The bounded-degree modeling FO-limit characterization conjecture

Let a modeling be a measurable graph equipped with a probability measure for which first-order definable sets are measurable. A graph has bounded degree if its vertex degrees are uniformly bounded; it is weakly uniform if its definable sets satisfy the paper's weak-uniformity condition, and it satisfies the Intrinsic Mass Transport Principle if mass sent along edges is balanced. It has the finite model property if every sentence it satisfies has a finite model. Bounded-degree modeling FO-limit characterization conjecture. A modeling is the Benjamini–Schramm limit of a bounded-degree graph sequence if and only if it is a graph with bounded degree, is weakly uniform, satisfies the Intrinsic Mass Transport Principle, and has the finite model property. This is proposed as a generalization of the Aldous–Lyons characterization problem to modeling limits; the paper presents it as a conjectural necessary-and-sufficient condition.