The *-Edge-Reinforced Random Walk

TL;DR

This paper introduces the *-Edge-Reinforced Random Walk, a generalized non-reversible process that extends previous models and provides explicit mixing laws for Yaglom reversible Markov chains.

Contribution

It defines a new class of non-reversible reinforced random walks and derives explicit mixing laws extending prior reversible models.

Findings

Provides explicit expression for the mixing law of *-ERRW

Extends the 'magic formula' to Yaglom reversible chains

Generalizes previous ERRW models to non-reversible cases

Abstract

We define a linearly reinforced process called the *-Edge-Reinforced Random Walk (*-ERRW ) which can be seen as a Yaglom reversible, hence non-reversible, extension of the Edge-Reinforced Random Walk (ERRW) introduced by Coppersmith and Diaconis in 1986. This family of processes also generalizes the r-dependent ERRW introduced by Bacallado (2009). Under some assumptions on the initial weights, the *-ERRW is partially exchangeable in the sense of Diaconis and Freedman (1980), and thus it is a random walk in a random environment. The main result of the paper gives the explicit expression of the mixing law, hence extending the "magic formula" of Coppersmith and Diaconis from the case of mixtures of reversible Markov chains to the case of mixtures of Yaglom reversible Markov chains.