Large N limit of the O(N) linear sigma model via stochastic quantization

Abstract

In this paper we continue the study of large N problems for the Wick renormalized linear sigma model, i.e. N -component Φ 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 Q with 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 quartic interaction term of the model has no effect on the large N limit of the field Φ, 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 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 Q. Contents√ N :Φ 2 : 29 6. Next order stationary dynamic 41 Appendix A. Besov spaces 48 References 52

A stochastic PDE approach to large N problems in quantum field theory: a survey

“…its meaning needs interpretation, see (5.10) below. We could actually prove this convergence result under more general conditions for initial data (see [77,Sec. 4]).…”

“…This is ultimately similar with the mechanism (3.4) in the earlier section for the ODE case. After considerably technical estimates, we can show that sup t∈[0,T ] v i (t) 2 L 2 → 0 in probability, as N → ∞, and Theorem 5.1 holds (see [77,Sec. 4] for details).…”

“…We can then iterate this procedure, and eventually bring down the power of N to 0 ! See [77,Sec. 6] for details.…”

McKean-Vlasov SDEs and SPDEs with Locally Monotone Coefficients

“…In [40] Kotelenez studied a class of quasilinear SPDEs (called the mezoscopic equation) of McKean-Vlasov type with mass conservation, and showed the existence and continuity of solutions to the mezoscopic SPDEs. Recently, Shen et al [62] investigated the large N limits of a coupled system of N interacting Φ 4 models on 2-dimensional torus T 2 . They considered the N → ∞ limit of each component and proved that a suitable distribution dependent nonlinear SPDE governs the limiting dynamics.…”