I will explain the central limit theorem for the focusing Φ^4 measure around the soliton manifold in the infinite volume limit. The focusing Φ^4 measure is the invariant Gibbs measure for the nonlinear Schrödinger equation, first studied by Lebowitz, Rose, and Speer  (1988), and later extended by Bourgain (1994), Brydges, Slade (1996), and Carlen, Fröhlich, and Lebowitz (2016). 

Rider previously showed that the measure exhibits increasingly sharp concentration around the soliton manifold which represents a family of minimizers of the Hamiltonian associated with the Gibbs measure. Our result shows that the scaled field under the Gibbs measure converges to white noise in the infinite volume limit, thereby identifying the next-order fluctuations, as predicted by Rider.

This talk is based on joint work with Philippe Sosoe (Cornell).