The Aldous–Lyons conjecture on limits of bounded-degree graphs

A graphing is a measure-preserving graph limit object for bounded-degree graphs, and a unimodular distribution is a distribution on rooted countable graphs satisfying the unimodularity condition. A bounded-degree graph sequence is convergent when all rooted-neighborhood probabilities converge. Aldous–Lyons conjecture. Every graphing is the limit of a convergent sequence of bounded-degree graphs. Equivalently, every unimodular distribution on rooted countable graphs with bounded degree is the limit of a bounded-degree graph sequence.

This conjecture asks whether every analytic local weak limit of bounded-degree graphs can be approximated by finite bounded-degree graphs. The preceding theorem records equivalent limit objects for convergent sequences, but does not establish the converse asserted here.