Scientists have shown that, when both the lattice spacing shrinks to zero and the simulation volume grows without bound, the probability measures describing the two‑dimensional Abelian–Higgs model form a tight family. This means that as the system approaches the continuum and infinite‑volume limits, the distributions do not wander off or spread out uncontrollably.

The proof relies on an iterated forward‑backward stochastic differential equation (FBSDE) representation. In this approach, the lattice measure is identified with the joint terminal law of a pair of FBSDEs for the gauge field and the Higgs field. The authors combine sophisticated frequency decompositions with covariant energy estimates for the Higgs field and high‑frequency force estimates for the gauge field. They also employ quantitative control of infrared behavior through a diamagnetic estimate.

While the analysis establishes tightness, it does not show that the limiting measure is unique. Consequently, multiple subsequential limits may exist in principle.

These results advance the mathematical understanding of gauge theories in two dimensions, laying groundwork for future work that might establish uniqueness or extend the methods to higher‑dimensional models.