# On discrete loop signatures and Markov loops topology

## TL;DR

This paper investigates the relationships between loop ensembles on finite graphs, combinatorial group theory, and topological invariants, providing new insights into the distribution of loop homotopy classes and homologies.

## Contribution

It introduces a novel analysis of loop measures and signatures in discrete paths, linking topological and algebraic properties of loops on graphs.

## Key findings

- Distribution of loop homotopy classes determined
- First and second homologies characterized
- Connections between loop measures and topological invariants established

## Abstract

Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central series of the fundamental group.