Uniqueness of the critical long-range percolation metrics

In this work, we study the random metric for the critical long-range percolation on ${\mathbb{Z}}^d$. A recent work by Bäumler [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.