Aldous–Lyons conjecture for graphings

A graphing is a bounded-degree Borel graph satisfying the Intrinsic Mass Transport Principle. A bounded-degree graph sequence has a Benjamini–Schramm limit when the distribution of rooted finite-radius neighborhoods converges. Aldous–Lyons conjecture. Every graphing is the Benjamini–Schramm limit of a bounded-degree graph sequence. Equivalently, every unimodular distribution on rooted countable graphs with bounded degree is the Benjamini–Schramm limit of a bounded-degree graph sequence. The conjecture concerns the characterization of Benjamini–Schramm limits in the bounded-degree setting. The paper notes that graphings represent such limits, while the converse realization statement remains open.