Uniqueness and dimension for the geodesic of the critical long-range percolation metric

TL;DR

This paper proves the uniqueness of geodesics in the critical long-range percolation metric and determines their Hausdorff dimension, advancing understanding of the metric's geometric properties.

Contribution

It establishes the uniqueness of geodesics and the Hausdorff dimension of these geodesics in the critical long-range percolation model.

Findings

Uniqueness of geodesics between fixed points

Continuity of the metric distribution

Hausdorff dimension of geodesics

Abstract

By recent works of B\"aumler [2] and of the authors of this paper [5], the (limiting) random metric for the critical long-range percolation was constructed. In this paper, we prove the uniqueness of the geodesic between two fixed points, for which an important ingredient of independent interest is the continuity of the metric distribution. In addition, we establish the Hausdorff dimension of the geodesics.