Quantum Hall edges beyond the plasma analogy

We demonstrate that the widely used plasma analogy is unreliable at predicting edge properties of quantum Hall states. This discrepancy arises from a fundamental difference between quantum Hall droplets and plasmas (Coulomb gases): the former are incompressible liquids subject to area-preserving deformations, while the latter are governed by electrostatics and thus involve conformal transformations. Consequently, the plasma analogy fails at the edge, except in fine-tuned geometries, as it does not account for the emergent local edge velocity. We quantitatively show how the analogy's failure affects physical quantities, such as fluctuations of local observables and absorption rates in microwave spectroscopy, measurable in both solid-state experiments and quantum simulators.