Large N limit and 1/N expansion of invariant observables in O(N) linear σ-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 Phi 4 model, in two spatial dimensions, using stochastic quantization methods and Dyson-Schwinger equations. We identify the large N limiting lawof a collection ofWick renormalized O(N) invariant observables. In particular, under a suitable scaling, the quadratic observables converge in the large N limit to amean-zero (singular) Gaussian field denoted byQwith an explicit covariance; and the observables which are 2n-th renormalized powers of the fields converge in the large N limit to suitably renormalized n-th powers of Q. The (Wick renormalized) quartic interaction term of the model has no effect on the large N limit of the field Phi, but has nontrivial contributions to the limiting law of the observables, and the renormalization of the n-th powers of Q in the limit has an interesting finite shift from the standard one. Furthermore, we derive the 1/N asymptotic expansion for the kpoint 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 Q.