Large $N$ limit and $1/N$ expansion of invariant observables in $O(N)$ linear $\sigma$-model via SPDE

TL;DR

This paper studies the large N behavior of invariant observables in the 2D Wick renormalized linear sigma model, revealing their convergence to Gaussian fields and deriving detailed 1/N expansions and corrections.

Contribution

It provides a detailed analysis of the large N limit and 1/N expansion of invariant observables in the 2D O(N) sigma model using stochastic quantization and Dyson-Schwinger equations.

Findings

Quadratic observables converge to a Gaussian field with explicit covariance.

Renormalized powers of the Gaussian field have finite shifts in the large N limit.

Derived 1/N asymptotic expansion for k-point functions of quadratic observables.

Abstract

In this paper, we continue the study of large $N$ problems for the Wick renormalized linear sigma model, i.e. $N$-component $\Phi^4$ model, in two spatial dimensions, using stochastic quantization methods and Dyson--Schwinger equations. We identify the large $N$ limiting law of a collection of Wick renormalized $O(N)$ invariant observables. In particular, under a suitable scaling, the quadratic observables converge in the large $N$ limit to a mean-zero (singular) Gaussian field denoted by $\mathcal{Q}$ with an explicit covariance; and the observables which are renormalized powers of order $2n$ converge in the large $N$ limit to suitably renormalized $n$-th powers of $\mathcal{Q}$. The quartic interaction term of the model has no effect on the large $N$ limit of the field, but has nontrivial contributions to the limiting law of the observables, and the renormalization of the $n$-th powers of $\mathcal{Q}$ in the limit has an interesting finite shift from the standard one. Furthermore, we derive the $1/N$ asymtotic expansion for the $k$-point functions of the quadratic observables by employing graph representations and analyzing the order of each graph from Dyson--Schwinger equations. Finally, turning to the stationary solutions to the stochastic quantization equations, with the Ornstein--Uhlenbeck process being the large $N$ limiting dynamic, we derive here its next order correction in stationarity, as described by an SPDE with the right-hand side having explicit fixed-time marginal law which involves the above field $\mathcal{Q}$.