…”

Linearly edge-reinforced random walks

“…The following representation of the edge-reinforced random walk on a finite graph as a mixture of reversible Markov chains is shown in Theorem 3.1 of [28]. This representation as a random walk in a random environment has been extremly useful in studying the edge-reinforced random walk on infinite ladders.…”