We study the infinite-volume limit of the Gibbs measure for the one-dimensional focusing, mass-subcritical nonlinear Schrödinger equation. Under the critical scaling, we show that the measure concentrates around a single soliton over a Gaussian background.

This program was initiated by Lebowitz-Rose-Speer (1988), who introduced the grand-canonical ensemble for the focusing NLS and conjectured a phase transition in the infinite-volume limit. This conjecture was partially confirmed in recent work by Tolomeo-Weber (2023). Building on the soliton-Gaussian decomposition, we complete the analysis of the critical regime, showing that in this case the measure exhibits a soliton resolution.

The talk is based on joint work with Justin Forlano (Monash University) and Leonardo Tolomeo (University of Edinburgh).