Phase transition for invariant measures of the focusing Schrödinger equation

We consider the Gibbs measure for the focusing nonlinear Schrödinger equation on the one-dimensional torus $\mathbb{T}$, that was introduced in a seminal paper by Lebowitz, Rose and Speer (1988). We show that in the large torus limit, the measure exhibits a phase transition, depending on the size of the nonlinearity. This phase transition was originally conjectured on the basis of numerical simulation by Lebowitz, Rose and Speer (1988). Its existence is however striking in view of a series of negative results by McKean (1995) and Rider (2002).