Observing Joinings: A Distance-Array Characterization of Furstenberg Disjointness

Abstract

Joinings are fundamental global objects in ergodic theory, yet in compact metric models one naturally observes only finite orbit-distance patterns. We bridge this gap by introducing multi-particle distance arrays, which sample finite orbit segments and record their joint metric evolution. In the anchored fixed-model setting, this framework yields a purely finite-observable characterization of Furstenberg disjointness: two systems are disjoint if and only if all their anchored multi-orbit distance-array projections are independent. The structural engine behind this criterion is a marked and colored version of the Gromov--Vershik reconstruction principle for exchangeable arrays; unanchored arrays reconstruct the intrinsic twin-free quotient, while anchors recover the actual joining in a fixed model. To quantify this independence, we introduce Wasserstein dependence coefficients, establishing an all-order zero criterion for disjointness, and show that weak neighborhoods of the product joining always admit finite distance-array certificates. Examples from compact rotations, Bernoulli and reversible Markov shifts, common factors, Kronecker factors, and weak mixing demonstrate the strict necessity of the multi-particle level and the broad scope of this approach.