Holder continuity of interfaces for scale-invariant Poisson stick soup

TL;DR

This paper investigates the regularity of interfaces in a scale-invariant Poisson stick soup model, establishing tightness and Holder continuity of the exploration paths' limits, with implications for planar percolation models.

Contribution

It proves tightness and Holder continuity of exploration paths in the Poisson stick soup, advancing understanding of its scaling limits and connections to planar percolation.

Findings

Established tightness of exploration path family.

Proved Holder continuity of limiting measures.

Linked the model to long-range percolation with critical parameter.

Abstract

We study the interface of covered and vacant sets in the subcritical phase of a scale-invariant Poisson stick soup on the plane. This model is a natural candidate for scaling limit of some planar models and has connections with long-range percolation on the plane with critical parameter $s=4$. We analyze a family of exploration paths on boxes and prove tightness for this family and Holder continuity for its limiting measures.