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REVIEW 3 major objections 4 minor 92 references

Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper recasts exact renormalization group flows as stochastic coarsening processes, preserving the distribution of the empirical magnetization by construction.

desk verdict Solid pedagogical review of stochastic RG; the new finite-volume scheme conserves, doesn't compute, p_L(m_e) despite the abstract's promise. read the letter →

arxiv 2502.09506 v3 pith:A7RALZZR submitted 2025-02-13 cond-mat.stat-mech math.PR

Cecile Monthus This is my paper · ORCID
classification cond-mat.stat-mechmath.PR MSC 82B2860J6060H1582C31 PACS 05.10.Gg05.40.-a64.60.-i
keywords exactrenormalizationgroupFokker-PlanckequationOrnstein-UhlenbeckprocessLangevinstochasticdifferentialcoarseningempiricalmagnetizationlargedeviationsEdwards-Wilkinsondynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that Wilson's exact renormalization group of 'incomplete integration' is, for the probability distribution of a field, a linear Fokker-Planck flow: every Fourier mode evolves as an independent Ornstein-Uhlenbeck process. Building on the Carosso interpretation, it reads the associated Langevin equation—the stochastic heat equation, also known as Edwards-Wilkinson dynamics—as a genuine infinitesimal coarse-graining of the field, with the noise representing the information discarded at each RG step. On a finite volume with periodic boundary conditions, the paper freezes the empirical magnetization (the zero-momentum Fourier coefficient) and adapts the Carosso RG to the remaining non-zero modes. It derives explicit RG flows for the intensive energy and for the generating function of the Fourier coefficients, so that the probability distribution of the empirical magnetization is conserved by construction and is obtained by gradually integrating out all non-zero momentum modes. If correct, this provides a stochastic-process representation of exact RG in which large-deviation properties of the magnetization emerge from a diffusion-like flow instead of a single high-dimensional integral.

What carries the argument

The central object is the Ornstein-Uhlenbeck process for each Fourier coefficient of the field, with drift coefficient $-\omega_L(k)\hat\phi$ and diffusion coefficient $D/(2L^d)$. The Laplacian eigenvalues $\omega_L(k)=4\pi^2 k^2/L^2$ provide the momentum-dependent damping; the zero-momentum mode is frozen by imposing the global constraint $\int \eta_t(x)\,dx=0$ on the noise, which removes the zero-momentum noise component. This turns the exact RG flow into a diffusion process in the space of non-zero Fourier modes: the Fokker-Planck generator is separable, the finite-time propagator is a product of Gaussian kernels, and averaging over the noise realizes the gradual integration over the eliminated modes. The same machinery yields explicit flows for the intensive energy (through the transformation $P_t=\exp(-L^d E_t)$) and for the generating function of the Fourier coefficients, making the conservation of $p_L(m_e)$ an identity at every RG time.

What would settle it

Run the Carosso flow for a simple interacting theory such as $\phi^4$ on a small finite-volume lattice and compare the exact marginal $p_L(m_e)$ with the marginal obtained by integrating the flow to $t=\infty$ with the assumed white noise. If the two disagree, the noise statistics are not those generated by the eliminated modes and the stochastic RG must be modified.

Watch

Extended reading notes

Core claim

The central claim is that the exact renormalization group of Wilson's incomplete integration is equivalent to a linear Fokker-Planck evolution for the renormalized probability distribution, in which the Fourier coefficients of the field undergo independent, non-identical Ornstein-Uhlenbeck processes. The corresponding Langevin stochastic differential equations—for the Carosso scheme, exactly the stochastic heat equation (Edwards-Wilkinson dynamics)—are not a mere change of variables in the partition function but genuine infinitesimal coarsening transformations, the continuous analog of block spins; the noise carries the information lost when modes are integrated out. On a finite volume $L^d$ with periodic boundary conditions, the paper imposes the constraint that the empirical magnetization $m_e$ remains frozen, so the zero-momentum coefficient is untouched and the RG flow acts only on the non-zero Fourier modes. The paper derives the resulting flows explicitly: the intensive energy $E_t$ satisfies the non-linear flow of Eq. 157, with its leading large-volume form in Eq. 158, and the generating function $W_t$ satisfies the linear flow of Eq. 166 with the finite-time expression of Eq. 163. These flows interpolate between the initial model and a factorized Gaussian steady state in which all non-zero modes have been integrated out and the only surviving information is the distribution $p_L(m_e)$ of the empirical magnetization itself.

Load-bearing premise

The load-bearing premise is that the Gaussian white noise in the Langevin equation faithfully represents the information lost when high-momentum modes are integrated out; the paper assumes this noise form rather than deriving it from the model's interactions.

Editorial extensions

If this is right

  • The probability distribution of the empirical magnetization $p_L(m_e)$ can be obtained by running the Carosso flow until all non-zero Fourier modes reach their Gaussian steady state; the marginal at $t=\infty$ is exactly $p_L(m_e)$, so the large-deviation rate function $i(m_e)$ follows from the stationary part of the flow.
  • At leading order in $1/L^d$, the RG flow for the intensive energy becomes a first-order Hamilton-Jacobi equation, so the rate function is obtained by optimizing the initial energy over the non-zero modes at fixed $m_e$.
  • The generating function of the Fourier coefficients obeys a linear first-order flow, meaning the scaled cumulants of the non-zero modes can be propagated exactly in time once the initial generating function is known.
  • The Markov property of the Ornstein-Uhlenbeck process gives a Chapman-Kolmogorov composition law for successive RG steps, so the whole flow can be built from elementary infinitesimal coarsening transformations—the continuous counterpart of iterative block-spin decimation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper postulates Gaussian white noise rather than deriving it from the model's interactions; for a small non-Gaussian model (e.g., $\phi^4$ theory on a finite lattice), one could measure the noise statistics produced by integrating out a thin momentum shell and check consistency with the assumed delta-correlated white noise.
  • The same construction should extend to freezing several low-lying Fourier coefficients, giving joint large-deviation functions for a set of intensive observables rather than only the zero mode.
  • Because the flow is linear in the generating function, it may be possible to reinterpret exact RG as a diffusion process whose stationary measure is the model's Gibbs measure, connecting to sampling schemes and generative diffusion models—connections the paper notes but does not develop.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper is a pedagogical review of stochastic formulations of exact renormalization group flows: the Wilson-Kogut "incomplete integration" scheme as a linear Fokker-Planck flow for independent Ornstein-Uhlenbeck modes, the Wegner-Morris continuity-equation perspective, and the Carosso interpretation of the associated Langevin dynamics as genuine field coarsening. After this review, the paper adapts the Carosso RG to a finite volume L^d with periodic boundary conditions, freezes the empirical magnetization m_e, imposes a zero-mean constraint on the noise, and derives explicit Langevin/Fokker-Planck equations for the nonzero Fourier modes, an effective-energy flow (Eq. 157), a finite-time saddle-point solution (Eq. 160), and a generating-function flow (Eq. 163). The abstract and conclusion claim that p_L(m_e) can thereby be obtained by gradual integration over all nonzero-momentum Fourier coefficients, yielding the large-deviation rate function i(m_e).

Significance. The pedagogical part of the manuscript is clearly written and largely internally consistent; the explicit propagators, Fokker-Planck equations, and finite-time flow equations (Eqs. 144, 147, 157, 160, 163) are useful reference results and the connections among Wilson, Wegner-Morris, and Carosso formulations are presented accessibly. The advertised new application, however, is not delivered: in Section VII the distribution p_L(m_e) is conserved by construction rather than computed by a gradual integration, and the derivation of the noise statistics from the underlying interacting model is explicitly deferred. The paper is therefore best assessed as a valuable exposition and a proposal for a class of conditional stochastic RG flows, not as a demonstration that the Carosso RG computes the large-deviation properties of p_L(m_e).

major comments (3)
  1. [Section VII F, Eqs. (130), (134), (153)] The central claim of the abstract and of Section VIII is that p_L(m_e) can be obtained by "gradual integration" over the nonzero-momentum Fourier coefficients via the Carosso stochastic RG. In Section VII F, however, the conservation of p_L[m_e] is imposed by construction: the zero-momentum mode is frozen (Eq. 130), the noise is constrained to have zero spatial integral (Eq. 134), and each Ornstein-Uhlenbeck propagator integrates to unity over the final nonzero Fourier coefficients for every t, giving Eq. 153 as an identity. Consequently, the method does not compute p_L(m_e) or the rate function i(m_e); Eq. 58 remains the original constrained minimization of E0, and Eq. 161 reduces at t = +infinity to min_{phi0} E0[m_e; phi0]. The advertised incremental marginalization is never actually performed.
  2. [Section VI D 2, Eqs. (119), (135)] The Gaussian white noise in the Carosso Langevin SDE is postulated, with amplitude D left as a free parameter, and the manuscript explicitly defers to future work the derivation of the noise statistics from the interactions of the eliminated modes. For a formal exact-RG flow this regulator freedom is acceptable, but for the claimed large-deviation application it is load-bearing: unless the noise amplitude and correlation structure are those generated by integrating out the nonzero modes of the phi^4-type model, the Carosso flow is an arbitrary stochastic deformation and the conserved p_L(m_e) is not shown to be the original model's marginal. Since Eq. 153 holds for any D, the parameter D carries no information about the original p_L(m_e); the manuscript should either derive D from the microscopic model or explicitly state that the construction defines a different conditional stochastic process.
  3. [Section VII G, Eq. (158)] The simplified RG flow for the intensive energy drops the O(1/L^d) terms present in Eq. 157, retaining only the leading-order contributions. This is a legitimate large-L saddle-point approximation, but the wording around Eq. 63 and the conclusion presents the procedure as an exact finite-volume RG. The manuscript should state clearly that Eq. 158 is a leading-order truncation and explain why the discarded O(1/L^d) terms do not affect the claimed relation between the flow and p_L(m_e). As written, the exactness of the finite-volume RG flow is overstated.
minor comments (4)
  1. [Section VII D heading] The heading contains a typo: "Orstein-Uhlenbeck" should be "Ornstein-Uhlenbeck."
  2. [Section VIII, paragraph (ii)] The conclusion says the magnetization application was discussed in "sections II, III and IV," but Section IV is the Wilson-Kogut review, not the finite-volume magnetization construction; this should be "sections II, III and VII."
  3. [Section VII F, Eq. (153)] The notation P_t[h; ...] with square brackets for a probability density over Fourier coefficients is nonstandard and may be confused with a functional or a generating function; a lowercase p_t would be clearer.
  4. [Section III and VII] The paper cites the recent literature on large deviations of the magnetization [13-18] but does not compare the conditional-OU construction with the exact functional RG methods used there; a short comparison would help the reader judge the claimed novelty.

Circularity Check

2 steps flagged · score 6.0 of 10

The Carosso adaptation conserves p_L(m_e) by construction, so the advertised gradual integration does not derive the rate function; the large-deviation result is the initial constrained minimization restated.

  1. self definitional [Section VII F, Eq. (153), with Eq. (130) and Eq. (134)]
    "The distribution p_L[m_e] of the empirical magnetization m_e of the initial condition P_0[m_e; phi_R(.,0); phi_I(.,0)] is conserved by construction and can be obtained at any time from the integration of the joint distribution P_t[m_e; phi(.)] over all the initial Fourier coefficients k > 0."

    The constraint of Eq. (130) freezes the zero-momentum coefficient m_e, and the noise constraint of Eq. (134) removes the zero-momentum noise component. The Ornstein-Uhlenbeck propagator for the remaining modes is normalized to unity, so Eq. (153) states that the marginal over nonzero modes at time t equals the initial marginal. Thus p_L(m_e) is not computed or derived from the stochastic flow; it is the original input marginal. The phrase 'can be obtained from gradual integration' is therefore a restatement of the conservation imposed by construction, not a derivation of p_L from the model.

  2. renaming known result [Section VII G, Eqs. (160)-(161), compared with Section III E, Eq. (58)]
    "For t → +∞, the variables {phi_0(.)} disappear from the first contribution and one recovers the intensive energy of the asymptotic steady state of Eq. 154 ... where the last term will be directly related to the rate function i(m_e) of the empirical magnetization m_e introduced in Eq. 33 up to the constant E_0 (see Eq. 58)."

    At t = +∞ the Gaussian kernel contributes only a positive term quadratic in the final nonzero Fourier coefficients, whose minimum is zero (attained at phi_alpha(k) = 0). The only m_e-dependent term left is min over initial nonzero modes of E_0[m_e; phi_0(.)], which is exactly the constrained minimization defining the rate function i(m_e) in Eq. (58). The stochastic RG therefore adds no new information about the large deviations of m_e; it merely repackages the original minimization of E_0 as the 'asymptotic' rate function.

full rationale

The pedagogical sections IV-VI are self-contained: they reproduce the Wilson-Kogut incomplete integration and the Carosso Langevin representation, and these are external, non-circular results. The circularity enters in the advertised application, Section VII. The paper explicitly defines the RG goal as p_L(m_e, t) = p_L(m_e, 0) in Eq. (63), then imposes m_e frozen (Eq. 130) and zero-mean noise (Eq. 134). Equation (153) then shows p_L is conserved by construction, so the 'gradual integration over all other Fourier coefficients' is an identity rather than a computation of p_L from the model. Correspondingly, the large-time effective energy in Eq. (161) reduces to the original constrained minimum of E_0, which is exactly the definition of the rate function i(m_e) in Eq. (58); the stochastic flow contributes only a positive Gaussian term with zero minimum. This makes the central application partially circular: the large-deviation result is the input, not an output. The paper itself flags the un-derived status of the noise statistics in Section VI D 2: deriving the diffusion coefficients D_t(q) from the model's interactions is left to future work. That is a load-bearing assumption about the physical interpretation of the noise, though it is a missing derivation rather than a circular step. Finally, there is no load-bearing self-citation: Carosso's work [39-41] is external, and the author's own citations appear only as background. Overall, the pedagogic stochastic-RG exposition is independent, but the advertised 'obtaining' of p_L(m_e) and its large-deviation rate function reduces by construction to the initial constrained minimization, giving a circularity score of 6.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central construction rests on standard probabilistic and field-theoretic assumptions plus one scheme-specific postulate: the noise in the Carosso Langevin SDE represents integrated-out degrees of freedom. The paper explicitly leaves deriving the noise statistics to future work in Section VI D 2.

free parameters (1)
  • D
    Noise amplitude in Eq. 119, with D_L equal to D divided by 2L^d in Eq. 141. Chosen by hand to define the Carosso scheme; not derived from the model, and scheme dependence is not analyzed.
assumptions (4)
  • standard math The functional measure Dphi and its Fourier-space image differ only by a constant Jacobian.
    Used implicitly in Eqs. 46 and 51 to translate real-space integrals to Fourier coefficients.
  • domain assumption The initial model energy is a translation-invariant local functional with a potential term and a gradient term.
    Eq. 5 defines the class of models for which the large-deviation formalism is set up.
  • domain assumption For large L, saddle-point evaluation of the integrals over Fourier coefficients is valid.
    Used in Eqs. 53, 57, and 160 to identify the rate function with the optimized intensive energy.
  • ad hoc to paper The Gaussian white noise in the Carosso Langevin SDE represents the degrees of freedom integrated out at each infinitesimal RG step.
    Section VI D 2 and Eq. 118; this physical identification is asserted and its derivation is left to future work.

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Pith. "Pith review of Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes." pith.science (2026). https://pith.science/paper/A7RALZZR

@misc{pith2026250209506,
  author       = {Pith},
  title        = {Pith review of: Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7RALZZR}},
  note         = {Machine review of arXiv:2502.09506}
}
abstract

Within the Wilson RG of 'incomplete integration' as a function of the effective RG-time $t$, the non-linear differential RG-flow for the energy $E_t[\phi(.)]$ translates for the probability distribution $P_t[\phi(.)] \sim e^{- E_t[\phi(.)]} $ into the linear Fokker-Planck RG-flow associated to independent non-identical Ornstein-Uhlenbeck processes for the Fourier modes. The corresponding Langevin stochastic differential equations for the real-space field $\phi_t(\vec x)$ have been recently interpreted by Carosso as genuine infinitesimal coarsening-transformations that are the analog of spin-blocking, and whose irreversible character is essential to overcome the paradox of the naive description of the Wegner-Morris Continuity-Equation for the RG-flow as a meaningless infinitesimal change of variables in the partition function integral. This interpretation suggests to consider new RG-schemes, in particular the Carosso RG where the Langevin SDE corresponds to the stochastic heat equation also known as the Edwards-Wilkinson dynamics. After a pedestrian self-contained introduction to this stochastic formulation of RG-flows, we focus on the case where the field theory is defined on the large volume $L^d$ with periodic boundary conditions, in order to distinguish between extensive and intensives observables while keeping the translation-invariance. Since the empirical magnetization $m_e \equiv \frac{1}{L^d} \int_{L^d} d^d \vec x \ \phi(\vec x) $ is an intensive variable corresponding to the zero-momentum Fourier coefficient of the field, its probability distribution $p_L(m_e)$ can be obtained from the gradual integration over all the other Fourier coefficients associated to non-vanishing-momenta via an appropriate adaptation of the Carosso stochastic RG, in order to obtain the large deviation properties with respect to the volume $L^d$.

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Works this paper leans on

92 extracted references · 70 canonical work pages

  1. [1]

    Wegner-Morris continuity Equation for the unnormalized densityρ t(ϕ) =e −Et(ϕ) The conservation of the partition functionZ t of Eq. 65 Zt = Z +∞ −∞ dϕe−Et(ϕ) = Z +∞ −∞ dϕρt(ϕ) =Z 0 for any timet(99) can be analyzed betweentand (t+dt) via an infinitesimal change of variables betweenϕ=ϕ t and ˜ϕ=ϕ t+dt ˜ϕ=ϕ+dtv t(ϕ) (100) with the corresponding infinitesima...

  2. [2]

    70 corresponds to the Wegner-Morris continuity Eq

    Discussion : choice of the velocityv t(ϕ)to include some irreversibility in the RG-flow for the densityρ t(ϕ) Let us first return to the Wilson-Kogut RG scheme where the Fokker-Planck evolution of Eq. 70 corresponds to the Wegner-Morris continuity Eq. 103 where the advective velocityv t(ϕ) depends on the unnormalized density ρt(ϕ) =e −Et(ϕ) via Wilson-Kog...

  3. [3]

    For diffusion processes, this decomposition into infinitesimal RG-steps leads to the path-integral representation of the propagatorP[ϕ(.), t|ϕ0(.),0] over the RG-time-window [0, t]

    Advantages of the stochastic formulation of exact RG-flows As stressed by Carosso [39–41], the stochastic formulation of exact RG-flows has a certain number of advantages, both conceptual and computational: •At the conceptual level, the first important point is that the Markovian character of the stochastic process, where the future depends only on the pr...

  4. [4]

    F ourier-series decomposition involving derivatives the fieldϕ(⃗ x) The Fourier-series of the derivatives of the fieldϕ(⃗ x) can be directly obtained from Eq. 38 ∂ϕ(⃗ x) ∂xµ = X ⃗k∈Zd ˆϕ(⃗k) i 2π L kµ ei 2π L ⃗k.⃗ x (A10) with the simple consequence for the Laplacian ∆ ∆ϕ(⃗ x)≡ dX µ=1 ∂2ϕ(⃗ x) ∂x2µ =− X ⃗k∈Zd ˆϕ(⃗k) 4π2 L2 dX µ=1 k2 µ ! ei 2π L ⃗k.⃗ x=− X...

    work page

  5. [5]

    82 translates into Wt(h) =h 2 D 2ω 1−e −2tω +W 0 h0 =he −ωt (87) where one can plug the series expansion of Eq

    Corresponding RG-flow for the generating functionW t(h)of the cumulantsw n(t) The corresponding RG-flow forW t(h) = ln Zt(h) Z0 reads ∂Wt(h) ∂t = ∂Zt(h) ∂t Zt(h) =−ωh ∂Wt(h) ∂h +Dh 2 (86) while the finite-time relation of Eq. 82 translates into Wt(h) =h 2 D 2ω 1−e −2tω +W 0 h0 =he −ωt (87) where one can plug the series expansion of Eq. 67 to obtain that t...

  6. [6]

    Corresponding RG-flow for the Legendre transformΓ t(Φ)ofW t(h) For the two single-variable-functions Γt(Φ) andW t(h), the Legendre transformations of Eqs 18 and 19 reduce to Γt(Φ) = max h hΦ−W t(h) = Φht(Φ)−W t(ht(Φ)) Φ = ∂Wt(h) ∂h h=ht(Φ) ht(Φ) = ∂Γt(Φ) ∂Φ (89) As a consequence, the RG-flow for Γ t(Φ) can be obtained from the RG-flow of Eq. 86 concerning...

  7. [7]

    70 for the unnormalized densityρ t(ϕ) =e −Et(ϕ), of Eq

    Discussion In summary, this single-variable toy-model is very useful to understand the idea of ’incomplete integration, and to see the relations between the corresponding RG-flows that appear for the various observables, namely: (i) the linear RG-flow of Eq. 70 for the unnormalized densityρ t(ϕ) =e −Et(ϕ), of Eq. 85 for the generalized partition functionZ...

  8. [8]

    Comparison of the three perspectives described in the three last sections Let us first summarize the similarities and the differences between the perspectives described the three last sections: •in section IV, we have explained how the Wilson RG-scheme of ’incomplete integration’ corresponds to a Fokker- Planck Evolution for the renormalized probability d...

Show all 92 references
  1. [9]

    Discussion on the role and on the physical meaning of the noise in the Langevin SDE of Carosso stochastic RG •As explained by Carosso [39–41], the role of the noise in the stochastic RG can be understood from the comparison with the block-spin-RG concerning discrete spins mode...

    1905

  2. [10]

    135, the Fourier decomposition analog to Eq

    Probability distribution of the Fourier coefficients[ˆη R t (⃗k),ˆηI t (⃗k)]of the real-space noiseη t(⃗ x) For the real-space noiseη t(⃗ x) of Eq. 135, the Fourier decomposition analog to Eq. 43 only involves the Fourier coefficients associated to ⃗k >0 (since the zero-moment...

  3. [11]

    43 satisfy the following Langevin Equations obtained from Eq

    Langevin SDE for the Fourier coefficients[ ˆϕR t (⃗k); ˆϕI t (⃗k)]of the real-space fieldϕ t(⃗ x) The Fourier coefficients of Eq. 43 satisfy the following Langevin Equations obtained from Eq. 118 ∂t ˆϕR t (⃗k) = 1 Ld Z Ld dd⃗ y ∆ϕt(⃗ x) +ηt(⃗ x) cos 2π L ⃗k.⃗ y =− 4π2 L2 ⃗k2 ˆ...

  4. [12]

    42 read ˆhR(⃗k) = 1 Ld Z Ld dd⃗ xh(⃗ x) cos 2π L ⃗k.⃗ x ˆhI (⃗k) = 1 Ld Z Ld dd⃗ xh(⃗ x) sin 2π L ⃗k.⃗ x (A4)

    F ourier-series decomposition of spatial-functions like the fieldϕ(⃗ x)or the magnetic fieldh(⃗ x) The Fourier-series decomposition of the fieldϕ(⃗ r) explained in detail in subsection III A of the main text, can be applied to any other spatial-function, in particular to the m...

  5. [13]

    Parseval-Plancherel identity for the scalar product of two spatial-functions The Parseval-Plancherel identity for the scalar product of two spatial-functions like the fieldϕ(⃗ x) and the magnetic fieldh(⃗ x) Z Ld dd⃗ xh(⃗ x)ϕ(⃗ x) = X ⃗ q∈Zd ˆh(⃗ q) X ⃗k∈Zd ˆϕ(⃗k) Z Ld dd⃗ xei...

  6. [14]

    11, the contribution associated to the local potentialU t[ϕ] of Eq

    Energy associated to the local potentialU t[ϕ]in terms of the F ourier coefficients In the energy functional of Eq. 11, the contribution associated to the local potentialU t[ϕ] of Eq. 12 E U t [ϕ(.)]≡ Z Ld dd⃗ xUt[ϕ(⃗ x)] = +∞X n=0 u2n(t) (2n)! Z Ld dd⃗ xϕ2n(⃗ x) = +∞X n=0 u2n...

  7. [15]

    12 reduces to the quadratic term with coefficientu 2 >0, so that the energy functional of Eq

    Real-space properties to illustrate the notations of section II Let us recall the simplest example of the Gaussian model where the local potentialU[ϕ] of Eq. 12 reduces to the quadratic term with coefficientu 2 >0, so that the energy functional of Eq. 5 is quadratic E Quadrati...

  8. [16]

    Independent Gaussian distributions for the Fourier coefficients For the Gaussian model of Eq

    F ourier-space properties to illustrate the notations of section III a. Independent Gaussian distributions for the Fourier coefficients For the Gaussian model of Eq. B1 where the only non-vanishing coefficient isu 2, the intensive energy of Eq. 45 in terms the intensive Fourie...

  9. [17]

    F ourier coefficients ˆϕα=R,I t (⃗k >0): OU-processes with time-dependent parameters[ω L(⃗k, t);DL(⃗k, t)] The OU-process of Eq. 143 with time-independent positive parameters [ω L(⃗k)>0;D L(⃗k)>0] can be generalized to the case of time-dependent positive parameters [ωL(⃗k, t)>...

  10. [18]

    Statistical properties of the F ourier coefficient ˆϕα t (⃗k)at timetfor a given initial condition ˆϕα 0 (⃗k) For a given initial condition ˆϕα t=0(⃗k) att= 0, the Langevin SDE of Eq. C1 for the Fourier coefficient ˆϕα t (⃗k) can be integrated to obtain the solution ˆϕα t (⃗k)...

  11. [19]

    Langevin SDE : from the F ourier-space towards the real-space a. Statistical properties of the real-space noiseη t(⃗ x) The real-space noiseη t(⃗ x) as reconstructed from the linear combination of its Fourier coefficients [ˆηR t (⃗k),ˆηI t (⃗k)] via Eq. 43 using ˆηR(⃗0, t) = 0...

  12. [20]

    F unctional RG-flows in real-space in terms of the two kernelsΩ L(⃗ x−⃗ y, t)andDL(⃗ x−⃗ y, t) The real-space Langevin SDE of Eq. C17 translates into the following functional Fokker-Planck RG-flow for the probability distributionP t[ϕ(.)] of the real-space field configurationϕ...

  13. [21]

    118 involving the Laplacian operator ∆ corresponds to the case ΩL(⃗ x−⃗ y, t) =−∆δ(d)(⃗ x−⃗ y) (C23) with its corresponding eigenvalues of Eq

    Special case of the Carosso RG-scheme for the volumeL d described in section VII of the main text The Carosso RG-scheme of Eq. 118 involving the Laplacian operator ∆ corresponds to the case ΩL(⃗ x−⃗ y, t) =−∆δ(d)(⃗ x−⃗ y) (C23) with its corresponding eigenvalues of Eq. 133 ωL(...

  14. [22]

    Further work is needed to see whether these generalizations can be useful for some purposes from the RG point of view

    Discussion In this Appendix, we have described how the calculations concerning the Carosso RG-scheme for the volume Ld described in section VII can be generalized to time-dependent OU-processes for the Fourier modes, and we have discussed their real-space interpretations in te...

  15. [23]

    Jona-Lasinio, Il Nuovo Cimento 26, 99(1975)

    G. Jona-Lasinio, Il Nuovo Cimento 26, 99(1975)

    1975

  16. [24]

    Cassandro and G

    M. Cassandro and G. Jona-Lasinio, Advances in Physics, 27:6, 913(1978)

    1978

  17. [25]

    Jona-Lasinio, Physics Reports 352, 439 (2001)

    G. Jona-Lasinio, Physics Reports 352, 439 (2001)

    2001

  18. [26]

    Calvo, J

    I. Calvo, J. C. Cuchi, J. G. Esteve and F. Falceto, J. Stat. Phys. 141, 409 (2010)

    2010

  19. [27]

    Bouchaud and A

    J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990)

    1990

  20. [28]

    Clusel and E

    M. Clusel and E. Bertin, International Journal of Modern Physics B Vol. 22, No. 20, pp. 3311 (2008)

    2008

  21. [29]

    Binder, Physical Review Letters

    K. Binder, Physical Review Letters. 47 (9): 693–696 (1981)

    1981

  22. [30]

    Binder, Zeitschrift f¨ ur Physik B: Condensed Matter

    K. Binder, Zeitschrift f¨ ur Physik B: Condensed Matter. 43 (2): 119–140.(1981)

    1981

  23. [31]

    Binder, J

    K. Binder, J. Computational Physics 59, l-55 (1985)

    1985

  24. [32]

    Oono, Progress of Theoretical Physics Supplement 99, 165 (1989)

    Y. Oono, Progress of Theoretical Physics Supplement 99, 165 (1989)

    1989

  25. [33]

    Ellis, Physica D 133, 106 (1999)

    R.S. Ellis, Physica D 133, 106 (1999)

    1999

  26. [34]

    Touchette, Phys

    H. Touchette, Phys. Rep. 478, 1 (2009); H. Touchette, Modern Computational Science 11: Lecture Notes from the 3rd International Oldenburg Summer School, BIS-Verlag der Carl von Ossietzky Universitat Oldenburg, (2011)

    2009

  27. [35]

    Balog, A

    I. Balog, A. Ran¸ con, B. Delamotte, Phys. Rev. Lett. 129, 210602 (2022)

    2022

  28. [36]

    S. Sahu, B. Delamotte, A. Ran¸ con, arXiv:2407.12603

    arXiv

  29. [37]

    Balog, B

    I. Balog, B. Delamotte, A. Ran¸ con, arXiv:2409.01250

    arXiv

  30. [38]

    Ran¸ con, B

    A. Ran¸ con, B. Delamotte, L.ˇSaravanja, I. Balog, arXiv:2501.04465

    arXiv

  31. [39]

    Sahu, arXiv:2501.18615

    S. Sahu, arXiv:2501.18615

  32. [40]

    G. Teza, A. L. Stella, arXiv:2407.07782

    arXiv

  33. [41]

    Wilson and J

    K.G. Wilson and J. Kogut, Phys. Rep. 12, 75 (1974)

    1974

  34. [42]

    E. K. Riedel, G. R Golner , K. E. Newman, Annals of Physics 161, 178 (1985)

    1985

  35. [43]

    F. J. Wegner, A. Houghton, Phys. Rev. A 8 , 401 (1973)

    1973

  36. [44]

    Polchinski, Nucl

    J. Polchinski, Nucl. Phys. B 231, 269 (1984)

    1984

  37. [45]

    Wetterich, Nucl

    C. Wetterich, Nucl. Phys. B 352, 529 (1991); C. Wetterich, Z. Phys. C 57, 451 (1993); C. Wetterich, Z. Phys. C 60, 461 (1993); C. Wetterich, Phys. Lett. B 301, 90 (1993); C. Wetterich, Int. J. Mod. Phys. A 9, 3571 (1994). 42

    1991

  38. [46]

    Aoki, Int

    K. Aoki, Int. J. Mod. Phys. B 14, 1249 (2000)

    2000

  39. [47]

    Bagnuls, C

    C. Bagnuls, C. Bervillier, Phys. Rep. 348, 91 (2001)

    2001

  40. [48]

    Berges, N

    J. Berges, N. Tetradis, C. Wetterich, Phys. Rep. 363, 223 (2002)

    2002

  41. [49]

    Polonyi, Central Eur

    J. Polonyi, Central Eur. J. Phys. 1, 1 (2004)

    2004

  42. [50]

    J. M. Pawlowski, Annals Phys. 322, 2831 (2007)

    2007

  43. [51]

    Osborn and D.E

    H. Osborn and D.E. Twigg, Annals of Physics 327, 29 (2012)

    2012

  44. [52]

    O. J. Rosten, Phys. Rept. 511, 177 (2012)

    2012

  45. [53]

    H.Gies, Introduction to the Functional RG and Applications to Gauge Theories, Lecture Notes in Physics 852, 287 Springer Berlin Heidelberg (2012)

    2012

  46. [54]

    B.Delamotte, Renormalization Group and Effective Field Theory Approaches to Many-Body Systems in Lecture Notes in Physics 852, 49, Eds A.Schwenk, J.Polonyi, Springer Berlin Heidelberg, (2012)

    2012

  47. [55]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J.M. Pawlowski, M. Tissier, N. Wschebor, Phys. Rept. 910, 1 (2021)

    2021

  48. [56]

    Wegner, J

    F.J. Wegner, J. Phys. C: Solid State Phys. 7, 2098 (1974)

    1974

  49. [57]

    T. R. Morris, Int.J.Mod.Phys. A9, 2411 (1994)

    1994

  50. [58]

    J. I. Latorre, T. R. Morris, Exact scheme independence, JHEP 11 (2000) 004

    2000

  51. [59]

    J. I. Latorre, T. R. Morris, Int. J. Mod. Phys. A16, 2071 (2001)

    2001

  52. [60]

    Arnone, A

    S. Arnone, A. Gatti, T. R. Morris, Exact scheme independence at one loop, JHEP 05 (2002) 059. arXiv:hep-th/0201237

    2002 arXiv

  53. [61]

    Carosso, J

    A. Carosso, J. High Energ. Phys. 2020, 172 (2020)

    2020

  54. [62]

    Carosso, A

    A. Carosso, A. Hasenfratz, E. T. Neil, arXiv:1912.01766

    1912 arXiv

  55. [63]

    Carosso, Novel Approaches to Renormalization Group Transformations in the Continuum and on the Lattice, PhD thesis, University of Colorado Boulder, arXiv:2006.07481

    A. Carosso, Novel Approaches to Renormalization Group Transformations in the Continuum and on the Lattice, PhD thesis, University of Colorado Boulder, arXiv:2006.07481

    2006 arXiv

  56. [64]

    Hairer, arXiv:0907.4178

    M. Hairer, arXiv:0907.4178

    arXiv

  57. [65]

    Barabasi and H.E

    A.L. Barabasi and H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, England (1995)

    1995

  58. [66]

    Halpin-Healy and Y.-C

    T. Halpin-Healy and Y.-C. Zhang, Phys. Rep. 254, 215 (1995)

    1995

  59. [67]

    Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences

    C. W. Gardiner, “Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences” (Springer Series in Synergetics), Berlin (1985)

    1985

  60. [68]

    Stochastic processes in physics and chemistry

    N.G. Van Kampen, “Stochastic processes in physics and chemistry”, Elsevier Amsterdam (1992)

    1992

  61. [69]

    The Fokker-Planck equation : methods of solutions and applications

    H. Risken, “The Fokker-Planck equation : methods of solutions and applications”, Springer Verlag Berlin (1989)

    1989

  62. [70]

    G. A. Pavliotis, ”Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations” (Texts in Applied Mathematics Book 60) Springer New-York (2014)

    2014

  63. [71]

    Quantum Mechanics and Path Integrals

    R. P. Feynman and A. R. Hibbs “Quantum Mechanics and Path Integrals” McGraw-Hill, New-York (1965)

    1965

  64. [72]

    Caticha, arXiv:1605.06366; A

    A. Caticha, arXiv:1605.06366; A. Caticha, PhD Changes of Variables and the Renormalization Group, California Institute of Technology (1985)

    1985 arXiv

  65. [73]

    D. Ron, R. H. Swendsen, A. Brandt, Phys. Rev. Lett. 89, 275701 (2002)

    2002

  66. [74]

    Bachtis, G

    D. Bachtis, G. Aarts, F. Di Renzo, B. Lucini, Phys. Rev. Lett. 128, 081603 (2022)

    2022

  67. [75]

    Marchand, M

    T. Marchand, M. Ozawa, G. Biroli, S. Mallat, Phys. Rev. X 13, 041038 (2023)

    2023

  68. [76]

    Bachtis, Phys

    D. Bachtis, Phys. Rev. B 110, L140202 (2024)

    2024

  69. [77]

    Bachtis, arXiv:2405.16288

    D. Bachtis, arXiv:2405.16288

    arXiv

  70. [78]

    D. S. Berman, M. S. Klinger, Entropy 2024, 26, 389

    2024

  71. [79]

    S Berman, M

    D. S Berman, M. S Klinger and A. G Stapleton, Mach. Learn.: Sci. Technol. 4, 045011 (2023)

    2023

  72. [80]

    J. N. Howard, M. S. Klinger, A. Maiti, A. G. Stapleton arXiv:2405.17538

    arXiv

  73. [81]

    Masuki, Y

    K. Masuki, Y. Ashida, arXiv:2501.09064

    arXiv

  74. [82]

    Cotler, S

    J. Cotler, S. Rezchikov, arXiv:2308.12355

    arXiv

  75. [83]

    Aarts, D

    G. Aarts, D. E. Habibi, L. Wang, K. Zhou, Mach.Learn.Sci.Tech. 6 (2025) 025004

    2025

  76. [84]

    L. Wang, G. Aarts, K. Zhou, JHEP 05 (2024) 060 ; Q. Zhu, G. Aarts, W. Wang, K. Zhou, L. Wang, arXiv:2502.05504

    2024

  77. [85]

    Bachtis, G

    D. Bachtis, G. Aarts, B. Lucini, Phys. Rev. D 103, 074510 (2021); D. Bachtis, G. Aarts and B. Lucini, J. Phys.: Conf. Ser. 2207 012056 (2022); D. Bachtis, G. Aarts, B. Lucini, arXiv:2109.07730

    2021 arXiv

  78. [86]

    Cotler and S

    J. Cotler and S. Rezchikov, Phys. Rev. D 108, 025003 (2023)

    2023

  79. [87]

    Le Doussal, Annals of Physics 325, 49 (2010)

    P. Le Doussal, Annals of Physics 325, 49 (2010)

    2010

  80. [88]

    Tarjus and M

    G. Tarjus and M. Tissier, European Physical Journal B 93, 50 (2020)

    2020

  81. [89]

    D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992); D. S. Fisher, Phys. Rev. B 51, 6411 (1995)

    1992

  82. [90]

    Igl´ oi and C

    F. Igl´ oi and C. Monthus, Physics Reports 412, 277 (2005); F. Igl´ oi and C. Monthus, European Physical Journal B 91, 290, (2018)

    2005

  83. [91]

    Monthus, B

    C. Monthus, B. Berche and C. Chatelain, J. Stat. Mech. (2009) P12002

    2009

  84. [92]

    Monthus and T

    C. Monthus and T. Garel, J. Stat. Mech. (2010) P06014

    2010

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