In this work, we study the random metric for the critical long-range percolation on Z d Zᵈ. A recent work by Bäumler (2022) 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 (2021b), and Ding and Gwynne (2023a) on the uniqueness of Liouville quantum gravity metrics.
Ding et al. (Fri,) studied this question.