Random interlacement is a factor of i.i.d.

The random interlacement point process (introduced by Sznitman, generalized Teixeira) is a Poisson on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift transient graph $G$. We show that any transitive $G$ factor i.i.d., i.e., it can be constructed from family i.i.d. variables indexed vertices via an equivariant measurable map. Our proof uses variant soft local time method Popov and to construct as almost sure limit sequence finite-length variants model with increasing length. also discuss more direct proving which works if only non-unimodular.