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Local limit of the focusing discrete NLS

Kesav Krishnan, Gourab Ray

TL;DR

This work analyzes the local limits of the invariant Gibbs measure for the focusing discrete NLS on a high-dimensional torus, in the regime p > 4 and d ≥ 3, by balancing the linear and nonlinear contributions in the Hamiltonian. The authors connect the NLS field to a tilted spherical model and identify three local-limit regimes determined by the phase diagram: a massive Gaussian free field, a massless Gaussian free field plus a random constant, and a potential mixture with mass partially lost to concentration; in the supercritical regime, the local limit is a massive GFF with a random mass M governed by a variational condition. The proofs are probabilistic and hinge on Fourier-mode analysis, domain Markov properties of Gaussian free fields, and precise control of the free-energy landscape via the functions I and W, together with detailed spherical-model estimates and boundary-conditioned limits. The results provide a detailed probabilistic description of concentration phenomena (solitons) in the infinite-volume limit and establish local weak convergence to well-understood Gaussian fields, with potential implications for soliton-resolution-type behavior in discrete NLS.

Abstract

We examine the behavior of a function sampled from the invariant measure associated to the focusing discrete Non Linear Schrödinger equation, defined on a discrete torus of dimension $d \geq 3$, and nonlinearity parameter $p>4$, in the infinite volume limit. The Gibbs measure has two parameters, the inverse temperature and the strength of the non linearity, and the scaling is such that the non linear and linear parts of the Hamiltonian contribute on the same scale. It was shown by Dey, Kirkpatrick and the first author that this measure undergoes a phase transition, concerning concentration of mass of a typical function {at the level of partition functions}. We prove that the three regions of the phase diagram yield three distinct local limits, a massive Gaussian free field, a massless Gaussian free field plus a random constant, and finally a (possibly trivial) mixture of massive Gaussian free fields, where some mass is ``lost'' to the region of concentration. Our proof relies on the analysis of the spherical model of a ferromagnet. The measure under consideration is an exponential tilt of the spherical model, and the additional tilt can be seen to induce an additional phase transition. Although our motivations come from the NLS equation and soliton resolution conjecture, our proofs are completely probabilistic, and can be read without any knowledge of PDE theory.

Local limit of the focusing discrete NLS

TL;DR

This work analyzes the local limits of the invariant Gibbs measure for the focusing discrete NLS on a high-dimensional torus, in the regime p > 4 and d ≥ 3, by balancing the linear and nonlinear contributions in the Hamiltonian. The authors connect the NLS field to a tilted spherical model and identify three local-limit regimes determined by the phase diagram: a massive Gaussian free field, a massless Gaussian free field plus a random constant, and a potential mixture with mass partially lost to concentration; in the supercritical regime, the local limit is a massive GFF with a random mass M governed by a variational condition. The proofs are probabilistic and hinge on Fourier-mode analysis, domain Markov properties of Gaussian free fields, and precise control of the free-energy landscape via the functions I and W, together with detailed spherical-model estimates and boundary-conditioned limits. The results provide a detailed probabilistic description of concentration phenomena (solitons) in the infinite-volume limit and establish local weak convergence to well-understood Gaussian fields, with potential implications for soliton-resolution-type behavior in discrete NLS.

Abstract

We examine the behavior of a function sampled from the invariant measure associated to the focusing discrete Non Linear Schrödinger equation, defined on a discrete torus of dimension , and nonlinearity parameter , in the infinite volume limit. The Gibbs measure has two parameters, the inverse temperature and the strength of the non linearity, and the scaling is such that the non linear and linear parts of the Hamiltonian contribute on the same scale. It was shown by Dey, Kirkpatrick and the first author that this measure undergoes a phase transition, concerning concentration of mass of a typical function {at the level of partition functions}. We prove that the three regions of the phase diagram yield three distinct local limits, a massive Gaussian free field, a massless Gaussian free field plus a random constant, and finally a (possibly trivial) mixture of massive Gaussian free fields, where some mass is ``lost'' to the region of concentration. Our proof relies on the analysis of the spherical model of a ferromagnet. The measure under consideration is an exponential tilt of the spherical model, and the additional tilt can be seen to induce an additional phase transition. Although our motivations come from the NLS equation and soliton resolution conjecture, our proofs are completely probabilistic, and can be read without any knowledge of PDE theory.

Paper Structure

This paper contains 21 sections, 36 theorems, 326 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Heuristics behind the phase transition
  3. Motivations: Solitions and NLS equation
  4. Outline of the proof
  5. Summary of notations
  6. GFF preliminaries
  7. Massive GFF with Dirichlet boundary condition
  8. Zero average GFF on the torus
  9. Infinite volume GFF
  10. Further estimates on the spherical model
  11. Local limit of focusing NLS
  12. Tempering the NLS field
  13. Free energy landscape
  14. Proof of \ref{['thm:main_summary']}
  15. Local limit of the subcritical phase $\nu<\nu_c$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Fix $p>4$ and $d \geqslant 3$. For every $\theta>0$ there exists a $\nu_c=\nu_c(\theta) \in [0,\infty)$ such that the following holds. All the GFFs mentioned below are complex GFFs with variance $\theta^{-1}$.

Figures (1)

  • Figure 1: The phase diagram for NLS field. The purple region corresponds the massive GFF limit for supercritical $\nu$ (\ref{['thm:main_summary']}, part c.), the red region corresponds to the massive GFF limit for subcritical $\nu$ (\ref{['thm:main_summary']}, part a.), and the green region corresponds to the (non-massive) GFF with the random global shift limit (\ref{['thm:main_summary']}, part b.) We do not know of the behaviour of the field at the critical curve (drawn in solid brown), see \ref{['sec:future']} for further discussions. Something curious happens if we fix $\nu$ to correspond to the horizontal dashed red line. As we vary $\theta$, the correlations are decaying exponentially at first (red part), then polynomially (green part), and then exponentially again (purple part).

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 65 more