A note on the recurrence of edge reinforced random walks

TL;DR

This paper provides a simplified proof that linearly edge reinforced random walks on locally finite graphs are recurrent if they return to their starting point almost surely, using exchangeability and De Finetti's theorem.

Contribution

It offers a shorter proof of recurrence for ERRW on graphs, avoiding the need for mixture of Markov chains argument used in prior work.

Findings

ERRW on locally finite graphs is recurrent iff it returns to start almost surely

The proof leverages exchangeability and De Finetti's theorem on finite graphs

Simplifies the understanding of ERRW recurrence properties

Abstract

We give a short proof of Theorem 2.1 from [MR07], stating that the linearly edge reinforced random walk (ERRW) on a locally finite graph is recurrent if and only if it returns to its starting point almost surely. This result was proved in [MR07] by means of the much stronger property that the law of the ERRW is a mixture of Markov chains. Our proof only uses this latter property on finite graphs, in which case it is a consequence of De Finetti's theorem on exchangeability.