Abstract
The random interlacement point process (introduced in [47], generalized in [50]) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph G. We show that the random interlacement point process on any transient transitive graph G is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced in [37]) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if G is non-unimodular.
Funding Statement
The work of M. Borbényi was partially supported by the UNKP-21-2 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund and the ERC Synergy under Grant No. 810115 - DYNASNET. The work of B. Ráth was partially supported by grants NKFI-FK-123962 and NKFI-KKP-139502 of NKFI (National Research, Development and Innovation Office) and the ERC Synergy under Grant No. 810115 - DYNASNET. The work of S. Rokob was partially supported by the ERC Consolidator Grant 772466 “NOISE”.
Acknowledgments
We thank Ádám Tímár for suggesting the idea of the proof of implication (B) (A) of Proposition 1.3 to us. We thank Gábor Pete for sketching the proof of Lemma 2.5 to us.
Citation
Márton Borbényi. Balázs Ráth. Sándor Rokob. "Random interlacement is a factor of i.i.d.." Electron. J. Probab. 28 1 - 45, 2023. https://doi.org/10.1214/23-EJP950
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