A perturbative microscopic derivation of the focusing $\Phi^6_1$ measure with rough cut-off

We give a derivation of the Gibbs measure for the focusing nonlinear Schrödinger equation (NLS) on the circle with rough cut-off.

This extends earlier work by Sohinger and the second author, which proved analogous results for smooth cut-offs.

Our proof is based on the perturbative expansion developed by Fröhlich, Knowles, Schlein, and Sohinger (2017), and provides an alternative proof of the recent derivation given in Lü, Nam, and Zhu (2026).

To prove convergence of the explicit terms, we employ a Wigner measure approach and an inductive argument to overcome the lack of smoothness for the cut-off.

In particular, we give a derivation of the Gibbs measure for the focusing quintic NLS with the optimal cut-off from Oh, Sosoe, and Tolomeo (2022).