Branching structure for the transient (1;R)-random walk in random environment and its applications

TL;DR

This paper uncovers a multitype branching structure in transient (1,R)-random walks in random environments, allowing explicit characterization of invariant measures and a new proof of the law of large numbers with drift.

Contribution

It introduces a novel multitype branching framework within the (1,R)-RWRE, enabling explicit analysis of invariant measures and drift, improving upon previous results.

Findings

Explicit density of the invariant measure derived

Reproves LLN with explicit drift in terms of environment

Establishes a new branching perspective for RWRE analysis

Abstract

An intrinsic multitype branching structure within the transient (1;R)-RWRE is revealed. The branching structure enables us to specify the density of the absolutely continuous invariant measure for the environments seen from the particle and reprove the LLN with an drift explicitly in terms of the environment, comparing with the results in Br\'emont (2002).