# Phase transitions for edge-reinforced random walks on the half-line

## TL;DR

This paper investigates phase transitions in edge-reinforced random walks on the half-line, highlighting how reinforcement strength and initial weights influence the process's behavior, extending previous foundational results.

## Contribution

It provides a comprehensive description of phase transitions in heterogeneous edge-reinforced random walks, completing and extending prior theoretical work.

## Key findings

- Identification of different behavioral regimes
- Description of phase transition conditions
- Extension of previous theoretical results

## Abstract

We study the behaviour of a class of edge-reinforced random walks {on $\mathbb{Z}_+$}, with heterogeneous initial weights, where each edge weight can be updated only when the edge is traversed from left to right. We provide a description for different behaviours of this process and describe phase transitions that arise as trade-offs between the strength of the reinforcement and that of the initial weights. Our result aims to complete the ones given by Davis~\cite{Davis89, Davis90}, Takeshima~\cite{Takeshima00, Takeshima01} and Vervoort~\cite{Vervoort00}.