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

Authors Márton Borbényi, Balázs Ráth, Sándor Rokob
Publication date 2023-04-21

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.

Keywords: random interlacements; factor of iid; random walk; unimodularity.

DOI: 10.1214/23-EJP950

Electron. J. Probab. 28 (2023), article no. 58, 1–45. ISSN: 1083-6489

Márton Borbényi (Alfréd Rényi Institute of Mathematics and Eötvös Loránd University)
Balázs Ráth (Alfréd Rényi Institute of Mathematics, MTA-BME Stochastics Research Group)
Sándor Rokob (Alfréd Rényi Institute of Mathematics, MTA-BME Stochastics Research Group)