Characterization and scaling limit of three-dimensional loop-erased random walk and random-walk loop-catchers

Abstract: Random-walk loop-catchers (RWLC) are a one-parameter family of random subsets of the trace of a random walk, recently introduced in [Cai 2026, arXiv:2607.18070], which interpolates between loop-erased random walk (LERW) and the full walk trace, satisfying a recovery property. We prove that the continuous version of this recovery property uniquely characterizes the scaling limit of LERW and RWLC in three dimensions, using a Green-function test via entangled multipath LERW, as outlined in [Cai 2026, arXiv:2607.18070]. For LERW, this resolves the 3D case of the conjecture of Sapozhnikov--Shiraishi [Sapozhnikov--Shiraishi 2018, Probab. Theory Related Fields 172, 615--662] and gives a new and axiomatic proof of the existence of the scaling limit, which was first proved in [Kozma 2007, Acta Math. 199, 29--152]. For RWLC, this gives the construction of 3D Brownian loop-catchers. The characterization also yields inversion invariance of the 3D LERW scaling limit.