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In this paper, we continue the study of large $N$ problems for the Wickrenormalized linear sigma model, i.e. $N$-component $\Phi^4$ model, in twospatial dimensions, using stochastic quantization methods and Dyson--Schwingerequations. We identify the large $N$ limiting law of a collection of Wickrenormalized $O(N)$ invariant observables. In particular, under a suitablescaling, the quadratic observables converge in the large $N$ limit to amean-zero (singular) Gaussian field denoted by $\mathcal{Q}$ with an explicitcovariance; 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 thelarge $N$ limit of the field, but has nontrivial contributions to the limitinglaw of the observables, and the renormalization of the $n$-th powers of$\mathcal{Q}$ in the limit has an interesting finite shift from the standardone.  Furthermore, we derive the $1/N$ asymtotic expansion for the $k$-pointfunctions of the quadratic observables by employing graph representations andanalyzing 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, wederive here its next order correction in stationarity, as described by an SPDEwith the right-hand side having explicit fixed-time marginal law which involvesthe above field $\mathcal{Q}$.

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