LARGE N LIMIT OF THE O(N) LINEAR SIGMA MODEL VIA STOCHASTIC QUANTIZATION

Hao Shen; Scott A. Smith; Rongchan Zhu; Xiangchan Zhu

doi:10.1214/21-AOP1531

LARGE N LIMIT OF THE O(N) LINEAR SIGMA MODEL VIA STOCHASTIC QUANTIZATION

Hao Shen, Scott A. Smith^*, Rongchan Zhu, Xiangchan Zhu

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

Abstract

This article studies large N limits of a coupled system of N interacting Φ⁴ equations posed over 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/ √ N with respect to the Wasserstein distance.

Original language	English
Pages (from-to)	131-202
Number of pages	72
Journal	Annals of Probability
Volume	50
Issue number	1
DOIs	https://doi.org/10.1214/21-AOP1531
Publication status	Published - Jan 2022

Keywords

Mean-field limit
O(n) linear sigma model
Space-time white noise
Stochastic quantization
Φ

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10.1214/21-AOP1531

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abstract = "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.",

keywords = "Mean-field limit, O(n) linear sigma model, Space-time white noise, Stochastic quantization, Φ",

author = "Hao Shen and Smith, \{Scott A.\} and Rongchan Zhu and Xiangchan Zhu",

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AU - Smith, Scott A.

AU - Zhu, Rongchan

AU - Zhu, Xiangchan

N1 - Publisher Copyright: © Institute of Mathematical Statistics, 2022

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N2 - 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.

AB - 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.

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KW - O(n) linear sigma model

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KW - Stochastic quantization

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LARGE N LIMIT OF THE O(N) LINEAR SIGMA MODEL VIA STOCHASTIC QUANTIZATION

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Keywords

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