January 2022 Large N limit of the O(N) linear sigma model via stochastic quantization
Hao Shen, Scott A. Smith, Rongchan Zhu, Xiangchan Zhu
Author Affiliations +
Ann. Probab. 50(1): 131-202 (January 2022). DOI: 10.1214/21-AOP1531


This article studies large N limits of a coupled system of N interacting Φ4 equations posed over Td 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/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.

Funding Statement

H.S. gratefully acknowledges financial support from NSF grant DMS-1712684/DMS-1909525 and DMS-1954091. R.Z. is grateful to the financial supports of the NSFC (No. 11922103). X.Z. is grateful to the financial supports in part by National Key R&D Program of China (No. 2020YFA0712700) and the NSFC (Nos. 11771037, 12090014, 11688101) and the support by key Lab of Random Complex Structures and Data Science, Youth Innovation Promotion Association (2020003), Chinese Academy of Science. The financial support by the DFG through the CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” is greatly acknowledged.


We would like to thank Antti Kupiainen for helpful discussion on large N problems and are grateful to Fengyu Wang and Xicheng Zhang for helpful discussion on distributional dependent SDE.

Rongchan Zhu is the corresponding author.

This work was completed while the second author was employed by UW-Madison.


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Hao Shen. Scott A. Smith. Rongchan Zhu. Xiangchan Zhu. "Large N limit of the O(N) linear sigma model via stochastic quantization." Ann. Probab. 50 (1) 131 - 202, January 2022. https://doi.org/10.1214/21-AOP1531


Received: 1 June 2020; Revised: 1 April 2021; Published: January 2022
First available in Project Euclid: 23 February 2022

MathSciNet: MR4385125
zbMATH: 1490.60192
Digital Object Identifier: 10.1214/21-AOP1531

Primary: 35R60 , 60H15

Keywords: Mean-field limit , O(N) linear sigma model , space-time white noise , Stochastic quantization , Φ4

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 1 • January 2022
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