# Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth

## TL;DR

This paper establishes laws of the iterated logarithm for random walks on covering graphs with polynomial volume growth, using moderate deviation principles and geometric analysis to characterize limit points.

## Contribution

It introduces a geometric approach to MDPs on covering graphs and derives laws of the iterated logarithm for such structures.

## Key findings

- MDPs characterized by quadratic forms via Albanese metric
- Laws of the iterated logarithm established for covering graphs
- Limit points of normalized random walks characterized

## Abstract

Moderate deviation principles (MDPs) for random walks on covering graphs with groups of polynomial volume growth are discussed in a geometric point of view. They deal with any intermediate spatial scalings between those of laws of large numbers and those of central limit theorems. The corresponding rate functions are given by quadratic forms determined by the Albanese metric associated with the given random walks. We apply MDPs to establish laws of the iterated logarithm on the covering graphs by characterizing the set of all limit points of the normalized random walks.