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

In this paper, we continue the study of large $N$ problems for the Wick renormalized linear sigma model, i.e. $N$-component ${\mathrm{\Phi }}^4$ 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 $1\mathrm{/}N$ 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 $\mathcal{Q}$.