Discrete RG flow of a hierarchical singular SPDE

Discrete RG ﬂow of a hierarchical singular SPDE L ´ eonard F erdinand * and Simon Gabriel † Abstract This paper illustrates the R enormalisation Group (RG) approach to singular SPDEs following a framework introduced by Kupiainen [K up16]. W e study a linear elliptic SPDE with a hierarchical Laplace operator and multiplicative noise, in two dimensions. Although this model is a signiﬁcant simpliﬁcation, it captures the core mechanisms of the RG method in a transparent setting. P articular emphasis is placed on the dynamical system governing the ﬂow of the effective force coefﬁcients, the central object of the RG method. K eywords. Renormalisation Group; Singular SPDEs; Anderson model. MSC classiﬁcation. Primary: 60H15, Secondary: 35R60; 81T17. Contents 1 Introduction 2 2 Background on hierarchical lattice operators 4 2.1 A veraging and ﬂuctuation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The hierarchical Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 The renormalised model on the unit lattice 10 3.1 The coarse grained solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 The effective force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 The enhanced noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Derivation of the perturbative dynamical system 16 5 Uniform control of the RG ﬂow 20 5.1 The dynamical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Stochastic estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3 Fixed point problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.4 Control of the renormalised solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6 Convergence in the UV limit 27 6.1 Convergence of the effective force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2 Convergence of the solution and proof of Proposition 3.5 . . . . . . . . . . . . . . . . . 29 A Moment calculations 30 * Email: leonard.ferdinand@mis.mpg.de, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany . † Email: s.gabriel@berkeley .edu, Department of Mathematics, University of California, Berkeley , USA. 1 L . F E R D I N A N D A N D S . G A B R I E L 2 1 Introduction In [Kup16], Kupiainen introduced a Renormalisation Group (RG) approach to singular SPDEs, pro- viding an alternative to the theories of R egularity Structures [Hai14] and P aracontrolled Distributions [GIP15]. More recently , interest in RG approaches to singular SPDEs has grown further , in particular after Duch introduced a continuous alternative [Duc25a]. Kupiainen’s original treatment focussed on the dynamic φ 4 3 equation [Kup16], and the method was subsequently extended to an anisotropic KPZ equation in [KM17]. Similar to R egularity Structures and P aracontrolled Distributions, the RG approach relies on the introduction of a new reference scale, which is larger than the regularisation scale, and facilitates the construction of the solution. In the context of RG, the strategy is to derive the effective equation solved by the restriction of the solution to this reference scale. Effective equations at different scales are then related through a dynamical system. Kupiainen’s work requires a familiarity with Cauchy estimates and rescalings which are common in the RG community , but less so in the SPDE community . Much of the technical difﬁculty stems from the presence of nonlocal terms in the evolution of the effective equation, which have to be disentangled from their local counterparts via localisation procedures. In the present work, we aim to clarify the strategy of the RG framework by setting aside the difﬁculties associated with any nonlocal behaviour . T o this end, we work with a hierarchical model, for which the effective equation remains local at every scale. More speciﬁcally , we study the hierarchical Anderson equation, which for g P R is given by 0 “ ∆ H u p x q ` p g ξ p x q ´ 8q u p x q ` 1 , x P T 2 , (1) where ∆ H denotes the hierarchical Laplace operator , which we introduce in the next section. More- over , T 2 def “ r´ 1 { 2 , 1 { 2 q 2 denotes the two–dimensional torus with periodic boundary conditions, and ξ is a white noise on T 2 on a ﬁxed probability space p Ω , F , P q . The hierarchic component of our model allows for major simpliﬁcation, avoiding any localisation procedure, or Cauchy estimates. While the SPDE (1) is linear , it allows us to keep notation to a min- imum, and still provide the key conceptual ideas of [K up16]. W e also unfold with great care how to obtain the dynamical system (54) that governs the evolution of the effective equation, when con- sidered on a unit reference scale. In K upiainen’s work, this structure is treated less explicitly . The formulation of the dynamical system (54) will be familiar to readers with a background in Renormal- isation Group theory . Indeed, we show that the discrete ﬂow of an effective equation across scales is analogous to the discrete ﬂow of an effective action in the context of the RG, see for example, [BBS19, Section 5.3.1]. W e therefore emphasise clarifying the relation between these two perspectives and facilitating communication across communities. W e now turn to the analytical difﬁculties posed by the equation itself. The SPDE (1) contains the singular product ξ u , which cannot be made sense of by classical means, and needs to given meaning through approximation. T o this end, we introduce the discrete tori Λ ´ N def “ ␣ ´ 1 2 p 1 ´ L ´ N q , . . . , ´ 2 L ´ N , ´ L ´ N , 0 , L ´ N , 2 L ´ N , . . . , 1 2 p 1 ´ L ´ N q ( d Ă “ ´ 1 2 , 1 2 ˘ d . The regularised version of (1) on the lattice Λ ´ N , referred to as the bare equation , then takes the form p´ ∆ H q u N “ p g ξ N ` r N q u N ` 1 , (2) with a postulated bare mass of the form r N ” r N p r , g q def “ r ´ p 1 ´ L ´ 2 q N g 2 , for some r ě 0 . Here, u N P C p Λ ´ N q and ∆ H ” ∆ H, ´ N denotes the hierarchical Laplace operator on Λ ´ N . The regularised noise ξ N is given in terms of i.i.d. Gaussian random variables ξ N p x q „ N p 0 , L 2 p 2 ´ α q N q , x P Λ ´ N , where α def “ 2 ´ d { 2 denotes the expected regularity of the solution u . Furthermore, we embed u N into T d by linear interpolation between lattice points. D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 3 Our main result is as follows. Theorem 1.1. Let d “ 2 and r ě 1 . There exists an odd L P N , sufﬁciently large and depending only on r , such that the following holds. F or every g P p 0 , 1 s , there exists an event Ω g Ă Ω such that P p Ω g q ě 1 ´ C e ´ cg ´ 2 . On Ω g , the SPDE (2) with renormalised mass r and renormalised coupling constant g admits a unique solution u N on Λ N , for every N P N . Moreover , there exists a function u ‹ P C 1 ´ p T 2 q such that u N Ñ u ‹ in C 1 ´ p T 2 q , as N Ñ 8 . The H ¨ older norm } ‚ } C 1 ´ κ is deﬁned in Deﬁnition 2.11, and the events Ω g are characterised in (31) . The bare mass r N is the sum of the renormalised mass r and the mass counterterm ´p 1 ´ L ´ 2 q N g 2 . The latter diverges as N Ò 8 , in agreement with the presence of the term ´8 u in (1). Moreover , we notice that while u in (1) appeared to be fully characterised by the renormalised coupling constant g , it is in fact necessary to specify two quantities to identify u N , and its limit u ‹ : The renormalised coupling constant g P R and the renormalised mass r P R . Since we only consider the massive equation, we assume that r ě 1 . Moreover , the condition on a small coupling g P p 0 , 1 s is necessary to deal with the large ﬁeld problem. Of course, taking g P r´ 1 , 0 q or r ď ´ 1 would not bring any modiﬁcation, but we choose to ﬁx the signs to simplify the notation. The proof of Theorem 1.1 will be given at the end of Section 3. The strategy is to derive the effective equation solved by a coarse grained version of u N at a scale L ´ n ě L ´ N . The key step is the derivation of a dynamical system which relates the effective equation at scale L ´ n to the effective equation at the subsequent scale L ´ n ` 1 . At each step of the RG , we rescale the effective equation back to the unit lattice, which is necessary to compare across different scales in a meaningful way , see also R emark 5.1. R emark 1.2 (Random coupling) . Theorem 1.1 is equivalent to saying that there exists a random coupling g p ω q , ω P Ω , with negative p -th moment growing like ? p , such that the equation (2) is almost surely well- posed. In fact, the true quantity that needs to be taken random, in order to avoid the large ﬁeld problem, is the quotient g { r . In other words, instead of a small random coupling g p ω q , it would equally be possible to take a large random mass r p ω q . W e refrain from doing so to keep the notation lighter , and because the former choice is more common [Duc25b]. The SPDE (1) is the elliptic (and hierarchical) analogue of the parabolic Anderson model pB t ´ ∆ q u “ p g ξ ´ 8q u , describing the evolution of a particle in a random medium, see for example [Hai14, GIP15, HL15] among others. In the Euclidean setting, the elliptic equation (1) has been studied previously for d “ 2 , 3 , see for example [AC15, Lab19, BCZ23]. In d “ 4 , the equation is scaling critical, and falls outside the reach of subcritical theories. Fluctuations of the critical (Euclidean) equation are investigated in [GR26]. In the present paper , we restrict attention to the case d “ 2 and do not treat nonlinearities of the form σ p u q ξ . However , the method should extend to the full subcritical regime ( α ą 0 ) under strong enough assumptions on the nonlinearity σ , at the expense of enlarging the noise structure and incorporating additional stochastic terms into the dynamical system. Lastly , our assumption that the noise ξ is Gaussian is not essential, however , convenient as we use hypercontractivity for the necessary moment estimates in Lemma 5.2. L . F E R D I N A N D A N D S . G A B R I E L 4 Outline of the paper W e begin in Section 2 by introducing the lattice operations, in particular the hierarchical Laplacian. In Section 3, we set the stage by rescaling the problem to the unit scale, which allows for comparison across scales, and prove Theorem 1.1. W e place particular emphasis on the derivation of the force co- efﬁcients and their connection to the effective equation. Next, we perturbatively derive the dynamical system of force coefﬁcients in Section 4, which is the key idea of RG. In Section 5 we will make this perturbative analysis rigorous. W e conclude the article by proving convergence of the effective force and the renormalised solutions in Section 6. Acknowledgements The authors express their gratitude towards Ajay Chandra and Hendrik W eber for suggesting the investigation of hierarchical models, as well as helpful discussions. This material is based upon work supported by the National Science F oundation under Grant No. DMS-1928930, while the authors were in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley , California, during the fall semester of 2025. Moreover , SG acknowledges ﬁnancial support through the ERC Grant 101045082, held by Hendrik W eber , and the DFG Excellence Cluster in M ¨ unster EXC 2044–390685587. 2 Background on hierarchical lattice operators In this section, we introduce the regularised lattice and the hierarchical Laplace operator , closely following [BBS19]. W e then set the framework by deﬁning rescaling operations and a hierarchical analogue of Besov spaces. As a ﬁrst application, we regularise the white noise ξ and show that it belongs to the natural Besov space, which also allows the reader to become familiar with the concepts. Deﬁnition 2.1 (Lattices) . W e ﬁx a coarse graining scale L ě 3 , which is an odd 1 integer , eventually taken sufﬁciently large. Then for any pair of integers m ď M , we set Λ M m def “ ´ L m Z L M Z ¯ d Ă “ ´ 1 2 L M , 1 2 L M ˘ d , to be the lattice of side-length L M and spacing L m , which we identify with Λ M m ” ␣ ´ 1 2 p L M ´ L m q , . . . , ´ 2 L m , ´ L m , 0 , L m , 2 L m , . . . , 1 2 p L M ´ L m q ( d . Importantly , we have Λ m Ă T d , and Λ M Ă L M T d . Whenever one of these indices equals zero, we drop it from the notation, and write Λ m ” Λ 0 m , for m ď 0 , and Λ M ” Λ M 0 , for M ě 0 . W e equip Λ M m with the product topology , therefore, we denote the set of bounded functions Λ M m Ñ R , which are trivially continuous, by C p Λ M m q . F or n P N , and for every x P Λ ´ n , we denote by ˝ ´ n p x q Ă T d the square of side-length L ´ n centred at x . F or example, Λ ´ 2 with L “ 3 can be represented in terms of 1 W e could also consider even integers, however , it is convenient to consider L odd such that each square ˝ has a centre point. Since L is chosen large eventually , we did not strive for the greatest generality here. D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 5 ˝ 0 ˝ ´ 1 p x q x “ ˝ ´ 2 p x q 2.1 A veraging and ﬂuctuation operators In this section, we introduce the operators needed to deﬁne the hierarchical Laplace operator . W e begin with the averaging operator acting on distributions on T d , which can be viewed as testing a distribution against a constant test function. Deﬁnition 2.2 (Averaging operators) . F or n P N , we deﬁne the averaging operator Q ´ n : D 1 p T d q Ñ C p Λ ´ n q with Q ´ n f p x q def “ L dn ż ˝ ´ n p x q f p y q d y , x P Λ ´ n . In particular , we have Q 0 “ ş T d f p y q d y . By convention, we also enforce Q n ” 0 for all n ą 0 . F or n ď N , Q ´ n restricts to an operator Q p N q ´ n : C p Λ ´ N q Ñ C p Λ ´ n q , which we also denote by Q ´ n whenever clear from context: Q ´ n f p x q “ L ´p N ´ n q d ÿ y P ˝ ´ n p x qX Λ ´ N f p y q , x P Λ ´ n . Note that Q 0 projects onto constants, while Q ´ N restricts to the identity on C p Λ ´ N q . F or convenience, we will also enforce Q p N q ´ n ” 0 whenever n ą N . Example 1 (R egularised white noise) . The averaging operator Q ´ n allows us to deﬁne white noise on Λ ´ n , which can be interpreted as a scale- n regularisation of the original white noise ξ on T d . Namely , we deﬁne ξ n p x q def “ Q ´ n ξ p x q „ N p 0 , L dn q , x P Λ ´ n , (3) which yields an independent and identically distributed family of centred normal random variables. Having introduced the averaging operator , we can describe ﬂuctuations across scales via the cor- responding ﬂuctuation operators. Deﬁnition 2.3 (Fluctuation operators) . F or n ě 1 , the ﬂuctuation operator P ´ n : C 8 p T d q Ñ C p Λ ´ n q X k er p Q ´ n ` 1 q is given by P ´ n def “ Q ´ n ´ Q ´ n ` 1 . The family p P ´ n q 1 ď n ď N allows for a decomposition of any function f P C p Λ ´ N q into its scale components. Namely , for any f P C p Λ ´ N q , we can write f “ Q 0 f ` N ÿ n “ 1 P ´ n f . (4) L . F E R D I N A N D A N D S . G A B R I E L 6 In fact, these projections are orthogonal, because P ´ m P ´ n “ P ´ m δ m,n , since Q ´ m Q ´ n “ Q ´p m ^ n q . (5) Rather than working directly on Λ ´ N directly , it will be convenient to rescale the problem to the unit lattice of size L N , Λ N “ L N Λ ´ N , which places us in the framework of [BB S19]. W e return to this point in R emark 2.7 at the end of this section. T o carry out such rescaling, we introduce scaling operators that allow functions to be pulled back and forth between the different lattices. Deﬁnition 2.4 (Operators on unit lattices) . F or k P Z , we introduce the scaling operator S k : C p Λ M m q Ñ C p Λ M ´ k m ´ k q which acts as S k f p ‚ q def “ f p L k ‚ q . W e will be mostly interested in the cases S N : C p Λ N q Ñ C p Λ ´ N q and S ´ N : C p Λ ´ N q Ñ C p Λ N q . Notably , the scaling operators allow us to scale back Q p N q ´ n and P p N q ´ n to Λ N , and deﬁne (a) Q p N q n : C p Λ N q Ñ C p Λ N n q with Q p N q n def “ S ´ N Q p N q ´ N ` n S N , (b) P p N q n : C p Λ N q Ñ C p Λ N n ´ 1 q X k er p Q 1 S n ´ 1 q with P p N q n def “ S ´ N P p N q ´ N ` n ´ 1 S N . Again, when no ambiguity can arise, we drop the dependency on N in the superscript. Lemma 2.5. W e have that Q 0 is the identity 2 , Q N is the projector onto constants, P n “ Q n ´ 1 ´ Q n , and Q n acts by averaging as Q n g p x q “ L ´ nd ÿ y P ˝ n p x qX Λ N g p y q , x P Λ N n , (6) where ˝ n p x q denotes unique square of side-length L n of L N T d centred at x . Proof . The ﬁrst two claims are immediate. Next, we compute for every x P Λ N Q n g p x q “ S ´ N Q ´ N ` n S N g p x q “ L ´ nd ÿ y P ˝ ´ N ` n p L ´ N x qX Λ ´ N g p L N y q . T o conclude (6), we notice that y P ˝ ´ N ` n p L ´ N x q X Λ ´ N is equivalent to L N y P ˝ n p x q X Λ N . T o prove the third claim, we compute P n “ S ´ N P ´ N ` n ´ 1 S N “ S ´ N Q ´ N ` n ´ 1 S N ´ S ´ N Q ´ N ` n S N “ Q n ´ 1 ´ Q n . This ﬁnishes the proof . Example 2 (White noise (continued)) . In Example 1, we deﬁned the regularised white noise ξ n on the lattice Λ ´ n . By means of the scaling operator S ´ n from Deﬁnition 2.4, we can pushforward ξ n onto a noise on the unit lattice Λ n . Namely , we deﬁne ξ ´ n p x q def “ L p´ 2 ` α q n p S ´ n ξ n qp x q ” L p´ 2 ` α q n p S ´ n Q ´ n ξ qp x q „ N p 0 , 1 q , x P Λ n , (7) 2 Contrary to Q , where Q ´ N played the role of the identity on Λ ´ N . D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 7 which still deﬁnes an independent family of random variables. The factor L p´ 2 ` α q n “ L ´ d 2 n is the natural scaling imposed by the regularity of white noise, as in the Euclidean case. Analogously , for every N P N and n ď N , we have ξ ´ n p x q “ L ´p 2 ´ α q n S N ´ n Q N ´ n S ´ N ξ N p x q “ L p 2 ´ α qp N ´ n q S N ´ n Q N ´ n ξ ´ N p x q , x P Λ n , bl abl (8) where the ﬁrst identity is a consequence of L ´p 2 ´ α q n S N ´ n Q N ´ n S ´ N ξ N “ L ´p 2 ´ α q n S ´ n Q p N q ´ n Q ´ N ξ “ L ´p 2 ´ α q n S ´ n Q ´ n ξ “ L ´p 2 ´ α q n S ´ n ξ n “ ξ ´ n . Here we used the expression of Q p N q n from Deﬁnition 2.4, (5) , and the deﬁnition of ξ in (3) . W e close this section with two useful identities. Lemma 2.6. F or every k, n P N , k ` n ď N , we have that (a) S k ` n Q p N q k ` n “ S k Q p N ´ n q k S n Q p N q n , (b) S k ` n P p N q k ` n “ S k P p N ´ n q k S n Q p N q n . Before we turn to the proof of this lemma, we ﬁrst verify that both sides are compatible. Indeed, we have Λ N Q n Ý Ñ Λ N n S n Ý Ñ Λ N ´ n Q k Ý Ñ Λ N ´ n k S k Ý Ñ Λ N ´p k ` n q , so that the right-hand side lies in C p Λ N ´p k ` n q q which is the image of S k ` n Q k ` n . Notice also, that Lemma 2.6 (a) and (b) imply Q 1 S n ´ 1 P n “ Q 1 P 1 S n ´ 1 Q n ´ 1 “ 0 , which justiﬁes that the image of P n lies inside k er p Q 1 S n ´ 1 q . Proof of Lemma 2.6. W e establish that S n Q n “ S 1 Q 1 S n ´ 1 Q n ´ 1 , the ﬁrst claim then follows by induc- tion, and the second claim is immediate from the ﬁrst one. W e compute for x P Λ N n S ´ n ` 1 Q 1 S n ´ 1 Q n ´ 1 g p x q “ L ´ dn ÿ y P ˝ 1 p L ´ n ` 1 x qX Λ N ÿ z P ˝ n ´ 1 p L n ´ 1 y qX Λ N g p z q “ Q n g p x q . Indeed, as y ranges over the square of unit mesh size and side length L centred at L ´ n ` 1 x , the rescaled points L n ´ 1 y range over the square of mesh size L n ´ 1 and side length L n centred at x . Summing over all z in the unit–mesh square of side length L n ´ 1 centred at L n ´ 1 y therefore reproduces the sum over ˝ n p x q . R emark 2.7. W e want to motivate the transition from the coarse lattices Λ ´ n Ă T d to the unit lattices Λ n . W orking on unit lattices guarantees that, after each RG step, one returns to a lattice with the same ﬁxed spacing. This renders the RG transformation homogeneous, in the sense that the induced dynamical system is independent of explicit occurrences of the scale n . In particular , the regularity properties of the random terms will be contained in the spectrum of the linearisation of the RG. F or instance, considering the evolution of the noise, we see that the trajectory p ξ ´ n q n P N of white noises on the unit lattices satisﬁes the recursion ξ ´ n ` 1 “ L 2 ´ α S 1 Q 1 ξ ´ n . Of course, this recursion is linear , and the noise blows up like L 2 ´ α at each step, which is consistent with the fact that the regularity of the noise is ´ 2 ` α ´ κ . In contrast, the trajectory p ξ ´ n q n P N of white noises on the coarse lattices satisﬁes the recursion ξ n ´ 1 “ S n Q 1 S ´ n ξ n . L . F E R D I N A N D A N D S . G A B R I E L 8 Here we see that the blow up L 2 ´ α is not explicit, but hidden inside the ﬂuctuations of the noise, and that the ﬂow still depends explicitly on n through the operators S n and S ´ n . This is normal: One compares quantities that should not be compared, since they do not live on the same lattice. F or this reason, we will always work on the unit lattice. 2.2 The hierarchical Laplace operator Finally , we introduce the hierarchical Laplace operator , which is the central simpliﬁcation in the SPDE (1) and enables a more tractable renormalisation analysis. Deﬁnition 2.8 (Hierarchical Laplace operator) . The hierarchical Laplace operator ∆ H, ´ N on C p Λ ´ N q is deﬁned as ´ ∆ H, ´ N def “ N ÿ n “ 1 L 2 n P ´ n . Moreover , we deﬁne the hierarchical Laplacian ∆ H,N on C p Λ N q by rescaling of ∆ H, ´ N ∆ H,N def “ L ´ 2 N S ´ N ∆ H, ´ N S N . (9) Whenever the value of N is clear from the context, we omit the subscript, and write ∆ H instead of ∆ H,N . The additional factor L ´ 2 N in (9) arises from imposing the same scaling behaviour as that of the classical Laplace operator . Note that when acting on g P C p Λ N q , the deﬁnition (9) reads ∆ H,N g p ‚ q “ L ´ 2 N ∆ H, ´ N ` g p L N ‚ q ˘ p L ´ N ‚ q . Furthermore, by explicit calculation, we may express ∆ H,N in terms of p P n q 1 ď n ď N : ´ ∆ H,N “ L ´ 2 N N ÿ n “ 1 L 2 n S ´ N P ´ n S N “ N ÿ n “ 1 L 2 p n ´ N q P N ´ n ` 1 “ N ÿ n “ 1 L ´ 2 p n ´ 1 q P n . (10) This representation agrees with [BBS19, Deﬁnition 4.1.7]. R emark 2.9. The deﬁnition of the hierarchical Laplacian is best understood in light of (4) , which shows that the operators P ´ n play the role of sharp Littlewood–P aley projections at scale L ´ n . At a spatial scale ℓ , the hierarchical Laplacian therefore behaves like multiplication by ℓ ´ 2 , in analogy with the usual Laplace operator . The operator ∆ H, ´ N is called hierarchical because its kernel depends on the hierarchical distance L ´ h p x,y q rather than on the Euclidean distance, where h p x, y q is the largest n P t 0 , . . . , N ´ 1 u such that x, y P ˝ ´ n p z q for some z P Λ ´ n if x ‰ y , with the convention h p x, x q “ 8 . Finally , also the inverted hierarchical Laplace operator admits an explicit representation, which we state without proof , as it may be veriﬁed by direct calculation. Lemma 2.10. Let g P C p Λ N q such that Q N g “ 0 (i.e. g is orthogonal to constants), then p´ ∆ H,N q ´ 1 g “ N ÿ n “ 1 L 2 p n ´ 1 q P n g . (11) In particular , we have the useful identity P 1 p´ ∆ H q ´ 1 “ p´ ∆ H q ´ 1 P 1 “ P 1 . D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 9 2.3 Besov spaces The ﬁnal technical ingredient in our framework is a notion of regularity . T o this end, we introduce hierarchical Besov spaces. W e ﬁrst formulate them on the torus T d , where they arise in a natural manner , and subsequently transfer the deﬁnition to functions deﬁned on the unit lattice. Deﬁnition 2.11 (Hierarchical Besov norms) . F or any β P R , we deﬁne the hierarchical H ¨ older–Besov space C β ” C β p T d q on T d as the completion of smooth functions under the norm } w } C β def “ | Q 0 w | ` sup n P N L β n sup x P T d | P ´ n w p x q| . (12) Indeed, recall that P ´ n is the analogue of a sharp Littlewood–P aley block at scale L ´ n . In view of this, (12) strongly resembles the deﬁnition of Besov spaces in the Euclidean space. Moreover , as in the Euclidean case, whenever β ă 0 , replacing P ´ n by Q ´ n yields an equivalent norm. Indeed, using Q ´ n “ Q 0 ` ř n k “ 1 P ´ k we have L β n | Q ´ n w p x q| ď L β n | Q 0 w | ` n ÿ k “ 1 L β p n ´ k ` k q | P ´ k w p x q| ď n ÿ k “ 0 L β p n ´ k q } w } C β ď p 1 ´ L β q ´ 1 } w } C β . The following lemma allows us to control the Besov norm (12) on a coarse lattice by that of its rescaled version on the unit lattice, where our estimates are carried out. Lemma 2.12. F or any β P R and w N P C p Λ ´ N q , the H ¨ older–Besov norm can be more conveniently recast as } w N } C β “ | Q 0 w N | ` max 1 ď n ď N L β n sup x P Λ n | P 1 S N ´ n Q N ´ n S ´ N w N p x q| . Proof . First, by deﬁnition, we have P ´ n w N “ 0 for n ą N . Then, using Lemma 2.6, for n ď N we have P ´ n “ S N P N ´ n ` 1 S ´ N “ S n P 1 S N ´ n Q N ´ n S ´ N . As an instructive example and to gain familiarity with the notation, let us show that white noise on T d indeed lies in the Besov spaces C ´ d { 2 ´ κ , for every κ ą 0 , almost surely . Lemma 2.13. Let p ξ ´ n q n P N be the family of regularised white noises in (7) . F or every κ ą 0 and p P p 1 , 8q , it holds E “ } ξ } p C ´ 2 ` α ´ κ p T d q ‰ “ E ”´ sup n P N L ´ κn sup x P Λ n | ξ ´ n p x q| ¯ p ı À p p { 2 . Proof . Let κ ą 0 . R ecall the deﬁnition of ξ ´ n on the unit lattice Λ n in terms of ξ , cf. (7). Since the noise has negative regularity , each occurrence of P ´ n in (12) can be replaced by Q ´ n . Therefore, } ξ } C ´ 2 ` α ´ κ p T d q “ sup n P N L p´ 2 ` α ´ κ q n sup x P T d | Q ´ n ξ p x q| “ sup n P N L ´ κn sup x P Λ n | ξ ´ n p x q| . W e proceed with a classical Kolmogorov argument. First, for every p ě 1 , we can crudely estimate both suprema in the previous display by sums, which yields E ”´ sup n P N L ´ κn sup x P Λ n | ξ ´ n p x q| ¯ p ı ď 8 ÿ n “ 0 L ´ pκn ÿ x P Λ n E ” | ξ ´ n p x q| p ı . L . F E R D I N A N D A N D S . G A B R I E L 10 Next, using hypercontractivity of the Gaussian variables ξ ´ n p x q , we see that E ”´ sup n P N L ´ κn sup x P Λ n | ξ ´ n p x q| ¯ p ı À p p { 2 8 ÿ n “ 0 L ´ pκn ÿ x P Λ n E r ξ ´ n p x q 2 s p { 2 “ p p { 2 8 ÿ n “ 0 L p d ´ pκ q n , (13) where we used E r ξ ´ n p x q 2 s “ 1 in the last identity . The right–hand side of (13) converges, provided we choose p large enough. By Markov’s inequality , we extend the moment bound to all p P p 1 , 8q . 3 The renormalised model on the unit lattice W e scale the solution u N to the regularised equation (2) to the unit lattice Λ N , deﬁning a function v ´ N P Λ N by v ´ N p x q def “ L αN S ´ N u N p x q ” L αN u N p L ´ N x q , x P Λ N , which now solves the equation p´ ∆ H,N q v ´ N “ p L ´ αN g ξ ´ N ` L ´ 2 N r N q v ´ N ` L ´p 2 ´ α q N . (14) Here, ξ ´ N p x q „ N p 0 , 1 q denote the independent standard normal random variables that were deﬁned in (7) for x P Λ N . F or convenience, we also deﬁne λ ´ N def “ L ´ αN g , µ ´ N def “ L ´ 2 N r N , and γ ´ N def “ L ´p 2 ´ α q N , which allows us to rewrite (14) as p´ ∆ H q v ´ N “ p λ ´ N ξ ´ N ` µ ´ N q v ´ N ` γ ´ N . (15) The reader may again wonder why we consider v ´ N on the unit lattice instead of studying u N directly on T d . This choice will become clear when we derive the dynamical system that describes how v ´ N changes across scales. W e refer to R emark 5.1 below for further explanation. Throughout this section, we analyse the SPDE (14) by examining its evolution across scales. T o this end, we introduce coarse-grained observables of v ´ N , which allow us to capture its behaviour at pro- gressively larger scales. W e will conclude the section with the proof of our main result, Theorem 1.1, for which the scale analysis of v ´ N is the key ingredient. 3.1 The coarse grained solution Henceforward, we enforce that d “ 2 , or equivalently that α “ 1 . Nevertheless, we keep the parameter α explicit, to be able to discuss the effect of the dimension. The central idea underlying renormalisation group techniques is to exploit the fact that the degrees of freedom of a system depend on the scale at which it is observed. In what follows, we analyse the evolution of the system parameters as the reference scale is varied. T o this end, we deﬁne for n P t 0 , . . . , N u the average of v ´ N at scale L ´ n , which we denote by v p N q ´ n def “ L ´ α p N ´ n q S N ´ n Q N ´ n v ´ N . (16) Notice that v p N q ´ n is deﬁned on the unit lattice of size L n . In particular , v p N q ´ N “ v ´ N and v p N q 0 “ Q 0 u N . W e will again drop the superscript and leave the dependency on the ultraviolet cutoff N implicit. The function v ´ n should be thought of as a coarse grained version of v ´ N , describing the solution u N of (2) at scale L ´ n . Indeed, we have v p N q ´ n ” L αn S N ´ n Q N ´ n S ´ N u N . (17) In particular , control on the trajectory p v p N q ´ n q 0 ď n ď N will be sufﬁcient to control u N . T o establish said control, we will rely on the fact that v ´ n solves again a linear SPDE, the effective equation , which is now driven by an effective force at scale n . D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 11 3.2 The effective force The bare force at scale L ´ N is given in terms of the right–hand side of (15), namely F ´ N r v ´ N s def “ p λ ´ N ξ ´ N ` µ ´ N q v ´ N ` γ ´ N . (18) This allows us rewrite (15) (in integrated form) as v ´ N “ p´ ∆ ´ 1 H,N q F ´ N r v ´ N s . Likewise, we can introduce a family p F ´ n q n ď N of effective forces at each scale L ´ n F ´ n r v ´ n s def “ L p 2 ´ α qp N ´ n q S N ´ n Q N ´ n F ´ N r v ´ N s , (19) such that the coarse grained solution v ´ n satisﬁes the effective equation v ´ n “ p´ ∆ ´ 1 H,n q F ´ n r v ´ n s . Indeed, this is a consequence of the following chain of identities v ´ n “ L ´ α p N ´ n q S N ´ n Q N ´ n v ´ N “ L ´ α p N ´ n q S N ´ n Q N ´ n p´ ∆ ´ 1 H,N q F ´ N r v ´ N s “ L p 2 ´ α qp N ´ n q p´ ∆ ´ 1 H,n q S N ´ n Q N ´ n F ´ N r v ´ N s “ p´ ∆ ´ 1 H,n q F ´ n r v ´ n s , where in the third equality we used S N ´ n Q N ´ n p´ ∆ ´ 1 H,N q “ N ÿ k “ N ´ n ` 1 L 2 p k ´ 1 q S N ´ n P k “ N ÿ k “ N ´ n ` 1 L 2 p k ´ 1 q P k ´ N ` n S N ´ n Q N ´ n “ L 2 p N ´ n q n ÿ k “ 1 L 2 p k ´ 1 q P k S N ´ n Q N ´ n “ L 2 p N ´ n q p´ ∆ ´ 1 H,n q S N ´ n Q N ´ n , (20) with S N ´ n P p N q k “ P p n q k ´p N ´ n q S N ´ n Q p N q N ´ n , see Lemma 2.6 (b), being used in the second equality . Note for future reference that (20) is equivalent to Q ´ n p´ ∆ ´ 1 H, ´ N q “ p´ ∆ ´ 1 H, ´ n q Q ´ n . (21) The aim of the renormalisation group is to establish a closed expression for F ´ n in terms of F ´ N . In turn, this will allow the construction of v ´ n at any scale, and therefore the solution u N itself . The difﬁculty with (19) lies in the fact that F ´ N is still evaluated at v ´ N , while we only want it to depend on v ´ n . T o remove this dependence, we introduce the ﬂuctuation ﬁeld f ´ n def “ Γ N ´ n F ´ N r v ´ N s , where Γ N ´ n def “ p 1 ´ Q N ´ n qp ∆ H,N q ´ 1 ” N ´ n ÿ k “ 1 L 2 p k ´ 1 q P k . (22) The operator Γ N ´ n is referred to as the ﬂuctuation propagator . W e can then reexpress v ´ N in terms of v ´ n and the f ´ n , using (4): v ´ N “ L α p N ´ n q S ´ N ` n v ´ n ` f ´ n . (23) Notably , the ﬂuctuation ﬁeld f ´ n can be obtained by solving a ﬁxed point problem driven by F ´ N only . Indeed, inserting (23) into (19) and (22), we have # F ´ n r ‚ s “ L p 2 ´ α qp N ´ n q S N ´ n Q N ´ n F ´ N “ L α p N ´ n q S ´ N ` n ‚ ` f ´ n r ‚ s ‰ , f ´ n r ‚ s “ Γ N ´ n F ´ N “ L α p N ´ n q S ´ N ` n ‚ ` f ´ n r ‚ s ‰ . (24) L . F E R D I N A N D A N D S . G A B R I E L 12 Let us return to the explicit expression of F ´ N in (18), and pretend for simplicity that F ´ N r v s “ λ ´ N ξ ´ N v . This is at least heuristically true, because the terms µ ´ N and γ ´ N play no essential role. Therefore, with this simpliﬁcation in (24), we can solve explicitly for f n r ‚ s in ` 1 ´ λ ´ N Γ N ´ n p ξ ´ N ‚ q ˘“ f ´ n ‰ r v ´ n s “ L α p N ´ n q λ ´ N p Γ N ´ n ξ ´ N q S ´ N ` n v ´ n , where we used that v ´ n is already averaged at scale L ´ n , and can be factorised from Γ N ´ n . Introducing λ ´ n def “ L α p N ´ n q λ ´ N , we can thus (formally) write f ´ n r v ´ n s “ 8 ÿ k “ 1 L ´ α p N ´ n qp k ´ 1 q λ k ´ n ` Γ N ´ n p ξ ´ N ‚ q ˘ ˝ k r S ´ N ` n v ´ n s , such that the corresponding effective force (19) is given in terms of F ´ n r v ´ n s “ 8 ÿ k “ 1 L p 2 ´ kα qp N ´ n q λ k ´ n S N ´ n Q N ´ n ` ξ ´ N ` Γ N ´ n p ξ ´ N ‚ q ˘ ˝ k ´ 1 ˘ r 1 s v ´ n . (25) The interest in (25) stems from the fact that it provides a formal expansion of the effective force into directions that expand or contract at different rates as n decreases. In fact, if α ą 0 (i.e. d ă 4 ), then only a ﬁnite number of directions expand. These contributions are called deterministically relevant . In d “ 2 , the only relevant contribution in (25) beyond the noise is the term of order λ 2 ´ n . This is why , in the sequel, we will make the assumption that the effective force is of the following form: F ´ n r v ´ n s “ ` λ ´ n ξ ´ n ` µ ´ n ` λ 2 ´ n ´ n ` λ 3 ´ n Ψ ´ n ˘ v ´ n ` γ ´ n , (26) where ´ n denotes the (centred) part of the λ 2 ´ n –term in (26), and Ψ ´ n gathers all the determin- istically irrelevant contributions of order at least λ 3 ´ n (or equivalently L p 2 ´ 3 α qp N ´ n q ). Indeed, in Section 3.3, we will carefully disentangle the deterministic contribution of the inhomogeneous chaos term of order λ 2 ´ n in (26), and collect it in µ ´ n . In turn, we are only left with the second homoge- neous chaos contribution ´ n . Such an operation corresponds to the extraction in the context of the renormalisation group. Now , because “ p ´ n q n ď N is centred, its evolution along the ﬂow will not be dictated by the scaling of its expectation, but by that of its variance. When looking at the variance of ´ n , we gain an entropic factor L ´ 2 p 2 ´ α qp N ´ n q , which is a consequence of the fact that E “ | Q 1 ξ ´ n p 0 q| 2 ‰ “ L ´ 2 d ÿ x P ˝ 1 p 0 q E “ | ξ ´ n p x q| 2 ‰ “ L ´ d “ L ´ 2 p 2 ´ α q . In combination with the square of the factor L p 2 ´ 2 α qp N ´ n q in (25), this leads the variance of p N q ´ n to ﬂow like L ´ 2 α p N ´ n q . F or this reason, we say that is probabilistically irrelevant . W e refer to the proof of Lemma 5.2 for a detailed calculation. The situation is analogous to that of the noise ξ . While the noise in (25) is ampliﬁed by a factor L p 2 ´ α qp N ´ n q along the ﬂow , it is the covariance that dictates its evolution. Indeed, we ﬁnd that the covariance of the noise is constant along the ﬂow , due to scale invariance: A covariance calculation shows that we gain an entropic factor L ´ d { 2 p N ´ n q which exactly balances the ampliﬁcation of the noise by L p 2 ´ α qp N ´ n q along the ﬂow , see also (7). The noise is therefore probabilistically marginal . Overall, in the subcritical regime α ą 0 , only a ﬁnite number of quantities are deterministically relevant, and only the noise is probabilistically marginal. D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 13 Therefore, provided expectations of deterministically relevant terms have been placed inside the evolution of the mass term, we are left with studying the ﬂow of a ﬁnite collection of probabilistically irrelevant quantities through covariance computations. Based on this heuristic, we deﬁne rigorously and Ψ in the next sections. Moreover , we show that satisﬁes good probabilistic estimates (see Lemma 5.2), while Ψ can be constructed deterministically given the noise ξ and (see Lemma 5.3). R emark 3.1. The characterisation (24) is slightly different from the equation for the effective force ob- tained in [K up16, Display (15)]. This is due to a fundamental difference between hierarchical and Eu- clidean models, related to the domain of the effective Green’s function G µ . As K upiainen [K up16], we consider a ﬁxed point problem of the form v “ GF r v s , where G denotes the associated Green’s function. Likewise, we introduce scales µ P L ´ N and a trajectory a regularised Green’s functions p G µ q µ start- ing from G , and we seek to deﬁne a trajectory of effective solutions p v µ q µ starting from v and solving v µ “ G µ F µ r v µ s , where p F µ q µ is a trajectory of effective forces starting from F . In the Euclidean case, we can assume that while the kernel of G µ grows with the scale µ , G µ has the same domain as G , namely it acts by convolution on all (say) bounded functions on the Euclidean space. In particular , G µ can act directly on F r v s , so a minimal choice of trajectory p v µ q µ satisfying v 0 “ v is given by v µ “ G µ F r v s . Making this choice and recalling the deﬁnition of F µ yields the relation F µ r v µ s ´ F r v s P Ker p G µ q . In fact, because shifting F µ r v µ s by an element of the kernel of G µ would not modify the equation satisﬁed by v µ , one has the freedom to enforce that F µ r v µ s “ F r v s is constant in µ . Consequently , for the ﬂuctuation ﬁeld we have f µ “ v ´ v µ “ GF r v s ´ G µ F µ r v µ s “ p G ´ G µ q F µ r v µ s , where the right–hand side is expressed directly in terms of F µ r v µ s . This yields the ﬁxed point problem for the effective force: F µ “ F r ‚ ` p G ´ G µ q F µ r ‚ ss . (27) In the hierarchical case, the situation is different. The effective Green’s function is given by G µ ” p´ ∆ H, log L µ q ´ 1 , and has domain C p Λ log L µ q , so F r v s P C p Λ ´ N q does not lie in the domain of G µ for all the scales µ ą L ´ N . On the other hand, in the hierarchical case we deﬁned v µ by v µ def “ ρ µ v “ ρ µ GF r v s where ρ µ ” Q log L µ , cf . (16) . In view of (21) , ρ µ and G satisfy the commutation relation ρ µ G “ G µ ρ µ , which implies that v µ “ G µ ρ µ GF r v s . Note the presence of ρ µ : C p Λ ´ N q Ñ C p Λ log L µ q , which brings back F r v s into the domain of G µ . Because of the necessary presence of ρ µ , we only have that F µ r v µ s “ ρ µ F r v s (in contrast to F µ r v µ s “ F r v s in the Euclidean case). In particular , the ﬂuctuation ﬁeld cannot be rewritten directly in terms of F µ r v µ s , since it also depends on p 1 ´ ρ µ q F r v s which cannot be expressed in terms of F µ r v µ s . W e observe that, with regard to analytical difﬁculty , the system (24) is equivalent to the ﬁxed-point problem (27) . Indeed, (24) has a triangular structure: W e ﬁrst solve for f µ in terms of F , which is the same as solving (27) . Once f µ is known, F µ is obtained directly . R emark 3.2. Y et another difference to the Euclidean RGs is that in the hierarchical model we are re- stricted to the discrete set of scales L ´ N . While in the Euclidean case, we may choose a continuous scale µ P r 0 , 1 s , and replace (27) by a differential equation. Moreover , such a continuous RG ﬂow allows to derive a differential equation solved by the cumulants of the effective equation. This yields an inductive construction of the stochastic terms, bypassing any tedious moment computations [Duc25a, Duc25b]. 3.3 The enhanced noise W e introduced the family p ξ ´ n q n P N of independent standard normal random variables in (7). The random ﬁeld ξ ´ n is the effective white noise at scale L ´ n , see also Lemma 2.13. W e denote by ξ p N q L . F E R D I N A N D A N D S . G A B R I E L 14 the truncation of ξ at n “ N , namely ξ p N q ´ n “ # ξ ´ n for n ď N , 0 otherwise . Moreover , given two linear functionals ξ p f q , ξ p g q of the noise, we deﬁne the W ick product ξ p f q ˛ ξ p g q def “ ξ p f q ξ p g q ´ E r ξ p f q ξ p g qs , to denote the projection of the product ξ p f q ξ p g q onto the second homogeneous Gaussian chaos. Let us now turn our attention to the term ´ n in the effective force (26). F or N P N and 0 ď n ď N , we deﬁne p N q ´ n def “ L p 2 ´ 2 α qp N ´ n q S N ´ n Q N ´ n p ξ ´ N ˛ Γ N ´ n ξ ´ N q , (28) which describes the evolution of the second homogeneous chaos of the effective force along the ﬂow , at scale n . F or n ă 0 or n ą N , we enforce p N q ´ n ” 0 , by convention. Note that contrary to the effective noise ξ p N q , which starts from ξ p N q ´ N “ ξ ´ N , the sequence p N q “ p p N q ´ n q n P N starts from 0 at ´ N . Indeed, is not present in the original equation, but is only generated by the ﬂow . Throughout the paper , we will often drop the dependence on the cutoff N and write p ξ , q for the sequence p ξ p N q ´ n , p N q ´ n q n ď N . W e call this tuple the enhanced noise . F or a small ﬁxed κ s ą 0 , we endow the stochastic terms ξ “ p ξ ´ n q n P N and “ p ´ n q n P N with the norm | | | ξ , | | | def “ sup n P N L ´ κ s n ´ sup x P Λ n | ξ ´ n p x q| _ sup x P Λ n | ´ n p x q| 1 { 2 ¯ . (29) Moreover , we set 8 ξ , 8 def “ sup N P N | | | ξ p N q , p N q | | | . (30) Throughout the rest of the article, we will assume that we are on a good event where p ξ p N q , p N q q is of ﬁnite size uniformly in N ě 0 . Namely , for every g P p 0 , 1 s , we deﬁne Ω g def “ ␣ 8 ξ , 8 ď p 2 g q ´ 1 ( . (31) The choice to work on Ω g is a way to eliminate the large ﬁeld problem, since occurrences of ξ and p N q will always appear with a g and g 2 , respectively , so that the size of these object does not pose any problem. The side effect is that we do not cover the full probability space, and that the event Ω g shrinks if one considers large g . On the other hand, for small g we have an exponentially vanishing bound of the tail probability , which is a consequence of moment bounds for (29) which we establish in Lemma 5.2. Lemma 3.3. There exist constants c, C ą 0 , such that for every g P p 0 , 1 s P p Ω g q ě 1 ´ C e ´ cg ´ 2 . W e defer the proof of the lemma to Section 5.2, just after the statement of Lemma 5.2. W orking on events of the form Ω g is analogous to choosing a small random time (or a small coupling constant) as in [Hai14, Kup16, Duc25a]. Indeed, one could equally have chosen to randomise the coupling constant setting g “ 8 ξ , 8 ´ 1 in the present paper . D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 15 R emark 3.4. Recall from Lemma 2.13 that the norm on the stochastic terms controls } ξ } C ´ 2 ` α ´ κ s ď 8 ξ , 8 . On the other hand, for the effective second chaos n def “ L p 2 ´ 2 α q n S n p N q ´ n on Λ ´ n , the norm enforces that uniformly in N , n and x P Λ ´ n | p N q n p x q| À L ´p´ 2 ` 2 α ´ 2 κ s q n . Therefore, ﬁniteness of the stochastic norm (29) ensures that on the unit lattice the noise is of regularity ´ 2 ` α ´ κ s , and the second chaos is of (better) regularity ´ 2 ` 2 α ´ 2 κ s “ ´ 2 κ s . All other chaoses are of positive regularity , and will be constructed deterministically as functions of the enhanced noise. 3.4 Proof of the main result In the previous sections, we introduced the effective force which led to the effective equation p´ ∆ H,N q v p N q ´ n “ ` λ ´ n ξ ´ n ` λ 2 ´ n p N q ´ n ` λ 3 ´ n Ψ p N q ´ n ` µ ´ n ˘ v p N q ´ n ` γ ´ n . (32) The right–hand side consists in particular of the enhanced noise p ξ p N q , p N q q “ p ξ p N q ´ n , p N q ´ n q n ď N , which we can control with high probability , see Lemma 3.3. Our goal in the subsequent sections is to describe the evolution of the constants p λ ´ n , µ ´ n , γ ´ n q n ď N , and prove that the UV limit p ξ ´ n , ‹ ´ n , Ψ ‹ ´ n q n P N def “ lim N Ñ8 ´ ξ p N q ´ n , p N q ´ n , Ψ p N q ´ n ¯ n ď N , exists, provided we are on the good event Ω g . The main result for the trajectory of solutions p v p N q ´ n q n “ 0 ,...,N on unit lattices, which is the main ingredient of the proof of Theorem 1.1, is summarised in the fol- lowing proposition. Proposition 3.5. Let d “ 2 . F or every r ě 1 , there exists an odd L P N , sufﬁciently large (and depending only on r ), such that for every g P p 0 , 1 s and every n P N , the limit v ‹ ´ n def “ lim N Ñ8 v p N q ´ n exists on Ω g , and solves the renormalised equation p´ ∆ H q v ‹ ´ n “ ` λ ´ n ξ ´ n ` λ 2 ´ n ‹ ´ n ` λ 3 ´ n Ψ ‹ ´ n ` µ ´ n ˘ v ‹ ´ n ` γ ´ n , with λ ´ n “ L ´ n g , γ ´ n “ L ´ n , and µ ´ n “ L ´ 2 n ` r ´ n p 1 ´ L ´ d q g 2 ˘ . Moreover , we have v ‹ ´ n “ L ´ α p m ´ n q S m ´ n Q m ´ n v ‹ ´ m , for all m ą n . The proof of Proposition 3.5 is the goal of Section 6, and will be stated at the end of said section. Moreover , we establish uniform control over the convergence of trajectories in a H ¨ older sense: Lemma 3.6. Let d “ 2 . F or every r ě 1 , there exists an odd L P N , sufﬁciently large (and depending only on r ), such that for every g P p 0 , 1 s and every κ P p 3 κ s , 1 q lim N Ñ8 sup n ě 1 L ´ κn sup x P Λ n ˇ ˇ P 1 ` v ‹ ´ n ´ v p N q ´ n ˘ p x q ˇ ˇ “ 0 , on Ω g . The proof of Theorem 1.1 is now a consequence of Proposition 3.5 and Lemma 3.6, by rescaling the solution v ´ n to the coarse grained lattice Λ ´ n Ă T 2 . L . F E R D I N A N D A N D S . G A B R I E L 16 Proof of Theorem 1.1. Let κ P p 3 κ s , 1 q . W e recall from (17) that P 1 v p N q ´ n “ L αn P 1 S N ´ n Q N ´ n S ´ N u N , and that v p N q 0 “ Q 0 u N . W e therefore set u ‹ n def “ L ´ αn S n v ‹ ´ n , where v ‹ is the limit from Proposi- tion 3.5, which provides a candidate trajectory of renormalised solutions. In particular , we have u ‹ n “ Q ´ m ` n u ‹ m for all m ą n . W e deﬁne the reconstruction of p u ‹ n q n P N to be the unique limit u ‹ “ lim n Ñ8 u ‹ n . Indeed, the limit exists because the sequence is Cauchy in C 1 ´ κ p T 2 q : F or m ą n } u ‹ m ´ u ‹ n } C 1 ´ κ “ sup n ă k ď m L p 1 ´ κ q k sup x P T d ˇ ˇ P ´ k u ‹ m p x q ˇ ˇ “ sup n ă k ď m L ´ κk sup x P Λ ´ k ˇ ˇ P ´ k S k v ‹ ´ k p x q ˇ ˇ ď L ´p κ ´ 3 κ s q n ´ sup n ă k ď m L ´ 3 κ s k sup x P Λ ´ k ˇ ˇ S k P p k q 1 v ‹ ´ k p x q ˇ ˇ ¯ , (33) where used the fact that Q ´ n u ‹ m “ u ‹ n in the ﬁrst equality , and in the last step that P ´ k S k “ S k P p k q 1 , cf . Deﬁnition 2.4 (b). The right–hand side can be chosen arbitrarily small by choosing n sufﬁciently large, because the term inside the parenthesis is uniformly bounded by Lemma 3.6. Thus, p u ‹ n q n P N is a Cauchy sequence. T o conclude the theorem, it is only left to show that u N indeed converges to u ‹ on Ω g , i.e. to prove that the following quantity vanishes in the large N limit: } u ‹ ´ u N } C 1 ´ κ ď › › u ‹ ´ u ‹ N › › C 1 ´ κ ` › › u ‹ N ´ u N › › C 1 ´ κ . (34) The ﬁrst term on the right–hand side vanishes by (33). On the other hand, for the second term we apply Lemma 2.12, which yields › › Q ´ N u ‹ ´ u N › › C 1 ´ κ “ | Q 0 p u ‹ ´ u N q| ` max 1 ď n ď N L p 1 ´ κ q n sup x P Λ n ˇ ˇ P 1 S N ´ n Q N ´ n S ´ N ` Q ´ N u ‹ ´ u N ˘ p x q ˇ ˇ “ | v ‹ 0 ´ v p N q 0 | ` max 1 ď n ď N L p 1 ´ κ q n sup x P Λ n ˇ ˇ P 1 ` L ´ αN S N ´ n Q N ´ n v ‹ ´ N ´ L ´ αn v p N q ´ n ˘ p x q ˇ ˇ “ | v ‹ 0 ´ v p N q 0 | ` max 1 ď n ď N L ´ κn sup x P Λ n ˇ ˇ P 1 ` v ‹ ´ n ´ v p N q ´ n ˘ p x q ˇ ˇ , where we used that S ´ N Q ´ N u ‹ “ S ´ N u ‹ N “ L ´ αN v ‹ ´ N . Applying Lemma 3.6 ﬁnishes the proof . 4 Derivation of the perturbative dynamical system In this section, we perform the ﬁrst coarse graining step, going from v ´ N to v ´ N ` 1 . W e will then state a candidate for the effective equation for v ´ N ` 1 , using a perturbative argument. While this derivation is non–rigorous, it sheds light on the main idea of the R enormalisation Group mechanism. In Sec- tion 5, we will then keep track of (and control) the remainder terms explicitly , providing the rigorous justiﬁcation for the steps outlined here. W e recall from (16) that v ´ N ` 1 p x q “ L ´ α S 1 Q 1 v ´ N p x q ” L ´ α Q 1 v ´ N p Lx q , x P Λ N ´ 1 . F or convenience, we also introduce w ´ N ` 1 def “ Q 1 v ´ N such that v ´ N ` 1 p x q “ L ´ α w ´ N ` 1 p Lx q . More- over , we decompose the solution into v ´ N “ w ´ N ` 1 ` z ´ N ` 1 , with z ´ N ` 1 def “ P 1 v ´ N . D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 17 In the jargon of the R enormalisation Group, w ´ N ` 1 is called the background ﬁeld , since it corre- sponds to the average of v ´ N ` 1 , and z ´ N ` 1 is called the ﬂuctuation ﬁeld . Observe that w ´ N ` 1 and z ´ N ` 1 satisfy the following useful identities Q 1 z ´ N ` 1 “ 0 and Q 1 p f w ´ N ` 1 q “ p Q 1 f q w ´ N ` 1 , @ f P C p Λ N q . (35) Since we are only performing a single coarse graining step below , it will be convenient to drop the subscripts and simply write w “ w ´ N ` 1 and z “ z ´ N ` 1 for the background and ﬂuctuation ﬁeld, respectively . Now , by application of Q 1 to both sides of (15), we see that w satisﬁes the equation p´ Q 1 ∆ H q w “ Q 1 ` p λ ´ N ξ ´ N ` µ ´ N q v ´ N ` γ ´ N ˘ , (36) where we used that ´ Q 1 ∆ H “ ř N n “ 2 L ´ 2 p n ´ 1 q P n Q 1 “ p´ Q 1 ∆ H q Q 1 . Using the decomposition v ´ N “ w ` z and (35), the right–hand side can then be cast into p´ Q 1 ∆ H q w “ λ ´ N p Q 1 ξ ´ N q w ` λ ´ N Q 1 p ξ ´ N z q ` µ ´ N w ` γ ´ N . (37) In particular , the right–hand side is expressed entirely in terms of w , except for the second term which depends on z . In order to replace z with a suitable expression in terms of w , we write the analogue of (37) for the ﬂuctuation ﬁeld z . T o this end, we apply P 1 to both sides of (15), which yields z “ P 1 ` p λ ´ N ξ ´ N ` µ ´ N qp w ` z q ` γ ´ N ˘ “ λ ´ N P 1 p ξ ´ N z q ` λ ´ N p P 1 ξ ´ N q w ` µ ´ N z , (38) where we used that P 1 p´ ∆ H q ´ 1 “ P 1 , see Lemma 2.10, and in the second step that P 1 projects constants onto zero, and that P 1 w “ 0 because P 1 Q 1 “ 0 . Next, we notice that w ÞÑ z r w s is linear , because we were considering an afﬁne equation. Therefore, it is convenient to introduce the random ﬁeld M such that z “ M w , which then satisﬁes the equation M “ λ ´ N P 1 p ξ ´ N M q ` λ ´ N p P 1 ξ ´ N q ` µ ´ N M , (39) which is independent of w . Notice that the independence of w is a consequence of the fact that the equation we are considering is linear . Thus, in order to stay close to the core idea of RG, we will not attempt to solve for M directly . Overall, we have successfully derived two coupled equations # p´ Q 1 ∆ H q w “ λ ´ N p Q 1 ξ ´ N q w ` λ ´ N Q 1 p ξ ´ N M q w ` µ ´ N w ` γ ´ N , M “ λ ´ N P 1 p ξ ´ N M q ` λ ´ N p P 1 ξ ´ N q ` µ ´ N M , (40) describing the coarse grained behaviour of v ´ N at scale L ´ N ` 1 , as well as the ﬂuctuation behaviour M that was lost when mapping from v ´ N to w . By formally expanding the quantities in the system (40) in the coupling constant λ ´ N , we expect that to leading order M is well approximated by M r 1 s def “ λ ´ N p P 1 ξ ´ N q , which forms the basis of the perturbative argument that is about to follow . Here, the superscript indicates that we only approximate using terms in a single power of λ N . W e then expect that up to lower–order corrections of order O p λ 3 ´ N q , w is well described in terms of w r 2 s , which solves the equation p´ Q 1 ∆ H q w r 2 s “ λ ´ N p Q 1 ξ ´ N q w r 2 s ` λ 2 ´ N Q 1 p ξ ´ N P 1 ξ ´ N q w r 2 s ` µ ´ N w r 2 s ` γ ´ N , (41) L . F E R D I N A N D A N D S . G A B R I E L 18 where we simply replaced M in (40) with M r 1 s . In other words, we perform a perturbative expansion up to order λ 2 ´ N , while neglecting all remaining terms that are of order O p λ 3 ´ N q . W e have found an approximate candidate equation for w which can now be scaled into an effective equation on Λ N ´ 1 using v r 2 s ´ N ` 1 def “ L ´ α S 1 w r 2 s : p´ ∆ H q v r 2 s ´ N ` 1 “ L 2 ´ α S 1 ´ ` λ ´ N p Q 1 ξ ´ N q ` λ 2 ´ N Q 1 p ξ ´ N P 1 ξ ´ N q ` µ ´ N ˘ w r 2 s ` γ ´ N ¯ (42) “ ` L α λ ´ N ξ ´ N ` 1 ` L 2 λ 2 ´ N S 1 Q 1 p ξ ´ N P 1 ξ ´ N q ` L 2 µ ´ N ˘ v r 2 s ´ N ` 1 ` L 2 ´ α γ ´ N , where we used the deﬁnition of ξ ´ N ` 1 in (8), the fact that S 1 p f ¨ g q “ p S 1 f qp S 1 g q for all f , g P C p Λ N q , and moreover that for every f P Λ N 1 S 1 Q p N q 1 p´ ∆ H,N q f “ N ÿ n “ 2 L ´ 2 p n ´ 1 q S 1 P p N q n f “ L ´ 2 N ´ 1 ÿ n “ 1 L ´ 2 p n ´ 1 q P p N ´ 1 q n S 1 f “ L ´ 2 p´ ∆ H,N ´ 1 q S 1 f , which gives rise to the extra factor L 2 . Next, we decompose the second inhomogeneous chaos term on the right–hand side of (42) into L 2 λ 2 ´ N S 1 Q 1 p ξ ´ N P 1 ξ ´ N q “ L 2 α λ 2 ´ N ´ N ` 1 ` L 2 λ 2 ´ N E “ S 1 Q 1 p ξ ´ N P 1 ξ ´ N q ‰ “ L 2 α λ 2 ´ N ´ N ` 1 ` L 2 λ 2 ´ N p 1 ´ L ´ d q , (43) where we used (28) and Lemma A.1. Hence, (42) can be brought into the form (32), such that v ´ N ` 1 approximately solves the equation p´ ∆ H q v r 2 s ´ N ` 1 “ ` λ ´ N ` 1 ξ ´ N ` 1 ` λ 2 ´ N ` 1 ´ N ` 1 ` µ ´ N ` 1 ˘ v r 2 s ´ N ` 1 ` γ ´ N ` 1 , (44) with $ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ % λ ´ N ` 1 “ L α λ ´ N , µ ´ N ` 1 “ L 2 ` µ ´ N ` p 1 ´ L ´ d q λ 2 ´ N ˘ , γ ´ N ` 1 “ L 2 ´ α γ ´ N , ´ N ` 1 “ L 2 ´ 2 α S 1 Q 1 p ξ ´ N ˛ P 1 ξ ´ N q . In other words, the coefﬁcients of the effective equation for v r 2 s ´ N ` 1 can be entirely expressed in terms of the coefﬁcients of the equation for v ´ N , on the lattice Λ N . Thus far , we have only performed the ﬁrst step of the renormalisation group procedure. In the same manner , we can now go from v r 2 s ´ N ` 1 to a v r 2 s ´ N ` 2 . The only difference to the ﬁrst perturbative step is that we have created a new noise term, namely ´p N ` 1 q , which in turn creates new terms of order O p λ 3 ´ N ` 1 q . Again, we may ignore those due to our current perturbative perspective. More generally , this algorithm allows to map v r 2 s ´ n to v r 2 s ´ n ` 1 , for any ﬁxed n ď N . The ﬂow of force coefﬁcients in the effective equation p´ ∆ H q v r 2 s ´ n “ ` λ ´ n ξ ´ n ` λ 2 ´ n ´ n ` µ ´ n ˘ v r 2 s ´ n ` γ ´ n , is then expressed in terms of the dynamical system $ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ % λ ´ n ` 1 “ L α λ ´ n , µ ´ n ` 1 “ L 2 ` µ ´ n ` p 1 ´ L ´ d q λ 2 ´ n ˘ , γ ´ n ` 1 “ L 2 ´ α γ ´ n , ´ n ` 1 “ L 2 ´ 2 α ` S 1 Q 1 ´ n ` S 1 Q 1 p ξ ´ n ˛ P 1 ξ ´ n q ˘ . (45) D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 19 Notice that (45) provides an alternative, recursive representation of , which indeed agrees with (28), see Lemma 4.1 below . Overall, we have derived a recursive characterisation of the effective force coefﬁcients in (32), albeit perturbative at this stage. W e recall from Section 3.2 that λ “ p λ ´ n q n ď N , µ “ p µ ´ n q n ď N and γ “ p γ ´ n q n ď N are (deterministically) relevant quantities, since they increase along the ﬂow . T o make sure that these relevant terms stay bounded along the RG trajectory , it is necessary to solve their ﬂows backwards, by prescribing a ﬁnite value at n “ 0 . Hence, we enforce that the renormalised couplings are given in terms of λ 0 “ g P p 0 , 1 s , µ 0 “ r ě 1 , and γ 0 “ 1 . Solving then the ﬂow equation (45) yields λ ´ n “ L ´ αn g , µ ´ n “ L ´ 2 n ` r ´ p 1 ´ L ´ d q n g 2 ˘ and γ ´ n “ L ´p 2 ´ α q n , (46) which is consistent with the bare equation (2). In particular , we recover the bare mass r N “ L 2 N µ ´ N . On the other hand, the probabilistically irrelevant term p ´ n q n ď N will be created by the ﬂow , and can be proven to remain bounded with high probability . In fact, we will show that p N q ´ n converges in the UV limit N Ñ 8 . In turn, convergence of both the deterministic and random force coefﬁcients allows us to conclude convergence of v p N q , r 2 s ´ n , as N Ñ 8 . W e thus formally recover the statement of Proposition 3.5 with Ψ ‹ ´ n ” 0 . The remainder of the paper makes this argument rigorous and provides all the necessary details, in particular the control of the remainder term. W e conclude the section by showing that the recursive expression of is compatible with (28). Lemma 4.1. F or every N P N , and n ď N , we have p N q ´ n “ N ´ n ÿ k “ 1 L p 2 ´ 2 α q k S k Q k p ξ ´ n ´ k ˛ P 1 ξ ´ n ´ k q . (47) Proof . W e verify the recursion relation using the deﬁnition of from (28). First, we notice that ´ n ` 1 “ L p 2 ´ 2 α qp N ´ n ` 1 q S N ´ n ` 1 Q N ´ n ` 1 p ξ ´ N ˛ Γ N ´ n ` 1 ξ ´ N q “ L p 2 ´ 2 α q L p 2 ´ 2 α qp N ´ n q S 1 Q 1 S N ´ n Q N ´ n ` ξ ´ N ˛ p Γ N ´ n ` L 2 p N ´ n q P N ´ n ` 1 q ξ ´ N ˘ , where we used Lemma 2.6 (a), and the deﬁnition of Γ N ´ n ` 1 (22). The ﬁrst term on the right–hand side recovers ´ n . Next, we apply Lemma 2.6 (b) which yields ´ n ` 1 “ L p 2 ´ 2 α q S 1 Q 1 ´ n ` L p 2 ´ 2 α q L p 4 ´ 2 α qp N ´ n q S 1 Q 1 S N ´ n Q N ´ n p ξ ´ N ˛ S ´ N ` n P 1 S N ´ n Q N ´ n ξ ´ N q “ L p 2 ´ 2 α q S 1 Q 1 ´ n ` L p 2 ´ 2 α q L p 4 ´ 2 α qp N ´ n q S 1 Q 1 p S N ´ n Q N ´ n ξ ´ N ˛ P 1 S N ´ n Q N ´ n ξ ´ N q “ L p 2 ´ 2 α q S 1 Q 1 ´ n ` L p 2 ´ 2 α q S 1 Q 1 p ξ ´ n ˛ P 1 ξ ´ n q . where in the second identity we used that S ´ N ` n P 1 S N ´ n Q N ´ n ξ ´ N is already averaged at scale N ´ n and can factorise from Q N ´ n . The last identity is a consequence of the deﬁnition of ξ ´ n (8). W e ﬁnish the proof , by noticing that the recursion in (45) yields precisely the right–hand side of (47). R emark 4.2 (A single large RG step) . The reader may have spotted that the chain of coarse grainings can be performed in a single (large) step using v ´ n “ L ´ α p N ´ n q S N ´ n Q N ´ n v ´ N by its deﬁnition in (16) : v ´ N v ´ N ` 1 v ´ N ` 2 ¨ ¨ ¨ v ´ n ´ 1 v ´ n Indeed, the same perturbative dynamical system (45) can be obtained by performing a single large RG step. However , we will leverage on the step–by–step procedure, which is central to the RG framework, when controlling the remainder term. This will be the focus of Section 5. L . F E R D I N A N D A N D S . G A B R I E L 20 5 Uniform control of the RG ﬂow In Section 4, we derived perturbatively an effective equation for v ´ n “ v r 2 s ´ n ` O p λ 3 ´ n q , of the form p´ ∆ H q v r 2 s ´ n “ ` λ ´ n ξ ´ n ` λ 2 ´ n ´ n ` µ ´ n ˘ v r 2 s ´ n ` γ ´ n . However , we entirely neglected control of the remainder term λ 3 ´ n Ψ ´ n “ O p λ 3 ´ n q . The goal of the present section is to expand our perturbative analysis to control this error . W e then establish the required stochastic estimates for the term . Finally , we present a ﬁxed point argument which yields uniform (in N ) control of the remainder Ψ p N q “ p Ψ p N q ´ n q 0 ď n ď N , and consequently control of the trajectory v p N q “ p v p N q ´ n q 0 ď n ď N . 5.1 The dynamical system In Section 4, we only performed the ﬁrst (perturbative) RG step, and then deduced the dynamical system for the subsequent steps. In the following, we will not treat the ﬁrst step differently , instead we present generally how to derive the effective equation for v ´ n ` 1 “ L ´ α S 1 Q 1 v ´ n from the effective equation for v ´ n , for 1 ď n ď N . The basis of the derivation is that v ´ n “ v p N q ´ n solves equation (32): p´ ∆ H q v ´ n “ p λ ´ n ` λ 2 ´ n ´ n ` λ 3 ´ n Ψ ´ n ` µ ´ n q v ´ n ` γ ´ n , Clearly , this holds true when n “ N , with ´ N , Ψ ´ N ” 0 , cf . (15). As in Section 4, we write v ´ n “ w ´ n ` 1 ` z ´ n ` 1 , where w ´ n ` 1 def “ Q 1 v ´ n , and z ´ n ` 1 def “ P 1 v ´ n . Then, analogously to (37), we see that w ´ n ` 1 solves the equation p´ Q 1 ∆ H q w ´ n ` 1 “ ` λ ´ n Q 1 ξ ´ n ` λ 2 ´ n Q 1 ´ n ` λ 3 ´ n Q 1 Ψ ´ n ` µ ´ n ˘ w ´ n ` 1 ` γ ´ n ` λ ´ n Q 1 ` ξ ´ n z ´ n ` 1 ˘ ` λ 2 ´ n Q 1 p ´ n z ´ n ` 1 q ` λ 3 ´ n Q 1 p Ψ ´ n z ´ n ` 1 q . (48) Furthermore, analogously to (38), z ´ n ` 1 solves the equation z ´ n ` 1 “ ` λ ´ n P 1 ξ ´ n ` λ 2 ´ n P 1 ´ n ` λ 3 ´ n P 1 Ψ ´ n ˘ w ´ n ` 1 ` λ ´ n P 1 ` ξ ´ n z ´ n ` 1 ˘ ` λ 2 ´ n P 1 p ´ n z ´ n ` 1 q ` λ 3 ´ n P 1 p Ψ ´ n z ´ n ` 1 q ` µ ´ n z ´ n ` 1 . (49) Here, we used again that Q 1 z ´ n ` 1 “ 0 , P 1 γ ´ n “ 0 , and P 1 w ´ n ` 1 “ 0 . Once more, we notice that w ÞÑ z ´ n ` 1 r w s is linear , and introduce the random ﬁeld M ´ n ` 1 such that z ´ n ` 1 “ M ´ n ` 1 w ´ n ` 1 . R eplacing z ´ n ` 1 with this new representation, (49) implies M ´ n ` 1 “ λ ´ n P 1 ξ ´ n ` λ 2 ´ n P 1 ´ n ` λ 3 ´ n P 1 Ψ ´ n ` λ ´ n P 1 ` ξ ´ n M ´ n ` 1 ˘ ` λ 2 ´ n P 1 p ´ n M ´ n ` 1 q ` λ 3 ´ n P 1 p Ψ ´ n M ´ n ` 1 q ` µ ´ n M ´ n ` 1 . (50) Combining (48) and (50) yields the non–perturbative equivalent of the coupled system (40). Next, we proceed differently than in the perturbative derivation. W e deﬁne the remainder λ 2 ´ n R ´ n ` 1 def “ M ´ n ` 1 ´ λ ´ n P 1 ξ ´ n , (51) which we neglected in Section 4. Combining (50) with the decomposition (51) and rearranging terms, we derive the following equation for R ´ n ` 1 : R ´ n ` 1 “ P 1 ´ n ` P 1 p ξ ´ n p P 1 ξ ´ n qq ` λ ´ n P 1 Ψ ´ n ` λ ´ n P 1 ` ξ ´ n R ´ n ` 1 q ` λ ´ n P 1 p ´ n P 1 ξ ´ n q ` λ 2 ´ n P 1 p ´ n R ´ n ` 1 q ` λ 2 ´ n P 1 p Ψ ´ n P 1 ξ ´ n q ` λ 3 ´ n P 1 p Ψ ´ n R ´ n ` 1 q ` µ ´ n λ ´ 1 ´ n P 1 ξ ´ n ` µ ´ n R ´ n ` 1 . (52) D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 21 Finally , inserting z ´ n ` 1 “ p λ ´ n P 1 ξ ´ n ` λ 2 ´ n R ´ n ` 1 q w ´ n ` 1 into (48), we can write the effective equa- tion for v ´ n ` 1 “ L ´ α S 1 w ´ n ` 1 in terms of p´ ∆ H q v ´ n ` 1 “ p λ ´ n ` 1 ξ ´ n ` 1 ` λ 2 ´ n ` 1 ´ n ` 1 ` λ 3 ´ n ` 1 Ψ ´ n ` 1 ` µ ´ n ` 1 q v ´ n ` 1 ` γ ´ n ` 1 , (53) with the following regrouping of terms $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % λ ´ n ` 1 “ L α λ ´ n , µ ´ n ` 1 “ L 2 ` µ ´ n ` p 1 ´ L ´ d q λ 2 ´ n ˘ , γ ´ n ` 1 “ L 2 ´ α γ ´ n , ξ ´ n ` 1 “ L 2 ´ α S 1 Q 1 ξ ´ n , ´ n ` 1 “ L 2 ´ 2 α ` S 1 Q 1 ´ n ` S 1 Q 1 p ξ ´ n ˛ P 1 ξ ´ n q ˘ , Ψ ´ n ` 1 “ L 2 ´ 3 α ` S 1 Q 1 Ψ ´ n ` S 1 Q 1 p ξ ´ n R ´ n ` 1 q ` S 1 Q 1 p ´ n P 1 ξ ´ n q ` λ ´ n S 1 Q 1 p ´ n R ´ n ` 1 q ` λ ´ n S 1 Q 1 p Ψ ´ n P 1 ξ ´ n q ` λ 2 ´ n S 1 Q 1 p Ψ ´ n R ´ n ` 1 q ˘ . (54) Notice that the above dynamical system is an extension of the one we derived perturbatively in (45), now incorporating the higher-order error term represented by Ψ . Moreover , we observe that the lin- ear evolution of the remainder term Ψ is given by multiplication with L 2 ´ 3 α “ L ´ 1 . This renders the heuristic discussion in Section 4 precise and conﬁrms that Ψ is indeed deterministically irrele- vant. As will be shown in the next section, this stands in clear contrast to the term , which is only probabilistically irrelevant. Indeed, given p ξ , q , the good prefactor L 2 ´ 3 α allows us to construct Ψ by deterministic means. R emark 5.1 (Why the unit lattice?) . Let us return to the question why we view the evolution of scales on the unit lattice, rather than the equivalent trajectory p u p N q n q 0 ď n ď N of effective solutions deﬁned directly on lattices Λ ´ n Ă T d , with u p N q n “ Q p N q ´ n u N . Notice that the system (54) is particularly transparent, as the spectrum of its linearisation reﬂects directly the regularity properties of the terms involved. It particular , it is immediate from (54) that the remainder Ψ contracts along the ﬂow , since the linearisation of its evolution is governed by the factor L 2 ´ 3 α ă 1 . In contrast, if we were to work with the trajectory p u p N q n q 0 ď n ď N , the situation would be different. In that case, the spectrum of the linearisation of the RG would be trivial, and the scaling properties of the different terms would be hidden within their deﬁnitions (see Remark 2.7 for a discussion of this phenomenon for the evolution of the noise). In other words, rescaling to the unit lattice at each step allows for direct comparison of the different quantities. 5.2 Stochastic estimates In this section, we prove the stochastic estimate of the enhanced noise p ξ p N q , p N q q “ ` ξ p N q ´ n , p N q ´ n ˘ n ď N . Namely , we prove the following lemma: Lemma 5.2 (Stochastic estimate) . F or every p P p 1 , 8q , it holds that E “ 8 ξ , 8 p ‰ “ E ” sup N P N | | | ξ p N q , p N q | | | p ı À p p { 2 , (55) with an implicit constant that is independent of p . In fact, we control p ξ , q in the full subcritical regime α ą 0 , which includes d “ 2 , 3 . Therefore, we keep the variable α explicit throughout the proof . L . F E R D I N A N D A N D S . G A B R I E L 22 Proof . W e begin by noticing from (30) that 8 ξ , 8 “ } ξ } C ´ 2 ` α ´ κ s p T d q _ sup N P N max 0 ď n ď N L ´ κ s n sup x P Λ n | p N q ´ n p x q| 1 { 2 . In Lemma 2.13, we already proved E “ } ξ } p C ´ 2 ` α ´ κ s p T d q ‰ À p p { 2 , for all p P p 1 , 8q . Therefore, it is only left to control , and show that (while writing κ “ 2 κ s ) E ”´ sup N P N max 0 ď n ď N L ´ κn sup x P Λ n | p N q ´ n p x q| ¯ p ı À p p . (56) In the following, it will be convenient to use the shorthand } p N q ´ n } 8 def “ sup x P Λ n | p N q ´ n p x q| . First, recalling that p K q ´ n “ 0 whenever K ă 0 or n ě K , we have the telescope representation p N q ´ n “ N ÿ K “ n ` p K q ´ n ´ p K ´ 1 q ´ n ˘ . W e then crudely estimate the maximum over n with a sum, which yields the upper bound max 0 ď n ď N L ´ κn } p N q ´ n } 8 ď N ÿ n “ 0 L ´ κn N ÿ K “ n } p K q ´ n ´ p K ´ 1 q ´ n } 8 ď 8 ÿ K “ 0 K ÿ n “ 0 L ´ κn } p K q ´ n ´ p K ´ 1 q ´ n } 8 , (57) uniformly in N . Next, by inserting (57) into (56), we see that › › › sup N P N max 0 ď n ď N L ´ κn } p N q ´ n } 8 › › › L p p Ω q ď 8 ÿ K “ 0 K ÿ n “ 0 L ´ κn › › › › › p K q ´ n ´ p K ´ 1 q ´ n › › 8 › › › L p p Ω q “ 8 ÿ K “ 0 K ÿ n “ 0 L ´ κn › › › sup x P Λ n ˇ ˇ p K q ´ n ´ p K ´ 1 q ´ n p x q ˇ ˇ p › › › 1 { p L 1 p Ω q ď 8 ÿ K “ 0 K ÿ n “ 0 L ´ κn › › › ÿ x P Λ n ˇ ˇ p K q ´ n ´ p K ´ 1 q ´ n ˇ ˇ p p x q › › › 1 { p L 1 p Ω q “ 8 ÿ K “ 0 K ÿ n “ 0 L ´ κn ´ ÿ x P Λ n E “ ˇ ˇ p K q ´ n ´ p K ´ 1 q ´ n ˇ ˇ p p x q ‰ ¯ 1 { p “ 8 ÿ K “ 0 K ÿ n “ 0 L p d { p ´ κ q n E “ ˇ ˇ p K q ´ n ´ p K ´ 1 q ´ n ˇ ˇ p p 0 q ‰ 1 { p , (58) where we ﬁrst applied the triangle inequality , and then estimated sup x P Λ n by the sum over Λ n , lastly , we used spatial invariance in law . Then, by means of Gaussian hypercontractivity , we arrive at › › › sup N P N max 0 ď n ď N L ´ κn } p N q ´ n } 8 › › › L p p Ω q À p 8 ÿ K “ 0 K ÿ n “ 0 L p d { p ´ κ q n E “` p K q ´ n ´ p K ´ 1 q ´ n ˘ 2 p 0 q ‰ 1 { 2 . (59) Thus, to conclude the proof , we only need to perform a covariance computation. W e start by noticing that by (47), p K q ´ n ´ p K ´ 1 q ´ n “ L p 2 ´ 2 α qp K ´ n q S K ´ n Q K ´ n ` p P 1 ξ ´ K q ˛ 2 ˘ , where we furthermore used the fact that Q K ´ n ` Q 1 ξ ´ K ˛ P 1 ξ ´ K ˘ “ 0 , which allowed us to replace ξ ´ K ˛ P 1 ξ ´ K with p P 1 ξ ´ K q ˛ 2 in the expression above. The scaling by S K ´ n does not affect the variance computation, since it only rescales the reference lattice. Hence, E “` p K q ´ n ´ p K ´ 1 q ´ n ˘ 2 p 0 q ‰ “ L p 4 ´ 4 α ´ 2 d qp K ´ n q ÿ x,y P Λ K ´ n E rp P 1 ξ ´ K q ˛ 2 p x qp P 1 ξ ´ K q ˛ 2 p y qs . D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 23 Therefore, using Corollary A.3, we compute E “` p K q ´ n ´ p K ´ 1 q ´ n ˘ 2 p 0 q ‰ “ 2 L p 4 ´ 4 α ´ 2 d qp K ´ n q ÿ x P Λ K ´ n ÿ y P ˝ 1 p x q p δ x,y ´ L ´ d q 2 “ 2 L p 4 ´ 4 α ´ 2 d qp K ´ n q ÿ x P Λ K ´ n ` 1 ´ 2 L ´ d ` L ´ d ˘ , and using d “ 4 ´ 2 α and κ ă α , we obtain the upper bound E “` p K q ´ n ´ p K ´ 1 q ´ n ˘ 2 p 0 q ‰ “ 2 p 1 ´ L ´ d q L ´ 2 α p K ´ n q ď 2 L ´ κ p K ´ n q . (60) Finally , inserting (60) into (59) yields › › › sup N P N max 0 ď n ď N L ´ κn } p N q ´ n } 8 › › › L p p Ω q À p 8 ÿ K “ 0 L ´ κ { 2 K K ÿ n “ 0 L p d { p ´ κ { 2 q n . The right-hand side is summable, provided p ą 2 d { κ , which concludes (56). The extension to all p P p 1 , 8q is then a consequence of Markov’s inequality . Having the stochastic bound from Lemma 5.2 at hand, we can prove the desired tail estimate. Proof of Lemma 3.3. Let g P p 0 , 1 s . W e apply Chebychev’s exponential inequality , which yields for every c ą 0 P ` Ω c g ˘ ď e ´ 4 cg ´ 2 E ” e c 8 ξ , 8 2 ı . Next, using Fubini’s theorem and Lemma 5.2 (recall that the implicit constant is independent of p ) E ” e c 8 ξ , 8 2 ı À 8 ÿ p “ 0 p p p ! p 2 c q p À 8 ÿ p “ 0 p 2 c ¨ e q p , where we used Stirling’s approximation in the last inequality . The series converges, provided we choose c small enough. 5.3 Fix ed point problem In this section, we will prove bounds for the remainder term Ψ p N q ´ n , uniformly in n and N . W e ﬁx a renormalised coupling constant g P p 0 , 1 s , alongside a renormalised mass r ě 1 . The estimates on the remainder Ψ will hold conditionally on Ω g . Lemma 5.3. Let d “ 2 . Then for every r ě 1 and large enough, odd L P N (depending only on r ), we have that for every g P p 0 , 1 s max 0 ď n ď N L ´ κ s n sup x P Λ n | Ψ p N q ´ n p x q| 1 { 3 ď p 2 g q ´ 1 , on Ω g , (61) where κ s ą 0 is the same as in the deﬁnition of | | | ‚ | | | in (29) . Proof . As a preliminary step, note that we have control of the effective mass. Indeed, recalling the expression of the effective mass in (46) and using λ ´ n “ L ´ αn g with g ď 1 , we obtain | µ ´ n | ď L p´ 2 ` κ s q n p r ` c q , (62) L . F E R D I N A N D A N D S . G A B R I E L 24 for c def “ sup n P N n 2 ´ κ s n which is independent of n , N and L . Again, it will be convenient to abbreviate all supremum norms by simply writing } f } 8 def “ sup x P Λ k | f p x q| @ k P N , f P C p Λ k q , leaving the dependency of the corresponding lattice implicit. Therefore, on Ω g (31), we observe that λ ´ n } ξ ´ n } 8 ď 1 2 L p´ α ` κ s q n and λ 2 ´ n } ´ n } 8 ď 1 4 L 2 p´ α ` κ s q n . (63) W e prove the uniform bound (61) by induction. F or this, we introduce | | | ξ , | | | n def “ sup k ě n L ´ κ s k ´ sup x P Λ k | ξ ´ k p x q| _ sup x P Λ k | ´ k p x q| 1 { 2 ¯ , (64) such that in particular | | | ‚ | | | 0 ” | | | ‚ | | | . Now , let N P N be arbitrary and note that Ψ p N q ´ N ” 0 . W e will construct the trajectory Ψ p N q ´ n for n ď N by induction, to which end we assume that for some 0 ď n ă N it holds that max n ď k ď N L ´ 3 κ s k sup x P Λ k | Ψ p N q ´ k p x q| ď p 2 g q ´ 1 | | | ξ p N q , p N q | | | 2 n ` 1 . (65) Note that our induction hypothesis is stronger than (61) (recall that on the event Ω g , | | | ξ p N q , p N q | | | 2 n ` 1 is indeed bounded by p 2 g q ´ 2 ). This will be convenient when proving convergence of Ψ p N q as the cutoff N is removed in Section 6. In the remainder of the proof we will drop the superscripts that indicate the dependency on N . Before proceeding, we recall the discussion at the bottom of R emark 3.1 about the triangular structure of the system (24): the remainder Ψ ´ n ` 1 in (54) is not determined solely by Ψ ´ n but also depends on the ﬂuctuation ﬁeld R ´ n ` 1 . Accordingly , each induction step is divided into two parts: we ﬁrst estimate R ´ n ` 1 via a ﬁxed–point problem, and then derive the desired bound (65) for Ψ ´ n ` 1 . 1. Remainder ﬂuctuation ﬁeld. Using (52) together with the fact that | P 1 f | ď 2 } f } 8 , we have } R ´ n ` 1 } 8 ď 2 ` λ ´ n } ξ ´ n } 8 ` λ 2 ´ n } ´ n } 8 ` λ 3 ´ n } Ψ ´ n } 8 ` µ ´ n ˘ } R ´ n ` 1 } 8 ` 4 ´ } ´ n } 8 ` } ξ ´ n } 2 ` λ ´ n } Ψ ´ n } 8 ` λ ´ n } ´ n } 8 } ξ ´ n } 8 ` λ 2 ´ n } Ψ ´ n } 8 } ξ ´ n } 8 ` µ ´ n λ ´ 1 ´ n } ξ ´ n } 8 ¯ . (66) Now , using the estimates (62), (63), and (65), we can upper bound the ﬁrst term on the right- hand side of (66): 2 ` λ ´ n } ξ ´ n } 8 ` λ 2 ´ n } ´ n } 8 ` λ 3 ´ n } Ψ ´ n } 8 ` µ ´ n ˘ } R ´ n ` 1 } 8 ď 2 ` 1 2 L p´ α ` κ s q n ` 1 4 L 2 p´ α ` κ s q n ` 1 8 L 3 p´ α ` κ s q n ` L p´ 2 ` κ s q n p r ` c q ˘ } R ´ n ` 1 } 8 ď p 1 ` 3 L ´ 1 { 2 C r q L p´ α ` κ s q n } R ´ n ` 1 } 8 ď p 1 ` 3 L ´ 1 { 2 C r q L ´ 1 { 2 } R ´ n ` 1 } 8 , where we introduced the notation C r def “ p 1 ` r ` c q ą 1 , used the induction hypothesis for Ψ ´ n in the ﬁrst inequality , and crude estimates such as p´ α ` κ s q n ď ´ 1 { 2 . Along the same lines, we can control the second terms on the right–hand side of (66): 4 ´ } ´ n } 8 ` } ξ ´ n } 2 ` λ ´ n } Ψ ´ n } 8 ` ` λ ´ n } ´ n } 8 ` λ 2 ´ n } Ψ ´ n } 8 ` µ ´ n λ ´ 1 ´ n ˘ } ξ ´ n } 8 ¯ D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 25 ď 2 ´ 2 L 2 κ s n ` 2 L p´ α ` 3 κ s q n ` L p´ 2 α ` 4 κ s q n ` L p´ 2 ` α ` 2 κ s q n p r ` c q ¯ g ´ 1 | | | ξ p N q , p N q | | | n ď 4 p 1 ` 4 L ´ 1 { 2 C r q L 2 κ s n g ´ 1 | | | ξ p N q , p N q | | | n , and ﬁnally arrive at } R ´ n ` 1 } 8 ď p 1 ` 3 L ´ 1 { 2 C r q L ´ 1 { 2 } R ´ n ` 1 } 8 ` 4 p 1 ` 4 L ´ 1 { 2 C r q L 2 κ s n g ´ 1 | | | ξ p N q , p N q | | | n ď 1 2 } R ´ n ` 1 } 8 ` 8 L 2 κ s n g ´ 1 | | | ξ p N q , p N q | | | n ď 16 L 2 κ s n g ´ 1 | | | ξ p N q , p N q | | | n , (67) where again we used the deﬁnition of C r and then took L large enough depending on r . In particular , R ´ n ` 1 only depends on the randomness at scales lower or equal to L ´ n . 2. Remainder effective force. W e perform the same type of estimate for Ψ ´ n ` 1 : From (54), we see that } Ψ ´ n ` 1 } 8 ď L 2 ´ 3 α ` 1 ` 2 λ ´ n } ξ ´ n } 8 ` λ 2 ´ n } R ´ n ` 1 } 8 ˘ } Ψ ´ n } 8 ` L 2 ´ 3 α ` 2 } ´ n } 8 } ξ ´ n } 8 ` } ξ ´ n } 8 } R ´ n ` 1 } 8 ` λ ´ n } ´ n } 8 } R ´ n ` 1 } 8 ˘ , bl abl abl (68) where we again used } P 1 f } 8 ď 2 } f } 8 . Inserting (63), (65) and (67), we deduce the upper bounds ` 1 ` 2 λ ´ n } ξ ´ n } 8 ` λ 2 ´ n } R ´ n ` 1 } 8 ˘ } Ψ ´ n } 8 ď L 3 κ s n g ´ 1 | | | ξ p N q , p N q | | | 2 n (69) and ` 2 } ´ n } 8 } ξ ´ n } 8 ` } ξ ´ n } 8 } R ´ n ` 1 } 8 ` λ ´ n } ´ n } 8 } R ´ n ` 1 } 8 ˘ ď 10 L 3 κ s n g ´ 1 | | | ξ p N q , p N q | | | 2 n . (70) Inserting (69) and (70) into (68) then yields } Ψ ´ n ` 1 } 8 ď ` 11 L 2 ´ 3 α ˘ L 3 κ s n g ´ 1 | | | ξ p N q , p N q | | | 2 n . (71) Once more, we may choose L large enough, such that 11 L 2 ´ 3 α ď 2 ´ 1 . Overall, this extends (65) to n ´ 1 ď k ď N and concludes the induction step. 5.4 Control of the renormalised solution In the previous section, we proved a uniform bound (in n and N ) of the remainder term Ψ p N q ´ n in the effective equation (32). This provides the last ingredient to control the coefﬁcients in the effective equation on the event Ω g , where the enhanced noise p ξ p N q , p N q q remains controlled by g . It is only left to push this control from the coefﬁcients onto the solution v ´ n , and derive that sup N P N ´ | v p N q 0 | _ max 1 ď n ď N L ´ κn sup x P Λ n ˇ ˇ P 1 v p N q ´ n p x q| ¯ ă 8 . The key ingredient of this control is an inductive scheme backwards along the ﬂow , which we present next. This is in contrast to the previous section, where we propagated bounds forward along the ﬂow . Lemma 5.4. Let d “ 2 . Then for every r ě 1 and large enough, odd L P N (depending only on r ), we have that for every g P p 0 , 1 s it holds that sup N P N ´ | v p N q 0 | _ max 1 ď n ď N sup x P Λ n ´ L ´p α ` κ s q n | Q 1 v p N q ´ n p x q| _ L ´ 3 κ s n | P 1 v p N q ´ n p x q| ¯¯ ď 8 , on Ω g . (72) L . F E R D I N A N D A N D S . G A B R I E L 26 Before turning to the proof , let us brieﬂy explain the strategy and the origin of the estimate (72). R ecall that v 0 is simply a scalar , determined by the algebraic equation (32) for n “ 0 , where the Laplacian term vanishes. It is therefore the ﬁrst object to be deﬁned. Our goal is then to construct the family p v ´ n q n ě 0 recursively by induction over n , passing from v ´ n to v ´ n ´ 1 . Assuming that v ´ n is already known, we leverage on the fact that Q 1 v ´ n ´ 1 is obtained from v ´ n by a simple rescaling. Consequently , it only remains to control P 1 v ´ n ´ 1 , where we will use (49) and the bound on Q 1 v ´ n ´ 1 . The estimate (72) involves a minor subtlety , arising from the fact that we are solving an equation whose solution has positive regularity . While P 1 v ´ n grows at most like L κn , the term Q 1 v ´ n exhibits a worse behaviour . This phenomenon is analogous to a classical observation: If f has H ¨ older–Besov regularity β ą 0 and p ∆ i q i ě 0 denotes a Littlewood–P aley decomposition, then } ∆ i f } 8 „ 2 ´ β i whereas › › ř i k “ 0 ∆ k f › › 8 „ 1 . Since we expect the solution on T d to have regularity α ´ κ , rescaling to the unit lattice suggests that Q 1 v ´ n should grow like L p α ` κ q n . Proof of Lemma 5.4. Let N P N . W e recall the notation from Section 5.1 and write w ´ n ` 1 “ Q 1 v ´ n and z ´ n ` 1 “ P 1 v ´ n . The idea of the proof is built on the fact that v ´ n “ L α S ´ 1 v ´ n ` 1 ` z ´ n ` 1 . In order to control v ´ n , it sufﬁces to control v ´ n ` 1 and the ﬂuctuations that were averaged out going from the lattice Λ n to the lattice Λ n ´ 1 . First, to control the ﬂuctuations, we make use of (49) which yields } z ´ n ` 1 } 8 ď 2 ` λ ´ n } ξ ´ n } 8 ` λ 2 ´ n } ´ n } 8 ` λ 3 ´ n } Ψ ´ n } 8 ˘ } w ´ n ` 1 } 8 ` 2 ` λ ´ n } ξ ´ n } 8 ` λ 2 ´ n } ´ n } 8 ` λ 3 ´ n } Ψ ´ n } 8 ` µ ´ n ˘ } z ´ n ` 1 } 8 where we again used } P 1 f } 8 ď 2 } f } 8 , as well as the notation } ‚ } 8 for the supremum–norm without specifying the underlying lattice. Once more inserting that λ ´ n “ L ´ αn g , (62), (63) and Lemma 5.3, we arrive at } z ´ n ` 1 } 8 ď ` L p´ α ` κ s q n ` L 2 p´ α ` κ s q n ` L 3 p´ α ` κ s q n ˘` } w ´ n ` 1 } 8 ` } z ´ n ` 1 } 8 ˘ ` 2 L p´ 2 ` κ s q n p r ` c q} z ´ n ` 1 } 8 ď ` 3 L ´ 1 { 2 ` 2 L ´ 1 p r ` c q ˘ } z ´ n ` 1 } 8 ` 2 L p´ α ` κ s q n } Q 1 v ´ n } 8 ď 1 2 } z ´ n ` 1 } 8 ` 2 L p´ α ` κ s q n } Q 1 v ´ n } 8 ď 4 L p´ α ` κ s q n } Q 1 v ´ n } 8 ď p 4 L ´ α ` κ s q n } Q 1 v ´ n } 8 , (73) for every n ě 1 , where again we took L large enough depending on r . Let us now proceed by induction over n . First, we observe that in view of the deﬁnition (10), ∆ H, 0 “ 0 . Therefore, equation (32) becomes (for n “ 0 ) ` g ξ 0 ` g 2 0 ` g 3 Ψ 0 ` r ˘ v 0 ` 1 “ 0 , which is an algebraic equation in v 0 P R . Here we used that λ 0 “ g , µ 0 “ r , and γ 0 “ 1 . Therefore, | v 0 | “ ˇ ˇ g ξ 0 ` g 2 0 ` g 3 Ψ 0 ` r ˇ ˇ ´ 1 ď p r ´ g | ξ 0 | ´ g 2 | 0 | ´ g 3 | Ψ 0 |q ´ 1 ď ` r ´ 1 2 ´ 1 4 ´ 1 8 ˘ ´ 1 ď 8 , where we used (63) and Lemma 5.3 in the second inequality , alongside the fact that r ě 1 . Hence, by deﬁnition of v ´ 1 and (73), we therefore conclude that } Q 1 v ´ 1 } 8 ď L α | v 0 | ď 8 L α and thus } z 0 } 8 ď 32 L κ s . D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 27 W e proceed by induction and assume that } Q 1 v ´ n } 8 ď p 8 L α q n and thus } z ´ n ` 1 } 8 ď p 32 L κ s q n , (74) for some 1 ď n ă N . Then, } Q 1 v ´ n ´ 1 } 8 ď L α } v ´ n } 8 ď L α ` } Q 1 v ´ n } 8 ` } z ´ n ` 1 } 8 ˘ ď 1 8 p 8 L α q n ` 1 ` L α p 32 L κ s q n ď p 8 L α q n ` 1 ` 1 8 ` 1 2 L ´ α ` κ s ˘ ď p 8 L α q n ` 1 , (75) where we used the induction hypothesis in the second line, and in the last inequality that L is large enough (independent of N ). T o conclude the induction step, it only remains to control the ﬂuctuation term } z ´ n } 8 , for which we use (73) and (75), which yields } P 1 v ´ n ´ 1 } 8 “ } z ´ n } 8 ď p 4 L ´ α ` κ s q n ` 1 } Q 1 v ´ n ´ 1 } 8 ď p 32 L κ s q n ` 1 . Finally , taking L large enough, we can enforce that 8 ď L κ s . This concludes the proof . 6 Convergence in the UV limit In the previous section, we proved control of the RG ﬂow of the enhanced noise, the remainder Ψ , and the effective solution. F ollowing, the same steps as in the proofs of Lemmas 5.2, 5.3, and 5.4, we can show that the sequences are in fact Cauchy , and thus converge. This extension is standard for readers familiar with subcritical SPDEs and may be skipped. However , we provide the complete argument in this section for any reader that is less familiar with the topic. 6.1 Convergence of the effective force First, we show that the enhanced noise converges P –almost surely in the UV limit. Lemma 6.1 (Convergence of the enhanced noise) . Let α ą 0 , i.e. d ă 4 , and L P N . Then for every g P p 0 , 8q the following limit exists P –almost surely lim N Ñ8 ` ξ p N q , p N q ˘ “ p ξ ´ n , ‹ ´ n q n P N , in the topology induced by | | | ‚ | | | . Proof . It is not difﬁcult to check that p ξ p N q q N P N converges almost surely to p ξ ´ n q n P N in | | | ‚ | | | . T o con- clude convergence of the enhanced noise, it is therefore only left to show that p p N q q N P N forms a Cauchy sequence. More precisely , we prove that P –almost surely sup M ,N P N L κ { 4 N max 0 ď n ď N ` M L ´ κn sup x P Λ n | p N ` M q ´ n ´ p N q ´ n |p x q ă 8 , (76) where κ : “ 2 κ s for convenience. The proof carries over almost verbatim from the proof of Lemma 5.2, however , we include the main steps for a complete account. Again, we start from the telescope representation p N ` M q ´ n ´ p N q ´ n “ N ` M ÿ K “p N _ n q` 1 ` p K q ´ n ´ p K ´ 1 q ´ n ˘ , L . F E R D I N A N D A N D S . G A B R I E L 28 which allows us to crudely replace suprema with sums (as we did in (57)), such that sup M ,N P N L κ { 4 N max 0 ď n ď N ` M L ´ κn } p N ` M q ´ n ´ p N q ´ n } 8 ď sup M P N 8 ÿ N “ 0 L κ { 4 N N ` M ÿ n “ 0 L ´ κn N ` M ÿ K “p N _ n q` 1 } p K q ´ n ´ p K ´ 1 q ´ n } 8 ď 8 ÿ N “ 0 L κ { 4 N 8 ÿ K “ N ` 1 K ÿ n “ 0 L ´ κn } p K q ´ n ´ p K ´ 1 q ´ n } 8 . Thus, in complete analogy to (58), we have › › › sup M ,N P N L κ { 4 N max 0 ď n ď N ` M L ´ κn } p N ` M q ´ n ´ p N q ´ n } 8 › › › L p p Ω q ď 8 ÿ N “ 0 L κ { 4 N 8 ÿ K “ N ` 1 K ÿ n “ 0 L ´ κn › › › sup x P Λ n | p K q ´ n ´ p K ´ 1 q ´ n |p x q › › › L p p Ω q ď 8 ÿ N “ 0 L κ { 4 N 8 ÿ K “ N ` 1 K ÿ n “ 0 L p d { p ´ κ q n E “ ˇ ˇ p K q ´ n ´ p K ´ 1 q ´ n ˇ ˇ p p 0 q ‰ 1 { p . Applying additionally hypercontractivity in combination with (60), we conclude › › › sup M ,N P N L κ { 4 N max 0 ď n ď N ` M L ´ κn } p N ` M q ´ n ´ p N q ´ n } 8 › › › L p p Ω q À p 8 ÿ N “ 0 L κ { 4 N 8 ÿ K “ N ` 1 K ÿ n “ 0 L p d { p ´ κ q n L ´ κ { 2 p K ´ n q ď p 8 ÿ N “ 0 L ´ κ { 4 N 8 ÿ K “ 1 L ´ κ { 2 K 8 ÿ n “ 0 L p d { p ´ κ { 2 q n À p , where we performed an index shift ( K ÞÑ K ´ N ) in the second inequality , and assumed that p is large enough such that the most inner sum converges. This moment estimate concludes in particular that (76) is ﬁnite, P –almost surely . Likewise, the remainder Ψ p N q forms a Cauchy sequence on Ω g , and converges. Lemma 6.2 (Convergence of the remainder) . Let d “ 2 . F or every r ě 1 , there exists an odd L P N , sufﬁciently large (and depending only on r ), such that for every g P p 0 , 1 s the limit Ψ ‹ “ p Ψ ‹ ´ n q n P N exists on Ω g , such that lim N Ñ8 max n ě 0 L ´ κ s n sup x P Λ n | Ψ p N q ´ n p x q ´ Ψ ‹ ´ n p x q| 1 { 3 “ 0 , on Ω g , where κ s ą 0 is the same as in (29) . Here we set Ψ p N q ´ n ” 0 for all n ą N . Proof . In the following, we assume to be on the good event Ω g . Again, we will prove that p Ψ p N q ´ n q n P N is Cauchy . More precisely , we show that max 0 ď n ď N ` M L ´ κ s n sup x P Λ n | Ψ p N ` M q ´ n ´ Ψ p N q ´ n | 1 { 3 p x q ď p 2 g q ´ 1 { 3 | | |p ξ p N ` M q , p N ` M q q ´ p ξ p N q , p N q q| | | 2 { 3 , (77) cf . Lemma 5.3. Notice that by Lemma 6.1, the right–hand side vanishes on Ω g , as N Ñ 8 . W e follow the same steps as in the proof of Lemma 5.3, using the norm | | | ‚ | | | n from (64) which only considers scales above n . First, we assume that the following hypothesis is true max n ď k ď N ` M L ´ 3 κ s k sup x P Λ k | Ψ p N ` M q ´ k p x q ´ Ψ p N q ´ k p x q| ď p 2 g q ´ 1 | | |p ξ p N ` M q , p N ` M q q ´ p ξ p N q , p N q q| | | 2 n ` 1 . for some 0 ď n ă N . Notice that for N ď n ď N ` M the statement reduces to (65), because in this case Ψ p N q ´ n ” 0 and p ξ p N q ´ k , p N q ´ k q ” 0 for every k ě n ` 1 . W e proceed inductively . D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 29 1. Remainder ﬂuctuation ﬁeld. Using again (52) together with the fact that | P 1 f | ď 2 } f } 8 , yields (after adding and subtracting multiple zero terms and applying the triangle inequality) } R p N ` M q ´ n ` 1 ´ R p N q ´ n ` 1 } 8 ď 2 ` λ ´ n } ξ p N q ´ n } 8 ` λ 2 ´ n } p N q ´ n } 8 ` λ 3 ´ n } Ψ p N q ´ n } 8 ` µ p N q ´ n ˘ } R p N ` M q ´ n ` 1 ´ R p N q ´ n ` 1 } 8 ` 2 ´ λ 2 ´ n } p N ` M q ´ n ´ p N q ´ n } 8 ` λ 3 ´ n } Ψ p N ` M q ´ n ´ Ψ p N q ´ n } 8 ¯ } R p N ` M q ´ n ` 1 } 8 ` 4 ´ } p N ` M q ´ n ´ p N q ´ n } 8 ` λ ´ n } Ψ p N ` M q ´ n ´ Ψ p N q ´ n } 8 ` λ ´ n } p N ` M q ´ n ´ p N q ´ n } 8 } ξ ´ n } 8 ` λ 2 ´ n } Ψ p N ` M q ´ n ´ Ψ p N q ´ n } 8 } ξ ´ n } 8 ¯ . Because each term can be upper bounded using the deﬁnition of Ω g , (65), or the induction hypothesis, we see that } R p N ` M q ´ n ` 1 ´ R p N q ´ n ` 1 } 8 ď 1 2 } R p N ` M q ´ n ` 1 ´ R p N q ´ n ` 1 } 8 ` 8 L 2 κ s n g ´ 1 | | |p ξ p N ` M q , p N ` M q q ´ p ξ p N q , p N q q| | | n ď 16 L 2 κ s n g ´ 1 | | |p ξ p N ` M q , p N ` M q q ´ p ξ p N q , p N q q| | | n , (78) which follows verbatim from the steps performed in (67). 2. Remainder effective force. Likewise, we can estimate the difference of the Ψ –terms using (54): } Ψ p N ` M q ´ n ` 1 ´ Ψ p N q ´ n ` 1 } 8 ď L 2 ´ 3 α ` 1 ` 2 λ ´ n } ξ ´ n } 8 ` λ 2 ´ n } R p N q ´ n ` 1 } 8 ˘ } Ψ p N ` M q ´ n ´ Ψ p N q ´ n } 8 ` L 2 ´ 3 α λ 2 ´ n } R p N ` M q ´ n ` 1 ´ R p N q ´ n ` 1 } 8 } Ψ p N ` M q ´ n } 8 ` L 2 ´ 3 α ` 2 } p N ` M q ´ n ´ p N q ´ n } 8 } ξ ´ n } 8 ` } ξ ´ n } 8 } R p M ` N q ´ n ` 1 ´ R p N q ´ n ` 1 } 8 ˘ ` L 2 ´ 3 α ` λ ´ n } p N ` M q ´ n ´ p N q ´ n } 8 } R p N q ´ n ` 1 } 8 ` λ ´ n } p N ` M q ´ n } 8 } R p N ` M q ´ n ` 1 ´ R p N q ´ n ` 1 } 8 ˘ . Therefore, using the induction hypothesis for increments of Ψ p N q ´ n , the upper bound (78), as well as the deﬁnition of Ω g and (65), we have extended the induction hypothesis to Ψ p N q ´ n ` 1 . This concludes the proof . 6.2 Convergence of the solution and proof of Proposition 3.5 Next, we conclude that also the trajectories p v p N q ´ n q n ď N form a Cauchy sequence. T o this end, recall that w p N q ´ n ` 1 “ Q 1 v p N q ´ n and z p N q ´ n ` 1 “ P 1 v p N q ´ n . Therefore, it sufﬁces to prove sup M ,N P N L κ { 4 N ´ max 1 ď n ď N ` M L ´ κn sup x P Λ n ˇ ˇ ` z p N ` M q ´ n ` 1 ´ z p N q ´ n ` 1 ˘ p x q ˇ ˇ ¯ ă 8 , (79) with κ “ 3 κ s . As we have done so far , we also follow here the proof of the analogue in Section 5, namely the proof of Lemma 5.4, to control the differences of the two ﬂuctuation ﬁelds. W e avoid repetition and keep the argument short. L . F E R D I N A N D A N D S . G A B R I E L 30 Proof of Lemma 3.6. First, by means of (49) and } P 1 f } 8 ď 2 } f } 8 , we have } z p N ` M q ´ n ` 1 ´ z p N q ´ n ` 1 } 8 ď 2 ` λ ´ n } ξ ´ n } 8 ` λ 2 ´ n } p N q ´ n } 8 ` λ 3 ´ n } Ψ p N q ´ n } 8 ` µ ´ n ˘` } z p N ` M q ´ n ` 1 ´ z p N q ´ n ` 1 } 8 ` } w p N ` M q ´ n ` 1 ´ w p N q ´ n ` 1 } 8 ˘ ` µ ´ n } z p N ` M q ´ n ` 1 ´ z p N q ´ n ` 1 } 8 ` 2 ` λ 2 ´ n } p N ` M q ´ n ´ p N q ´ n } 8 ` λ 3 ´ n } Ψ p N ` M q ´ n ´ Ψ p N q ´ n } 8 ˘` } z p N ` M q ´ n ` 1 } 8 ` } w p N ` M q ´ n ` 1 } 8 ˘ . Using Lemmas 5.2, 5.3 and 5.4, and rearranging the terms appropriately , yields } z p N ` M q ´ n ` 1 ´ z p N q ´ n ` 1 } 8 ď 2 ´ 4 L p´ α ` κ s q n } w p N ` M q ´ n ` 1 ´ w p N q ´ n ` 1 } 8 ` 16 L p´ α ` 3 κ s q n g 2 ` } p N ` M q ´ n ´ p N q ´ n } 8 ` λ ´ n } Ψ p N ` M q ´ n ´ Ψ p N q ´ n } 8 ˘ ¯ . Notice that the second line can be bounded by an arbitrary small constant for N large enough, see Lemmas 6.1 and 6.2. Thus, we have an estimate analogous to (73). The remainder of the proof follows by the same induction argument as for Lemma 5.4. Finally , we can summarise our ﬁndings in the following proof: Proof of Proposition 3.5. R estricting ourselves to the event Ω g , we proved that v p N q converges to v ‹ “ p v ‹ ´ n q n P N , see Lemma 3.6. Moreover , we established convergence of the force coefﬁcients lim N Ñ8 p N q ´ n “ ‹ ´ n , and lim N Ñ8 Ψ p N q ´ n “ Ψ ‹ ´ n , in Lemmas 6.1 and 6.2. It is only left to notice that v ‹ ´ n solves indeed the limiting equation p´ ∆ H q v ‹ ´ n “ ` λ ´ n ξ ´ n ` λ 2 ´ n ‹ ´ n ` λ 3 ´ n Ψ ‹ ´ n ` µ ´ n ˘ v ‹ ´ n ` γ ´ n , since the equation can be written in terms of ﬁnite linear combinations of all quantities involved. Notice that the constants λ ´ n , µ ´ n , and γ ´ n were independent of N . Lastly , we note that v p N q ´ n “ L ´ α p m ´ n q S m ´ n Q p N q m ´ n v p N q ´ m for every n ă m ď N , which remains true when taking limits. A Moment calculations In this appendix, we collect several moment calculations which are useful throughout the paper . Lemma A.1. Let N P N , then for every n ă N E “ Q N ´ n p ξ ´ N Γ N ´ n ξ ´ N qp x q ‰ “ p 1 ´ L ´ d q N ´ n ÿ k “ 1 L p 2 ´ d qp k ´ 1 q , @ x P Λ N N ´ n , where Γ N ´ n was deﬁned in (22) . In particular , E “ Q 1 p ξ ´ n P 1 ξ ´ n q ‰ “ p 1 ´ L ´ d q . Proof . First, we notice that due to spatial homogeneity (in law), for every x P Λ N N ´ n E “ Q N ´ n p ξ ´ N Γ N ´ n ξ ´ N qp x q ‰ “ E “ ξ ´ N p 0 q Γ N ´ n ξ ´ N p 0 q ‰ “ N ´ n ÿ k “ 1 L 2 p k ´ 1 q E r ξ ´ N p 0 q P k ξ ´ N p 0 qs . Next, we observe that E r ξ ´ N p 0 q P k ξ ´ N p 0 qs “ E r ξ ´ N p 0 qp Q k ´ 1 ´ Q k q ξ ´ N p 0 qs “ L ´ d p k ´ 1 q p 1 ´ L ´ d q , (80) D I S C R E T E R G F L O W O F A H I E R A R C H I C A L S I N G U L A R S P D E 31 where we used that ξ ´ N is a ﬁeld of independent unit variance random variables, which implies E r ξ ´ N p 0 q Q k ξ ´ N p 0 qs “ L ´ dk ÿ x P ˝ k p 0 q E r ξ ´ N p 0 q ξ ´ N p x qs “ L ´ dk . (81) R ecall that ˝ k p 0 q denotes the block of side–length L k centred at 0 . Combining (80) and (81) yields E “ Q N ´ n p ξ ´ N Γ N ´ n ξ ´ N qp x q ‰ “ p 1 ´ L ´ d q N ´ n ÿ k “ 1 L p 2 ´ d qp k ´ 1 q , which concludes the ﬁrst statement. Lastly , we notice that (80) implies the second statement of the lemma by choosing k “ 1 and N “ n . Lemma A.2. F or every n P N and x P Λ n , we have E “ P 1 ξ ´ n p x q P 1 ξ ´ n p 0 q ‰ “ # δ x, 0 ´ L ´ d , if x P ˝ 1 p 0 q , 0 , otherwise . Proof . Clearly , if x and 0 do not lie in the same box of side–length L , i.e. x R ˝ 1 p 0 q , then the two random variables are independent. On the other hand, if x P ˝ 1 p 0 q , then ˝ 1 p x q “ ˝ 1 p 0 q and E “ P 1 ξ ´ n p x q P 1 ξ ´ n p 0 q ‰ “ E „ ´ L ´ d ÿ y P ˝ 1 p 0 qX Λ n ξ ´ n p y q ´ ξ ´ n p x q ¯´ L ´ d ÿ z P ˝ 1 p 0 qX Λ n ξ ´ n p z q ´ ξ ´ n p 0 q ¯ ȷ “ L ´ 2 d ÿ y ,z P ˝ 1 p 0 qX Λ n E r ξ ´ n p y q ξ ´ n p z qs ` E r ξ ´ n p x q ξ ´ n p 0 qs ´ 2 L ´ d ÿ y P ˝ 1 p 0 qX Λ n E r ξ ´ n p y q ξ ´ n p 0 qs “ L ´ d ` δ x, 0 ´ 2 L ´ d “ δ x, 0 ´ L ´ d , where we simply used that p ξ ´ n p x qq x P Λ n are independent standard normal random variables. Corollary A.3. F or every n P N and x P Λ n , E “ p P 1 ξ ´ n q ˛ 2 p x qp P 1 ξ ´ n q ˛ 2 p 0 q ‰ “ # 2 p δ x, 0 ´ L ´ d q 2 , if x P ˝ 1 p 0 q , 0 , otherwise . (82) Proof . The statement is an immediate consequence of the deﬁnition of the Wick ordering: E “ p P 1 ξ ´ n q ˛ 2 p x qp P 1 ξ ´ n q ˛ 2 p 0 q ‰ “ 2 ` E “ p P 1 ξ ´ n qp x qp P 1 ξ ´ n qp 0 q ‰˘ 2 , and Lemma A.2. R eferences [AC15] R. Allez and K. Chouk. The continuous Anderson hamiltonian in dimension two. arXiv preprint arXiv:1511.02718 , 2015. [BBS19] R. Bauerschmidt, D.C. Brydges, and G. Slade. Introduction to a renormalisation group method , volume 2242 of Lecture Notes in Mathematics . Springer , Singapore, 2019. [BCZ23] L. Broux, F . Caravenna, and L. Zambotti. An example of singular elliptic stochastic PDE. Mat. Contemp. , 58:3–67, 2023. L . F E R D I N A N D A N D S . G A B R I E L 32 [Duc25a] P . Duch. Flow equation approach to singular stochastic PDEs. Probab. Math. Phys. , 6(2):327–437, 2025. [Duc25b] P . Duch. R enormalization of singular elliptic stochastic PDEs using ﬂow equation. Probab. Math. Phys. , 6(1):111–138, 2025. [GIP15] M. Gubinelli, P . Imkeller , and N. P erkowski. P aracontrolled distributions and singular PDEs. F orum Math. Pi , 3:e6, 75, 2015. [GR26] S. Gabriel and T. R osati. Fluctuations in the weakly coupled 4D Anderson Hamiltonian. arXiv preprint arXiv:2602.22509 , 2026. [Hai14] M. Hairer . A theory of regularity structures. Inventiones mathematicae , 198(2):269–504, Nov 2014. [HL15] M. Hairer and C. Labb ´ e. A simple construction of the continuum parabolic Anderson model on R 2 . Electron. Commun. Probab. , 20:no. 43, 11, 2015. [KM17] A. K upiainen and M. Marcozzi. Renormalization of generalized KPZ equation. J. Stat. Phys. , 166(3-4):876–902, 2017. [Kup16] A. K upiainen. Renormalization group and stochastic PDEs. Ann. Henri P oincar ´ e , 17(3):497– 535, 2016. [Lab19] C. Labb ´ e. The continuous Anderson Hamiltonian in d ď 3 . J. Funct. Anal. , 277(9):3187– 3235, 2019.

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