# Activated Random Walks on $\mathbb{Z}^d$

## TL;DR

This paper reviews the complex behavior of Activated Random Walks on integer lattices, highlighting recent progress in understanding their critical phenomena despite analytical challenges.

## Contribution

It provides a comprehensive overview of existing results and techniques developed for studying Activated Random Walks, emphasizing recent methodological advances.

## Key findings

- Progress in understanding self-organized criticality in the model
- Development of new analytical tools for conservative stochastic systems
- Clarification of the mathematical structure of Activated Random Walks

## Abstract

Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.