Large $N$ Limit of the $O(N)$ Linear Sigma Model via Stochastic Quantization

TL;DR

This paper investigates the large N behavior of the O(N) linear sigma model, establishing uniform bounds, convergence to a mean-field SPDE, and analyzing invariant measures and fluctuations.

Contribution

It provides the first rigorous analysis of the large N limit for the coupled Phi^4 equations, including convergence results and fluctuation descriptions.

Findings

Uniform bounds on dynamics for all N

Convergence to a mean-field singular SPDE

Invariant measures converge to Gaussian free field at rate 1/√N

Abstract

This article studies large $N$ limits of a coupled system of $N$ interacting $\Phi^4$ equations posed over $\mathbb{T}^{d}$ for $d=2$, known as the $O(N)$ linear sigma model. Uniform in $N$ bounds on the dynamics are established, allowing us to show convergence to a mean-field singular SPDE, also proved to be globally well-posed. Moreover, we show tightness of the invariant measures in the large $N$ limit. For large enough mass, they converge to the (massive) Gaussian free field, the unique invariant measure of the mean-field dynamics, at a rate of order $1/\sqrt{N}$ with respect to the Wasserstein distance. We also consider fluctuations and obtain tightness results for certain $O(N)$ invariant observables, along with an exact description of the limiting correlations.