# Biased random walk on the trace of biased random walk on the trace of...

## TL;DR

This paper investigates the behavior of a sequence of biased random walks on evolving graphs, proving transience, laws of large numbers, and criteria for ballisticity, with examples illustrating different limiting graph structures.

## Contribution

It introduces a novel analysis of biased random walks on trace graphs, establishing transience, LLN, and criteria for ballisticity in this complex setting.

## Key findings

- Proves transience and LLN for the sequence of walks.
- Provides criteria distinguishing ballistic and sub-ballistic behavior.
- Constructs examples with various limiting graph structures.

## Abstract

We study the behaviour of a sequence of biased random walks X(i), i>=0 on a sequence of random graphs, where the initial graph is Zd and otherwise the graph for the i-th walk is the trace of the (i - 1)-st walk. The sequence of bias vectors is chosen so that each walk is transient. We prove the aforementioned transience and a law of large numbers, and provide criteria for ballisticity and sub-ballisticity. We give examples of sequences of biases for which each X(i), i>=1 is (transient but) not ballistic, and the limiting graph is an infinite simple (self-avoiding) path. We also give examples for which each X(i), i>=1 is ballistic, but the limiting graph is not a simple path.