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Hyperbolic $O (N)$ linear sigma model and its mean-field limit

Ruoyuan Liu, Shao Liu, Tadahiro Oh

TL;DR

This work analyzes the large-$N$ behavior of the hyperbolic $O(N)$ linear sigma model on the 2D torus, establishing pathwise global well-posedness for the coupled SdNLW system and proving convergence to a mean-field SdNLW with a nonlocal nonlinear term. The authors develop a renormalized framework via Wick powers, construct enhanced data sets, and implement a hybrid $I$-method to obtain uniform-in-$N$ global control and convergence results, including a law-of-large-numbers mechanism that replaces empirical averages by expectations. They also extend the analysis to invariant Gibbs dynamics, proving convergence to the mean-field Gibbs dynamics and demonstrating propagation of chaos, with a convergence rate of $N^{-1/2+\varepsilon}$ on large time intervals and under higher moment assumptions a rate $N^{-1/2+\varepsilon_0}$. The results contribute a rigorous dynamical understanding of large-$N$ limits for hyperbolic stochastic quantization models and their Gibbsian equilibria, connecting stochastic damped wave dynamics with mean-field and propagation-of-chaos phenomena in a dispersive setting.

Abstract

We study large $N$ limits of the hyperbolic $O(N)$ linear sigma model ($\text{HLSM}_N$) on the two-dimensional torus $\mathbb T^2$, namely, a system of $N$ interacting stochastic damped nonlinear wave equations (SdNLW) with coupled cubic nonlinearities. After establishing (pathwise) global well-posedness of $\text{HLSM}_N$ and the limiting equation, called the mean-field SdNLW, we first establish convergence of $\text{HLSM}_N$ to the mean-field SdNLW with general initial data (under a suitable assumption). In particular, for the local-in-time convergence, we obtain a convergence rate of order $N^{- \frac 12 + \varepsilon}$ for any $ \varepsilon > 0$. We then show that the invariant Gibbs dynamics for $\text{HLSM}_N$) converges to that for the mean-field SdNLW with a convergence rate of order $N^{- \frac 12 + \varepsilon}$ on any large time intervals.

Hyperbolic $O (N)$ linear sigma model and its mean-field limit

TL;DR

This work analyzes the large- behavior of the hyperbolic linear sigma model on the 2D torus, establishing pathwise global well-posedness for the coupled SdNLW system and proving convergence to a mean-field SdNLW with a nonlocal nonlinear term. The authors develop a renormalized framework via Wick powers, construct enhanced data sets, and implement a hybrid -method to obtain uniform-in- global control and convergence results, including a law-of-large-numbers mechanism that replaces empirical averages by expectations. They also extend the analysis to invariant Gibbs dynamics, proving convergence to the mean-field Gibbs dynamics and demonstrating propagation of chaos, with a convergence rate of on large time intervals and under higher moment assumptions a rate . The results contribute a rigorous dynamical understanding of large- limits for hyperbolic stochastic quantization models and their Gibbsian equilibria, connecting stochastic damped wave dynamics with mean-field and propagation-of-chaos phenomena in a dispersive setting.

Abstract

We study large limits of the hyperbolic linear sigma model () on the two-dimensional torus , namely, a system of interacting stochastic damped nonlinear wave equations (SdNLW) with coupled cubic nonlinearities. After establishing (pathwise) global well-posedness of and the limiting equation, called the mean-field SdNLW, we first establish convergence of to the mean-field SdNLW with general initial data (under a suitable assumption). In particular, for the local-in-time convergence, we obtain a convergence rate of order for any . We then show that the invariant Gibbs dynamics for ) converges to that for the mean-field SdNLW with a convergence rate of order on any large time intervals.

Paper Structure

This paper contains 23 sections, 32 theorems, 369 equations.

Table of Contents

  1. Introduction
  2. Hyperbolic $O (N)$ linear sigma model
  3. Large $N$ limit of the hyperbolic $O (N)$ linear sigma model
  4. Large $N$ limit of the invariant Gibbs dynamics
  5. Notations and preliminary lemmas
  6. Notations
  7. Product estimates
  8. $I$-operator and basic commutator estimates
  9. Stochastic convolutions and their Wick powers
  10. Well-posedness of the hyperbolic $O (N)$ linear sigma model
  11. Local well-posedness
  12. Preliminary estimates
  13. Global well-posedness
  14. Well-posedness of the mean-field SdNLW
  15. Local well-posedness
  16. ...and 8 more sections

Key Result

Theorem 1.1

(i) Let $m > 0$ and $\frac{1}{2} \le s < 1$. Then, given $N \in \mathbb{N}$, $\text{HLSM}_N$NLW4 is locally well-posed in $(\mathcal{H}^s (\mathbb{T}^2))^{\otimes N}$. More precisely, given $(\mathbf{v}_N^{(0)}, \mathbf{v}_N^{(1)}) = \{(v^{(0)}_{N, j}, v^{(1)}_{N, j}) \}_{j = 1}^N \in (\mathcal{H}^ to NLW3 with $(\mathbf{v}_N, \partial_t \mathbf{v}_N)|_{t = 0} = (\mathbf{v}_N^{(0)}, \mathbf{v}_N^

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 56 more