Central limit theorem for the focusing $\Phi^4$-measure in the infinite volume limit

TL;DR

This paper investigates the fluctuations of the focusing -measure on a one-dimensional torus, demonstrating that scaled fields converge to white noise, thus revealing the nature of second-order fluctuations around soliton solutions.

Contribution

It establishes the convergence of scaled -fields to white noise, providing a detailed analysis of second-order fluctuations in the focusing -measure in the infinite volume limit.

Findings

Scaled -fields converge to white noise

Fluctuations around solitons are characterized as white noise

Supports Rider's predictions on second-order behavior

Abstract

We study the fluctuations of the focusing $\Phi^4$-measure on the one-dimensional torus in the infinite volume limit. This measure is an invariant Gibbs measure for the nonlinear Schr\"odinger equation. It had previously been shown by B. Rider that the measure is strongly concentrated around a family of minimizers of the Hamiltonian associated with the measure. These exhibit increasingly sharp spatial concentration, resulting in a trivial limit to first order. We study the fluctuations around this soliton manifold. We show that the scaled field under the Gibbs measure converges to white noise in the limit, identifying the next order fluctuations predicted by B. Rider.