{'fraction_non_alphanumeric': 0.04748910009096327, 'fraction_numerical': 0.016373388538628023, 'mean_word_length': 4.478948272899124, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Towards Functional Flows for Hierarchical Models

The recursion relations of hierarchical models are studied and contrasted with functional renormalisation group equations in corresponding approximations. The formalisms are compared quantitatively for the Ising universality class, where the spectrum of universal eigenvalues at criticality is studied. A significant correlation amongst scaling exponents is pointed out and analysed in view of an underlying optimisation. Functional flows are provided which match with high accuracy all known scaling exponents from Dyson's hierarchical model for discrete block-spin transformations. Implications of the results are discussed.


Introduction
Renormalisation group methods [1], and in particular Wilson's renormalisation group [2], play an important role in the study of physical systems at strong coupling and/or large correlations lengths. Differential implementations of Wilson's idea [3][4][5][6][7][8][9][10] rely on an appropriately-introduced momentum cutoff leading to flow equations for running couplings and N -point functions, which can be studied with a large variety of analytical and numerical methods. Numerical stability and reliability in the results is ensured through powerful control and optimisation techniques [11][12][13][14]. A different implementation of Wilson's idea is realised in hierarchical models of lattice scalar theories [15][16][17][18][19]. Hierarchical renormalisation group transformations are often discrete rather than continuous. Here, sophisticated numerical methods have been developed to extract the relevant physics, most notably for high-accuracy studies of scaling exponents for scalar models at criticality [20][21][22] and related theories (see [19] and references therein).
Given the close similarity of the underlying principles, it is natural to ask whether Wilsonian (functional) flows can be linked explicitly, and on a fundamental level, to hierarchical models. If so, this link would provide a number of benefits. It will make powerful functional and numerical methods available to the study of hierarchical models. Vice versa, the numerical tools for hierarchical models could be employed for functional flows in specific approximations. Furthermore, an explicit link may lead to a path integral representation of hierarchical models, allowing for systematic improvements beyond a standard kinetic term. Finally, well-developed optimisation techniques for functional flows could be taken over for hierarchical models as well.
In the limit of continuous hierarchical block-spin transformations, an explicit link between Dyson's hierarchical model [15,17] and the Wilson-Polchinski flow [23] in the local potential approximation has been established long ago by Felder [24]. Following a conjecture of [13], this link has been extended [25,26] to include optimised versions [11,12] of Wetterich's flow for the effective average action [27]. These interrelations have recently been backed-up by extensive numerical studies of critical potentials and scaling exponents to high accuracy from either formalism [28].
In this paper, we evaluate the more general case and ask whether hierarchical models for discrete block-spin transformations are linked to functional flows with continuous renormalisation group transformations. We first contrast the basic setups for functional flows (Sec. 2), background field flows (Sec. 3), and hierarchical models (Sec. 4). At a Wilson-Fisher fixed point, underlying similarities and differences are worked out and compared for the leading scaling exponent (Sec. 5). An extensive numerical study of the eigenvalue spectrum of the Ising universality class from functional flows is performed (Sec. 6). A strong correlation of scaling exponents is established and analysed (Sec. 7). It is shown that specific functional flows match the leading and subleading scaling exponent from Dyson's hierarchical models for discrete transformation parameter to high accuracy (Sec. 8). We close with a discussion of the results and further implications (Sec. 9).

Functional Flows
Wilsonian (functional) flows integrate-out quantum fluctuations within a path integral representation of quantum field theory. In their simplest form, they are generated through a cutoff term quadratic in the field added to the Schwinger functional, where the (classical) action is replaced by S → S+∆S k and ∆S k ∼ dqφ(q)R k (q 2 )φ(−q). The infrared momentum cutoff R k (q 2 ) ensures that the propagation of small momentum modes q 2 ≪ k 2 is suppressed, while the large momentum modes q 2 ≫ k 2 remain unaffected. Under an infinitesimal change in the Wilsonian (infrared) cutoff scale k, the effective action Γ k changes according to its functional flow, which reads (t = ln k) in the form put forward by Wetterich [27]. The trace denotes a momentum integration and a summation over fields. The factor ∂ t R k in the integrand is peaked in the vicinity of q 2 ≈ k 2 . The cutoff function R k obeys R k (q 2 ) → 0 as k 2 /q 2 → 0, R k (q 2 ) > 0 as q 2 /k 2 → 0, and R k (q 2 ) → ∞ as k → Λ, and can be chosen freely elsewise, e.g. [11]. It ensures that the flow is well-defined, thereby interpolating between an initial action S at k = Λ in the ultraviolet (UV) and the full quantum effective action Γ ≡ Γ k=0 in the infrared k → 0.
Below, we are interested in 3d scalar theories at criticality, where we can sent the ultraviolet scale Λ → ∞. To leading order in the derivative expansion, the effective action reads Γ k = Introducing r(y) = R k (q 2 )/q 2 with y = q 2 /k 2 , we find with u(ρ) = U k (ρ)/k 3 and ρ =ρ/k. An irrelevant constant originating from the angular integration has been rescaled into the potential and the fields. For the optimal cutoff R opt = (k 2 − q 2 )θ(k 2 − q 2 ) with r opt = (1/y − 1)θ(1 − y), the flow reads [12] ∂ t u = −3u + ρu ′ + 1 after an additional rescaling. This flow is integrated analytically in the limit of a large number of scalar fields [35]. We note that the universal content of the flow (3) is equivalent to the Wilson-Polchinski flow in the local potential approximation [13,25,26].

Background Field Flows
A different form of the flow (1) is obtained for momentum cutoffs which depend additionally on a background fieldφ. Background fields are most commonly used for the study of gauge theories [36], see [32,37,38] for applications. They have also been employed for a path integral derivation of (generalised) proper-time flows [39,40].
In the presence of background fields, the functional Γ k [φ] turns into a functional of both fields, Γ k [φ,φ]. In order to maintain the one-loop exactness of (1), the momentum cutoff can only depend on the background field, but not on the propagating field. Following [39], , which depend now on the background field. Here, the regulator cuts off both large momentum modes q 2 ≫ k 2 and large field amplitudes with Γ (2,0) k ≫ k 2 . The full advantage of background fields becomes visible once they are identified with the physical mean, leading to the functional Γ The resulting flow is closed provided Γ For scalars, this relation becomes exact in the infrared limit studied below (for gauge fields, see [38]). Using the momentum cutoffs [39] we are lead to the background field flow If the term ∼ ∂ t x on the right hand side is dropped -meaning that additional flow terms originating from the implicit scale dependence in the momentum cutoff are neglected over the leading term -the flow (4) reduces to the proper-time flow of Liao [41]. A general proper-time flow is a linear combination of the first term in (4) for various m [40]; see [42,43] for applications.
Next, we specialise to the proper-time approximation to leading order in the derivative expansion. The flow equation for the effective potential takes the very simple form where m parameterises the momentum cutoff, and an irrelevant constant factor has been rescaled into the potential and the fields. For m ∈ [1, 5 2 ], the flow (5) is mapped onto the flow (2) [39]. At m = 5 2 , the flow (5) is equivalent to (3), modulo a trivial rescaling. As a final remark, we note that this proper-time flow is also obtained from linear combinations of higher scale-derivatives of Callan-Szymanzik flows, without relying on background fields [40]. In this representation, the approximation leading to (5) consists in the neglection of higher order flow terms ∼ ∂ n t Γ k .

Hierarchical Models
Several hierarchical models for an effective potential v(ϕ) of a lattice scalar field have been introduced in the literature [15][16][17] (see also [19]). The hierarchical transformation laws relate the potential v(ϕ) at momentum scale k/ℓ with an average in field space over v(ϕ) at momentum scale k, where ℓ ≥ 1 is the renormalisation group step parameter. We restrict ourselves to the three-dimensional case; the generalisation to arbitrary dimensions is straightforward.
In Dyson's model [15,17], the renormalisation group step k → k/ℓ for the potential is expressed as The details of the averaging procedure are encoded in the measure factor µ ℓ (ξ), in the ξdependence of the potential on the right-hand side of (6), and in the choice for the decimation parameter ℓ. As is evident from (6), a decimation parameter ℓ = 2 1/3 -employed for most numerical studies [19][20][21][22] -corresponds to a volume decimation of ℓ 3 = 2 at each iteration. For Dyson's model, the measure is chosen as . By definition, (6) describes a flow towards the infrared for decimation parameters ℓ ≥ 1.
A different version of a hierarchical model has been introduced by Wilson [16]. Here, the recursion relation is written as In Wilson's original model, the ξ-dependence of the measure is µ ℓ (ξ) = N ℓ exp(−ξ 2 ), where the normalisation factor N ℓ is ξ-independent [19]. The measure factor is different from the one in Dyson's model, because the Gaussian width is ℓ-independent. If instead we employ the measure of Dyson's model, the limit ℓ → 1 can be performed analytically. 2 Up to a trivial rescaling, we find In contrast to the Wilson-Polchinski flow (7), the non-linear term (v ′ ) 2 is absent. This comes about because the integrand of (8) -as opposed to the integrand of (6) -is manifestly symmetric under ξ → −ξ. Numerical evaluations of (8) have been reported in [16,44]. For other representations of hierarchical models we refer to [19] and references therein.

Matching Hierarchical Models
In order to match hierarchical models by functional flows, we have to detail the scheme dependences of physical observables in either formalism. In the functional RG framework, the fully integrated flow is independent of the momentum cutoff R k (q 2 ) chosen for the integration. Scheme dependences, which enter as a consequence of truncations of Γ k [φ], have been discussed extensively in the literature [11-13, 29, 45]. Their origin is easily understood. Since the momentum cutoff R in (1) couples to all operators in the theory, the missing back-coupling of operators neglected in a given truncation can result in a spurious dependence of physical observables evaluated either from Γ 0 [φ], or from a fixed point solution Γ * [φ]. The scheme dependence is reduced by identifying those momentum cutoffs, which, in a given truncation, lead to an improved convergence and stability of the flow.
In Fig. 1, we discuss the scheme dependence quantitatively for the leading order scaling exponent ν at a fixed point of the 3d Ising universality class [29]. Within exact flows (2), the full R k -dependence has been studied in [30] by evaluating the fixed points of (2) for general cutoffs (Fig. 1, first column). The main result is that the range of achievable values is bounded from above and from below. The upper bound is attained for Callan-Symanzik type flows with R k ∼ k 2 . The lower bound with ν = ν opt is attained with the optimal flow (3), and hence equivalent to the Wilson-Polchinski flow. The sharp cutoff result is indicated for comparison.
The proper-time flow (5) rests on an intrinsically different truncation, because implicit dependences on the background field have been neglected as well as higher order flow terms proportional to the flow of Γ (2) ; see Sec. 3. Therefore scheme dependences are quantitatively different. In the approximation (4), the m-dependence of scaling exponents from (5) has been studied in [39,42] (Fig. 1, second column). The range of values is again bounded from above by a Callan-Symanzik flow. The lower bound is achieved for m → ∞. We note that the range of values exceeds those achievable within (standard) exact flows. The lower bound 2 The variance of the Gaussian measure in (8) can be changed by an explicit rescaling of the fields as ϕ → ϕ/ √ σ for finite σ, see [19]. Rescaling also the integration variable ξ → ξ/ √ σ, and denoting the potential in terms of the rescaled fields again as v(ϕ), we obtain (8)   may be overcome once the additional flow terms, neglected here, are taken into account [39]. This is indicated by the dashed line.
Next we consider scheme dependences of hierarchical models. Based on their construction, we expect that physical observables depend on the averaging procedure, on the measure factor µ ℓ , and on the decimation parameter ℓ. It has proven difficult to systematically include wave function renormalisations and higher order operators in hierarchical models, and it is therefore not known whether the scheme dependence vanishes upon higher order corrections [18,19]. Still, the scheme dependence should give a reasonable estimate for the underlying error in the model assumptions, in particular in comparison with functional methods.
The ℓ-dependence of Wilson's model (8), originally constructed for ℓ ≈ 2, has been studied in [44] in the range ℓ ∈ [2 1/3 , 2] (Fig. 1, third column). The full line covers the range of values obtained in the literature, while the dashed lines indicate that the underlying parameter space has not been exhausted. Because of (9) being linear in the potential as opposed to (7), we expect a strong ℓ-dependence, possibly a discontinuity, in the limit ℓ → 1. The slope ℓ∂ ℓ ν(ℓ) along the data points with ℓ > 2 1/3 is negative, meaning that ν(ℓ) increases for smaller ℓ.
We stress that Wilson's HM has an overlap both with exact flows and proper-time flows. Therefore, it is possible to map ν(ℓ) of Wilson's model for certain decimation parameters ℓ onto ν(R) from functional flows with appropriately chosen R. On the other hand, for some decimation parameter ℓ, Wilson's model can only be mapped onto proper-time flows but not on exact flows, while for some decimation parameter it cannot be mapped onto either of them.
The ℓ-dependence of Dyson's model (6) is displayed in Fig. 1, fourth column. The full line connects the known results at ℓ = 1 [28,30], ℓ = 2 1/3 [20,22] and ℓ = 2 [20]. The dashed line towards larger values for ν indicates that the parameter space ℓ ≥ 1 has not been exhausted. We note that the ℓ-dependence is very weak, with a tiny slope in the range of ℓ-values covered. The important observation is that the slope ℓ∂ ℓ ν(ℓ) is positive in the vicinity of ℓ ≈ 1 − 2, implying that ν(ℓ) > ν(1) for ℓ > 1. Consequently, it is possible to map the scaling exponent ν(ℓ) at discrete block-spin transformation ℓ > 1 onto ν(R) from functional flows for specific momentum cutoff R, both within the standard exact flows and within proper-time flows. This supports the conjecture that Dyson's model can be mapped onto functional flows.

Spectrum of Eigenvalues
Whether the observations of the preceeding section can be promoted to a full map between the formalisms crucially depends on further observables including the subleading scaling exponents. Here and in the following section, we study the spectrum of universal eigenvalues (scaling exponents) form functional flows (2) to high accuracy. A fixed point solution u * = const. of (2) is characterised by the universal eigenvalues of eigenperturbations in its vicinity. We denote the ordered set of eigenvalues as O(R) = {ω i (R), i = 0, · · · , ∞}, with ω i < ω j for i < j. 3 In addition to the leading exponent ν(R) ≡ −1/ω 0 , we study the first three subleading scaling exponents ω(R) ≡ ω 1 (R), ω 2 (R) and ω 3 (R) within the exact flow (2) for various cutoffs and coarse graining parameters.
For the numerical analysis, we introduce several classes of momentum cutoffs defined through r mexp = b/((b+ 1) y − 1), r exp = 1/(exp cy b − 1); r mod = 1/(exp[c(y + (b− 1)y b )/b]− 1), with c = ln 2; and r opt,n = b(1/y − 1) n θ(1 − y). These cutoffs include the sharp cutoff (b → ∞) and asymptotically smooth Callan-Symanzik type cutoffs R k ∼ k 2 as limiting cases. The larger the parameter b, for each class, the 'sharper' the corresponding momentum cutoff. The cutoff r opt,n probes a two-dimensional parameter space in the vicinity of r opt to which it reduces for b = 1 and n = 1. For integer n, r opt,n is a C (n+1) function. In addition, we consider the cutoffs , which obey r mix (1/y) = 1/r mix (y). Note that we have covered a large variety of qualitatively different momentum cutoffs including exponential, algebraic, power-law, sharp cutoffs and cutoffs with compact support. Except for r opt,n , all cutoffs are C (∞) -functions. We employ the numerical techniques developed in [28,30].  Our results for the universal eigenvalues at criticality are displayed in Fig. 2 for the six two-dimensional projections of the four-dimensional subspace {−ω 0 , ω, ω 2 , ω 3 } of observables. The plot contains roughly 10 3 data points, the different classes of cutoffs are colour-coded. We focus on the relevant 10%-vicinity of the Wilson-Polchinski result with scaling exponents O opt ≡ O(R opt ) from (3), indicated by a large black dot, see [28] for the high-accuracy numerical values. The central result of Fig. 2 is that scaling exponents are very strongly correlated. Despite having probed the space of observables by many qualitatively different momentum cutoffs, we find that only a small subset of values can actually be achieved. The correlations increase the closer the eigenvalues O move towards O opt . In the immediate vicinity of the Wilson-Polchinski result, we only find a very narrow "throat" connecting observables O(R) with O opt . For the sub-leading scaling exponents ω i , the throat remains very narrow even further away from O opt . This is seen most clearly in the correlation of ω 2 with ω 3 in Fig. 2e), as well as in the correlations of ω with both ω 2 and ω 3 in Fig. 2d) and f). In turn, for the leading exponents ν, the throat opens up more rapidly once its value is further away from ν opt , see Fig. 2a), b) and c).

Correlations of Eigenvalues
The strong correlation of scaling exponents is a structural fingerprint of Wilsonian flows (2). Since the Wilson-Polchinski result is distinguished in the space of scaling exponents, it is natural to normalise the data of Fig. 2 with respect to it. We introduce the distance of any pair of scaling exponents (x, y) from the optimal result (x opt , y opt ) as We have chosen a standard metric in the space of observables (other choices can be applied as well). In this representation, full agreement with the (optimal) Wilson-Polchinski result is achieved for ρ → 0 and N ρ → ∞. We also introduce the angles ϕ(x, y) = arctan(x/y) .
In the representation (10), we can study the close vicinity of the Wilson-Polchinski result. In Fig. 3, we display our data points as functions of the angles ϕ, and their distance from ρ opt in a semi-logarithmic basis. It is noteworthy that only a very narrow range of angles ϕ is actually achieved by the data, despite the fact that large fractions of the underlying space of momentum cutoffs is covered. Also, and in contrast to Fig. 2a), many data points are degenerate in the representation (N ρ , ϕ). A priori, the Wilson-Polchinski value (ρ opt , ϕ opt ) could have been approached along many different paths. Instead, we find that only a narrow range of (N ρ , ϕ)-values is achieved for arbitrary momentum cutoff.  (10), (11). Only a narrow range of angles ϕ(ν, ω) in the vicinity of ϕ ≈ π/4 is achieved by the data. Many data points are nearly degenerate. Data from functional flows in the local potential approximation (2); same data sets and colour coding as in Fig. 2, plus further high resolution data points from r opt,1 in the close vicinity of (ν opt , ω opt ). Results from the sharp cutoff limit, the Callan-Symanzik type flow (with R k ∼ k 2 ) and Dyson's hierarchical model (with ℓ = 2 1/3 ) are also indicated (black dots). The Wilson-Polchinski (optimal flow) result corresponds to ϕ = ϕ opt and N ρ → ∞.
This pattern is further highlighted in Fig. 4, where we have magnified the non-trivial range of data sets from functional flows (2). In addition, we have added data points from the background field flow (5) using the cutoff r PT,m for m < 5 2 . It is remarkable that these data sets display the same pattern as the data from (2). Our results from this and the preceeding section are summarised as follows: Extremum.-The Wilson-Polchinski (optimal flow) result in Fig. 2 corresponds to an extremum in the space of physical observables with |ω i | ≤ |ω i,opt | for all obervables in the vicinity of O opt . The extremum is local, because the exponents approach ω i = 2i − 1, i ≥ 0, for very soft (Callan-Symanzik-type) momentum cutoffs [30]. For the eigenvalue products Π n i=0 (ω i /ω i,opt ), the Wilson-Polchinski extremum is a global one.   Fig. 3 in the vicinity of ϕ ≈ π/4 where ν/ω ≈ 1. The data points for the distance N ρ (ν, ω) as a function of ν/ω remain highly degenerate. Same data sets and colour coding as in Fig. 3, plus additional high resolution data points from background field flows (5) using r PT,m with m < 5 2 . Results from the sharp cutoff limit, the Callan-Symanzik type flow (with R k ∼ k 2 ) and Dyson's hierarchical model (with ℓ = 2 1/3 ) are also indicated (black dots). The Wilson-Polchinski (optimal flow) result corresponds to ν opt /ω opt = 0.9905692 · · · and N ρ → ∞.
Redundancy.-The eigenvalue correlations are so strong that the first two scaling exponents ν(R) and ω(R), for a given R, contain enough information to fix the remaining observables on the percent level or below. These 'dynamical' constraints point at a major redundancy of (2) with respect to the underlying momentum cutoffs R. A relevant parameter has been identified previously. The gap min y≥0 y(1 + r) for normalised cutoffs [11], when maximised, leads towards the Wilson-Polchinski result [12,13,30].
Optimised observables.-Previous reasonings in favour of an optimisation only invoked properties of the underlying flow (1), e.g. its convergence, locality, stability and boundedness, allowing for improved physical predictions. This has been exemplified quantitatively for the observable ν(R) which obeys 1 ≥ ν(R) ≥ ν opt [30], where the lower bound ν opt is closest to the physical result [25]. Fig. 2 now shows that this pattern extends to subleading eigenvalues. This equally extends to asymmetric corrections-to-scaling [46]. Therefore, one may turn the original reasoning around and argue that -because of the extremum property of the observables O(R) -an extremisation of the functional flow along the lines discussed in [11,12,14], or similar, should naturally lead towards the values O opt . Stated differently, Figs. 2-4 show that observables derived from (1) admit an optimisation.  Finally, we note that the data point from Dyson's hierarchical model with ℓ = 2 1/3as plotted in Figs. 3 and 4 -nicely fits into the set of data points covered by functional flows, extending the link observed in Sec. 5 beyond the leading exponent. This observation is addressed quantitatively in the following section.

Matching Beyond the Leading Exponent
To further substantiate our conjecture that hierarchical models could be mapped onto functional flows, we have to show quantitatively that results from hierarchical model are reproduced by specific functional flows. Here, we study the close vicinity of the Wilson-Polchinski result O opt , where the correlations are strongest, see Figs. 2 and 3. We have to restrict our search to Dyson's hierarchical model, where high-accuracy data for the first sub-leading scaling exponent ω is available. No subleading exponents have been computed for Wilson's model.
For the numerical analysis, we introduce additional classes of momentum cutoffs R which contain the optimal flow (3) in some limit. In addition to the two-parameter family of cutoffs r opt,n , we also study the cutoff 1], which is effective for a finite interval of momenta q 2 ∈ [bk 2 , k 2 ]. In the limit b → 0 (b → 1), the corresponding flows are equivalent to (3). Hence, r opt,n , r compact and r int parametrise substantially different classes of cutoffs. More generally, there are infinitely many cutoffs R k leading to scaling exponents identical with O opt , and the examples provided above serve to illustrate this.
At ℓ = 1, Dyson's hierarchical transformation is continuous, and the scaling exponents are equivalent to those from the optimal flow (3) and the Wilson-Polchinski flow (7). In Tab. 1, we compare exponents from different functional flows. We confirm numerically, and with high accuracy, that the cutoffs r opt,n , r compact and r int lead to the Wilson-Polchinski result for specific parameter values. At ℓ = 2 1/3 , Dyson's hierarchical transformation is discrete. The reference data reads ν DHM = 0.649570 and ω DHM = 0.655736 [22]. 4 These values differ only at the order 10 −5 from the optimal (Wilson-Polchinski) result, and are therefore sufficiently close to O opt to confirm or refute the correlations observed in the previous section. Figs. 3 and 4 indicate that the result from Dyson's hierarchical model is fully matched by functional flows. Our numerical results are given in Tab. 2; brackets indicate that a digit is possibly affected by numerical errors. We have found several sets of parameter values, such that the scaling exponents agree with Dyson's model to order 10 −6 . More importantly, the momentum cutoffs are quite different. Hence, our analysis also confirms the strong correlation of scaling exponents in the immediate vicinity of the Wilson-Polchinski result. Based on the eigenvalue correlations within functional flows, we conjecture that the subleading eigenvalues ω i with i ≥ 2 of Dyson's model at ℓ = 2 1/3 also agree to the corresponding accuracy with the values implied through the functional flows in Tab. 2.
Our results based on the proper-time flow (5) with r PT has also been given in Tab. 2. Full agreement is achieved either with the exponent ν DHM or the subleading exponent ω DHM , but not with both of them. Once one of them is matched, the deviation in the other observable is of the order 10 −5 . The relevant parameters are m < 5 2 , the regime where (5) is mapped onto (2) [39]. Therefore, the values in Tab. 2 reflects well the range covered by standard Wilsonian flows (2). We expect that full agreement is achieved for proper-time flows which are linear combinations of (5) for different m, but we did not attempt to do so here.
In summary, we have provided numerical maps from several functional flows onto Dyson's model at a non-trivial ℓ = 1, with an accuracy of the order 10 −6 . The set of achievable values for scaling exponents from functional flows in the close vicinity of the optimal result is just wide enough to accommodate for the data from Dyson's model. This is a non-trivial result, also showing that the ℓ-dependence of Dyson's model and the R k -dependence of functional flows are very intimately related. Based on our results for ν and ω at ℓ = 1, and on continuity in ℓ, we expect that this map extends to other universal quantities in the same approximation, analogous to the full map which is known for ℓ = 1. Data for further symmetric and asymmetric corrections-to-scaling exponents, once available, will allow for additional checks of this picture. Full equivalence is guaranteed as soon as an explicit link in the form ℓ = ℓ(R k ) or R k (q 2 ) = R k (q 2 , ℓ) is furnished. For the local potential approximation, our results indicate that this map, if it exists, is not unique.

Discussion and Conclusions
Establishing equivalences between implementations of Wilson's renormalisation group as different as discrete hierarchical models of lattice scalar fields on one side and continuous functional flows on the other, allows for new views and insights on the respective formalisms and on the underlying physics. Previously, equivalences were known only in the limit where the hierarchical transformation becomes continuous. In this paper, based on similarities in the dependences related to the underlying coarse-graining, we have extended this link towards discrete hierarchical transformations. This correspondence shows that continuous RG flows (1) are sensitive to implicit discretisation effects via the momentum cutoff.
Specifically, for the 3d Ising universality class, we have compared the formalisms on the level of scaling exponents. Their dependence on the step-size parameter ℓ within Dyson's hierarchical model (6) is qualitatively and quantitatively similar to their dependence on the momentum cutoff R within functional flows (2). In either case, scaling exponents are bounded by the Wilson-Polchinski values obtained for ℓ → 1 and R → R opt . Once the hierarchical transformations are discrete, ℓ = 1, slight variations in all known scaling exponents from Dyson's model are matched by functional flows with non-optimal momentum cutoffs R = R opt . This is quite remarkable, particularily in view of the strong eigenvalue correlations found amongst functional flows. In this light, the optimisation of functional flows with R → R opt can now, alternatively, be viewed as the removal of discretisation effects, at least to leading order in a derivative expansion as studied here. It will be interesting to contrast these findings with the construction of improved or perfect actions on the lattice. More generally, it is conceivable that Dyson's model for arbitrary ℓ is mapped by functional flows on a fundamental level beyond the numerical map provided for ℓ = 2 1/3 . An explicit map would be very welcome, also in view of linking hierarchical models to a path integral representation of the theory. In Wilson's hierarchical model (8), the range covered by the leading scaling exponent indicates that a partial map onto functional flows exists, though only for a restricted domain of ℓ-values. Interestingly, the overlap with background field flows is even larger. Whether these maps extend beyond the leading exponent cannot be settled presently due to a lack of data for subleading exponents from Wilson's model.
In addition, we found a distinct correlation of scaling exponents from functional flows (2). The eigenvalue spectrum, a fingerprint of the physics in a local potential approximation, is severely constrained and achieves the Wilson-Polchinski values as an extremum. Furthermore, the full space of physical observables is described by very few parameters only, instead of the infinitely many moments of the momentum cutoff. Resolving this redundancy should prove useful for studies of e.g. non-trivial momentum structures and higher orders in the derivative expansion. Finally this pattern amongst physical observables highlights the extremum property of cutoffs leading to the optimal Wilson-Polchinski result [11][12][13][14]. We expect that the intimate link between optimised flows on one side, and extremum points in the space of observables on the other, persists in more complex theories. This observation will prove useful for studies in QCD and quantum gravity, where an appropriate optimisation is even more important to extract the relevant physics.