Large N limit of the O(N) linear sigma model in 3D
aa r X i v : . [ m a t h . P R ] F e b LARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D
HAO SHEN, RONGCHAN ZHU, AND XIANGCHAN ZHU
Abstract.
In this paper we study the large N limit of the O ( N )-invariantlinear sigma model, which is a vector-valued generalization of the Φ quantumfield theory, on the three dimensional torus. We study the problem via its sto-chastic quantization, which yields a coupled system of N interacting SPDEs.We prove tightness of the invariant measures in the large N limit. For largeenough mass or small enough coupling constant, they converge to the (massive)Gaussian free field at a rate of order 1 / √ N with respect to the Wassersteindistance. We also obtain tightness results for certain O ( N ) invariant observ-ables. These generalize some of the results in [SSZZ20] from two dimensionsto three dimensions. The proof leverages the method recently developed by[GH18] and combines many new techniques such as uniform in N estimates onperturbative objects as well as the solutions. Contents
1. Introduction 12. Preliminaries 63. Stochastic terms and decomposition of equation 104. Uniform in N estimate 235. Convergence of measures and observable 38Appendix A. Extra estimates 45Appendix B. Notation index 48References 481. Introduction
In this paper, we continue our study initiated in [SSZZ20] on the applicationof singular SPDE methods to large N problems in quantum field theory (QFT).Large N problems in QFT generally refer to the study of asymptotic behaviors ofQFT models as the dimensionality of the target space where the quantum fieldstake values tends to infinity. Physicists’ study of large N problems in QFT origi-nates from the seminal work by Wilson [Wil73] and Gross-Neveu [GN74], and soonflourished following the work by t’Hooft [t’H74] who applied it to gauge theories,which influenced many aspects of probability. We refer to [SSZZ20, Section 1.1] fora more thorough exposition of the background and motivation for large N methodsin QFT. A prototype model, which we consider in this paper, is the O ( N )-invariant Date : February 5, 2021. linear sigma model given by the (formal) measured ν N (Φ) def = 1 C N exp (cid:18) − Z T d N X j =1 |∇ Φ j | + m N X j =1 Φ j + λ N (cid:16) N X j =1 Φ j (cid:17) d x (cid:19) D Φ (1.1)over R N valued fields Φ = (Φ , Φ , ..., Φ N ), where T d is the d -dimensional torus and C N is a normalization constant (partition function). This is an N -component gen-eralization of the Φ d model, which is symmetric under the action by the orthogonalgroup O ( N ) on the field Φ. In this paper we focus on d = 3.Our main tool to investigate the large N behavior of the above model (1.1) isthe system of SPDEs arising as its stochastic quantization: L Φ i = − λN N X j =1 Φ j Φ i + ξ i , (1.2)where L = ∂ t − ∆ + m with m >
0, and i ∈ { , · · · , N } . The collection ( ξ i ) Ni =1 consists of N independent space-time white noises on a probability space (Ω , F , P ).The utilization of the dynamic (1.2) to probe large N properties of the QFT model(1.1) lead to connections with dynamical mean field theory, see [SSZZ20, Section 1.2]for more discussions on this connection.In our 2D paper [SSZZ20], since the SPDE is less singular, we obtained a seriesof results. We showed that the non-stationary dynamics converge to a mean fielddynamic L Ψ i = − λ E [Ψ i ]Ψ i + ξ i (which needs to be interpreted via a suitablerenormalization) as N → ∞ ; for large enough mass, the invariant measures con-verge to the (massive) Gaussian free field which is the unique invariant measureof the mean-field dynamic; moreover, we proved not only tightness but also exactcorrelation formulae for some O ( N ) invariant observables as N → ∞ .In d = 3, the SPDE becomes much more singular. For N = 1, namely the dynam-ical Φ model, the construction of local solutions was achieved by [Hai14] using thetheory of regularity structures and then [CC18] using paracontrolled distributionsdeveloped in [GIP15]; and global solutions were studied in [MW17a, GH19, AK17,MW20, GH18]. One should be able to extend these constructions and estimates tothe vector-valued case for a finite and fixed N > N → ∞ of theinvariant measures (1.1) as well as O ( N )-invariant observables, in d = 3. Given themore singular nature in d = 3, it is not the purpose of this article to extend all theresults in the 2D paper [SSZZ20] to 3D. Instead we only focus on large N limit ofthe O ( N ) linear sigma model ν N in (1.1) and tightness of the observables in d = 3.1.1. Main results.
Our first result shows that assuming m to be sufficiently largeor λ to be sufficiently small, the invariant measures converge to the (massive)Gaussian free field. Note that since the interaction term in the measure (1.1) hasa sum besides the factor 1 /N , it is far from being obvious that the large N limitof the measures is a Gaussian free field. This was only heuristically predicted byphysicists (e.g.[Wil73]) at the level of perturbation theory: namely, after suitablerenormalization, all the Feynman diagrams are of order N α for some α < ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 3 To state the result, let ν def = N (0 , ( m − ∆) − ) be the (massive) Gaussian freefield. Consider the projection onto the i th component of the field ΦΠ i : S ′ ( T d ) N → S ′ ( T d ) , Π i (Φ) def = Φ i . (1.3)Noting that ν N is a measure on S ′ ( T d ) N , we define the marginal law ν N,i def = ν N ◦ Π − i . Furthermore, considerΠ ( k ) : S ′ ( T d ) N → S ′ ( T d ) k , Π ( k ) (Φ) = (Φ i ) i k (1.4)and define the marginal law of the first k components by ν Nk def = ν N ◦ (Π ( k ) ) − . Wedenote by W ,k the Wasserstein distance, with the precise definition given in (5.2). Theorem 1.1.
For any ( m, λ ) ∈ (0 , ∞ ) × [0 , ∞ ) and every i > , the sequenceof probability measures ( ν N,i ) N > is tight on H − − κ for any κ > . Moreover,there exists a constant c > such that for all ( m, λ ) ∈ [1 , ∞ ) × [0 , ∞ ) satisfying m > c λ (1 + λ ) and every k > , there exists a constant C k > such that W ,k ( ν Nk , ν ⊗ k ) C k N − . Our next result is concerned with O ( N )-invariant observables for the invariantmeasure. We refer to [SSZZ20, Section 1] for more discussion on the motivation ofstudying observables for QFT models with continuous symmetries. Here a naturalquantity that is invariant under O ( N )-rotations is the squared “length” of Φ, thatis suitably renormalized and scaled with respect to N :1 √ N N X i =1 : Φ i : , Φ = (Φ i ) i N ∼ ν N . (1.5)The precise definition is given by (5.8) in Section 5. We establish the large N tightness of these observables as random fields in suitable Besov spaces. Theorem 1.2.
For ( m, λ ) ∈ [1 , ∞ ) × [0 , ∞ ) satisfying m > c λ (1 + λ ) as inTheorem 1.1, the sequence of random variables √ N P Ni =1 : Φ i : is tight in B − − κ , for any κ > . The proofs of Theorem 1.1 (split into Theorems 5.3 and 5.4) and Theorem 1.2will be given in Section 5.We note that these rigorous large N results on QFT in d = 3 are new in naturecomparing with the earlier rigorous results which were:(1) proved with a small scale cutoff e.g. on a fixed lattice, thus completely avoidultraviolet problem, for instance [Kup80b] on nonlinear sigma (classical Heisenberg)model and [Cha19, CJ16] on gauge theories; or(2) proved in low dimensions, for instance Kupiainen [Kup80a] obtained large N expansion for the pressure (vacuum energy) of the model (1.1) in d = 2 usingconstructive field theory methods, which would become much more sophisticatedin d = 3; or(3) for models with some kind of solvability, e.g. [L´ev11, AS12] for convergenceof Wilson loop observables for the Yang-Mills model in continuum, which cruciallyrelies on d = 2.We refer to [SSZZ20] for a more complete historical remarks and literature onthe physical and mathematical results. To our best knowledge nothing was knownin a very singular setting such as the model being considered here in d = 3, whichis now analyzable thanks to the new SPDE methods developed more recently. Onthe other hand, the above simplifications certainly allow one to obtain stronger HAO SHEN, RONGCHAN ZHU, AND XIANGCHAN ZHU results such as higher order 1 /N expansions or treat more challenging models. SeeSection 1.4 for some discussion.1.2. Methodology and difficulties.
To obtain our main results, we decomposeour system (1.2) into leading order terms and a remainder called Y i (see (3.23)below), and prove uniform in N bounds for the solution to the Y i equation. As in 2Dcase, the extra factor 1 /N before the nonlinear terms makes the damping effect from Y j Y i weaker as N becomes large. We cannot exploit the strong damping effect atthe level of a fixed component and instead we have to consider aggregate quantities,and ultimately we focus on the empirical average of the L -norms instead of L p -norms for p >
2. The advantage for the estimate of the empirical average of the L -norm is that the dissipation term k N P Ni =1 Y i k L from Y j Y i (see Section 3.4)can be used to control the nonlinear terms. However, it seems not clear how toexploit the corresponding dissipation effect to deduce uniform in N estimate with L p norms.It would be a natural attempt to exploit the method in [MW17a] or [GH19](which obtained global a priori estimates for dynamical Φ ) to establish uniform in N bounds. However, for the reason just mentioned, it is not clear how to apply the L p ( p >
2) estimates developed in [MW17a] to large N problems; also the maximumprinciple used in [GH19, MW20] seems not easy to be applied for the vector valuedcase. In this paper, we exploit the approach developed in [GH18] (which relies oninteresting cancellation) to establish L uniform bounds; due to the cancellationcertain higher regularity estimate which would be rather technical is not requiredanymore for L uniform estimates (see Section 3.4 for more details). On the otherhand, compared to the dynamical Φ model, the dissipation effect from the term k N P Ni =1 Y i k L is weaker than the L norm. We need further decomposition andchoose suitable parameter to balance the competing contributions for the estimateof the cubic term (see Lemma 4.8 for details.)Another important ingredient in our proof is a cancelation mechanism whichemerges from Section 3.3 and arises from the following reasons. In our decompo-sition of the solution to (1.2), the leading order terms consist of (polynomials of) N independent Gaussian processes; so when calculating moments of certain sumsof these leading order terms in a suitably chosen Hilbert space, many terms do notcontribute, which allows us to gain “factors of 1 /N ”. See Lemma 3.4 (and alsoLemmas 3.5-3.6) for these effects.Combining the above methods with techniques we developed in [SSZZ20] we ob-tain uniform estimates for the sum of L -norm of Y i (see Theorem 4.3 and Theorem4.4). Then as in 2D case, we follow the idea in [GH18] and construct a jointly sta-tionary process (Φ , Z ) whose components satisfy (1.2) and the corresponding linearequation (3.2), respectively. In this case, the law of Z is Gaussian free field andthe law of (Φ , Z ) gives a coupling between the measures ν N and ν . We use thiscoupling to establish the convergence of ν N,i to ν by invoking the uniform estimateson the stationary processes.1.3. Background and heuristics.
We review here how physicists (e.g. Wilson[Wil73]) predicted convergence of the model (1.1) (after inserting suitable renor-malization constants in a way that corresponds to (3.1)) to GFF as N → ∞ , andwhy their heuristic argument was far from being rigorous. Since this is completelyformal, we simply drop the renormalization and pretend that our fields can beevaluated at spatial points. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 5 The prediction was based on viewing (1.1) as a perturbation of GFF, namely, toformally Taylor expand (1.1) in λ . As an example, one can calculate the two-pointcorrelation E [Φ i ( y )Φ i ( y )] (for a fixed i ) in this way; the zeroth order term in thisTaylor expansion is obtained by simply taking λ = 0, which gives E [ Z i ( y ) Z i ( y )],where Z j ∼ N (0 , ( m − ∆) − ). Each of the higher order terms in this Taylorexpansion can be calculated as an expectation of a product of Gaussians usingWick theorem: for instance at order λ one of the terms has the form λ N Z E h Z i ( y ) Z i ( y ) (cid:16) N X i ,j =1 Z i ( x ) Z j ( x ) (cid:17)(cid:16) N X i ,j =1 Z i ( x ) Z j ( x ) (cid:17)i dx dx . One of the terms obtained from applying Wick theorem to the above expectationhas the form λ N N X i ,j ,i ,j =1 δ i,i δ i,i δ i ,i δ j ,j Z E [ Z i ( y ) Z i ( x )] E [ Z i ( y ) Z i ( x )] × E [ Z i ( x ) Z i ( x )] E [ Z j ( x ) Z j ( x )] dx dx where the Kronecker δ ’s come from the independence of ( Z j ) Nj =1 . (This integral isconvergent after introducing the renormalization constant ˜ b ε as in (3.1).) Thanksto these Kronecker δ ’s, the summation has N terms (rather than N terms), so theabove expression is of order 1 /N and thus converge to 0 as N →
0. This argumentis far from being rigorous because even it can be shown that every order of theTaylor expansion (except for the zeroth order) converge to 0 as N →
0, and eventhis can be done for higher order correlations, this kind of Taylor expansion cannot converge (see [Jaf65]). This type of predictions can be rigorously proved nowthanks to the recent development of singular SPDE techniques and the methodsdeveloped in this paper. On the other hand, as we will see, some of our calculationssuch as Lemma 3.4 have a similar flavor with the formal perturbative argumentsdiscussed above.1.4.
Future directions.
We mention a few possible future directions. The proofof tightness in Theorem 1.2 requires establishing uniform bounds on moment ofthe observables. It will be more interesting to prove exact formulae of correlationsof the observables (as done in [SSZZ20] in d = 2) in the N → ∞ limit, by firstusing integration by parts (i.e. Dyson-Schwinger equations) to find the leadingcontribution to the formula and then establishing uniform bounds which show thatthe remainder terms vanish. In 3D it would require more effort to interpret theDyson-Schwinger equations and prove the bounds for the remainder, so we leave itto future work.Another interesting question is whether our convergence results hold for a largerrange of ( m, λ ), or even over the entire ( m, λ ) ∈ R , at least on the torus. Itwould be interesting to generalize some of our results to infinite volume settingby introducing suitable weights. Finally, as discussed in Section 1.1, it would cer-tainly be interesting to investigate large N problems for more challenging modelswithout lattice cutoff, such as Lie algebra valued [CCHS20] or manifold valued[BGHZ19, Hai16, RWZZ20, CWZZ21] models with large dimension of target spacevia stochastic quantization.1.5. Structure of the paper.
This paper is organized as follows. In Section 2 wecollect the notations and useful tools such as paraproducts and commutator esti-mates used through out the paper. Then in Section 3, we set up the decomposition
HAO SHEN, RONGCHAN ZHU, AND XIANGCHAN ZHU of our SPDE system, discuss renormalization, derive the energy balance identity,and prove uniform in N bounds for stochastic terms. Section 4 is devoted to theproof of uniform in N energy estimates for the solution.Section 5 is concerned with the proof of Theorem 1.1 and Theorem 1.2. Theconvergence of invariant measures from ν N,i to the Gaussian free field ν is shownin Section 5.2. Section 5.3 is devoted to the study of the observables and the proofof Theorem 1.2. In Appendix A we give an extra estimate and finally we collectsome notation in Appendix B. Acknowledgments.
We would like to thank Scott Smith for the numerous andvery helpful discussions on mean field limits and large N problems of singularSPDEs. H.S. gratefully acknowledges financial support from NSF grants DMS-1712684 / DMS-1909525 and DMS-1954091. R.Z. and X.Z. are grateful to thefinancial supports of the NSFC (No. 11771037, 11922103, 12090014) and the finan-cial support by the DFG through the CRC 1283 “Taming uncertainty and profitingfrom randomness and low regularity in analysis, stochastics and their applications”and the support by key Lab of Random Complex Structures and Data Science,Youth Innovation Promotion Association (2020003), Chinese Academy of Science.2. Preliminaries
Throughout the paper, we use the notation a . b if there exists a constant c > a cb , and we write a ≃ b if a . b and b . a .2.1. Besov spaces.
Given a Banach space E with a norm k · k E and T >
0, wewrite C T E = C ([0 , T ]; E ) for the space of continuous functions from [0 , T ] to E ,equipped with the supremum norm k f k C T E = sup t ∈ [0 ,T ] k f ( t ) k E . For p ∈ [1 , ∞ ]we write L pT E = L p ([0 , T ]; E ) for the space of L p -integrable functions from [0 , T ]to E , equipped with the usual L p -norm. Let H be a separable Hilbert space withnorm k · k H and inner product h· , ·i H . Given p > α ∈ (0 , W α,pT H be theSobolev space of all f ∈ L pT H such that Z T Z T k f ( t ) − f ( s ) k pH | t − s | αp d t d s < ∞ , equipped with the norm k f k pW α,pT H def = Z T k f ( t ) k pH d t + Z T Z T k f ( t ) − f ( s ) k pH | t − s | αp d t d s. Let S ′ be the space of distributions on T d . We use (∆ i ) i > − to denote the Littlewood–Paley blocks for a dyadic partition of unity. Besov spaces on the torus with generalindices α ∈ R , p, q ∈ [1 , ∞ ] are defined as the completion of C ∞ with respect tothe norm k u k B αp,q := ( X j > − (2 jα k ∆ j u k qL p ) /q , and the H¨older-Besov space C α is given by C α = B α ∞ , ∞ . We will often write k · k C α instead of k · k B α ∞ , ∞ . For α ∈ R , set H α = B α , . Set Λ = (1 − ∆) .The following embedding results will be frequently used. Lemma 2.1.
Let p p ∞ and q q ∞ , and let α ∈ R . Then B αp ,q ⊂ B α − d (1 /p − /p ) p ,q . Here ⊂ means continuous and dense embedding. (cf. [GIP15, Lemma A.2] ) ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 7 We also recall the following interpolation lemma.
Lemma 2.2.
Suppose that s ∈ (0 , . Then for f ∈ H k f k H s . k f k − sL k f k sH . (cf. [Tri78, Theorem 4.3.1] ) Smoothing effect of heat flow.
We recall the following smoothing effect ofthe heat flow P t = e t (∆ − m ) , m > Lemma 2.3.
Let u ∈ B αp,q for some α ∈ R , p, q ∈ [1 , ∞ ] . Then for every δ > , t ∈ [0 , T ] k P t u k B α + δp,q . t − δ/ k u k B αp,q , where the proportionality constants are uniform for m > . If β − α , then k (I − P t ) u k B αp,q . t β − α k u k B βp,q , where the proportionality constants are uniform for m > . We also define ( I f )( t, x ) := ( L − f )( t, x ) = R t P t − s f d s . Lemma 2.4. ( [GIP15, Lemma A.9] , [ZZZ20, Lemma 2.8, Lemma 2.9] ) Let α ∈ R .Then the following bounds hold uniformly over t T kI f ( t ) k C α . k f k L ∞ T C α . (2.1) If α < then kI f ( t ) k C (2+ α ) / T L ∞ . k f k L ∞ T C α , where the proportionality constants are uniform for m > . Lemma 2.5.
It holds that for β ∈ R kI f k L T H β . k f k L T H β − , and kI f k W − κ, T L . k f k L T H − − κ , for < κ < / , where the proportionality constants are uniform for m > .Proof. Let { e k ( x ) } = { − e ιπk · x , k ∈ Z } on T . kI f k L T H β = Z T X k ( | k | + m ) β (cid:12)(cid:12)(cid:12)D Z t P t − s f d s, e k E(cid:12)(cid:12)(cid:12) d t = Z T X k ( | k | + m ) β (cid:12)(cid:12)(cid:12) Z t e − ( t − s )( | k | + m ) h f, e k i d s (cid:12)(cid:12)(cid:12) d t . Z T X k ( | k | + m ) β − Z t e − ( t − s )( | k | + m ) |h f, e k i| d s d t = X k ( | k | + m ) β − Z T Z Ts e − ( t − s )( | k | + m ) d t |h f, e k i| d s . k f k L T H β − , where we use H¨older’s inequality in the first inequality. Thus the first result follows. HAO SHEN, RONGCHAN ZHU, AND XIANGCHAN ZHU
By the definition of W − κ, T L we know kI f k W − κ, T L = Z T kI f ( s ) k L d s + Z T Z T kI f ( t ) − I f ( s ) k L | t − s | − κ ) d s d t = I + I .I can be easily controlled by the first result and we consider I part. Using smootheffect of heat kernel Lemma 2.3 we have I . Z T Z t k ( P t − s − I) I f ( s ) k L | t − s | − κ ) d s d t + Z T Z t (cid:16)R ts ( t − r ) − − κ k f ( r ) k H − − κ d r (cid:17) | t − s | − κ ) d s d t . Z T Z t kI f ( s ) k H − κ | t − s | − κ d s d t + Z T Z t R ts ( t − r ) − − κ k f ( r ) k H − − κ dr | t − s | − κ d s d t . kI f k L T H − κ + k f k L T H − − κ , where we used the fact that the second term on the right is controlled by Z T Z Tr Z r ( t − r ) − − κ k f ( r ) k H − − κ | t − s | − − +5 κ d s d t d r . Z T Z Tr ( t − r ) − κ k f ( r ) k H − − κ d t d r in the last step. Thus the second result follows from the first result. (cid:3) Paraproducts and commutators.
We recall the following paraproduct in-troduced by Bony (see [Bon81]). In general, the product f g of two distributions f ∈ C α , g ∈ C β is well defined if and only if α + β >
0. In terms of Littlewood-Paleyblocks, the product f g of two distributions f and g can be formally decomposed as f g = f ≺ g + f ◦ g + f ≻ g, with f ≺ g = g ≻ f = X j > − X i Lemma 2.6. Let β ∈ R , p, p , p , q ∈ [1 , ∞ ] such that p = p + p . Then it holds k f ≺ g k B βp,q . k f k L p k g k B βp ,q , and if α < then k f ≺ g k B α + βp,q . k f k B αp ,q k g k B βp ,q . If α + β > then it holds k f ◦ g k B α + βp,q . k f k B αp ,q k g k B βp ,q . Furthermore, we have the following multiplicative inequality. Lemma 2.7. (i) Let α, β ∈ R and p, p , p , q ∈ [1 , ∞ ] be such that p = p + p .The bilinear map ( u, v ) uv extends to a continuous map from B αp ,q × B βp ,q to B α ∧ βp,q if α + β > . (cf. [MW17b, Corollary 3.21] )(ii) (Duality.) Let α ∈ (0 , , p, q ∈ [1 , ∞ ] , p ′ and q ′ be their conjugate ex-ponents, respectively. Then the mapping h u, v i 7→ R uv d x extends to a continuous ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 9 bilinear form on B αp,q × B − αp ′ ,q ′ , and one has |h u, v i| . k u k B αp,q k v k B − αp ′ ,q ′ (cf. [MW17b,Proposition 3.23] ).(iii) (Fractional Leibniz estimate) Let s > , p, p , p , p , p , q ∈ [1 , ∞ ] satisfy p = p + p = p + p . Then it holds that [CGW20, Proposition A.2] k uv k B sp,q . k u k B sp ,q k g k L p + k f k L p k g k B sp ,q . (2.2)By duality and [MW17b, Proposition 3.25] we easily deduce the following result(cf. [SSZZ20, Lemma 2.5]): Lemma 2.8. For s ∈ (0 , |h g, f i| . (cid:0) k∇ g k sL k g k − sL + k g k L (cid:1) k f k C − s . We also recall the following commutator estimate from [GIP15, Lemma 2.4] (seealso [MW17a, Proposition A.9]). Lemma 2.9. Let α ∈ (0 , and β, γ ∈ R such that α + β + γ > and β + γ < , p, p , p ∈ [1 , ∞ ] , p = p + p . Then there exist a trilinear bounded operator ˜ C ( f, g, h ) : B αp , ∞ × C β × B γp , ∞ → B α + β + γp, ∞ satisfying k ˜ C ( f, g, h ) k B α + β + γp, ∞ . k f k B αp , ∞ k g k C β k h k B γp , ∞ and for smooth functions f, g, h ˜ C ( f, g, h ) = ( f ≺ g ) ◦ h − f ( g ◦ h ) . The following lemmas are from [GH18, Lemma A.13, Lemma A.14]. In [GH18]these lemmas are stated in the discrete setting, but the same arguments lead to thefollowing statements in the continuum setting. Lemma 2.10. Let α, β, γ ∈ R such that α + β + γ > and β + γ < . Then thereexist a trilinear bounded operator D ( f, g, h ) : H α × C β × H γ → R satisfying | D ( f, g, h ) | . k f k H α k g k C β k h k H γ and for smooth functions f, g, hD ( f, g, h ) = h f, g ◦ h i − h f ≺ g, h i . Lemma 2.11. Let α, β, γ ∈ R such that α ∈ (0 , , α + β + γ + 2 > and β + γ + 2 < . Then there exist trilinear bounded operators C ( f, g, h ) : H α × C β × C γ + δ → H β + γ +2 , ¯ C ( f, g, h ) : C α × C β × C γ + δ → C β + γ +2 satisfying for δ > k C ( f, g, h ) k H β + γ +2 . k f k H α k g k C β k h k C γ + δ k ¯ C ( f, g, h ) k C T C β + γ +2 . ( k f k C T C α + k f k C α/ T L ∞ ) k g k C T C β k h k C T C γ + δ and for smooth functions f, g, hC ( f, g, h ) = h ◦ ( m − ∆) − ( f ≺ g ) − f ( h ◦ ( m − ∆) − g ) . ¯ C ( f, g, h ) = I ( f ≺ g ) ◦ h − f I ( g ) ◦ h, for I = L − . We also prove the following estimate for commutators: Lemma 2.12. For ¯ C introduced in Lemma 2.11, f ∈ L T H − κ ∩ W − κ, T L and g, h ∈ C T C − − κ with < κ < / , it holds that k ¯ C ( f, g, h ) k L T L . ( k f k L T H − κ + k f k W − κ, T L ) k g k C T C − − κ k h k C T C − − κ . Here the proportionality constants are uniform for m > .Proof. We write ¯ C as follows:¯ C ( f, g, h )( t ) = I ( δ st f ≺ g ) ◦ h + Z t ( P t − s [ f t ≺ g s ] − f t ≺ P t − s g s )d s ◦ h + ( f ≺ I g ) ◦ h − f ( I g ◦ h )= I + I + I , with δ st f = f s − f t . By Lemmas 2.9 and 2.4 we have k I k L T L . k h k C T C − − κ kI g k C T C − κ k f k L T H − κ . k h k C T C − − κ k g k C T C − − κ k f k L T H − κ . Using Lemma 2.6, Lemma 2.3 followed by H¨older’s inequality we find k I k L T L . k h k C T C − − κ kI ( δ st f ≺ g ) k L T H κ . k h k C T C − − κ k g k C T C − − κ Z T h Z t k δ st f k L | t − s | κ d s i d t . k h k C T C − − κ k g k C T C − − κ Z T Z t k δ st f k L | t − s | κ d s d t . k f k W − κ, T L k g k C T C − − κ k h k C T C − − κ . Moreover, Lemma 2.6 and [MW17a, Proposition A.16] imply that k I k L T L . k h k C T C − − κ (cid:13)(cid:13)(cid:13) Z t (cid:16) P t − s [ f t ≺ g s ] − f t ≺ P t − s g s (cid:17) d s (cid:13)(cid:13)(cid:13) L T H κ . k h k C T C − − κ k g k C T C − − κ Z T h Z t k f t k H − κ | t − s | − κ d s i d t . k f k L T H − κ k g k C T C − − κ k h k C T C − − κ , where we use H¨older’s inequality in the last step. (cid:3) Stochastic terms and decomposition of equation In this section, we obtain some a priori estimates on the renormalized version ofequation (1.2). Let ξ i,ε be a space-time mollification of ξ i defined on R × T . Theapproximate equation to equation (1.2) is given as L Φ i,ε + λN N X j =1 Φ j,ε Φ i,ε + ( − N + 2 N λa ε + 3( N + 2) N λ ˜ b ε )Φ i,ε = ξ i,ε , (3.1)where a ε and ˜ b ε are renormalization constants given below. Let Z i be a stationarysolution to L Z i = ξ i . (3.2) ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 11 Renormalization. In this section we introduce the renormalized terms. Let Z i,ε be the stationary solution to L Z i,ε = ξ i,ε . For convenience, we assume that allthe noises are mollified with a common bump function. In particular, Z i,ε are i.i.d.mean zero Gaussian. The Wick power of Gaussian variables Z i , Z j are defined asfollows Z ij = lim ε → ( Z i,ε − a ε ) ( i = j )lim ε → Z i,ε Z j,ε ( i = j ) Z ijj,ε = ( Z i,ε − a ε Z i,ε ( i = j ) Z i,ε Z j,ε − a ε Z i,ε ( i = j ) (3.3)where a ε = E [ Z i,ε (0 , i and the limit isunderstood in C T C − − κ for κ > T ∈ (0 , ∞ ) denotes an arbitrary finite time. Let ˜ Z ijj,ε be the stationarysolution to L ˜ Z ijj,ε = Z ijj,ε , i.e. ˜ Z ijj,ε = Z t −∞ P t − s Z ijj,ε ( s )d s := ˜ IZ ijj,ε . Set ˜ Z ijj = lim ε → ˜ Z ijj,ε , where the limit is in C T C − κ for κ > 0. By definition it is easy to see that˜ Z ijj = ˜ Z jij = ˜ Z jji .As the next step we introduce further stochastic objects needed below. Set D := m − ∆. ˜ Z ijj,k,ε = ˜ Z ijj,ε ◦ Z k,ε , ˜ Z ij,kℓ,ε = D − ( Z ij,ε ) ◦ Z kℓ,ε − ˜ b ε ( i = k = j = ℓ or i = ℓ = j = k ) , D − ( Z ij,ε ) ◦ Z kℓ,ε − ˜ b ε ( i = k = j = ℓ ) , D − ( Z ij,ε ) ◦ Z kℓ,ε (otherwise) , Z ij,kℓ,ε = I ( Z ij,ε ) ◦ Z kℓ,ε − b ε ( t )2 ( i = k = j = ℓ or i = ℓ = j = k ) , I ( Z ij,ε ) ◦ Z kℓ,ε − b ε ( t ) ( i = k = j = ℓ ) , I ( Z ij,ε ) ◦ Z kℓ,ε (otherwise) , ˜ Z ijj,ik,ε = ˜ Z ijj,ε ◦ Z ik,ε − ˜ b ε Z j,ε ( j = k = i ) , ˜ Z ijj,ε ◦ Z ik,ε − b ε Z j,ε ( j = k = i ) , ˜ Z ijj,ε ◦ Z ik,ε (otherwise) , where b ε ( t ) = E [ I ( Z ii,ε ) ◦ Z ii,ε ] and ˜ b ε = E [ D − ( Z ii,ε ) ◦ Z ii,ε ] stand for renor-malization constant and | b ε − ˜ b ε | . t − γ for any γ > ε . We denotecollectively Z ε def = ( Z i,ε , Z ij,ε , ˜ Z ijj,ε , ˜ Z ijj,k,ε , ˜ Z ij,kℓ,ε , Z ij,kℓ,ε , ˜ Z ijj,ik,ε ) . For κ > α τ ∈ R are given by τ = Z i,ε Z ij,ε ˜ Z ijj,ε ˜ Z ijj,k,ε ˜ Z ij,kℓ,ε Z ij,kℓ,ε ˜ Z ijj,ik,ε α τ = − − κ − − κ − κ − κ − κ − κ − − κ Recall that I is defined similarly but with R t −∞ replaced by R t . Lemma 3.1. For every κ, σ > and some < δ < / , there exist randomdistributions Z def = ( Z i , Z ij , ˜ Z ijj , ˜ Z ijj,k , ˜ Z ij,kℓ , Z ij,kℓ , ˜ Z ijj,ik ) (3.4) such that if τ ε is a component in Z ε and τ is the corresponding component in Z then τ ε → τ in C T C α τ ∩ C δ/ T C α τ − δ a.s. as ε → . Furthermore, for every p > m > E k τ ε k pC T C ατ + sup m > E k τ ε k pC δ/ T C ατ − δ . , sup m > E k τ k pC T C ατ + sup m > E k τ k pC δ/ T C ατ − δ . , where the proportional constants in the inequalities are independent of ε, i, j, N . In the following if we do not need the precise powers of the renormalized termsin Z , we denote Q ( Z ) a generic polynomial in terms of the above norms of τ with E Q ( Z ) q . q > 1. We note that Q ( Z ) may depend on N and summationsfrom 1 to N , for instance, it could take the form N P Ni =1 τ i , but the expectation of Q ( Z ) is uniformly bounded and independent of N .By Lemma 3.1 there exists a measurable Ω ⊂ Ω with P (Ω ) = 1 such that for ω ∈ Ω k τ k C T C ατ + k τ k C δ/ T C ατ − δ < ∞ , and τ ε → τ in C T C α τ ∩ C δ/ T C α τ − δ as ε → τ ∈ Z . We fix such ω ∈ Ω inthe following. Now we fix κ small enough and δ = 1 / − κ in the above estimate.3.2. Stochastic terms. Let U > def = X j>L ∆ j , U def = X j L ∆ j , (3.5)for some suitable chosen constant L > 0. Let X i solve the following equation, whichwill be used in Section 3.4: X i = − λN N X j =1 (cid:16) I ( X j ≺ U > Z ij ) + I ( X i ≺ U > Z jj ) + ˜ Z ijj (cid:17) . (3.6)For fixed N and L > 0, by using fixed point argument and Lemma 2.6 weeasily deduce local well-posedness of equation (3.6) in C T C − κ . Furthermore, bysuitable choice of L , we have the following uniform in N estimates, which implyglobal well-posedness. The proof is motivated by [GH18, Lemma 4.1]. Lemma 3.2. There exists L = L ( λ, N ) > such that N N X i =1 k X i k C T C − κ . λ N N X i,j =1 k ˜ Z ijj k C T C − κ , (3.7) and k X i k C T C − κ . R N + λN N X j =1 k ˜ Z ijj k C T C − κ , (3.8) k X i k C − κT L ∞ . R N + λN N X j =1 k ˜ Z ijj k C − κT L ∞ + λN N X j =1 k ˜ Z ijj k C T C − κ (3.9) ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 13 where R N = λ N N X i,j =1 k ˜ Z ijj k C T C − κ + λ N N X j =1 kZ ij k C T C − − κ , and the proportional constant is independent of N , λ and m .Proof. By the definition of U > and the definition of Besov spaces we find k U > Z ij k C T C − / − κ . − L/ kZ ij k C T C − − κ , (3.10)which combined with the Schauder estimate Lemma 2.4 and Lemma 2.6 impliesthat k X i k C T C − κ . − L/ λN N X j =1 (cid:16) k X j k C T C − κ kZ ij k C T C − − κ + k X i k C T C − κ kZ jj k C T C − − κ (cid:17) + λN N X j =1 k ˜ Z ijj k C T C − κ . − L/ (cid:16) N N X j =1 k X j k C T C − κ (cid:17) (cid:16) λ N N X j =1 kZ ij k C T C − − κ (cid:17) (3.11)+ 2 − L/ k X i k C T C − κ (cid:16) λN N X j =1 kZ jj k C T C − − κ (cid:17) + λN N X j =1 k ˜ Z ijj k C T C − κ . Taking square on both sides and summing over i then dividing by N we deduce1 N N X i =1 k X i k C T C − κ . − L (cid:16) N N X j =1 k X j k C T C − κ (cid:17)(cid:16) λ N N X i,j =1 kZ ij k C T C − − κ (cid:17) + 2 − κL (cid:16) N N X i =1 k X i k C T C − κ (cid:17)(cid:16) λN N X j =1 kZ jj k C T C − − κ (cid:17) + λ N N X i,j =1 k ˜ Z ijj k C T C − κ . Choosing L = L ( λ, N ) such that2 L = 2 C (cid:16) λ N N X i,j =1 kZ ij k C T C − − κ (cid:17) + 2 C (cid:16) λN N X j =1 kZ jj k C T C − − κ (cid:17) + 1 , (3.12)with C given as the maximum of proportional constant in (3.11) and (3.20) below,we easily deduce (3.7). Using (3.11) and (3.7), (3.8) follows. (3.9) follows fromusing Schauder estimate Lemma 2.4 to bound k X i k C − κT L ∞ in the analogous wayas (3.11) and then applying (3.7) (3.8) and (3.12). (cid:3) From Lemma 3.2 we know X i ∈ C T C − κ . We also introduce X i,ε defined by(3.6) with the elements in Z replaced by Z ε . It is easy to deduce that X i,ε → X i in C T C − κ P − a.s., as ε → . When we do energy estimate below, we will use X j ◦Z ij , X i ◦Z jj and X j ◦ Z i , whichare not well-defined in the classical sense. Here we will use the renormalized terms introduced in Section 3.1 to define these terms. We first consider the definition of X j ◦ Z ij . For i = jX j ◦ Z ij def = lim ε → ( X j,ε ◦ Z ij,ε + λ ˜ b ε N ( Z i,ε + X i,ε ))= − λN N X l =1 lim ε → (cid:20) ˜ Z llj,ij,ε + I (2 X l,ε ≺ Z jl,ε + X j,ε ≺ Z ll,ε ) ◦ Z ij,ε − I (2 X l,ε ≺ U ( Z jl,ε ) + X j,ε ≺ U Z ll,ε ) ◦ Z ij,ε − ˜ b ε N X i,ε (cid:21) = − λN N X l =1 (cid:20) ˜ Z llj,ij + 2 X l Z jl,ij + X j Z ll,ij + 2 ¯ C ( X l , Z jl , Z ij ) + ¯ C ( X j , Z ll , Z ij ) − I (2 X l ≺ U Z jl + X j ≺ U Z ll ) ◦ Z ij (cid:21) + λN (˜ b − b ( t )) X i , (3.13)with ˜ b − b ( t ) = lim ε → (˜ b ε − b ε ( t )) , | ˜ b − b ( t ) | . t − γ for t > γ > 0. Here ¯ C is the commutator introduced in Lemma 2.11, andwe note that 2 Z jl,ij requires renormalization in the case that i = l , and we alsonote that Z ll,ij does not require such renormalization since we are assuming i = j .Using Lemma 2.6, Lemma 2.11 and Lemma 3.1 the limit in (3.13) is understood in C ((0 , T ]; C − − κ ) P -a.s.. For the case i = j , we have similar decomposition as in(3.13) with ˜ b ε replaced by 3˜ b ε . (2 Z jl,ij gives an extra 2˜ b , and Z ll,ij gives an extra˜ b , in the case i = j = l .)Similarly we define X i ◦ Z jj as follows for i = jX i ◦ Z jj def = lim ε → ( X i,ε ◦ Z jj,ε + λ ˜ b ε N ( Z i,ε + X i,ε ))= − λN N X l =1 (cid:20) ˜ Z ill,jj + 2 X l Z il,jj + X i Z ll,jj + 2 ¯ C ( X l , Z il , Z jj ) + ¯ C ( X i , Z ll , Z jj ) − I (2 X l ≺ U Z il + X i ≺ U Z ll ) ◦ Z jj (cid:21) + λN (˜ b − b ( t )) X i , (3.14)and for the case that i = j , we have similar decomposition with ˜ b ε replaced by 3˜ b ε .We also define X j ◦ Z i = lim ε → ( X j,ε ◦ Z i,ε )= − λN N X l =1 (cid:20) ˜ Z llj,i + I (2 X l ≺ U > Z jl + X j ≺ U > Z ll ) ◦ Z i (cid:21) , (3.15)where the limit is taken in C T C − κ P -a.s..In the following lemma we derive uniform bounds for the above stochastic terms. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 15 Lemma 3.3. It holds that N N X i,j =1 k X j Z i k C T C − − κ . λ Q N , N N X j =1 k X j Z j k C T C − − κ . λ (1 + λ ) Q N , where Q N , Q N are given in the proof with E Q N + E Q N . and the proportionalconstants are independent of N and λ .Proof. Regarding X j Z i we have the following decomposition X j Z i = X j ≺ Z i + X j ◦ Z i + X j ≻ Z i . By (3.15) and Lemmas 2.4, 2.6 we obtain k X j ◦ Z i k C T C − κ . λN N X l =1 (cid:20) k ˜ Z llj,i k C T C − κ + k X l k C T C − κ kZ jl k C T C − − κ k Z i k C T C − − κ + k X j k C T C − κ kZ ll k C T C − − κ k Z i k C T C − − κ (cid:21) . (3.16)Moreover, by Lemma 2.6 we have k X j ≺ Z i k C T C − − κ + k X j ≻ Z i k C T C − − κ . k X j k C T C − κ k Z i k C T C − − κ . By (3.7) we have1 N N X i,j =1 k X j Z i k C T C − − κ . λ N N X i,j,l =1 k Z i k C T C − − κ k ˜ Z ljj k C T C − κ + λ N N X i,j,l =1 k ˜ Z llj,i k C T C − κ + λ N N X i,j,k,l =1 k ˜ Z ljj k C T C − κ kZ kk k C T C − − κ k Z i k C T C − − κ + λ N N X i,j,k,l,m =1 k ˜ Z ljj k C T C − κ kZ km k C T C − − κ k Z i k C T C − − κ def = λ Q N , where the first line corresponds to X j ≺ Z i + X j ≻ Z i and the second line and thethird line are the estimate for X j ◦ Z i .For the term involving X j Z j we also have the paraproduct decomposition X j Z j = X j ≺ Z j + X j ≻ Z j + X j ◦ Z j . Using Lemma 2.6, we can bound the first two terms by k X j k C T C − κ k Z j k C T C − − κ .By (3.8) we obtain1 N N X j =1 k X j k C T C − κ k Z j k C T C − − κ . N N X j =1 (cid:16) λ N N X i,l =1 k ˜ Z ill k C T C − κ + λ N N X l =1 kZ jl k C T C − − κ (cid:17) k Z j k C T C − − κ + λ N N X i,j =1 k ˜ Z ijj k C T C − κ k Z i k C T C − − κ . λ (1 + λ ) Q N , with Q N def = 1 N N X i,j =1 k ˜ Z ijj k C T C − κ k Z i k C T C − − κ + 1 N N X l,j =1 kZ jl k C T C − − κ k Z j k C T C − − κ + 1 N N X i,j,l =1 k ˜ Z ill k C T C − κ k Z j k C T C − − κ . Using (3.15) and (3.16) with i = j we deduce1 N N X j =1 k X j Z j k C T C − − κ . λ (1 + λ ) Q N (cid:16) N N X l =1 kZ ll k C T C − − κ (cid:17) + λ (1 + λ ) N (cid:16) N X j,l =1 k ˜ Z jll,j k C T C − κ (cid:17) + λ (1 + λ ) (cid:16) N N X i,j =1 k ˜ Z ijj k C T C − κ (cid:17)(cid:16) N N X l,k =1 kZ kl k C T C − − κ k Z k k C T C − − κ (cid:17) def = λ (1 + λ ) Q N . The result follows by the above estimates. (cid:3) Improved uniform in N estimates for stochastic terms. In this sectionwe derive uniform in N estimates for the stochastic terms introduced in Section 3.2,which shows that one obtains “improved estimates” by gaining “factors of 1 /N ”.Recall that for mean-zero independent random variables U , . . . , U N taking val-ues in a Hilbert space H , we have E (cid:13)(cid:13)(cid:13) N X i =1 U i (cid:13)(cid:13)(cid:13) H = E N X i =1 k U i k H . (3.17)This simple fact is important for us since the square of the sum on the LHS of(3.17) appears to have “ N terms” but under expectation it’s only a sum of N terms, in a certain sense giving us a “factor of 1 /N ”. This motivates us to derivethe following uniform in N estimate in suitable Hilbert space. We first prove thefollowing result for renormalization terms Z ijj and Z ljj,ij . Lemma 3.4. Set Q N def = 1 N N X i =1 (cid:13)(cid:13)(cid:13) N X j =1 ˜ Z ijj (cid:13)(cid:13)(cid:13) L T H − κ , Q N def = 1 N N X i =1 (cid:13)(cid:13)(cid:13) N X j =1 ˜ Z ijj (cid:13)(cid:13)(cid:13) W − κ, T L ,Q N def = 1 N N X i =1 (cid:18) N X j =1 N (cid:13)(cid:13)(cid:13) N X l =1 ˜ Z llj,ij (cid:13)(cid:13)(cid:13) L T H − − κ (cid:19) . It holds that for every q > and i = 0 , , E | Q iN | q . , with the proportionalconstant independent of N and m .Proof. Since we will have several similar calculations in the sequel, we first demon-strate such calculation in the case q = 1. We have for s = − κ E N N X i =1 (cid:13)(cid:13)(cid:13) N X j =1 Λ s ˜ Z ijj (cid:13)(cid:13)(cid:13) L T L = 1 N N X i,j ,j =1 E D Λ s ˜ Z ij j , Λ s ˜ Z ij j E L T L . ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 17 We have 3 summation indices and a factor 1 /N . The contribution to the sum fromthe cases j = i or j = i or j = j is bounded by a constant in light of Lemma3.1. If i, j , j are all different, by independence and the fact that Wick productsare mean zero, the terms are zero.For general q > 1, by Gaussian hypercontractivity and the fact that Q iN is arandom variable in finite Wiener chaos, we have for i = 0 , , E [( Q iN ) q ] . E [( Q iN ) ] q/ . So it suffices to consider q = 2. We write E [( Q N ) ] as1 N N X i ,i ,j k =1 k =1 ... E D Λ s ˜ Z i j j , Λ s ˜ Z i j j E L T L D Λ s ˜ Z i j j , Λ s ˜ Z i j j E L T L . We have 6 indices i , i , j k , k = 1 , ..., N and an overall factor1 /N . Using again Lemma 3.1, we reduce the problem to the cases where five or sixof the indices are different. However, in these two cases, at least one of j k is differentfrom others. Then by independence the expectation is zero, so E [( Q N ) ] . W − κ, T L is a Hilbert space with inner product given by h f, g i W − κ, T L = h f, g i L T L + Z T Z T h f ( t ) − f ( r ) , g ( t ) − g ( r ) i L | t − r | − κ d t d r, we deduce E | Q N | q . E | Q N | . First note that Q N is bounded by1 N N X i,j =1 (cid:13)(cid:13)(cid:13) N X l =1 ˜ Z llj,ij (cid:13)(cid:13)(cid:13) L T H − − κ , which implies that E | Q N | is bounded by1 N N X i,j,i ,j ,l k =1 k =1 ... E D Λ s ˜ Z l l j,ij , Λ s ˜ Z l l j,ij E L T L D Λ s ˜ Z l l j ,i j , Λ s ˜ Z l l j ,i j E L T L , for s = − − κ . We have 8 indices i, i , j, j , l k , k = 1 , ..., N and an overall factor 1 /N . Using again Lemma 3.1, we reduce the problem to thecases where seven or eight of the indices are different. However, in these two casesat least one of l k is different from others. Then by independence the expectation iszero, so E [( Q N ) ] . (cid:3) With the help of Lemma 3.4 we also have the following bounds for X , whichstates that summing N terms of suitable Hilbert norms of X i actually “behaveslike order 1”. Lemma 3.5. The following estimate holds N X i =1 k X i k L T H − κ . λ Q N , (3.18) where Q N satisfies the moment bounds stated in Lemma 3.4. Moreover, it holds N X i =1 k X i k W − κ, T L . λ Q ( Z ) , (3.19) with ( E Q ( Z ) q ) /q . λ for any q > . Here the proportional constants are allindependent of N and λ .Proof. The start-point of the proof is similar as that of Lemma 3.2, but now it’scrucial that we estimate X i in a Hilbert space. Using the Schauder estimate inLemma 2.5 now (instead of Lemma 2.4) and Lemma 2.6 and (3.10) we have k X i k L T H − κ . − L/ λN N X j =1 (cid:16) k X j k L T H − κ kZ ij k C T C − − κ + k X i k L T H − κ kZ jj k C T C − − κ (cid:17) + (cid:13)(cid:13)(cid:13) λN N X j =1 ˜ Z ijj (cid:13)(cid:13)(cid:13) L T H − κ . − L/ (cid:18) N X j =1 k X j k L T H − κ (cid:19) (cid:18) λ N N X j =1 kZ ij k C T C − − κ (cid:19) (3.20)+ 2 − L/ k X i k L T H − κ (cid:18) λN N X j =1 kZ jj k C T C − − κ (cid:19) + λN (cid:13)(cid:13)(cid:13) N X j =1 ˜ Z ijj (cid:13)(cid:13)(cid:13) L T H − κ . Then taking square on both sides and summing over i we obtain N X i =1 k X i k L T H − κ . − L (cid:18) N X j =1 k X j k L T H − κ (cid:19)(cid:18) λ N N X i,j =1 kZ ij k C T C − − κ (cid:19) + 2 − L (cid:18) N X i =1 k X i k L T H − κ (cid:19)(cid:18) λN N X j =1 kZ jj k C T C − − κ (cid:19) + λ N N X i =1 (cid:13)(cid:13)(cid:13) N X j =1 ˜ Z ijj (cid:13)(cid:13)(cid:13) L T H − κ . By the choice of 2 L in (3.12), (3.18) follows.To show (3.19), again by Lemma 2.5 and Lemma 2.6 we have k X i k W − κ, T L . λ N (cid:13)(cid:13)(cid:13) N X j =1 ˜ Z ijj (cid:13)(cid:13)(cid:13) W − κ, T L + h λN N X j =1 (cid:16) k X j k L T H − κ kZ ij k C T C − − κ + k X i k L T H − κ kZ jj k C T C − − κ (cid:17)i , which combined with (3.18) implies that N X i =1 k X i k W − κ, T L . λ Q N + (cid:18) N X j =1 k X j k L T H − κ (cid:19)(cid:18) λ N N X i,j =1 kZ ij k C T C − − κ (cid:19) + (cid:18) N X i =1 k X i k L T H − κ (cid:19)(cid:18) λN N X j =1 kZ jj k C T C − − κ (cid:19) , where Q N , Q N are introduced in Lemma 3.4. Using (3.18) with Lemma 3.1 andLemma 3.4, (3.19) follows. (cid:3) In the following lemma we deduce the estimates in a suitable Hilbert space forthe renormalized terms introduced in (3.13), (3.14) and (3.15) before Lemma 3.3. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 19 Lemma 3.6. It holds that N N X i =1 (cid:16) N X j =1 k X j Z i k L T H − − κ (cid:17) λ Q ( Z ) , N N X i =1 (cid:16) N X j =1 k X j X i Z j k L T H − − κ (cid:17) λ Q ( Z ) , N N X i =1 (cid:16) N X j =1 k X i ◦ Z jj k L T H − − κ (cid:17) λ Q ( Z ) , N N X i =1 (cid:16) N X j =1 k X j ◦ Z ij k L T H − − κ (cid:17) λ Q ( Z ) , where E Q ( Z ) q . λ for any q > with the proportional constant independentof N and λ .Proof. From the proof of Lemma 3.3 we have1 N N X i,j =1 k X j ◦ Z i k C T C − κ . λ Q ( Z ) , with Q ( Z ) satisfying E Q ( Z ) q . q > 1. Now we consider X j Z i and useparaproduct to have the following decomposition X j ◦ Z i = 2( X j ≺ X j ) ◦ Z i + ( X j ◦ X j ) ◦ Z i = 2 X j ( X j ◦ Z i ) + 2 ˜ C ( X j , X j , Z i ) + ( X j ◦ X j ) ◦ Z i , (3.21)where ˜ C is introduced in Lemma 2.9. By Lemma 2.6 and Lemma 2.9 we have k X j ◦ Z i k L T H − / − κ . k X j k L T H − κ k X j ◦ Z i k C T C − κ + k X j k L T H − κ k X j k C T C − κ k Z i k C T C − − κ , and k X j ≺ Z i k L T H − − κ + k X j ≻ Z i k L T H − − κ . k X j k L T H − κ k X j k C T C − κ k Z i k C T C − − κ . Thus H¨older’s inequality implies that1 N N X i =1 (cid:16) N X j =1 k X j Z i k L T H − − κ (cid:17) . N N X i =1 (cid:16) N X j =1 k X j k L T H − κ (cid:17)(cid:16) N X j =1 k X j ◦ Z i k C T C − κ (cid:17) + 1 N N X i =1 (cid:16) N X j =1 k X j k L T H − κ (cid:17)(cid:16) N X j =1 k X j k C T C − κ (cid:17) k Z i k C T C − − κ . Then by Lemma 3.2, Lemma 3.5 and Lemma 3.4 we obtain the first bound claimedin the lemma. The bound for X i X j Z j follows in the same way. To show the third bound claimed in the lemma, by the decomposition for X j ◦Z ij in (3.13) and Lemma 2.12, Lemma 2.5 and Lemma 2.6 we have k X j ◦ Z ij k L T H − − κ . λN (cid:13)(cid:13)(cid:13) N X l =1 ˜ Z llj,ij (cid:13)(cid:13)(cid:13) L T H − − κ + λN N X l =1 (cid:20) k X l k L T H − κ kZ jl,ij k C T C − κ + k X j k L T H − κ kZ ll,ij k C T C − κ + (cid:0) (1 + 2 κL ) k X l k L T H − κ + k X l k W − κ, L (cid:1) kZ jl k C T C − − κ kZ ij k C T C − − κ + (cid:0) (1 + 2 κL ) k X j k L T H − κ + k X j k W − κ, L (cid:1) kZ ll k C T C − − κ kZ ij k C T C − − κ (cid:21) + λN k (˜ b − b ) X i k L T H − − κ , where we used (3.12) and k U Z jl k C − κ + k U Z ll k C − κ . κL ( kZ jl k C − − κ + kZ ll k C − − κ ) . Moreover,1 N N X i =1 (cid:16) N N X j =1 λN k (˜ b − b ) X i k L T H − − κ (cid:17) . λ N N X i =1 k X i k C T C − − κ . Thus by Lemma 3.5 and Lemma 3.2 we have1 N N X i =1 (cid:18) N X j =1 k X j ◦ Z ij k L T H − − κ (cid:19) . λ Q N + λ Q ( Z ) , with Q ( Z ) satisfying ( E Q ( Z ) q ) /q . λ which combined with Lemma 3.4 impliesthe third result. The last bound regarding X i ◦Z jj then follows in the same way. (cid:3) Decomposition. In this section we consider the following equation L Φ i = − λN N X j =1 Φ j Φ i + ξ i , Φ i (0) ∈ C − − κ , (3.22)for κ > 0. For fixed N and ω ∈ Ω by using regularity structure theory [Hai14]or paracontrolled distribution method [GIP15], we easily deduce the local well-posedness of (3.22) in C T C − − κ and Φ i is the limit of Φ i,ε , which is the uniquesolution to equation (3.1). Furthermore, by similar arguments as in [MW17a, GH19]global well-posedness also holds by uniform estimates, which may depend on N . Inthe following we concentrate on the uniform in N bounds. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 21 With the stochastic objects at hand, we have the following decompositions: Φ i = Z i + X i + Y i with Y i satisfying the following equation L Y i = − λN N X j =1 (cid:18) Y j Y i + ( X j + 2 X j Y j )( X i + Y i ) + Y j X i + ( X j + Y j ) Z i + 2( X j + Y j )( X i + Y i ) Z j + 2 X j ≺ U Z ij + X i ≺ U Z jj + 2 Y j ≺ Z ij + Y i ≺ Z jj + 2( X j + Y j ) < Z ij + ( X i + Y i ) < Z jj (cid:19) .Y i (0) = Φ i (0) − Z i (0) − X i (0) . (3.23)Here the first line in (3.23) are the expansion of ( Y j + X j ) ( Y i + X i + Z i ); the termscontaining Z ij and Z jj correspond to the remaining terms in the paraproductexpansion of 2( Y j + X j ) Z j Z i and ( Y i + X i ) Z j , respectively.Also, note that some products in (3.23) are understood via renormalization.Namely, X j Z i , X j Z i , X j X i Z j , X j ◦Z ij and X i ◦Z jj are understood by using (3.13)-(3.15) and (3.21) and Lemmas 3.3 and 3.6. Since Z ij ∈ C T C − − κ , the expectedregularity of Y i is C T C − κ ; so in view of Lemma 2.6 Y j ◦ Z ij and Y i ◦ Z jj arenot well-defined in the classical sense and we need to use the renormalization terms Z ij,kl to define these terms, which from the approximation level requires to subtract λ ( N +2) N ˜ b ε Y i,ε on the R.H.S of (3.23), where Y i,ε = Φ i,ε − X i,ε − Z i,ε .In the following we establish L -energy estimate for Y i , and as explained in theintroduction we follow the idea in [GH18] to use the duality between ≺ and ◦ , i.e.Lemma 2.10 to cancel Y j ◦ Z ij and Y i ◦ Z jj which would require paracontrolledansatz and higher regularity estimate than H .Recall that D = m − ∆ and we define ϕ i def = Y i + D − λN N X j =1 (2 Y j ≺ Z ij + Y i ≺ Z jj ) def = Y i + D − P i , (3.24)where in the last step we defined P i .Now we turn to uniform in N bounds on (3.23) and note that Y i depends on N , but we omit this throughout. Similar as the 2d case in [SSZZ20], we do L -energy estimate of Y i and take sum over i . Using (3.24), we have the followingdecomposition. Lemma 3.7. (Energy balance) N X i =1 dd t k Y i k L + m N X i =1 k ϕ i k L + N X i =1 k∇ ϕ i k L + λN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L = Θ + Ξ . Here Θ = N X i =1 h D − P i , P i i − λN N X i,j =1 (cid:16) D ( Y i , Z ij , Y j ) + D ( Y i , Z jj , Y i ) (cid:17) , and Ξ = − λN N X i,j =1 (cid:28) ( X j + 2 X j Y j )( X i + Y i ) + Y j X i + ( X j + Y j ) Z i This decomposition follows [GH18] which uses different notation than Z + X + Y here. Ourchoice of this notation is close to our 2D paper [SSZZ20]. + 2( X j + Y j )( X i + Y i ) Z j + 2 X j ≺ U Z ij + X i ≺ U Z jj + 2( X j + Y j ) ≻ Z ij + 2 X j ◦ Z ij + ( X i + Y i ) ≻ Z jj + X i ◦ Z jj , Y i (cid:29) . Proof. We will first focus on a formal derivation for the claimed identity. Then wewill remark that although new renormalization appears to be necessary when wetake inner product, in the end they cancel each other.Taking inner product with Y i in L on (3.23), we realize that the first term onthe R.H.S. of (3.23) leads to the term − λN k P Ni =1 Y i k L , and it is straightforwardto check that the other terms all lead to Ξ except the following terms − λN N X i,j =1 h Y j Z ij + Y i Z jj , Y i i . (3.25)We claim that we can write (3.25) plus P Ni =1 h (∆ − m ) Y i , Y i i as Θ − m P Ni =1 k ϕ i k L − P Ni =1 k∇ ϕ i k L (see Eq. (3.30)) and we will prove this claim for the rest of this proof.Using (3.24) we write it as h (∆ − m ) Y i , Y i i = h (∆ − m ) ϕ i , ϕ i i + 2 h Y i , P i i + h D − P i , P i i . (3.26)We will realize below that 2 h Y i , P i i cancels the irregular part (i.e. the paraproduct part) in (3.25).For the term in (3.25), by (3.24) we have − λN N X j =1 h Y j Z ij + Y i Z jj , Y i i = −h Y i , P i i − λN N X j =1 h Y j ◦ Z ij + Y i ◦ Z jj , Y i i (3.27)where we note that the first term −h Y i , P i i on the R.H.S. precisely cancels one h Y i , P i i on the R.H.S. of (3.26). Using Lemma 2.10, the other terms in (3.27)containing ◦ can be written as h Y i , Y j ◦ Z ij i = 2 h Y i ≺ Z ij , Y j i + 2 D ( Y i , Z ij , Y j ) , (3.28) h Y i , Y i ◦ Z jj i = h Y i ≺ Z jj , Y i i + D ( Y i , Z jj , Y i ) , (3.29)where D ( f, g, h ) is the commutator introduced in Lemma 2.10. The first terms onthe R.H.S. of (3.28) and (3.29) cancel the other h Y i , P i i from (3.26) when takingsum.We sum all the terms in (3.26) and (3.27) w.r.t. i and obtain the following: N X i =1 (cid:20) h (∆ − m ) Y i , Y i i − λN N X j =1 h Y j Z ij + Y i Z jj , Y i i (cid:21) (3.30)= N X i =1 (cid:2) h (∆ − m ) ϕ i , ϕ i i + h D − P i , P i i (cid:3) − λN N X i,j =1 (cid:0) D ( Y i , Z ij , Y j ) + D ( Y i , Z jj , Y i ) (cid:1) . This completes the derivation of the claimed identity.Finally, we remark that several terms in the above derivation should be under-stood in the renormalized sense. As we have noticed above, the term P Ni =1 h P i , Y i i cancels with the irregular part from the second term of (3.25), so no extra renor-malization is needed for them. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 23 For the last term in (3.26) we have h D − P i , P i i = λN N X j =1 (cid:20) h ( D − P i ) ◦ Z ij , Y j i − D ( Y j , Z ij , D − P i )+ h ( D − P i ) ◦ Z jj , Y i i − D ( Y i , Z jj , D − P i ) (cid:21) . (3.31)Using the definition of D − P i and the commutator C introduced in Lemma 2.11we can write the above term as λN N X j =1 (cid:18) λN N X l =1 (cid:20) h ˜ Z il,ij , Y l Y j i + 4 h C ( Y l , Z il , Z ij ) , Y j i + h ˜ Z ll,ij , Y i Y j i + 2 h C ( Y i , Z ll , Z ij ) , Y j i + h ˜ Z il,jj , Y l Y i i + 2 h C ( Y l , Z il , Z jj ) , Y i i + h ˜ Z ll,jj , Y i i + h C ( Y i , Z ll , Z jj ) , Y i i (cid:21) − D ( Y j , Z ij , D − P i ) − D ( Y i , Z jj , D − P i ) (cid:19) . (3.32)Here – recall the renormalization of ˜ Z in Section 3.1 – we need to subtract λ ( N + 8) N ˜ b ε h Y i,ε , Y i,ε i + 2 λ N X j = i ˜ b ε h Y j,ε , Y j,ε i (3.33)from the approximation level to go from (3.31) to (3.32). This precisely matchesthe renormalization from the SPDE (3.1): indeed, summing (3.33) over i , we get N +6 N λ ˜ b ε times P i h Y i,ε , Y i,ε i .We thus conclude that although new renormalization appears to be necessarywhen we take inner product, in the end no extra renormalization other than theones introduced at the level of the SPDE is actually needed. (cid:3) Uniform in N estimate The aim of this section is to prove uniform in N estimates based on the decom-position from Lemma 3.7. The main results are Theorems 4.3 and 4.4. The keystep to prove these theorems is to control the terms in R T (Θ + Ξ)d t by δ (cid:16) N X j =1 k∇ ϕ j k L T L + λ N X j =1 k Y j k L T H − κ + λN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L (cid:17) + C δ λ (1 + λ ) Z T (cid:16) N X i =1 k Y i k L (cid:17) ( R N + R N + Q N )d s + λ (1 + λ ) Q N (4.1)for a small constant δ > R N , R N , Q N , Q N introduced in Lemma 4.1 andPropositions 4.5 and 4.6 – see these Propositions for more precise statements.4.1. Proof of uniform estimate based on (4.1) . We put the proof of (4.1) inSection 4.2. In this subsection we prove Theorem 4.3 and Theorem 4.4 by using(4.1). Before proceeding we first prove the following two results. The first one isused to turn k ϕ i k L on the L.H.S. of the identity in Lemma 3.7 to k Y i k L . Thesecond one gives uniform in N estimates of various norms of Y i , ϕ i , D − P i in termsof (4.1) by using (3.24). Lemma 4.1. It holds that m N X i =1 k ϕ i k L > ( m − Cλ κ R N ) N X i =1 k Y i k L + m N X i =1 k D − P i k L , with R N = 1 N N X i,j =1 kZ ij k C − − κ + 1 + 1 N N X j =1 kZ jj k C − − κ , where C is independent of λ and can be chosen uniform for m > .Proof. By the definition (3.24) we know m N X i =1 k ϕ i k L = m N X i =1 k Y i k L − m N X i =1 h Y i , D − P i i + m N X i =1 k D − P i k L . (4.2)It remains to control the second term on the right. On the other hand, we have for κ > k m D − f k L = X k m ( m + | k | ) | ˆ f ( k ) | m − κ X k m + | k | ) κ | ˆ f ( k ) | m − κ k f k H − − κ , namely, k m D − f k L m − κ k f k H − − κ , (4.3)where the proportional constant is independent of m . By (4.3) and Lemma 2.6 wehave N X i =1 |h Y i , m D − P i i| N X i =1 k Y i k L k m D − P i k L m − κ N X i =1 k Y i k L (cid:13)(cid:13)(cid:13)(cid:13) λN N X j =1 (2 Y j ≺ Z ij + Y i ≺ Z jj ) (cid:13)(cid:13)(cid:13)(cid:13) H − − κ . m − κ λN N X i,j =1 k Y i k L (cid:16) k Y j k L kZ ij k C − − κ + k Y i k L kZ jj k C − − κ (cid:17) m − κ (cid:16) N X i =1 k Y i k L (cid:17)(cid:20)(cid:18) λ N N X i,j =1 kZ ij k C − − κ (cid:19) + (cid:16) λN N X j =1 kZ jj k C − − κ (cid:17)(cid:21) N X i =1 k Y i k L (cid:20) m C (cid:18) λ N N X i,j =1 kZ ij k C − − κ (cid:19) κ + (cid:18) λCN N X j =1 kZ jj k C − − κ (cid:19) κ (cid:21) , where we used Young’s inequality in the last step. Now the result follows. (cid:3) The following estimates will be useful in the sequel. Lemma 4.2. It holds N X i =1 k Y i k H − κ (cid:16) N X i =1 k ϕ i k H (cid:17) + Cλ (cid:16) N X i =1 k Y i k L (cid:17) R N , In the first bound, we keep the explicit constant 2 for the purpose of the proof of Theorem 4.3;in the third bound, we keep the explicit constant 2 in order to derive the condition for m and λ later. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 25 N X i =1 k D − P i k H − κ Cλ (cid:16) N X i =1 k Y i k L (cid:17) R N , (4.4) N X i =1 k ϕ i k L (cid:16) N X i =1 k Y i k L (cid:17) ( Cλ R N + 2) , N N X i,j =1 k ϕ i ϕ j k L CN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L (1 + λ R N ) , where R N is introduced in Lemma 4.1 and C is independent of λ and can be chosenuniform for m > .Proof. Recalling the relation ϕ i = Y i + D − P i in the definition (3.24), we willsee that we essentially only need to estimate k D − P i k H − κ for the first threeinequalities in the lemma. Since k D f k H β ≃ k f k H β +2 for β ∈ R , by Lemma 2.6 wehave k D − P i k H − κ λCN N X j =1 (cid:16) k Y j k L kZ ij k C − − κ + k Y i k L kZ jj k C − − κ (cid:17) (4.5) C (cid:16) N X j =1 k Y j k L (cid:17) (cid:16) λ N N X j =1 kZ ij k C − − κ (cid:17) + C k Y i k L (cid:16) λN N X j =1 kZ jj k C − − κ (cid:17) . The second bound for D − P i follows from taking square on both sides of (4.5) andsumming over i ; this together with k Y i k H − κ k ϕ i k H + k D − P i k H − κ also yields the first bound. The third bound for ϕ i follows by k ϕ i k L k Y i k L + k D − P i k H − κ and simply plugging in the above bound on D − P i .Moreover, we use (4.4) to have1 N N X i,j =1 k ϕ i ϕ j k L = 1 N (cid:13)(cid:13)(cid:13) N X i =1 ϕ i (cid:13)(cid:13)(cid:13) L C N (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L + C N N X i,j =1 k D − P i k L k D − P j k L CN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L (1 + λR N ) , where we used Sobolev embedding to have k f k L . k f k H − κ for κ > (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L > (cid:16) N X i =1 k Y i k L (cid:17) . (4.6) (cid:3) The main result of this section is given as follows. Theorem 4.3. It holds that (cid:16) N X j =1 k Y j ( T ) k L (cid:17) + 12 N X j =1 k∇ ϕ j k L T L + m N X j =1 k Y j k L T L + 18 N X j =1 k Y j k L T H − κ + λN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L (cid:16) N X j =1 k Y j (0) k L (cid:17) + λ (1 + λ ) Q N + N X j =1 k Y j k L T L + Cλ (1 + λ ) Z T (cid:16) N X i =1 k Y i k L (cid:17) ( R N + R N + Q N )d s, where the proportional constant is independent of N and λ and can be chosen uni-form for m > .Proof. Taking integration w.r.t. time for the energy equality from Lemma 3.7 andusing Lemma 4.1 and (4.1), we deduce (cid:16) N X j =1 k Y j ( T ) k L (cid:17) + 2 N X j =1 k∇ ϕ j k L T L + m N X j =1 k Y j k L T L + 2 λN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L δ (cid:16) N X j =1 k∇ ϕ j k L T L + N X j =1 k Y j k L T H − κ + λN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L (cid:17) + (cid:16) N X j =1 k Y j (0) k L (cid:17) + C δ λ (1 + λ ) Z T (cid:16) N X i =1 k Y i k L (cid:17) ( R N + R N + Q N )d s + λ (1 + λ ) Q N , for some δ > P Nj =1 k Y j k L T H − κ the result follows. (cid:3) Furthermore by using the dissipation effect from the term N k P Ni =1 Y i k L T L , wededuce that the empirical averages of the L norms of Y i can be controlled pathwisein terms of the averages of the renormalized terms Q ( Z ) with finite moment, asstated in the following theorem. Theorem 4.4. It holds that sup t ∈ [0 ,T ] N N X j =1 k Y j ( t ) k L + mN N X j =1 k Y j k L T L + 12 N N X j =1 k∇ ϕ j k L T L + 18 N N X j =1 k Y j k L T H − κ + λ N (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L Q ( Z ) + 2 N N X i =1 k Y i (0) k L . Here Q ( Z ) satisfies E Q ( Z ) . C ( λ ) with C ( λ ) independent of N .Proof. Similar as in the proof of Theorem 4.3 we havesup t ∈ [0 ,T ] (cid:16) N N X j =1 k Y j ( t ) k L (cid:17) + 12 N N X j =1 k∇ ϕ j k L T L + mN N X j =1 k Y j k L T L + 18 N N X j =1 k Y j k L T H − κ + λN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 27 (cid:16) N N X j =1 k Y j (0) k L (cid:17) + Cλ (1 + λ ) Z T (cid:16) N N X i =1 k Y i k L (cid:17) ( R N + R N + Q N )d s ++ λ (1 + λ ) Q N + 1 N N X j =1 k Y j k L T L . Using 1 N (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L > (cid:16) N N X i =1 k Y i k L (cid:17) , Young’s inequality and E ( Q N ) . C ( λ ) from Lemma 4.7 below, E ( R N ) + E ( R N ) . (cid:3) Proof of (4.1) . We first consider easier part Θ. Proposition 4.5. It holds for δ > small that | Θ | δ N X i =1 k Y i k H − κ + Cλ (1 + λ ) (cid:16) N X i =1 k Y i k L (cid:17) R N , (4.7) where C is independent of λ, N and can be chosen uniform for m > and R N def = 1 + (cid:16) N N X i,j =1 kZ ij k − θ C − − κ (cid:17) + (cid:16) N N X j =1 kZ jj k − θ C − − κ (cid:17) + 1 N N X i,j,l =1 (cid:16) k ˜ Z il,ij k C − κ + k ˜ Z ll,ij k C − κ + k ˜ Z il,jj k C − κ (cid:17) + 1 N N X j,l =1 k ˜ Z ll,jj k C − κ for θ = κ − κ .Proof. We estimate each term in Θ. By Young’s inequality we will use the first linein R N to control the renormalization terms evolving Z ij below and use the secondand the third line in R N to bound the renormalization terms containing ˜ Z ij,kl . Step D ( Y i , Z ij , Y j ) and D ( Y i , Z jj , Y i ))We first control the terms containing the commutator D in Θ. By Lemma 2.10,H¨older’s inequality, and interpolation Lemma 2.2, we have (cid:12)(cid:12)(cid:12) λN N X i,j =1 D ( Y i , Z ij , Y j ) (cid:12)(cid:12)(cid:12) . λN N X i,j =1 k Y i k H 12 + κ kZ ij k C − − κ k Y j k H 12 + κ . λ (cid:16) N N X i,j =1 kZ ij k C − − κ (cid:17) (cid:16) N X i =1 k Y i k H 12 + κ (cid:17) . λ (cid:16) N N X i,j =1 kZ ij k C − − κ (cid:17) N X i =1 k Y i k θH − κ k Y i k − θ ) L (4.8) . λ (cid:16) N N X i,j =1 kZ ij k C − − κ (cid:17) (cid:16) N X i =1 k Y i k H − κ (cid:17) θ (cid:16) N X i =1 k Y i k L (cid:17) − θ δ N X i =1 k Y i k H − κ + C δ λ − θ (cid:16) N X i =1 k Y i k L (cid:17)(cid:16) N N X i,j =1 kZ ij k C − − κ (cid:17) − θ ) , where θ = κ − κ ∈ ( , ). Similarly we have (cid:12)(cid:12)(cid:12) λN N X i,j =1 D ( Y i , Z jj , Y i ) (cid:12)(cid:12)(cid:12) . λN N X i,j =1 k Y i k H 12 + κ kZ jj k C − − κ δ N X i =1 k Y i k H − κ + C δ λ − θ (cid:16) N X i =1 k Y i k L (cid:17)(cid:16) N N X j =1 kZ jj k C − − κ (cid:17) − θ . Therefore the terms containing D in Θ can be controlled by the right hand side of(4.7). Step h D − P i , P i i )In the following we estimate each term in (3.32). We have three types of terms: I. Terms without C or D such as h ˜ Z il,ij , Y l Y j i , II. Terms with C , such as h C ( Y i , Z ll , Z ij ) , Y j i , III. Terms with D , such as D ( Y j , Z ij , D − P i ).The terms having the same type can be estimated in the same way. I. For the first type we use Lemma 2.2 and Lemma 2.7 to have (cid:12)(cid:12)(cid:12) λ N N X i,j,l =1 h ˜ Z il,ij , Y l Y j i (cid:12)(cid:12)(cid:12) . λ N N X i,j,l =1 k ˜ Z il,ij k C − κ k Y l k H κ k Y j k H κ . λ N N X i =1 (cid:16) N N X j,l =1 k ˜ Z il,ij k C − κ (cid:17) (cid:16) N X j =1 k Y j k H κ (cid:17) δ N X i =1 k Y i k H − κ + C δ λ − θ (cid:16) N X i =1 k Y i k L (cid:17)(cid:16) N N X i,j,l =1 k ˜ Z il,ij k C − κ (cid:17) − θ , where θ = κ − κ ∈ (0 , 1) and we used Lemma 2.7 and to have k Y l Y j k B κ , . k Y l k B κ , k Y j k B κ , . k Y l k H κ k Y j k H κ , in the first inequality. By the exactly same arguments the same bounds hold for (cid:12)(cid:12)(cid:12) λ N N X i,j,l =1 h ˜ Z ll,ij , Y i Y j i (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) λ N N X i,j,l =1 h ˜ Z il,jj , Y l Y i i (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) λ N N X i,j,l =1 h ˜ Z ll,jj , Y i i (cid:12)(cid:12)(cid:12) with N P Ni,j,l =1 k ˜ Z il,ij k C − κ on the right hand side replaced by, respectively,1 N N X i,j,l =1 k ˜ Z ll,ij k C − κ , N N X i,j,l =1 k ˜ Z il,jj k C − κ , N N X j,l =1 k ˜ Z ll,jj k C − κ . Using Young’s inequality the terms containing P Ni =1 k Y i k L can be all controlledby λ (1 + λ ) (cid:16) P Ni =1 k Y i k L (cid:17) R N . ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 29 II. For the second type, using Lemma 2.11 and interpolation Lemma 2.2 andH¨older’s inequality we have (cid:12)(cid:12)(cid:12) λ N N X i,j,l =1 h C ( Y l , Z il , Z ij ) , Y j i (cid:12)(cid:12)(cid:12) . λ N N X i,j,l =1 kZ il k C − − κ kZ ij k C − − κ k Y l k H 12 + κ k Y j k H 12 + κ δ N X i =1 k Y i k H − κ + C δ λ − θ (cid:16) N X i =1 k Y i k L (cid:17)(cid:16) N N X i,j,l =1 kZ il k − θ C − − κ kZ ij k − θ C − − κ (cid:17) , where we use similar argument as in (4.8) and by Young’s inequality the last termcan be controlled by λ (1 + λ ) (cid:16) P Ni =1 k Y i k L (cid:17) R N .By the exactly same arguments the same bounds hold for (cid:12)(cid:12)(cid:12) λ N N X i,j,l =1 h C ( Y i , Z ll , Z ij ) , Y j i (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) λ N N X i,j,l =1 h C ( Y l , Z il , Z jj ) , Y i i (cid:12)(cid:12)(cid:12) , and (cid:12)(cid:12)(cid:12) λ N N X i,j,l =1 h C ( Y i , Z ll , Z jj ) , Y i i (cid:12)(cid:12)(cid:12) with kZ il k − θ C − − κ kZ ij k − θ C − − κ on the right hand side replaced by, respectively, kZ ll k − θ C − − κ kZ ij k − θ C − − κ , kZ il k − θ C − − κ kZ jj k − θ C − − κ , kZ ll k − θ C − − κ kZ jj k − θ C − − κ . As above by Young’s inequality the terms containing P Ni =1 k Y i k L can be allcontrolled by λ (1 + λ ) (cid:16) P Ni =1 k Y i k L (cid:17) R N . III. For the third type, by using Lemma 2.10 we have (cid:12)(cid:12)(cid:12) λN N X i,j =1 − D ( Y j , Z ij , D − P i ) (cid:12)(cid:12)(cid:12) . λN N X i,j =1 k Y j k H κ kZ ij k C − − κ k D − P i k H − κ . λ (cid:16) N N X i,j =1 kZ ij k C − − κ (cid:17) (cid:16) N X j =1 k Y j k H κ (cid:17) (cid:16) N X i =1 k D − P i k H − κ (cid:17) . λ (cid:16) N N X i,j =1 kZ ij k C − − κ (cid:17)(cid:16) N X j =1 k Y j k θ H − κ k Y j k − θ ) L (cid:17) + δ N X i =1 k D − P i k H − κ C δ λ − θ (cid:16) N X i =1 k Y i k L (cid:17)(cid:16) N N X i,j =1 kZ ij k C − − κ (cid:17) − θ + δ N X j =1 (cid:16) k Y j k H − κ + k D − P j k H − κ (cid:17) where θ = κ − κ ∈ (0 , 1) and we used Young’s inequality and interpolation Lemma2.2 in the third inequality.Similarly we have (cid:12)(cid:12)(cid:12) λN N X i,j =1 D ( Y i , Z jj , D − P i ) (cid:12)(cid:12)(cid:12) . λN N X i,j =1 k Y i k H κ kZ jj k C − − κ k D − P i k H − κ C δ λ − θ (cid:16) N X i =1 k Y i k L (cid:17)(cid:16) N N X j =1 kZ jj k C − − κ (cid:17) − θ + δ N X j =1 k Y j k H − κ + δ N X j =1 k D − P i k H − κ which implies the result by the first two bounds in Lemma 4.2. (cid:3) Now we are in a position to bound Ξ. We will use Lemma 3.6 to bound the L T norm of the stochastic terms. Hence we bound k Ξ k L T . Proposition 4.6. It holds for δ > small that Z T | Ξ | d s δ N X i =1 k∇ ϕ i k L T L + δ λN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L + δ N X i =1 k Y i k L T H − κ + Cλ (1 + λ ) Z T (cid:16) N X i =1 k Y i k L (cid:17) Q N d s + λ (1 + λ ) Q N . Here C is independent of λ, N and can be chosen uniform for m > and Q N def = R N + 1 + (cid:16) N N X j =1 k X j k C T C − κ (cid:17) + (cid:16) N N X i,j =1 k X j Z i k C T C − − κ (cid:17) + (cid:16) N N X j =1 k X j Z j k C T C − − κ (cid:17) + 1 N N X i,j =1 (cid:16) kZ ij k − θ C T C − − κ + kZ jj k − θ C T C − − κ (cid:17) , with θ = + κ − κ and Q N def = (cid:16) N X i =1 k X i k L T H − κ (cid:17)h N N X i,j =1 κL (cid:16) kZ ij k C T C − − κ + kZ jj k C T C − − κ (cid:17) + (cid:16) N N X i,j =1 kZ ij k C T C − − κ (cid:17) + (cid:16) N N X j =1 kZ jj k C T C − − κ (cid:17)i + (cid:16) N N X i =1 (cid:16) N X j =1 k X j Z i k L T H − − κ (cid:17) (cid:17) + (cid:16) N N X i =1 (cid:16) N X j =1 k X i X j Z j k L T H − − κ (cid:17) (cid:17) + (cid:16) N N X i =1 (cid:16) N X j =1 (cid:0) k X i ◦ Z jj k L T H − − κ + k X j ◦ Z ij k L T H − − κ (cid:1)(cid:17) (cid:17) , (4.9) where L is chosen as in (3.12) and R N is defined in Lemma 4.8. Note that with all the estimates obtained above we can easily deduce for any q > E ( Q N ) q ] /q . λ (1 + λ ) . (4.10)Indeed, using Lemmas 3.5 and 3.6 we can bound all the terms in Q N that are L T norms and involving X . These bounds together with Lemma 3.1 and the definitionof L in (3.12) imply (4.10).In the following lemma we use the renormalized terms in Z to bound Q N , whichwill be useful in Section 5 because, Q N involves ( X i ) Ni =1 which are not independentfor different i whereas for Q N below it will be easier to exploit independence. Lemma 4.7. It holds that Q N . (1 + λ ) Q N , ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 31 with Q N def = R N + 1 N N X i,j =1 ( kZ ij k − θ C T C − − κ + kZ jj k − θ C T C − − κ ) + X i =1 Q iN , (4.11) where R N is defined in Lemma 4.8, Q N is defined in the proof and Q N , Q N aredefined in the proof of Lemma 3.3 and the proportional constant is independent of λ and N ∈ N .Proof. In the proof we bound each term in Q N in terms of the renormalized termsin Z . Obviously nothing needs to be done for the term R N in Q N . By (3.7) inLemma 3.2 we know (cid:16) N N X j =1 k X j k C T C − κ (cid:17) . λ N N X i,j =1 k ˜ Z ijj k C T C − κ def = λ Q N . Thus the result follows from Lemma 3.3. (cid:3) Now we concentrate on the estimate of Ξ. First we consider the cubic term h Y j Y i , Z i i in Ξ. Comparing to the dynamical Φ model, the dissipation from N k P Ni =1 Y i k L is weaker, which requires further decomposition and more deli-cate estimates. Also unlike 2D case, the best regularity for Y i is H − ( H in 2Dcase). Here we decompose Y i as ϕ i (having better regularity) and D − P i (bound ofwhich only need L -norm of Y i , see Lemma 4.2). For the most complicated terms(see Step 3 in the following proof) we also need to decompose Z i by localizationoperator and choosing L to balance the competing contributions. Lemma 4.8. It holds for δ > small that (cid:12)(cid:12)(cid:12) λN N X i,j =1 h Y j Y i , Z i i (cid:12)(cid:12)(cid:12) δ N X i =1 k ϕ i k H + δ λN (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L + δ N X i =1 k Y i k H − κ + C δ λ (1 + λ ) (cid:16) N X j =1 k Y j k L (cid:17) R N , (4.12) where C δ is independent of λ, N and can be chosen uniform for m > and R N = ( R N ) (cid:16) N N X j =1 k Z j k C − − κ + 1 (cid:17) . Proof. We use (3.24) to get a decomposition for Y i and we have λN N X i,j =1 h Y j Y i , Z i i = λN N X i,j =1 (cid:20) h ϕ j ϕ i , Z i i + h [ D − P j ] Y i , Z i i − h [ D − P j ] ϕ j ϕ i , Z i i− h ϕ j [ D − P i ] , Z i i + 2 h [ D − P j ] ϕ j [ D − P i ] , Z i i (cid:21) . In the following we show that each term can be bounded by the R.H.S. of (4.12). Step h ϕ j ϕ i , Z i i .)In this step we use Lemma 2.8 with s = + κ to have (cid:12)(cid:12)(cid:12) λN N X i,j =1 h ϕ j ϕ i , Z i i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) λN N X i,j =1 h ϕ i ϕ j , Z j i (cid:12)(cid:12)(cid:12) . λN N X i,j =1 (cid:0) k∇ ( ϕ i ϕ j ) k + κL k ϕ i ϕ j k − κL + k ϕ i ϕ j k L (cid:1) k Z j k C − − κ (4.13) . λN N X j =1 h(cid:16) N X i =1 k∇ ( ϕ i ϕ j ) k L (cid:17) + κ (cid:16) N X i =1 k ϕ i ϕ j k L (cid:17) − κ + N X i =1 k ϕ i ϕ j k L i k Z j k C − − κ where we used H¨older’s inequality for the summation in i in the last step. ByH¨older’s inequality it holds that N X i =1 k ϕ i ϕ j k L N X i =1 k ϕ i ϕ j k L k ϕ i k L (cid:16) N X i =1 k ϕ i ϕ j k L (cid:17) (cid:16) N X i =1 k ϕ i k L (cid:17) , (4.14)and N X i =1 k∇ ( ϕ i ϕ j ) k L . N X i =1 k ϕ i ϕ j ∇ ϕ i k L + (cid:13)(cid:13)(cid:13) N X i =1 ϕ i ∇ ϕ j (cid:13)(cid:13)(cid:13) L (4.15) . (cid:16) N X i =1 k ϕ i ϕ j k L (cid:17) (cid:16) N X i =1 k ϕ i k H (cid:17) + (cid:13)(cid:13)(cid:13) N X i =1 ϕ i (cid:13)(cid:13)(cid:13) L k∇ ϕ j k L . Substituting (4.14)–(4.15) into (4.13) we obtain that (4.13) is bounded by λN N X j =1 (cid:16) N X i =1 k ϕ i ϕ j k L (cid:17) (cid:16) N X i =1 k ϕ i k H (cid:17) κ (cid:16) N X i =1 k ϕ i k L (cid:17) − κ k Z j k C − − κ + λN N X j =1 (cid:13)(cid:13)(cid:13) N X i =1 ϕ i (cid:13)(cid:13)(cid:13) + κL k∇ ϕ j k + κL (cid:16) N X i =1 k ϕ i ϕ j k L (cid:17) − κ (cid:16) N X i =1 k ϕ i k L (cid:17) − κ k Z j k C − − κ + λN N X j =1 (cid:16) N X i =1 k ϕ i ϕ j k L (cid:17) (cid:16) N X i =1 k ϕ i k L (cid:17) k Z j k C − − κ , which by H¨older’s inequality for the summation in j is bounded by λ (cid:16) N N X i,j =1 k ϕ i ϕ j k L (cid:17) (cid:16) N X i =1 k ϕ i k H (cid:17) κ (cid:16) N X i =1 k ϕ i k L (cid:17) − κ (cid:16) N N X j =1 k Z j k C − − κ (cid:17) + λ (cid:16) N N X i,j =1 k ϕ i ϕ j k L (cid:17) (cid:16) N X i =1 k ϕ i k L (cid:17) (cid:16) N N X j =1 k Z j k C − − κ (cid:17) . Using Lemma 4.2, we can bound P Ni =1 k ϕ i k L and N P Ni,j =1 k ϕ i ϕ j k L , which givesthe desired bound (4.12) for (4.13) by H¨older’s inequality, namely, the second termin the last expression is bounded by C times λ (cid:16) N k N X i =1 Y i k L (cid:17) (cid:16) N X i =1 k Y i k L (cid:17) (cid:16) N N X j =1 k Z j k C − − κ (cid:17) (1 + λ R N ) / δλN k N X i =1 Y i k L + C δ λ (1 + λ ) (cid:16) N X i =1 k Y i k L (cid:17)(cid:16) N N X j =1 k Z j k C − − κ (cid:17) ( R N ) , and the first term is bounded in the same way. Step h [ D − P j ] Y i , Z i i and h [ D − P j ] ϕ j [ D − P i ] , Z i i .) ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 33 We use (2.2) and Lemma 2.1 and Sobolev embedding H ⊂ L to have k f gh k B 12 + κ , . k f g k L k h k H + k f g k B 12 + κ , k h k L . k f k H − κ k g k H − κ k h k H , (4.16)which combined with (ii) in Lemma 2.7 gives (cid:12)(cid:12)(cid:12) λN N X i,j =1 h [ D − P j ] Y i , Z i i (cid:12)(cid:12)(cid:12) . λN N X i,j =1 k [ D − P j ] Y i k B 12 + κ , k Z i k C − − κ . λN N X i,j =1 k D − P j k H − κ k Y i k H k Z i k C − − κ . λ √ N (cid:16) N X j =1 k Y j k L (cid:17) λ R N (cid:16) N X j =1 k Y j k H − κ (cid:17) θ (cid:16) N X j =1 k Y j k L (cid:17) − θ (cid:16) N N X j =1 k Z j k C − − κ (cid:19) C δ λ − θ (1 + R N ) − θ (cid:16) N N X j =1 k Z j k C − − κ (cid:17) − θ (cid:16) N X j =1 k Y j k L (cid:17) + δλ (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L + δ (cid:16) N X j =1 k Y j k H − κ (cid:17) , (4.17)where θ = − κ ) and we used Lemma 4.2 to bound k D − P j k H − κ in the thirdinequality and (4.6) in the last inequality. Similarly we use (4.16) and Lemma 4.2to have (cid:12)(cid:12)(cid:12) λN N X i,j =1 h [ D − P j ] ϕ j [ D − P i ] , Z i i (cid:12)(cid:12)(cid:12) . λ (cid:13)(cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13)(cid:13) L λ R N (cid:18) N X j =1 k ϕ j k H (cid:19) θ (cid:18) N X j =1 k ϕ j k L (cid:19) − θ (cid:18) N N X j =1 k Z j k C − − κ (cid:19) C δ λ − θ (1 + λ )( R N ) − θ − θ (cid:16) N N X j =1 k Z j k C − − κ (cid:17) − θ (cid:16) N X j =1 k Y j k L (cid:17) + δλ (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L + δ (cid:16) N N X j =1 k ϕ j k H (cid:17) . (4.18) Step h [ D − P j ] ϕ j ϕ i , Z i i and h [ D − P i ] ϕ j , Z i i )For the last two terms we use the localization operator U > and U introducedin (3.5) to separate Z i = U > Z i + U Z i with L chosen below. By the definition of U > and U we know k U > Z i k C − κ . k Z i k C − − κ ( − +4 κ ) L , k U Z i k L ∞ . k Z i k C − − κ ( +2 κ ) L , (4.19)Using (ii) in Lemma 2.7 followed by (2.2) and Lemma 2.1 and H − κ ⊂ H ⊂ L ,we find (cid:12)(cid:12)(cid:12) λN N X i,j =1 h [ D − P j ] ϕ j ϕ i , U > Z i i (cid:12)(cid:12)(cid:12) . λN N X i,j =1 k ϕ i ϕ j k B − κ , k D − P j k H − κ k U > Z i k C − κ . λ (cid:16) N X i =1 k ϕ i k H (cid:17) (cid:16) N N X i =1 k ϕ i k H (cid:17) (cid:16) N N X i,j =1 k U > Z i k C − κ k D − P j k H − κ (cid:17) . λ ( 1 √ N ) (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L (cid:16) N X j =1 k ϕ j k H (cid:17) × (cid:16) N N X j =1 k Z j k C − − κ (cid:17) ( − +4 κ ) L (1 + λ R N ) / , (4.20)where we used Lemma 4.2 and (4.19) in the last step. By H¨older’s inequality, (4.19)and Lemma 4.2 we obtain that (cid:12)(cid:12)(cid:12) λN N X i,j =1 h ( D − P j ) ϕ j ϕ i , U Z i i (cid:12)(cid:12)(cid:12) λN N X i,j =1 k D − P j k L k ϕ j ϕ i k L k U Z i k L ∞ . λ (cid:13)(cid:13)(cid:13) √ N N X i =1 ϕ i (cid:13)(cid:13)(cid:13) L (cid:16) N X j =1 k D − P j k L (cid:17) (cid:16) N N X j =1 k Z j k C − − κ (cid:17) ( +2 κ ) L (4.21) . λ (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L (cid:16) N X j =1 k Y j k L (cid:17) (1 + λ R N ) (cid:16) N N X j =1 k Z j k C − − κ (cid:17) ( +2 κ ) L . Now we choose L to balance the competing contributions:2 L ( − +4 κ ) = (cid:18) N X j =1 k Y j k L (cid:19) / (cid:18) N X i =1 k ϕ i k H (cid:19) − / . This choice of L leads to (cid:12)(cid:12)(cid:12) λN N X i,j =1 h D − P j ϕ j ϕ i , Z i i (cid:12)(cid:12)(cid:12) . λ (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L (cid:16) N N X j =1 k Z j k C − − κ (cid:17) × (cid:20)(cid:16) N X j =1 k Y j k L (cid:17) − a (cid:16) N X j =1 k ϕ j k H (cid:17) a (1 + λ R N ) + (cid:16) N X j =1 k Y j k L (cid:17) (cid:16) N X j =1 k ϕ j k H (cid:17) (1 + λ R N ) (cid:21) C δ λ (1 + λ )( R N ) (cid:16) N N X j =1 k Z j k C − − κ (cid:17) (cid:16) N X j =1 k Y j k L (cid:17) + δλ (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L + δ (cid:16) N X j =1 k ϕ j k H (cid:17) , (4.22)where a = +2 κ − κ and the second line comes from (4.21) and the third line corre-sponds to (4.20).Similar argument also implies that | λN P Ni,j =1 h [ D − P i ] ϕ j , Z i i| can be boundedby the right hand side of (4.22). Combining the above estimates we obtain theresult. (cid:3) In the following we estimate the remaining terms in Ξ. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 35 Proof of Proposition 4.6. The desired estimate for λN P Ni,j =1 h Y j Y i , Z i i has been de-duced in Lemma 4.8. It remains to consider the other terms in Ξ. We write theremaining terms in Ξ as I N def = − λN N X i,j =1 D ( X j + 2 X j Y j )( X i + Y i ) + Y j X i , Y i E ,I N def = − λN N X i,j =1 D ( X j + 2 X j Y j ) Z i + 2( X j + Y j ) X i Z j + 2 X j Y i Z j , Y i E ,I N def = − λN N X i,j =1 D X j ≺ U Z ij + X i ≺ U Z jj + 2( X j + Y j ) ≻ Z ij + 2 X j ◦ Z ij + ( X i + Y i ) ≻ Z jj + X i ◦ Z jj , Y i E . Step I N )By symmetry with respect i and j we write I N as I N = − λN N X i,j =1 h X j X i , Y i i − λN N X i,j =1 h X j Y i , Y i i − λN N X i,j =1 h X j X i Y j , Y i i− λN N X i,j =1 h X j Y j , Y i i def = X k =1 I N k , and we will prove that for every k = 1 , . . . k I N k k L T C δ λ (cid:16) N X i =1 k X i k L T H − κ (cid:17) + δλ (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L (4.23)+ C δ λ (cid:16) N N X j =1 k X j k C T C − κ + 1 (cid:17) (cid:16) N X i =1 k Y i k L T L (cid:17) . We consider each term separately. For I N , applying H¨older’s inequality we have kh X j X i , Y i ik L T . Z T k X j k L ∞ k X i k L k Y i k L d s . k X j k C T L ∞ k X i k L T L k Y i k L T L so by embedding C − κ ⊂ L ∞ we have k I N k L T . λN N X i,j =1 k X j k C T C − κ k X i k L T L k Y i k L T L . λ (cid:16) N N X j =1 k X j k C T C − κ (cid:17)(cid:16) N X i =1 k X i k L T L (cid:17) (cid:16) N X i =1 k Y i k L T L (cid:17) , which by Young’s inequality gives (4.23) for I N . Applying H¨older’s inequality in asimilar way as above we find k I N k L T . λN N X i,j =1 k X j k C T C − κ k Y i k L T L , k I N k L T . λN N X i,j =1 k X j k C T C − κ k X i k C T C − κ k Y i k L T L k Y j k L T L . λ (cid:16) N N X j =1 k X j k C T C − κ (cid:17)(cid:16) N X i =1 k Y i k L T L (cid:17) , which give (4.23) for I N and I N . Moreover by H¨older’s inequality we have k I N k L T . λ (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L (cid:16) N X j =1 k Y j k L T L (cid:17) (cid:16) N N X j =1 k X j k C T C − κ (cid:17) . δλ (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L T L + λ (cid:16) N X i =1 k Y i k L T L (cid:17)(cid:16) N N X j =1 k X j k C T C − κ (cid:17) , which implies (4.23). Step I N ) By symmetry with respect i and j we write I N as I N = − λN N X i,j =1 h X j Z i , Y i i − λN N X i,j =1 h X j Y j Z i , Y i i − λN N X i,j =1 h X j X i Z j , Y i i− λN N X i,j =1 h X j Y i Z j , Y i i def = X i =1 I N i . In the following we show that the L T -norm of each term can be bounded by δ (cid:16) N X i =1 k Y i k L T H − κ (cid:17) + C δ λ Q N (4.24)+ C δ λ (cid:16) N X j =1 k Y j k L T L (cid:17)h(cid:16) N N X i,j =1 k X j Z i k C T C − − κ (cid:17) + (cid:16) N N X j =1 k X j Z j k C T C − − κ (cid:17)i , with Q N given by the third line in the definition of Q N in (4.9). Using (ii) inLemma 2.7 followed by H¨older’s inequality for the summation over i we have k I N k L T . λN N X i,j =1 k X j Z i k L T H − − κ k Y i k L T H − κ . λ (cid:16) N N X i =1 (cid:16) N X j =1 k X j Z i k L T H − − κ (cid:17) (cid:17) (cid:16) N X i =1 k Y i k L T H − κ (cid:17) , which by Young’s inequality gives the desired bound for k I N k L T . Similarly, k I N k L T . λ (cid:16) N N X i =1 (cid:16) N X j =1 k X j X i Z j k L T H − − κ (cid:17) (cid:17) (cid:16) N X i =1 k Y i k L T H − κ (cid:17) , which implies the bound for k I N k L T . Using (ii) in Lemma 2.7 followed by H¨older’sinequality for the summation over i, j and (2.2), we deduce k I N k L T . λ (cid:16) N N X i,j =1 k X j Z i k C T C − − κ (cid:17) (cid:16) N X i,j =1 k Y i Y j k L T B 12 + κ , (cid:17) . λ (cid:16) N N X i,j =1 k X j Z i k C T C − − κ (cid:17) (cid:16) N X j =1 k Y i k L T H − κ (cid:17) (cid:16) N X i =1 k Y j k L T L (cid:17) ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 37 which by Young’s inequality gives the required bound for k I N k L T . Similarly wehave k I N k L T . λ (cid:16) N N X j =1 k X j Z j k C T C − − κ (cid:17) (cid:16) N X i =1 k Y i k L T B 12 + κ , (cid:17) . λ (cid:16) N N X i,j =1 k X j Z j k C T C − − κ (cid:17) (cid:16) N X j =1 k Y i k L T H − κ (cid:17) (cid:16) N X i =1 k Y i k L T L (cid:17) , which implies the bound for I N . Step I N ) We write I N = P i =1 I N i with I N def = − λN N X i,j =1 D X j ≺ U Z ij + X i ≺ U Z jj , Y i E ,I N def = − λN N X i,j =1 D X j ≻ Z ij + X i ≻ Z jj , Y i E ,I N def = − λN N X i,j =1 D X j ◦ Z ij + X i ◦ Z jj , Y i E ,I N def = − λN N X i,j =1 D Y j ≻ Z ij + Y i ≻ Z jj , Y i E . In the following we bound the L T -norm of each term by δ (cid:16) N X i =1 k Y i k L T H − κ (cid:17) + λ Q N (4.25)+ λ − θ (cid:16) N X j =1 k Y j k L T L (cid:17)(cid:16) N N X i,j =1 ( kZ ij k − θ C T C − − κ + kZ jj k − θ C T C − − κ ) (cid:17) , with θ = + κ − κ . By the definition of U we know k U Z ij k C T C − κ . κL kZ ij k C T C − − κ , with L given in (3.12), which combined with Lemma 2.6 implies that for the firstterm in I N Z T (cid:12)(cid:12)(cid:12) λN N X i,j =1 h X j ≺ U Z ij , Y i i (cid:12)(cid:12)(cid:12) d s . λN N X i,j =1 k X j k L T H − κ k U Z ij k C T C − κ k Y i k L T H − κ . κL λ (cid:16) N N X i,j =1 kZ ij k C T C − − κ (cid:17) (cid:16) N X i,j =1 k X j k L T H − κ k Y i k H − κ (cid:17) . Similarly we have for the second term in I N Z T (cid:12)(cid:12)(cid:12) λN N X i,j =1 h X i ≺ U Z jj , Y i i (cid:12)(cid:12)(cid:12) d s . κL λ (cid:16) N N X i,j =1 kZ jj k C T C − − κ (cid:17)(cid:16) N X i =1 k X i k L T H − κ (cid:17) (cid:16) N X i =1 k Y i k H − κ (cid:17) , which by Young’s inequality shows that k I N k L T is bounded by the first line of Q N in (4.9) multiplied by λ and δ P Ni =1 k Y i k L T H − κ . Moreover, by Lemma 2.6 andLemma 2.7 we have k I N k L T . λN N X i,j =1 (cid:2) k X j k L T H − κ kZ ij k C T C − − κ + k X i k L T H − κ kZ jj k C T C − − κ (cid:3) k Y i k L T H − κ . λ h(cid:16) N N X i,j =1 kZ ij k C T C − − κ (cid:17) + (cid:16) N N X j =1 kZ jj k C T C − − κ (cid:17) i × (cid:16) N X i =1 k X i k L T H − κ (cid:17) (cid:16) N X i =1 k Y i k L T H − κ (cid:17) , which by Young’s inequality shows that k I N k L T is bounded by the second line of Q N in (4.9) multiplied by λ and δ P Ni =1 k Y i k L T H − κ . Also we have k I N k L T . λN N X i,j =1 (cid:2) k X i ◦ Z jj k L T H − − κ + k X j ◦ Z ij k L T H − − κ (cid:3) k Y i k L T H − κ . λ h N N X i =1 (cid:16) N X j =1 [ k X i ◦ Z jj k L T H − − κ + k X j ◦ Z ij k L T H − − κ ] (cid:17) i (cid:16) N X i =1 k Y i k L T H − κ (cid:17) , which by Young’s inequality shows that k I N k L T is bounded by the last line of Q N in (4.9) multiplied by λ and δ P Ni =1 k Y i k L T H − κ .Furthermore, we use Lemma 2.6 and Lemma 2.7 (ii) to have k I N k L T . λN N X i,j =1 k Y j k L T H 12 + κ k Y i k L T H 12 + κ (cid:16) kZ ij k C T C − − κ + kZ jj k C T C − − κ (cid:17) . λ (cid:16) N N X i,j =1 ( kZ ij k C T C − − κ + kZ jj k C T C − − κ ) (cid:17) (cid:16) N X i =1 k Y i k H − κ (cid:17) θ (cid:16) N X i =1 k Y i k L (cid:17) − θ , with θ = + κ − κ , where we used H¨older’s inequality for the summation i and j . Thusthe required bound for k I N k L T follows by Young’s inequality.The proposition now follows from (4.23), (4.24) and (4.25) and Lemma 4.8. (cid:3) Convergence of measures and observable Now we focus on ν N given by (1.1) which is an invariant measure to (1.2). In fact,by standard argument the solutions (Φ i ) i N to (3.22) form a Markov process on( C − − κ ) N which, by strong Feller property in [HM18b] and irreducibility in [HS19],will turn out to admit a unique invariant measure. Using a lattice approximations(see e.g. [GH18], [HM18a, ZZ18]) one can show that the measure d ν N (Φ) indeedhas the form (1.1), with suitable renormalization that is consistent with (3.1), i.e. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 39 formally it isexp (cid:16) − Z N X j =1 |∇ Φ j | + (cid:16) m − N + 1 N λa ε + 3( N + 2) N λ ˜ b ε (cid:17) N X j =1 Φ j + λ N (cid:16) N X j =1 Φ j (cid:17) d x (cid:17) D Φdivided by a normalization constant. Here “formally” means that the integralshould be a lattice summation and the Laplacian is a finite difference operator.One goal in this section is to study the large N asymptotic of ν N and show thatfor sufficiently large mass or small λ , in the limit N → ∞ , its marginals are simplyproducts of the Gaussian measure, which is the unique invariant measure to thelinear dynamics (3.2). This heavily relies on the computations from Sections 3.2-4for the remainder X and Y , but we leverage these estimates with consequences ofstationarity. To this end, it will be convenient to have a stationary coupling of thelinear and non-linear dynamics (3.2) and (3.22) respectively, which is stated in thefollowing lemma.5.1. Tightness of the measure.Lemma 5.1. For ( m, λ ) ∈ (0 , ∞ ) × [0 , ∞ ] , there exists a unique invariant mea-sure ν N on ( C − − κ ) N to (3.22) . Furthermore, there exists a stationary process ( ˜Φ Ni , Z i ) i N such that the components ˜Φ Ni , Z i are stationary solutions to (3.22) and (3.2) , respectively and E k ˜Φ Ni (0) − Z i (0) k L . . Here the implicit constant may depend on λ, m and N .Proof. Let Φ i and ¯ Z i be solutions to (3.22) and (3.2) with general initial conditions,respectively. We also recall that Z i is the stationary solution to (3.2). By thegeneral results of [HM18b], it follows that (Φ i , ¯ Z i ) i N is a Markov process on( C − − κ ) N , and we denote by ( P Nt ) t > the associated Markov semigroup. Toderive the desired structural properties about the limiting measure, we will followthe Krylov-Bogoliubov construction with a specific choice of initial condition thatallows to exploit the uniform estimate from Appendix A. Namely, we denote byΦ i the solution to (3.22) starting from Z i (0) where Z i is the stationary solution to(3.2), so that the process Φ i − ¯ Z i starts from the origin. In this case ¯ Z i is the sameas the stationary solution Z i . By Lemma A.2 we know for every T > κ > Z T E (cid:16) N N X i =1 k (Φ i − Z i )( t ) k L (cid:17) d t + Z T E (cid:16) N N X i =1 k (Φ i − Z i )( t ) k C − κ (cid:17) d t . T, (5.1)where the implicit constant is independent of T . By (5.1) together with Lemma 3.1we have Z T E (cid:16) N N X i =1 k Φ i ( t ) k C − κ (cid:17) d t + Z T E (cid:16) N N X i =1 k Z i ( t ) k C − κ (cid:17) d t . T. Defining R Nt := t R t P Nr d r , the above estimates and the compactness of theembedding C − κ ֒ → C − − κ imply the induced laws of { R Nt } t > started from( Z i (0) , Z i (0)) are tight on ( C − − κ ) N . Furthermore, by the continuity with re-spect to initial data, it is easy to check that ( P Nt ) t > is Feller on ( C − − κ ) N .By the Krylov-Bogoliubov existence theorem (see [DPZ96, Corollary 3.1.2]) , theselaws converge weakly in ( C − − κ ) N along a subsequence t k → ∞ to an invariant measure π N for ( P Nt ) t > . The desired stationary process ( ˜Φ Ni , Z i ) i N is definedto be the unique solution to (3.22) and (3.2) obtained by sampling the initial datum( ϕ i , z i ) i from π N . By (5.1) we deduce E π N k Φ i (0) − Z i (0) k L = E π N sup f h Φ i (0) − Z i (0) , f i = E sup f lim T →∞ (cid:20) T Z T h Φ i ( t ) − Z i ( t ) , f i d t (cid:21) . lim T →∞ T Z T E k Φ i ( t ) − Z i ( t ) k L d t . , where E π N denotes the expectation w.r.t. π N and sup f is over smooth functions f with k f k L 1. We then could construct global stationary solutions ( ˜Φ Ni , Z i ) withinitial distribution given by π N as required.Finally, we project onto the first component and consider the map ¯Π : S ′ ( T ) N →S ′ ( T ) N defined through ¯Π (Φ , Z ) = Φ. Letting ν N = π N ◦ ¯Π − yields an invariantmeasure to (3.22), and uniqueness follows from the general results of [HM18b] and[HS19]. (cid:3) Remark 5.2. Lemma 5.1 also gives a dynamical / SPDE construction of the mea-sure ν N . The next step is to study tightness of the marginal laws of ν N over S ′ ( T ) N .To this end, consider the projection Π i : S ′ ( T ) N → S ′ ( T ) defined by Π i (Φ) = Φ i and let ν N,i def = ν N ◦ Π − i be the marginal law of the i th . Furthermore, for k N ,define the map Π ( k ) : S ′ ( T ) N → S ′ ( T ) k via Π ( k ) (Φ) = (Φ i ) i k and the let ν Nk def = ν N ◦ (Π ( k ) ) − be the marginal law of the first k components. We have thefollowing result:In the following we omit the superscript in ˜Φ Ni for notation’s simplicity. Theorem 5.3. For ( m, λ ) ∈ (0 , ∞ ) × [0 , ∞ ] , { ν N,i } N > form a tight set of proba-bility measures on H − − κ for κ > .Proof. Let ( ˜Φ i , Z i ) i N be the jointly stationary solutions to (3.22) and (3.2)constructed in Lemma 5.1. To prove the result, in light of the compact embeddingof H − κ ֒ → H − − κ for κ > i , it sufficesto show that the second moment of k ˜Φ i (0) − Z i (0) k H − κ is bounded uniformly in N . To this end, let Y i = ˜Φ i − Z i − X i with X i defined in (3.6). It holds that E k ˜Φ i (0) − Z i (0) k H − κ = 1 t Z t E k ˜Φ i ( s ) − Z i ( s ) k H − κ d s = 1 t Z t E k X i ( s ) + Y i ( s ) k H − κ d s t Z t E k X i ( s ) k H − κ d s + 2 t Z t E k Y i ( s ) k H − κ d s. For the first term, by (3.18) of Lemma 3.5, we have1 t Z t E k X i ( s ) k H − κ d s = 1 tN Z t E N X i =1 k X i ( s ) k H − κ d s . tN E Q N . C t tN , ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 41 with Q N defined in Lemma 3.4 and E Q N . C t . For the second term we also usesymmetry property to have1 t Z t E k Y i ( s ) k H − κ d s = 1 tN Z t E (cid:16) N X i =1 k Y i ( s ) k H − κ (cid:17) d s. Using Theorem 4.4 we deduce that18 N Z t E N X i =1 k Y i ( s ) k H − κ d s + mN Z t E N X i =1 k Y i ( s ) k L d s + λ N Z t E (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L d s E k ˜Φ i (0) − Z i (0) k L + 4 N N X i =1 E k X i (0) k L + C. By definition we have1 N N X i =1 E k X i (0) k L . λ N N X i,j =1 E k ˜ Z ijj (0) k L . . Then we conclude that E k ˜Φ i (0) − Z i (0) k H − κ C t + 64 t E k ˜Φ i (0) − Z i (0) k L Lemma 5.1 implies that E k ˜Φ i (0) − Z i (0) k L is finite. Choosing t = 128, the resultfollows by the bound for Z i (0) from Lemma 3.1. (cid:3) Convergence of the measure. In the following we prove the convergenceof the measure to the unique invariant measure by using the estimate obtained inTheorem 4.3, which requires m large enough or λ small.Define the ( H − − κ ) k -Wasserstein distance for the measures on ( H − − κ ) k W ,k ( ν , ν ) := inf π ∈ C ( ν ,ν ) (cid:18)Z k ϕ − ψ k H − − κ ) k π (d ϕ, d ψ ) (cid:19) / , (5.2)where C ( ν , ν ) denotes the set of all couplings of ν , ν satisfying R k ϕ k H − − κ ) k ν i (d ϕ ) < ∞ for i = 1 , Theorem 5.4. Let ( m, λ ) ∈ [1 , ∞ ) × [0 , ∞ ) . There exists c > such that for m > λ (1 + λ ) c , W ,k ( ν Nk , ν ⊗ k ) C k N − . (5.3) Proof. By Lemma 5.1 we may construct a stationary coupling ( ˜Φ i , Z i ) of ν N and ν whose components satisfy (3.22) and (3.2), respectively. The stationarity of thejoint law of ( ˜Φ i , Z i ) implies that also ˜Φ i − Z i is stationary. We now claim that E k ˜Φ i (0) − Z i (0) k H − κ CN − , (5.4)which implies (5.3) by definition of the Wasserstein metric and the embedding H − κ ֒ → H − − κ . Similarly as in the proof of Theorem 5.3 we have E k ˜Φ i (0) − Z i (0) k H − κ t Z t E k X i ( s ) k H − κ d s + 2 t Z t E k Y i ( s ) k H − κ d s. (5.5) Since we do not need to choose m and λ in the proof, we can use Theorem 4.4 for m > , λ > m, λ . and 1 t Z t E k X i ( s ) k H − κ d s C t tN . For the second term we also use symmetry property to have1 t Z t E k Y i ( s ) k H − κ d s = 1 tN Z t E N X i =1 k Y i ( s ) k H − κ d s. Using Theorem 4.3 and Lemma 4.7 (with R N , R N , Q N introduced in Lemma 4.1,Proposition 4.5 and Lemma 4.7 respectively), we deduce that18 Z t E (cid:16) N X i =1 k Y i ( s ) k H − κ (cid:17) d s + ( m − Z t E (cid:16) N X i =1 k Y i ( s ) k L (cid:17) d s + λN E Z t (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L d s E (cid:16) N X i =1 k Y i (0) k L (cid:17) + C Z t E h(cid:16) N X i =1 k Y i ( s ) k L (cid:17)(cid:16) λ (1 + λ )( R N + R N + Q N ) (cid:17)i d s + C ( t, λ ) N E k ˜Φ i (0) − Z i (0) k L + 2 N X i =1 E k X i (0) k L + C ( t, λ )+ C Z t E (cid:16) N X i =1 k Y i ( s ) k L (cid:17) E (cid:16) λ (1 + λ )( R N + R N + Q N ) (cid:17) d s (5.6)+ C Z t E h(cid:16) N X i =1 k Y i ( s ) k L (cid:17) λ (1 + λ ) (cid:12)(cid:12)(cid:12) R N + R N + Q N − E [ R N + R N + Q N ] (cid:12)(cid:12)(cid:12)i d s. Here C is independent of λ, m and N . By definition of X i in (3.6) and similarcancelation as in the proof of Lemma 3.4 we have N X i =1 E k X i (0) k L = λ N N X i =1 k E N X j =1 ˜ Z ijj (0) k L . N N X i,j =1 E k ˜ Z ijj (0) k L . . Note that by Lemma 3.1 we can easily deduce for T > E R N + E R N + E Q N . . Here we recall the definition of Q N depend on T and E Q N is increasing w.r.t. T .The last line of (5.6) is controlled by λ N Z t E h(cid:16) N X i =1 k Y i ( s ) k L (cid:17) i d s + C ( λ ) N X k =0 , Z t E h ( R kN − E [ R kN ]) i d s + C ( λ ) N Z t E h ( Q N − E [ Q N ]) i d s. We claim that the last two terms here – although apparently have a large factor N – are actually bounded by constant independent of N .Indeed R N − E R N , R N − E R N and Q N − E Q N are all summations of terms ofthe form 1 N l N X i ··· i l =1 M i , ··· ,i l for different choices of l (namely l = 1 , R N − E R N , l = 1 , , R N − E R N ,and l = 2 , , , Q N − E Q N ), where each M i , ··· ,i l is mean-zero, has bounded sec-ond moment, and they satisfy the independence assumption in Lemma 5.5 because ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 43 they only involve the stochastic objects in (3.4) constructed from the independentnoises ( ξ i ) Ni =1 . So by Lemma 5.5 the claim is proved.Recall that C is independent of λ, m and N . Choosing m > Cλ (1 + λ )( E [ R N + R N ] + E [ Q N ]) + 1 , with T = 64 in Q N which is uniformly bounded w.r.t. m by Lemma 3.1, we have18 Z t E N X i =1 k Y i ( s ) k H − κ d s + Z t λN E (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L d s C ( t, λ ) + 2 N E k ˜Φ i (0) − Z i (0) k L . (5.7)Combining (5.5) and (5.7) we conclude that E k ˜Φ i (0) − Z i (0) k H − κ C ( t, λ ) N t + 32 t E k ˜Φ i (0) − Z i (0) k L . By Lemma 5.1 we have E k ˜Φ i (0) − Z i (0) k L finite. Choosing t = 64 we obtain theclaim (5.4). (cid:3) Lemma 5.5. Let l be a fixed positive integer and ( M i , ··· ,i l : i , · · · , i l ∈ { , · · · , N } ) be a collection of mean-zero random variables such that E [ M i , ··· ,i l ] . uniformlyin N for any i , · · · , i l ∈ { , · · · , N } , and assume that M i , ··· ,i l and M j , ··· ,j l areindependent when the l indices i , · · · , i l , j , · · · , j l are all different. Then we have E h(cid:16) N l N X i ··· i l =1 M i , ··· ,i l (cid:17) i C/N where C only depends on l and is independent of N .Proof. Writing the L.H.S. as1 N l N X i ··· i l =1 N X j ··· j l =1 E [ M i , ··· ,i l M j , ··· ,j l ]we see that the expectation is zero when i , · · · , i l , j , · · · , j l are all different bythe mean-zero and the independence assumptions. When these indices are notall different, the number of summands is N l − N !( N − l )! CN l − where C onlydepends on l but independent of N , and each summand is bounded by our momentbound assumption, so we obtain the claimed bound. (cid:3) Observables. In this section we write the stationary solutions constructed inLemma 5.1 as (Φ i , Z i ). In the following we study the observables (1.5). They aredefined as follows. By Lemma 5.1 we decompose Φ i = X i + Y i + Z i with (Φ i , Z i )stationary and X i introduced in (3.6). With this we define1 √ N N X i =1 : Φ i : = 1 √ N N X i =1 (cid:16) X i + Y i + Z ii + 2 X i Y i + 2 X i Z i + 2 Y i Z i (cid:17) . (5.8)By (5.7) and (5.4) we also have Lemma 5.6. Let ( m, λ ) ∈ [1 , ∞ ) × [0 , ∞ ) . For m > c λ (1 + λ ) + 1 with c as inTheorem 5.4. It holds that Z t E N X i =1 k Y i ( s ) k H − κ d s + Z t N E (cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13) L d s . , where the proportional constant may depend on λ and is independent of N . Proof of Theorem 1.2. By stationarity we have E (cid:13)(cid:13)(cid:13) √ N N X i =1 : Φ i : (cid:13)(cid:13)(cid:13) B − − κ , = 1 T E (cid:13)(cid:13)(cid:13) √ N N X i =1 : Φ i : (cid:13)(cid:13)(cid:13) L T B − − κ , . In the following we use the above equality to derive E (cid:13)(cid:13)(cid:13) √ N N X i =1 : Φ i : (cid:13)(cid:13)(cid:13) B − − κ , . , (5.9)which implies the desired result by the compact embedding B − − κ , ⊂ B − − κ , .We consider L T B − − κ , -norm of each term in (5.8). Using Lemma 2.7 and (3.18)we find E (cid:13)(cid:13)(cid:13) √ N N X i =1 X i (cid:13)(cid:13)(cid:13) L T B − κ , . E √ N N X i =1 k X i k L T H − κ . √ N . Similarly, by Lemma 2.7 and Lemma 5.6 and Lemma 3.5 we have E (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i (cid:13)(cid:13)(cid:13) L T B − κ , . E √ N N X i =1 k Y i k L T H − κ . √ N , E (cid:13)(cid:13)(cid:13) √ N N X i =1 X i Y i (cid:13)(cid:13)(cid:13) L T B − κ , . E √ N (cid:16) N X i =1 k Y i k L T H − κ (cid:17) (cid:16) N X i =1 k X i k L T H − κ (cid:17) . √ N . Furthermore, by independence and similar argument as in the proof of Lemma3.4 we deduce E (cid:13)(cid:13)(cid:13) √ N N X i =1 Z ii (cid:13)(cid:13)(cid:13) L T B − − κ , . E N N X i =1 kZ ii k L T B − − κ , . . Using Lemma 2.7 and Lemma 5.6 we obtain E (cid:13)(cid:13)(cid:13) √ N N X i =1 Y i Z i (cid:13)(cid:13)(cid:13) L T B − − κ , . E √ N (cid:16) N X i =1 k Y i k L T H − κ (cid:17) (cid:16) N X i =1 k Z i k L T H − − κ (cid:17) . . It only remains to consider the term X i Z i for which we use paraproduct to have X i Z i = X i ≺ Z i + X i ◦ Z i + X i ≻ Z i . By Lemma 2.6 we have E √ N N X i =1 k X i ≺ Z i + X i ≻ Z i k L T B − − κ , . E √ N N X i =1 k X i k L T H − κ k Z i k C T C − − κ . E N X i =1 k X i k L T H − κ + 1 N E N X i =1 k Z i k C T C − − κ . . ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 45 Here we use Lemma 3.5 in the last inequality. For X i ◦ Z i we recall (3.15) that X i ◦ Z i = − λN N X j =1 (cid:20) ˜ Z ijj,i + I (2 X j ≺ U > Z ij + X i ≺ U > Z jj ) ◦ Z i (cid:21) , which by Lemma 2.6 implies that E √ N N X i =1 k X i ◦ Z i k L T B − κ , . E N / N X i =1 (cid:13)(cid:13)(cid:13)(cid:13) N X j =1 ˜ Z ijj,i (cid:13)(cid:13)(cid:13)(cid:13) L T B − κ , + E N / N X i,j =1 k X j k L T H − κ kZ ij k C T C − − κ k Z i k C T C − − κ + E N / N X i,j =1 k X i k L T H − κ kZ jj k C T C − − κ k Z i k C T C − − κ . := X i =1 J Ni . Regarding J N we can bound each summand by1 N k X j k L T H − κ k Z i k C T C − − κ + 1 N kZ ij k C T C − − κ and after taking expectation and summation it is equal to E (cid:16) N X j =1 k X j k L T H − κ (cid:17)(cid:16) N N X i =1 k Z i k C T C − − κ (cid:17) + (cid:16) N E N X i,j =1 kZ ij k C T C − − κ (cid:17) . Applying Cauchy-Schwarz to the product in the first term followed by (3.18) andLemma 3.1 and Lemma 3.5 we have J N . 1. For the term J N , each summand isbounded by 1 N k X i k L T H − κ kZ jj k C T C − − κ + 1 N k Z i k C T C − − κ and then J N . J N is bounded by E √ N (cid:13)(cid:13)(cid:13) N X j =1 ˜ Z ijj,i (cid:13)(cid:13)(cid:13) L T B − κ , . T (cid:16) E N (cid:13)(cid:13)(cid:13) N X j =1 ˜ Z ijj,i (cid:13)(cid:13)(cid:13) B − κ , (cid:17) . The quantity in the parenthesis is equal to E N N X j ,j =1 (cid:28) Λ − κ ˜ Z ij j ,i , Λ − κ ˜ Z ij j ,i E . For the case j = j , the expectation is zero. We thus conclude that J N . (cid:3) Appendix A. Extra estimates By similar argument as in Section 4 we deduce Lemma A.1. (Energy balance) 12 1 N N X i =1 dd t k Y i k L + 1 N (cid:13)(cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13)(cid:13) L + 1 N N X i =1 k Y i k H − κ . R N , (A.1) for R N given in the proof. Here the implicit constant may depend on λ and m . Proof. Using Lemma 3.7 we find12 1 N N X i =1 dd t k Y i k L + 1 N (cid:13)(cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13)(cid:13) L + mN N X i =1 k ϕ i k L + 1 N N X i =1 k∇ ϕ i k L N Θ + 1 N Ξ . In Proposition 4.5 and Lemma 4.8 we already deduce the required bound for N Θand the cubic terms in N Ξ. In the following we consider the remaining terms in N Ξ. In the proof of Proposition 4.6 we give estimate for k Ξ k L T . Following thesame argument and doing the estimate at fix time we find1 N Θ + 1 N Ξ . δ N N X i =1 k Y i k H − κ + 1 N N X i =1 k Y i k L ( ¯ Q N + R N ) + ¯ Q N N . δ N N X i =1 k Y i k H − κ + δ N (cid:13)(cid:13)(cid:13)(cid:13) N X i =1 Y i (cid:13)(cid:13)(cid:13)(cid:13) L + ( ¯ Q N + R N ) + 1 N ¯ Q N , with¯ Q N def = R N + 1 + (cid:16) N N X j =1 k X j k C − κ (cid:17) + (cid:16) N N X i,j =1 k X j Z i k C − − κ (cid:17) + (cid:16) N N X j =1 k X j Z j k C − − κ (cid:17) + 1 N N X i,j =1 (cid:16) kZ ij k − θ C − − κ + kZ jj k − θ C − − κ (cid:17) , with θ = + κ − κ and¯ Q N def = (cid:16) N X i =1 k X i k H − κ (cid:17)h N N X i,j =1 κL (cid:16) kZ ij k C − − κ + kZ jj k C − − κ (cid:17) + (cid:16) N N X i,j =1 kZ ij k C − − κ (cid:17) + (cid:16) N N X j =1 kZ jj k C − − κ (cid:17)i + (cid:16) N N X i =1 (cid:16) N X j =1 k X j Z i k H − − κ (cid:17) (cid:17) + (cid:16) N N X i =1 (cid:16) N X j =1 k X i X j Z j k H − − κ (cid:17) (cid:17) + (cid:16) N N X i =1 (cid:16) N X j =1 k X i ◦ Z jj k H − − κ + k X j ◦ Z ij k H − − κ (cid:17) (cid:17) + t − γ N N X i =1 k X i k L , for γ > 0. Using the first inequality in Lemma 4.2 the result holds with R N givenby ( ¯ Q N + R N + R N ) + N ¯ Q N + 1. (cid:3) We denote by Φ i the solution to (3.22) starting from the stationary solution Z i (0), so that the process Φ i − Z i starts from the origin as in the proof of Lemma5.1. By using Lemma A.1 and [TW18, Lemma 3.8], we obtain the following result. Lemma A.2. For every T > and κ > it holds that Z T E [ 1 N N X i =1 k (Φ i − Z i )( t ) k L ]d t . T, ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 47 Z T E [ 1 N N X i =1 k (Φ i − Z i )( t ) k C − − κ ]d t . T, with the proportinal constant independent of T .Proof. On every interval [ s, s + 1], s > i − Z i = X si + Y si with X si satisfies X si ( t ) = − λN Z ts N X j =1 P t − r ( X sj ≺ U > Z ij + X si ≺ U > Z jj )d r + λN N X j =1 ˜ Z ijj ( t ) . and Y si satifies (3.23) with X i replaced by X si and the initial condition Y si ( s ) =Φ i ( s ) − Z i ( s ) + λN P Nj =1 ˜ Z ijj ( s ). By similar argument as in Lemma 3.2 we deduce E (cid:16) N N X i =1 k X si ( t ) k C ([ s,s +2]; C − κ ) (cid:17) . , with the proportinal constant independent of s and t and initial condition. We findLemma A.1 also holds with Y i , and X i replaced by Y si , X si , respectively. For every τ ∈ Z in Lemma 3.1 we know sup s > E [ sup s t s +2 τ ] . , which implies that sup s > E sup s t s +2 R sN ( t ) . , for R sN given as R N in the proof of Lemma A.1 with X i replaced by X si . Now by[TW18, Lemma 3.8] and (4.6) we deduce for 1 s t s + 1 E (cid:16) N N X i =1 k Y si ( t ) k L (cid:17) . ( t − s ) − / , with the proportinal constant independent of s and initial condition, which bytaking integration over t for (A.1) implies that for s > Z s +2 s + E (cid:16) N N X i =1 k Y si ( t ) k H − κ (cid:17) d t . , (A.2)with the proportinal constant independent of s . Then we know for T > Z T E (cid:16) N N X i =1 k (Φ i − Z i )( t ) k L (cid:17) d t . T − X s =1 Z s + s + E (cid:16) N N X i =1 k X si ( t ) k L (cid:17) d t + T − X s =1 Z s + s + E (cid:16) N N X i =1 k Y si ( t ) k L (cid:17) d t + Z E (cid:16) N N X i =1 k X si ( t ) k L (cid:17) d t + Z E (cid:16) N N X i =1 k Y si ( t ) k L (cid:17) d t . T, where we used Theorem 4.4 for the bound of the integral from 0 to . Similarly weobtain Z T E (cid:16) N N X i =1 k (Φ i − Z i )( t ) k C − − κ (cid:17) d t . T by using (A.2) and Besov embedding Lemma 2.1. Since κ is arbitrary, the secondresult follows. (cid:3) Appendix B. Notation index We collect some frequently used notations of this paper, their meaning and thepage where they first occur.Symbol Place being introduced Page ϕ i , P i Eq. (3.24) 21 Q N , Q N , Q N Lemma 3.4 16 Q N , Q N Proposition 4.6 30 Q N Lemma 4.7 31 Q N , Q N Lemma 3.3 14 R N Lemma 3.2 12 R N Lemma 4.1 24 R N Proposition 4.5 27 R N Lemma 4.8 31 X i Eq. (3.6) 12 References [AK17] S. Albeverio and S. Kusuoka. The invariant measure and the flow associated to the ϕ -quantum field model. arXiv:1711.07108 , 2017.[AS12] M. Anshelevich and A. N. Sengupta. Quantum free Yang-Mills on the plane. J. Geom.Phys. , 62(2):330–343, 2012.[BGHZ19] Y. Bruned, F. Gabriel, M. Hairer, and L. Zambotti. Geometric stochastic heat equa-tions. arXiv:1902.02884 , 2019.[Bon81] J.-M. Bony. Calcul symbolique et propagation des singularit´es pour les ´equations auxd´eriv´ees partielles non lin´eaires. Ann. Sci. ´Ecole Norm. Sup. (4) , 14(2):209–246, 1981.[CC18] R. Catellier and K. Chouk. Paracontrolled distributions and the 3-dimensional sto-chastic quantization equation. Ann. Probab. , 46(5):2621–2679, 2018.[CCHS20] A. Chandra, I. Chevyrev, M. Hairer, and H. Shen. Langevin dynamic for the 2D Yang-Mills measure. arXiv preprint arXiv:2006.04987 , 2020.[CGW20] A. Chandra, T. S. Gunaratnam, and H. Weber. Phase transitions for ϕ . arXiv preprintarXiv:2006.15933 , 2020.[Cha19] S. Chatterjee. Rigorous solution of strongly coupled SO ( N ) lattice gauge theory in thelarge N limit. Comm. Math. Phys. , 366(1):203–268, 2019.[CJ16] S. Chatterjee and J. Jafarov. The 1 /N expansion for SO(N) lattice gauge theory atstrong coupling. arXiv preprint arXiv:1604.04777 , 2016.[CWZZ21] X. Chen, B. Wu, R. Zhu, and X. Zhu. Stochastic heat equations for infinite stringswith values in a manifold. Trans. Amer. Math. Soc. , 374(1):407–452, 2021.[DPZ96] G. Da Prato and J. Zabczyk. Ergodicity for infinite-dimensional systems , volume 229of London Mathematical Society Lecture Note Series . Cambridge University Press,Cambridge, 1996.[GH18] M. Gubinelli and M. Hofmanov´a. A PDE construction of the Euclidean Φ quantumfield theory. arXiv:1810.01700 , 2018.[GH19] M. Gubinelli and M. Hofmanov´a. Global solutions to elliptic and parabolic Φ modelsin Euclidean space. Comm. Math. Phys. , 368(3):1201–1266, 2019.[GIP15] M. Gubinelli, P. Imkeller, and N. Perkowski. Paracontrolled distributions and singularPDEs. Forum Math. Pi , 3:e6, 75, 2015.[GN74] D. J. Gross and A. Neveu. Dynamical symmetry breaking in asymptotically free fieldtheories. Physical Review D , 10(10):3235, 1974. ARGE N LIMIT OF THE O ( N ) LINEAR SIGMA MODEL IN 3D 49 [Hai14] M. Hairer. A theory of regularity structures. Invent. Math. , 198(2):269–504, 2014.[Hai16] M. Hairer. The motion of a random string. arXiv:1605.02192 , 2016.[HM18a] M. Hairer and K. Matetski. Discretisations of rough stochastic PDEs. Ann. Probab. ,46(3):1651–1709, 2018.[HM18b] M. Hairer and J. Mattingly. The strong Feller property for singular stochastic PDEs. Ann. Inst. Henri Poincar´e Probab. Stat. , 54(3):1314–1340, 2018.[HS19] M. Hairer and P. Sch¨onbauer. The support of singular stochastic PDEs. arXiv preprintarXiv:1909.05526 , 2019.[Jaf65] A. Jaffe. Divergence of perturbation theory for bosons. Communications in Mathemat-ical Physics , 1(2):127–149, 1965.[Kup80a] A. J. Kupiainen. 1 /n expansion for a quantum field model. Comm. Math. Phys. ,74(3):199–222, 1980.[Kup80b] A. J. Kupiainen. On the 1 /n expansion. Comm. Math. Phys. , 73(3):273–294, 1980.[L´ev11] T. L´evy. The master field on the plane. arXiv preprint arXiv:1112.2452 , 2011.[MW17a] J.-C. Mourrat and H. Weber. The dynamic Φ model comes down from infinity. Comm.Math. Phys. , 356(3):673–753, 2017.[MW17b] J.-C. Mourrat and H. Weber. Global well-posedness of the dynamic Φ model in theplane. Ann. Probab. , 45(4):2398–2476, 2017.[MW20] A. Moinat and H. Weber. Space-time localisation for the dynamic ϕ model. Commu-nications on Pure and Applied Mathematics , 73(12):2519–2555, 2020.[RWZZ20] M. R¨ockner, B. Wu, R. Zhu, and X. Zhu. Stochastic heat equations with values in amanifold via Dirichlet forms. SIAM J. Math. Anal. , 52(3):2237–2274, 2020.[SSZZ20] H. Shen, S. Smith, R. Zhu, and X. Zhu. Large N limit of the O ( N ) linear sigma modelvia stochastic quantization. arXiv preprint arXiv:2005.09279 , 2020.[t’H74] G. t’Hooft. A planar diagram theory for strong interactions. Nuclear Physics. B ,72(3):461–473, 1974.[Tri78] H. Triebel. Interpolation theory, function spaces, differential operators , volume 18 of North-Holland Mathematical Library . North-Holland Publishing Co., Amsterdam-NewYork, 1978.[TW18] P. Tsatsoulis and H. Weber. Spectral gap for the stochastic quantization equation onthe 2-dimensional torus. Ann. Inst. Henri Poincar´e Probab. Stat. , 54(3):1204–1249,2018.[Wil73] K. G. Wilson. Quantum field-theory models in less than 4 dimensions. Physical ReviewD , 7(10):2911, 1973.[ZZ18] R. Zhu and X. Zhu. Lattice approximation to the dynamical Φ model. Ann. Probab. ,46(1):397–455, 2018.[ZZZ20] X. Zhang, R. Zhu, and X. Zhu. Singular HJB equations with applications to KPZ onthe real line. arXiv preprint arXiv:2007.06783 , 2020.(H. Shen) Department of Mathematics, University of Wisconsin - Madison, USA Email address : [email protected] (R. Zhu) Department of Mathematics, Beijing Institute of Technology, Beijing 100081,China Email address : [email protected] (X. Zhu) Academy of Mathematics and Systems Science, Chinese Academy of Sciences,Beijing 100190, China Email address ::