# A density version for H\"aggstr\"om's theorem

## TL;DR

This paper investigates whether, in invariant percolation on regular trees, there always exists an infinite self-avoiding path with a higher density of open edges than the overall edge probability p.

## Contribution

It introduces a density version of H"aggstr"om's theorem, exploring the existence of paths with greater local density of open edges in invariant percolation.

## Key findings

- Identifies conditions under which such paths exist
- Provides counterexamples or proofs for specific cases
- Advances understanding of percolation path densities

## Abstract

Given invariant percolation on a regular tree, where the probability of an edge to be open equals $p$, is it always possible to find an infinite self-avoiding path along which the density of open edges is bigger then $p$?