Uniqueness of the critical long-range percolation metrics

TL;DR

This paper proves the uniqueness of the scaling limit of the critical long-range percolation metric on b^d, establishing that it is characterized by a natural set of axioms, inspired by quantum gravity metrics.

Contribution

It demonstrates that the subsequential scaling limit of the critical long-range percolation metric is uniquely determined by a natural list of axioms, extending previous work on similar geometric models.

Findings

The subsequential limit is uniquely characterized by axioms.

The proof is inspired by methods used in Liouville quantum gravity.

The work confirms the robustness of the scaling limit in critical long-range percolation.

Abstract

In this work, we study the random metric for the critical long-range percolation on $\mathbb{Z}^d$. A recent work by B\"aumler [3] implies the subsequential scaling limit, and our main contribution is to prove that the subsequential limit is uniquely characterized by a natural list of axioms. Our proof method is hugely inspired by recent works of Gwynne and Miller [42], and Ding and Gwynne [25] on the uniqueness of Liouville quantum gravity metrics.