We experimentally investigate the statistics of zero-height isolines in gravity wave turbulence as physical candidates for conformal invariant curves. We present direct evidence that they can be described by the family of conformal invariant curves called stochastic Schramm-Löwner evolution (or ${\mathrm{SLE}}_{\kappa }$), with diffusivity $\kappa =2.88(8)$. A higher nonlinearity in the height fields is shown destroy this symmetry, though scale invariance is retained.

• Received 30 August 2023
• Accepted 25 January 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.094001