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

Kihoon Seong, Philippe Sosoe

TL;DR

The paper proves a central limit-type theorem for the focusing $Φ^4$ Gibbs measure in the infinite-volume limit on the one-dimensional torus. It shows that after centering around the soliton manifold and scaling by $L$, the fluctuations converge to white noise on a weighted Besov Sobolev space, distinguishing these fluctuations from the Gaussian free field typical in defocusing settings. The authors develop a conditional Ornstein–Uhlenbeck framework around the soliton manifold, deriving a density-based representation and rigorous tail and partition-function estimates to control errors. The results provide a refined understanding of the soliton-plus-radiation structure for the NLS-invariant Gibbs ensemble and point toward extensions to other continuum focusing Gibbs measures in low dimensions or discretized models.

Abstract

We study the fluctuations of the focusing $Φ^4$-measure on the one-dimensional torus in the infinite volume limit. This measure is an invariant Gibbs measure for the nonlinear Schrödinger 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.

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

TL;DR

The paper proves a central limit-type theorem for the focusing Gibbs measure in the infinite-volume limit on the one-dimensional torus. It shows that after centering around the soliton manifold and scaling by , the fluctuations converge to white noise on a weighted Besov Sobolev space, distinguishing these fluctuations from the Gaussian free field typical in defocusing settings. The authors develop a conditional Ornstein–Uhlenbeck framework around the soliton manifold, deriving a density-based representation and rigorous tail and partition-function estimates to control errors. The results provide a refined understanding of the soliton-plus-radiation structure for the NLS-invariant Gibbs ensemble and point toward extensions to other continuum focusing Gibbs measures in low dimensions or discretized models.

Abstract

We study the fluctuations of the focusing -measure on the one-dimensional torus in the infinite volume limit. This measure is an invariant Gibbs measure for the nonlinear Schrödinger 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.

Paper Structure

This paper contains 37 sections, 37 theorems, 556 equations.

Table of Contents

  1. Introduction
  2. Implications of the main result
  3. Main result
  4. Related results
  5. Structure of the proof
  6. Comments on the literature
  7. Infinite volume limit of Gibbs measures
  8. Gaussian Fluctuations in Euclidean $\Phi^4$ QFT
  9. Higher dimensional focusing $\Phi^4$ measure
  10. Invariance of the Gibbs ensemble under Hamiltonian PDEs
  11. Long time behavior of the dynamics
  12. Notations and preliminary lemmas
  13. Notations
  14. Weighted Besov spaces
  15. Concentration of measure around a family of minimizers
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $s<-\frac{1}{2}$, $D>0$, and let $F$ be a bounded and continuous function on $H^{s}_\mu(\mathbb{R})$. Then, we have as $L \to \infty$. Here, $\pi_L$ is the projection map in $H^1(\mathbb{T}_L)$ onto the soliton manifold $\mathcal{M}_L$ defined in project, where $\eta$ is the cutoff functionHere, the cutoff function is necessary to ensure that the solitons preserve the periodic boundary condit

Theorems & Definitions (92)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Remark 2.4
  • ...and 82 more