Universal Finite-Size Scaling for Percolation Theory in High Dimensions

arXiv:1606.00315v1 [cond-mat.stat-mech] 1 Jun 2016

Universal Finite-Size Scaling for Percolation Theory in High Dimensions Ralph Kenna1,3 and Bertrand Berche2,3 1

Applied Mathematics Research Centre, Coventry University, CV1 5FB, England

2

Universit´e de Lorraine, Statistical Physics Group, IJL, UMR CNRS 7198, Campus de Nancy, B.P. 70239, 54506 Vandœuvre l`es Nancy Cedex, France

3

L4 Collaboration & Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry, D-04009 Leipzig, Germany

June 2, 2016 Abstract We present a unifying, consistent, finite-size scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions. Behaviour at the critical point is non-universal; cluster proliferation is responsible for the breakdown of hyperscaling there when free boundary conditions are used but not when the boundary conditions are periodic. Universality is instead manifest at the pseudocritical point, where the non-proliferation scenario is independent of boundary conditions. The failure of hyperscaling in its traditional form there is universally ascribed to the growth in sizes of clusters rather than the growth of their number.

1

Introduction

Percolation theory has been the subject of extensive mathematical and simulational studies and is of relevance in a broad range of fields including physics, chemistry, network science, sociology, epidemiology, and geology. The existing picture of scaling close to the critical point is as follows. Below the upper critical dimension dc , where hyperscaling holds in its standard form, the average number of incipient spanning clusters is independent of finite-lattice size [1, 2]. The upper critical dimensionality is dc = 6, above which mean-field theory holds as a predictor of exponents governing critical behaviour [3–7]. Above dc , although there is a unique infinite cluster in the thermodynamic limit, the spanning clusters on finite lattices may proliferate and this phenomenon is associated with the breakdown of hyperscaling in its traditional form [8–10]. This picture was established over 30 years ago [8] in parallel with the formulation of the scaling picture for thermal phase transitions. More recently it has been proven that scaling in high dimensions is dependent on boundary conditions at the infinite-volume percolation threshold. While the proliferation phenomenon is expected for free boundary conditions (FBCs), where the largest clusters have fractal dimension D = 4 there, rigorous proofs have established that, under certain conditions, D = 2d/3 for periodic boundary conditions (PBCs), d being the dimension of the underlying lattice. The same value of D has been established for the expected largest cluster at the pseudocritical point. Thus the range of validity of the original finite-size scaling (FSS) picture of percolation in high dimensions has been severely reduced in favour of another, which is not universal at the critical point. The original picture was built on renormalization-group concepts such as Kadanoff’s length rescaling, Fisher’s dangerous irrelevant variables, as well as Binder’s thermodynamic length. The question therefore arises as to the status of these concepts under the new picture. The non-universality of FSS at the critical point also brings into question the proliferation mechanism for the failure of hyperscaling under different boundary conditions. Additionally, there are paradoxes associated with incipient spanning and infinite clusters [9] and there are apparent discrepancies between theory and simulations [11]. These need to be explained in a manner compatible with the scaling picture, the foundation for which is the renormalization group. That picture for percolation in high dimensions was reviewed in Refs. [9–12] and summarised in Ref. [13] as “incomplete, leaving an open challenge for future research.” Here we seek to meet that challenge. Recently, we introduced a new formulation for scaling and finite-size scaling (FSS) above the upper critical dimension [14–19]. The new framework is variously termed Q-scaling or Q-FSS or simply Q-theory to distinguish it from the older scaling scenario and its modifications [20]. In Refs. [14–19], the new set-up was expressed in terms of φ4 -theory and tested for spin models. Here we show that extending Q-scaling to percolation theory above the upper critical dimension fully resolves a number of important puzzles and unifies the above pictures. In particular, it shows that the proliferation picture holds only at the infinite-volume critical point for systems with FBCs where it explains the breakdown of hyperscaling in its standard form. For PBCs, the number of critical clusters is independent of lattice 1

volume and the failure of hyperscaling is instead due to the average radius of clusters scaling as a power of the lattice extent. This picture also holds at the pseudocritical point irrespective of boundary conditions and this is where universality is located. In the next section, we briefly review the evolution of percolation theory above six dimensions. We discuss how it parallels the progress of φ4 -theory above d = 4 and we summarize the recent development of Q-scaling there [14–19]. Then, in Section 3, we integrate the scaling theory [8–10] into the Q-framework, and show that it also accommodates the more recent results of Refs. [21–30]. In Section 4 we discuss the status of concepts and resolution of puzzles mentioned above and we conclude by making proposals for future research.

2

Scaling theories for percolation in high dimensions

We consider percolation on lattices of linear extent L and volume V = Ld . The probability that a site or bond is occupied is p. For small values of p, one typically has many small clusters of occupied sites or bonds and no infinite cluster in the thermodynamic limit. For sufficiently large p, on the other hand, there is typically also a spanning cluster which becomes infinite in size as L becomes infinite. The percolation threshold is typically characterised by pL in finite volume and p∞ in the infinite-volume limit. There, it is believed that there is no infinite cluster precisely at p∞ . The extension to long-range or spread-out models is straightforward [17, 19]. We refer to pL as the pseudocritical point and p∞ as the critical point. We represent by nL the density of clusters of size or mass s (the number of sites occupied by the cluster). In the infinite-volume limit this becomes n∞ and at the critical point this follows a power law n∞ ∼ s−τ . The mean size of the largest clusters diverges there as s ∼ |p − p∞ |−1/σ . In the counterpart FSS window, whose width is L−λ , the correlation length is is ξL ∼ Lϙ and the NL ∼ LX largest clusters scale in size as s ∼ LD .

2.1

Review of percolation theory in high dimensions

Prior to the 1980s, it was generally believed that there can only be a single infinite cluster at the percolation threshold [1, 31]. Then Newman and Schulman showed rigorously that the number of infinite clusters in infinite volume could actually be zero, one or infinity [32, 33]. In particular, in high dimensions, an infinite number of infinite clusters could, in principle, exist. It was expected, however, that at most a single infinite cluster exists in dimensions below dc . In 1984, Aharony, Gefen and Kapitulnik gave an argument linking the behaviour of percolation clusters to dangerous irrelevant variables and the breakdown of hyperscaling above the upper critical dimension [4]. They claimed that the fractal dimensionality D of the infinite cluster at the percolation threshold for dimensionalities d > 6 is stuck at D = 4 rather than the naive finite-size scaling prediction D = d − 2 which comes from inserting mean-field critical exponents into the d < dc scaling predictions. In 1985, Coniglio incorporated these developments to give a geometrical interpretation 2

for the breakdown in hyperscaling above dc [8]. According to this theory, the validity of hyperscaling below dc is equivalent to the presence of only a single critical cluster. Its breakdown in high dimensions is then supposed to be related to the proliferation and interpenetration of large clusters, and to the appearance at criticality of an infinite number of infinite clusters of zero density. In finite volume, the theory predicts there are NL ∼ Ld−6 such spanning clusters in the critical region, each with a mass LD with fractal dimension D = 4. An early numerical study of the topic came in 1987 when de Arcangelis provided evidence that there is indeed a unique critical cluster spanning the whole system at p∞ for d < dc , but multiple fragile spanning clusters when d > dc [31]. The numerics for small sizes did not quite match up with quantitative predictions, however. Ray and Klein numerically verified some predictions in the thermodynamic limit for long-range bond percolation [34]. In 1995 Aharony and Stauffer extended the scaling theory to consider the shift in the percolation threshold from the critical point p = p∞ in infinite volume to the pseudocritical point pL in finite volume. Their theoretical considerations were based on the arguments of Ref. [35] with new length scales introduced into the problem by dangerous irrelevant variables [36]. In the terminology of spin models, the relevant new length scale is the thermodynamic length [4, 37]. They predicted |p∞ − pL | ∼ AL−2 + BL−d/3 with A vanishing for PBCs [38]. This prediction received some numerical support in Ref. [39]. Since the rounding is given by L−d/3 for both PBCs and FBCs, but the shifting is L−2 in the latter case, the rounding is smaller than the shifting there. The apparent demarcation between d < dc and d > dc , in that the former manifests a single spanning cluster while the latter can have an infinite number of them, was undermined in 1996 when Hu, Lin and Hsu produced numerical evidence for numerous spanning clusters in two dimensions [40, 41]. Aizenman then gave a non-zero probability for k > 1 spanning clusters in 2D [9]. That there can be more than one spanning cluster at p∞ for d < dc has since been confirmed by Sen, Shchur, Cardy and others in the late 1990’s [12, 42–46]. However, in Ref. [47] it was shown that when there is percolation, the infinite cluster has to be unique. The single dominant cluster is referred to as the incipient infinite cluster [21, 30, 48]. Aizenman also produced a number of rigorous results confirming the picture of Refs. [4, 8] for d > 6 [9, 49, 50]. He considered percolation in a box of width L under bulk boundary conditions, meaning that the system is viewed as part of a larger domain; clusters in the box can be connected through sites outside of it. Assuming vanishing anomalous dimension above the upper critical dimension (established in certain circumstances in Refs. [6, 27, 28]), it was found that there are typically Ld−6 spanning clusters of size L4 . This was in line with the renormalization theory [4, 8] and similar scaling was expected for FBCs [9]. A different scaling behaviour was conjectured for the PBC case, namely D = 2d/3, which corresponds to the scaling law for the complete graph [51]. Indeed, it had long been known that at a certain probability, the size of the largest component of the Erd˝os-R´enyi random graph of V sites is V 2/3 . This was proven by Bollob´as [52] and Luczak [53] who also found that the scaling window has size V −1/3 . If percolation on finite graphs is similar to that in a Erd˝os-R´enyi random graph, it would 3

mean that the largest cluster has fractal dimension D = 2d/3 and that this holds inside a scaling window given by λ = d/3. This was the scaling window derived by Aharony and Stauffer in Ref. [38] (see also Ref. [54]) in the case of PBCs. In Refs. [22, 23], the connection with the Erd˝os-R´enyi random graph was further investigated for percolation on complete graphs, tori, hypercubes as well as finite-range percolation. Assuming that a quantity called the triangle diagram is sufficiently small (to ensure that the model corresponds to the mean-field theory) the largest connected component was found to indeed scale as V 2/3 inside a window of width V −1/3 . This was further developed for the torus in Refs. [24, 25], where it was shown that the critical point of the infinite lattice lies inside this scaling window. According to this picture, the diameter of large clusters is of order V 1/3 [26, 29, 55]. To summarise, on the one hand we have the Kadanoff-scaling (based ultimately on the renormalization group) and geometrical theories which tell us that D = 4 and X = d − 6 above the upper critical dimension. These results were proven when the correlation function decays with vanishing anomalous dimension, and this in turn is a prediction of mean-field theory [6, 27, 28]. In this context, Aharony and Stauffer give that λ = d/3 and λ = 2 for PBCs and FBCs, respectively. On the other hand the suggestion from the Erd˝os-R´enyi picture is that D = 2d/3 and that λ = d/3. This has been proven rigorously, subject to certain conditions, at least for PBCs in Refs. [22–25]. A number of questions arise. Since the Kadanoff renormalization picture, which explains the breakdown of hyperscaling through the proliferation scenario, manifestly only applies to the situation of FBCs at p∞ , what is the counterpart explanation for hyperscaling collapse for PBCs? What are the values of the other critical exponents τ , σ and X? Hyperscaling notwithstanding, what is the reason for the apparent breakdown of universal FSS in percolation theory above the upper critical dimension? Numerical simulations have hitherto delivered unsatisfactory results. What is a sensible direction for future simulations aimed at testing the above scenarios? In what follows, we provide answers to all of these questions and more. Our approach follows the recent developments of Refs. [14–19]. To explain and contextualise these, we have to give a brief review of φ4 -theory, which has evolved in parallel to percolation theory, and of recent developments there.

2.2

Universal and non-universal FSS for φ4 -theory in high dimensions

To place the theoretical problems associated with percolation in high dimensions in the context of this paper, we look at the parallel situation in spin models above dc . Dangerous irrelevant variables, which were integral to the percolation theory developed by Aharony et al. [4], were first introduced by Fisher for infinite-volume spin systems in Ref. [36]. The theory was formulated to repair a mismatch between some of the critical exponents coming from the Gaussian fixed point and those predicted by mean-field theory or Landau theory. These were the exponents α for the specific heat; β for the temperature dependence of the magnetisation and δ for its dependency on external field. Each of these is derivable from the free energy, and the danger associated with an irrelevant scaling 4

field in the equivalent φ4 -theory therefore applies to this sector. Because the values of the correlation-length exponent ν and the anomalous dimension η coming from the Gaussian fixed point coincide with those from Landau theory, it was not expected that the irrelevant term presents danger for the correlation sector. Since the susceptibility is linked to the correlation function by the fluctuation-dissipation theorem, one expects it to escape the danger of irrelevant variables too. Indeed, the value γ = 1 coming from the Gaussian fixed point is the same as that from mean-field theory, and this seemed to confirm said belief. In 1985, Binder, Nauenberg, Privman and Young applied Fisher’s concept of dangerous irrelevant variables to the finite-volume case [35]. Crucially, their theory demanded that the correlation length ξL cannot exceed the actual length L of the system, an assumption explicitly inputted on physical grounds. A consequence was that the standard formulation of FSS in terms of a ratio of length scales ξ∞ /L failed to deliver the correct size dependency of thermodynamic functions above the upper critical dimension. To achieve this, Binder introduced another length scale: the thermodynamic length [37]. Secondly, not only did hyperscaling and FSS break down above dc but universality did too; the finite-size behaviour seemed to depend strongly on boundary conditions. In particular, free boundaries appeared to deliver different results to periodic boundaries [14, 56–58]. The standard paradigm was developed in a rather ad hoc manner. Although it delivered the correct exponents by construction, it had a number of shortcoming. For example, it was unexplained a priori why the dangerous irrelevant variable mechanism would apply to one sector and not the other and, the Ginzburg criterion notwithstanding, there was no explanation for the collapse of universality above dc . The theory also involved a distinction between the behaviour of correlation functions over “short long” distances and “long long” distances which has parallels in percolation theory. Under the new Q-scheme [14–19], the thermodynamic length is obviated and hyperscaling (in a new form), finite-size scaling as well as universality all hold above the upper critical dimension. A crucial ingredient that differentiates Q-theory from the earlier set-up is that dangerous irrelevant variables are also allowed to affect the correlation sector of the model. In particular, the restriction that the correlation length be bounded by the length no longer applies [59]. This necessitates the introduction of a new exponent governing the FSS of the correlation length: ξL ∼ Lϙ .

(1)

The new exponent ϙ (“koppa”) is properly called a pseudocritical exponent rather than a critical one since it controls FSS inside the scaling window rather than scaling with temperature. It takes the value ϙ = 1 for d < dc where the standard theory of FSS holds sway. However, above the upper critical dimension, ϙ = d/dc (at least at pseudocriticality) so that the correlation length scales faster than the length of the system. For spin models with PBCs, the FSS window in which Eq.(1) holds includes both the critical and pseudocritical points. For FBCs, on the other hand, it includes only the pseudocritical point; because of the largeness of the shifting the critical point lies outside the region in which Eq.(1) holds. Thus ϙ is physical and universal, like the other exponents. The new 5

Table 2.1: Summary of scaling theory for percolation, expressed in terms of FSS as ξL ∼ Lϙ , s ∼ LD , NL ∼ LX and |p∞ − pL | ∼ L−λ . Regime of validity

ϙ

d < 6; at pL and p∞

1

at p∞

1

D β+γ ν β+γ =4 ν

at pL

d 6 d 6

(β + γ)ϙ 2d = ν 3 2d (β + γ)ϙ = ν 3

d > 6; FBCs

d > 6; PBCs at pL , p∞

X 0

λ 1 (usually) ν

d−6 1 =2 ν 0 0

ϙ d = ν 3

form for hyperscaling, which holds in all dimensions is [14] νd = 2 − α. ϙ

(2)

The breakdown in the old version of hyperscaling (νd = 2 − α), at least for thermal phase transitions, is because ϙ 6= 1 for d > dc . These results are consistent with explicit calculations and numerics in the case of the φ4 -theory [14–17, 19, 56]. Moreover, the old picture, in which ξL is bounded by L, is by now firmly falsified [18]. In Section 3 we bring percolation theory into the fold of Q-theory, resolving all of the above-mentioned outstanding issues above the upper critical dimension there. For convenience, in the next subsection we summarise some of the essential predictions of the new scheme.

2.3

Summary

In Table 2.1, we summarise the essential outcomes of the unifying Q-FSS theory for percolation in terms of the values of ϙ, D, X and λ. The critical exponents β, γ and ν, in the table, are associated with the density of sites in the largest critical clusters, the mean cluster size and the correlation length (also called connectedness length), respectively. In all cases, τ = 2 + β/(β + γ) and σ = 1/(β + γ). (Note that, below the upper critical dimension λ is usually given by 1/ν as in field theory, but this is not a requirement and one can construct boundary conditions which give alternative values of the shift exponent [60]. But any such alternatives arise through mechanisms different to that discussed in this paper, so we just give the usual value in the table.) Above the upper critical dimension, where the mean-field critical exponents are β = 1, γ = 1, ν = 1/2, these give τ = 5/2 and σ = 1/2. (The other mean-field critical exponents are, in standard notation, α = −1, δ = 2, η = 0.) Some results in the second and fourth columns are new, as are the scaling 6

relations involving ϙ in the third and fifth columns as well as the unifying framework which gives consistency between disparate results discussed above.

3

Q-Scaling theory for percolation

In this section we develop a heuristic Q-scaling theory for percolation by taking into account the possibility that ϙ 6= 1 in Eq.(1). We recognise that setting ϙ = 1 recovers the starting point of previous theories above dc [8–10]. The basic observables are defined in a standard manner. The density per unit volume (or per site if the lattice spacing is taken to be unity) of clusters of size s is nL (p, s). We then define X nL (p, s) (3) KL (p) = s

as the average number of clusters per site. The density of sites in the largest critical clusters is represented by PL (p). Now, if a site is active, it is either in the largest or non-largest clusters, so [13] X′ snL (p, s), (4) PL (p) = p − s

where the prime indicates that the largest clusters are omitted from the sum. The order parameter for the phase transition is P∞ (p). Finally, the probability Pthat an arbitrary site (whether active or not) belongs to a size-s cluster is w = snL (p, s)/ snL (p, s). Therefore the mean size for a cluster selected by randomly choosing sites is X X SL (p) = sw(p, s) ∝ s2 nL (p, s). (5) s

s

As the percolation threshold is approached in the infinite-volume system, these various moments behave in manners analogous to scaling at a temperature-driven phase transition: K∞ (p) ∼ |p − p∞ |2−α , P∞ (p) ∼ |p − p∞ |β , S∞ (p) ∼ |p − p∞ |−γ .

(6) (7) (8)

The correlation length also diverges [in percolation theory, ξL (p) represents the typical cluster radius on a lattice of extent L] and we write ξ∞ (p) ∼ |p − p∞ |−ν .

(9)

As in the case of a thermal phase transition, there are two scaling relations between the four critical exponents α, β, γ and ν. Our aim is to understand the origin and status of these and other scaling relations in the percolation context as well as the FSS of critical clusters. Also, as for thermal phase transitions, we may define the pseudocritical point as the value pL of p at which the susceptibility SL (p) is maximised for a given value of L. 7

We follow Coniglio and start with Kadanoff rescaling of the lattice lengths by a factor b [8, 10]; define L ξL s L′ = , ξL′ = ϙ , s′ = D , (10) b b b where D is the fractal dimension of the largest scaling clusters responsible for critical behaviour. Following Refs. [14–16], rescaling of the correlation length gives ξL (p, s) = bϙ Ξ[b−1 L, b−ϙ (p − pL )−ν , b−D s].

(11)

Here the scaling prefactor is bϙ , correcting an explicit assumption made in Ref. [35] and elsewhere. In the second argument of the unspecified function Ξ we have written the explicit p-dependence as a correlation length, meaning that it picks up a factor b−ϙ on rescaling. Also, we have used p − pL for its functional form rather than p − p∞ after Ref. [61]. As in Refs. [8, 10], if we assume that the large clusters do not interpenetrate, their number between s and s + ds is the same before and after rescaling: Ld nL (p, s)ds = L′ d nL′ (p′ , s′ )ds′ . Interpenetration picks up an extra scaling factor so that [8, 10] d

Ld nL (p, s)ds = bX L′ nL′ (p′ , s′ )ds′ ,

(12)

  nL (p, s) = bX−d−D f b−1 L, b−ϙ (p − pL )−ν , b−D s .

(13)

which we re-express as

3.1

Rescaling the correlation length

Setting b = s1/D in Eq.(11) and defining σ= we obtain

ϙ , νD

(14)

  1 ϙ ξL (p, s) = s D Ξ sL−D , s|p − pL | σ .

(15)

[The function Ξ(v, w) in Eq.(15) is clearly not the same as Ξ(x, y, z) in Eq.(11) but here, and throughout, we use the same notation loosely, to indicate a function of its arguments when precise knowledge of the function is unavailable and unnecessary.] Taking the limit L → ∞ so that the first argument of the function on the right hand side drops out,   1 −ν σ ξ∞ (p, s) = |p − p∞ | Ξ s|p − p∞ | . (16) Then ξ∞ [p, s(p)] ∼ |p − p∞ |−ν ,

(17)

recovering Eq.(9), provided 1

s(p) ∼ |p − p∞ |− σ .

8

(18)

Setting p = pL in Eq.(15), while keeping L finite, so that we are sitting at the pseudocritical point; ξL [pL , s(pL )] = Lϙ Ξ(sL−D ). (19) This delivers ξL [pL , sL (pL )] ∼ Lϙ ,

(20)

consistent with Eq.(1), provided sL (pL ) ∼ LD . Finally we consider Eq.(15) at the critical point p = p∞ ; ϙ

1

ξL [p∞ , s(p∞ )] = s D Ξ(sL−D , s|p∞ − pL | σ ).

(21)

Define the shift exponent λ through |p∞ − pL | ∼ L−λ , so that

(22) λ

ξL [p∞ , s(p∞ )] = Lϙ Ξ(LD− σ ).

(23)

This is Lϙ provided λ = Dσ = ϙ/ν. But if ϙ = 1, then ξL (p∞ ) ∼ L provided λ = 1/ν. According to Q-FSS, the value of λ indeed depends on the boundary conditions. In Ref. [14] we found that λ = ϙ/ν in the PBC case, so that ξL (p∞ , s) = Lϙ , i.e., the pseudocritical-type FSS extends as far as the critical point. This result was also found in Refs. [24, 25] for sufficiently long-range interactions (spread-out percolation). For FBCs, however, λ = 1/ν and the shifting exceeds the rounding [14, 38, 61]. Therefore, although FSS at the pseudocritical point and FSS at the critical point come from the same scaling function in the FBC case (as they do in the PBC case), the Q-FSS window, which is centred on the pseudocritical point, does not reach the critical point there [14, 38, 56, 61].

3.2

Rescaling critical cluster numbers

Set b = s1/D in Eq.(13) to find   1 −D σ nL (p, s) = s f sL , s|p − pL | , −τ

(24)

where σ is defined in Eq.(14) and τ=

D+d−X . D

(25)

We start by letting L → ∞ so that the first argument on the right hand side of Eq.(24) drops out;   1 −τ σ n∞ (p, s) = s f s|p − p∞ | . (26) Therefore the distribution of the scaling clusters follows a power law at criticality, n∞ (p∞ , s) ∼ s−τ . 9

(27)

Keeping L finite, and setting p = pL in Eq.(24),
nL [pL , s(pL )] = s−τ f sL−D ∼ LX−D−d ,

(28)

provided s(pL ) ∼ LD . Next, with L still finite, set p = p∞ in Eq.(24);   λ nL [p∞ , s(p∞ )] = s−τ f sL−D , sL− σ ,

(29)

Therefore, if s(p∞ ) ∼ LD ,   λ nL [p∞ , s(p∞ )] = LX−D−d f LD− σ ∼ LX−D−d ,

(30)

provided λ = Dσ = ϙ/ν. But if ϙ = 1, we find λ = 1/ν, again.

3.3

Non-proliferation and proliferation of critical clusters

We identify the number of critical clusters (scaling as s ∼ LD in the critical region) to be Z d nL (p, s)ds ∼ Ld nL s. (31) NL = L From Eq.(28) or (30), we identify NL ∼ LX ,

(32)

so that X is the exponent governing the number of critical clusters. Following Refs. [8,10], we assume that these are responsible for critical behaviour so that X−d

KL ∼ L−d NL ∼ nL s ∼ LX−d ∼ ξL ϙ .

(33)

Next, Eq.(4) gives PL ∼

Z

X+D−d

nL sds ∼ nL s2 = L−d NL s ∼ LX+D−d ∼ ξL ϙ

,

(34)

which expresses the fact that the probability for a site to be in any of the largest clusters equals the number of such clusters per unit volume times their size. Finally, Eq.(5), gives Z X+2D−d SL ∼ nL s2 ds ∼ nL s3 = L−d NL s2 ∼ LX+2D−d ∼ ξL ϙ , (35) which gives two ways to calculate the average cluster size. Taking the thermodynamic limit, Eqs.(33), (34) and (35) become X−d

K∞ (p) ∼ ξ∞ϙ

ν

∼ |p − p∞ |− ϙ (X−d)

X+D−d

P∞ (p) ∼ ξ∞ ϙ

ν

∼ |p − p∞ |− ϙ (X+D−d)

X+2D−d

S∞ (p) ∼ ξL ϙ

ν

∼ |p − p∞ |− ϙ (X+2D−d) . 10

(36) (37) (38)

Comparing these with Eqs.(6), (7) and (8), we obtain α + 2β + γ = 2, ϙ(β + γ) , D = ν ν(d − X) = 2 − α. ϙ

(39) (40) (41)

The first of these is the Essam-Fisher relation, which always holds. the second gives the fractal dimension of the largest clusters corrected for Q-theory. The third is a hyperscaling relation. We will show that it extends the traditional form of hyperscaling to the full scaling window above the upper critical dimension (encompassing both the pseudocritical and critical points). Before that, we comment that Eqs.(14) and (25) give 1 , β+γ β τ = 2+ , β+γ

σ =

(42) (43)

each of which is independent of ϙ. We now examine three scenarios for the ϙ-dependent Eqs.(40) and (41). Scenario 1: Eq.(41) recovers the traditional form of hyperscaling (νd = 2 − α) when X = 0 and ϙ = 1. This is the situation below the upper critical dimension and Eqs.(40) then gives β+γ , (44) D= ν recovering the standard result there. Scenario 2: When the traditional form of hyperscaling (νd = 2 − α) fails and ϙ = 1, we have the situation above the upper critical dimension considered in Refs. [8–10]. Eq.(44) holds there with D “stuck” at 4, independent of d > 6, as observed Ref. [4]. In this case, Eq.(41) gives ν(d − X) = 2 − α, (45) or X = d − dc .

(46)

Eq.(45) mean that νX is an additive correction term, extending hyperscaling above the upper critical dimension. Eqs.(46) means that the clusters whose fractal dimension is D = 4 proliferate above d = dc , just as predicted in Ref. [8]. But as we have seen, ϙ = 1 [and therefore Eq.(46)] only holds at the critical point p = p∞ for FBCs. This means that the scaling relation (45) and the associated value of X is non-universal. We can refer to such proliferating clusters as spanning clusters to follow previous terminology [9]. Scenario 3: When ϙ = d/dc and X = 0, 11

(47)

Eq.(41) recovers Eq.(2) in which ϙ multiplicatively extends hyperscaling above the upper critical dimension. Eq.(47) means that the critical clusters do not proliferate. As we have seen, this is valid universally at the pseudocritical point as well as at the critical point for PBCs. Eq.(40) gives 2d β+γ = , (48) D=ϙ ν 3 meaning that D is dimension dependent and is not “stuck” at its mean-field value D = 4 as earlier thought [4]. To adhere to the terminology of previous theories, we may refer to these non-proliferating clusters as incipient infinite clusters [21, 30, 48].

4

Discussion and Conclusions

Here we presented a unified and consistent FSS picture for percolation theory by placing it within a recently developed renormalization-group-based framework for scaling above the upper critical dimension [14–19]. The new picture stems from Eq.(11), which itself comes from extending the danger of irrelevant variables to the correlation sector in the renormalization-group approach. It does not require the notion of a thermodynamic length as discussed in various literature [4, 11, 62] and we suggest that this concept is now obsolete [14, 16]. The previous explanation [8–10] for the breakdown of hyperscaling above dc , namely the proliferation of spanning clusters there, survives in Q-theory but only at the critical point for systems with PBCs, for which ϙ = 1. This picture does not apply at the pseudocritical point and therefore it is not universal. Instead, the value of ϙ is universally d/dc at the pseudocritical point and it can take either this value or the value 1 at the infinite-volume critical point. This value for ϙ explains the collapse of hyperscaling in its traditional form above dc and extends it via Eq.(2) there. In universal Q-theory, the correction to hyperscaling is multiplicative, in contrast to the additive correction coming from non-universal older theories [8–10]. Although the typical cluster radius (the correlation length) scales faster than the system length, since the correlation volume is ξLdc , critical clusters can fit into the lattice volume Ld with ϙ = d/dc [14]. The Q-FSS prediction for the fractal dimensionality D = 2d/3 of the largest clusters at pseudocriticality is universal and agrees with rigorous results established under certain conditions [22, 23, 52, 53]. The previous rigorous result D = 4 for FBCs at the critical point [9, 49, 50] is based on an assumption that the anomalous dimension η vanishes above the upper critical dimension. As shown theoretically and confirmed numerically in Ref. [16], this is not true on the scale of L at the pseudocritical point. It is, however, correct on the scale of ξL there. Indeed, replacing L by ξL in the old theory of Refs. [8–10], gives NL ∼ ξL4 ∼ L2d/3 , in agreement with Eq.(48). This is the only length scale which survives in the infinite-volume limit, explaining previous results there [34]. In this context too, we recall the Stauffer paradox discussed in Ref. [9]. According to Refs. [32,33], there is almost surely zero, one or an infinite number of clusters in an infinite system. If spanning clusters are the finite-volume counterparts of infinite clusters, there 12

should be zero or one of them (since we can’t have an infinite number on a finite lattice). But this is in contradiction with a theorem which states that there can be more than one incipient spanning cluster [9]. Thus there appeared to be a paradoxical contradiction between one infinite cluster and more than one spanning clusters. The previous view was that this is resolved by differentiating short long scales from long long scales and whether one takes a local or macroscopic view of cluster numbers. But this led to another paradox wherein the infinite system represented just a small part of the finite one [9]. This paralleled the situation in ferromagnetic systems where short long range order was defined on the infinite system and long long range corresponded to distances between spins which scale along with the lattice size. This paradox was fully resolved in Ref. [16]. There is no need to distinguish between short long and long long ranges. Instead there are two different measures; length L and correlation length ξL . Measured on the scale of ξL , the anomalous dimension vanishes. But on the scale of the system size L, it does not. Thus the Staufer paradox is resolved and the distinction between incipient infinite clusters and incipient spanning clusters is merely one of sitting at the pseudocritical and critical points, respectively (in the FBC case). Finally, we have seen that the average cluster radius is ξL ∼ Lϙ at pseudocriticality and also at criticality in the PBC case. This is related to the average cluster mass SL ∼ Lϙβ/ν = L2ϙ via the usual expression of the susceptibility as the square of the correlation length above dc . We have also seen that the largest critical clusters scale as s ∼ LD = L2d/3 . Assuming the same relationship between cluster mass and radius, this suggests that the linear extent of the largest clusters is Ld/3 . This is in agreement with explicit calculations on the Erd˝os-R´enyi random graph and recently proven in Refs. [26, 29, 55]. As we have seen, throughout the long history of renormalization-founded approaches to the statistical physics of percolation theory, theoretical developments have been followed by numerical confirmation efforts. These, however, have only had mixed success above dc [11–13, 31, 34, 38, 39, 54]. We hope that this paper will stimulate renewed efforts on the numerical side, clearly differentiating between criticality and pseudocriticality as well as between FBCs and PBCs. Also, because of the connection between percolation and critical behaviour in the Ising model, for example, we suggest that the breakdown of hyperscaling in the latter case is also associated with rapidly diverging radii of Fortuin-Kasteleyn clusters, rather than an infinite multiplicity of percolating clusters.

Acknowledgements: We thank Ferenc Igl´oi for comments at the start of this work which was supported by EU FP7 Projects No. 295302, “Statistical Physics in Diverse Realisations,” No. 612707, “Dynamics of and in Complex Systems,” and by the Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry at the Franco-German University.

13

References [1] J.W. Essam, Percolation Theory, Rep. Prog. Phys. 43 (1980) 833-912. [2] D. Stauffer, Introduction to Percolation Theory, Taylor and Francis, London, 1985. [3] A.B. Harris, T.C. Lubensky, W. Holcomb and C. Dasgupta, Renormalization-Group Approach to Percolation Problems, Phys. Rev. Lett. 35 (1975) 327-330; Erratum ibid 1397. [4] A. Aharony, Y. Gefen and A. Kapitulnik, Scaling at the Percolation Threshold above Six Dimensions, J. Phys. A 17 (1984) L197-L202. [5] M. Aizenman and C. Newman, Tree Graph Inequalities and Critical Behavior in Percolation Models, J. Stat. Phys. 36 (1984) 107-143. [6] T. Hara and G. Slade, Mean-Field Behavior for Percolation in High Dimensions, Commun. Math. Phys. 128 (1990) 333-391. [7] D. Barsky and M. Aizenman, Percolation Critical Exponents under the Triangle Condition, Ann. Probab. 19 (1991) 1520-1536. [8] A. Coniglio, Shapes, Surfaces and Interfaces in Percolation Clusters, in “Springer Proceedings in Physics, Vol. 5: Physics of Finely Divided Matter”, by M. Daoud, N. Boccara (Eds.), Proc. of Les Houches Cong. on Physics of Finely Divided Matter, Springer Verlag, Berlin (1985) pp 84-101. [9] M. Aizenman, On the Number of Incipient Spanning Clusters, Nucl. Phys. B 485 (1997) 551-582. [10] A. Coniglio, Geometrical Approach to Phase Transitions in Frustrated and Unfrustrated Systems, Physica A 281 (2000) 129-146. [11] S. Fortunato, D. Stauffer and A. Coniglio, Percolation in High Dimensions is Not Understood, Physica A 334 (2004) 307-311. [12] D. Stauffer, Minireview: New Results for Old Percolation, Physica A 242 (1997) 1-7. [13] S. Fortunato, A. Aharony, A. Coniglio and D. Stauffer, Number of Spanning Clusters at the High-Dimensional Percolation Thresholds, Phys. Rev. E 70 (2004) 056116. [14] B. Berche, R. Kenna and J.C. Walter, Hyperscaling above the Upper Critical Dimension, Nucl. Phys. B 865 (2012) 115-132. [15] R. Kenna and B. Berche, A New Critical Exponent ϙ and its Logarithmic Counterpart ˆϙ, Cond. Matter Phys. 16 (2013) 23601. [16] R. Kenna and B. Berche, Fisher’s Scaling Relation above the Upper Critical Dimension, EPL 105 (2014) 26005. 14

[17] E.J. Flores-Sola, B. Berche, R. Kenna, M. Weigel, Finite-Size Scaling above the Upper Critical Dimension in Ising Models with Long-Range Interactions, Eur. Phys. J. B 88 (2015) 28. [18] R. Kenna and B. Berche, Scaling and Finite-Size Scaling above the Upper Critical Dimension, in “Order, Disorder, and Criticality: Advanced Problems of Phase Transition Theory” Vol. 4, Yu. Holovatch (Ed.), World Scientific (Singapore, 2015) pp 1-54. [19] E.J. Flores-Sola, B. Berche, R. Kenna, M. Weigel, Role of Fourier Modes in Finite-Size Scaling above the Upper Critical Dimension, Phys. Rev. Lett. 116 (2016) 11570. [20] J.G. Brankov, D.M. Danchev and N.S. Tonchev, Theory of Critical Phenomena in Finite-Size Systems: Scaling and Quantum Effects, World Scientific, Singapore, 2000. [21] R. van der Hofstad and A. J´arai, The Incipient Infinite Cluster for High-Dimensional Unoriented Percolation, J. Statist. Phys. 114 (2004) 625-663. [22] C. Borgs, J.T. Chayes, R. van der Hofstad, G. Slade and J. Spencer, Random Subgraphs of Finite Graphs: I. The Scaling Window under the Triangle Condition, Random Structures and Algorithms 27 (2005) 137-184. [23] C. Borgs, J.T. Chayes, R. van der Hofstad, G. Slade and J. Spencer, Random Subgraphs of Finite Graphs: II. The Lace Expansion and the Triangle Condition, Ann. Probab. 33 (2005) 1886-1944. [24] M. Heydenreich and R. van der Hofstad, Random Graph Asymptotics on High-Dimensional Tori , Comm. Math. Phys. 270 (2007) 335-358. [25] M. Heydenreich and R. van der Hofstad, Random Graph Asymptotics on High-Dimensional Tori II: Volume, Diameter and Mixing time, Probab. Theory Related Fields 149 (2011) 397-415. [26] R. van der Hofstad and A. Sapozhnikov, Cycle Structure of Percolation on High-Dimensional Tori , Ann. Inst. H. Poincar´e Probab. Statist. 50 (2014) 999-1027. [27] T.Hara, R. van der Hofstad and G. Slade, Critical Two-Point Functions and the Lace Expansion for Spread-Out High-Dimensional Percolation and Related Models, Ann. Probab. 31 (2003) 349-408 [28] T. Hara, Decay of Correlations in Nearest-Neighbor Self-Avoiding Walk, Percolation, Lattice Trees and Animals, Ann. Probab. 36 (2008) 530-593. [29] A. Nachmias and Y. Peres, Critical Random Graphs: Diameter and Mixing Time, Ann. Probab. 36 (2008) 1267-1286. [30] M. Heydenreich, R. van der Hofstad, and T. Hulshof, High-Dimensional Incipient Infinite Clusters Revisited , J. Stat. Phys. 155 (2014) 966-1025, . 15

[31] L. de Arcangelis, Multiplicity of Infinite Clusters in Percolation above Six Dimensions, J. Phys. A 20 (1987) 3057-3061. [32] C.M. Newman and L.S. Schulman, Infinite Clusters in Percolation Models, J. Stat. Phys. 26 (1981) 613-628. [33] C.M. Newman and L.S. Schulman, Number and Density of Percolating Clusters, J. Phys. A 14 (1981) 1735-1743. [34] T.S. Ray, W. Klein, Crossover and the Breakdown of Hyperscaling in Long-Range Bond Percolation, J. Stat. Phys. 53 (1988) 773-794. [35] K. Binder, M. Nauenberg, V. Privman, and A.P. Young, Hyperscaling, Phys. Rev. B 31 (1985) 1498-1502.

Finite Size Tests of

[36] M.E. Fisher, Scaling, Universality and Renormalization Group Theory, in “Lecture Notes in Physics 186, Critical phenomena,” ed F.J.W. Hahne, (Springer Verlag, Berlin, 1983) pp 1-139. [37] K. Binder, Critical Properties and Finite-Size Effects of the Five-Dimensional Ising Model , Z. Phys. B 61 (1985) 13-23. [38] A. Aharony and D. Stauffer, Finite-size scaling for percolation above six dimensions, Physica A 215 (1995) 242-246. [39] D. Stauffer and R.M. Ziff, Reexamination of seven-dimensional site percolation thresholds, Int. J. Mod. Phys. C 11 (2000) 205-209. [40] C.-K. Hu and C.-Y. Lin, Universal Scaling Functions for Numbers of Percolating Clusters on Planar Lattices, Phys. Rev. Lett. 77 (1996) 8-11. [41] H.-P. Hsu, C.-Y. Lin and C.-K. Hu, Universal Scaling Functions for Bond Percolation on Planar-Random and Square Lattices with Multiple Percolating Clusters, Phys. Rev. E 64 (2001) 016127. [42] P. Sen, Non-Uniqueness of Spanning clusters in Two to Five Dimensions, Int. J. Mod. Phys. C 7 (1996) 603-607. [43] P. Sen, Probability Distribution and Sizes of Spanning Clusters at the Percolation Thresholds, Int. J. Mod. Phys. C 8 (1997) 229-237. [44] L.N. Shchur, S.S. Kosyakov, Probability of Incipient Spanning Clusters in Critical Square Bond Percolation, Int. J. Mod. Phys. C 8 (1997) 473-481. [45] L.N. Shchur, S.S. Kosyakov, Probability of Incipient Spanning Clusters in Critical Two-Dimensional Percolation, Nucl. Phys. B (Proc. Suppl.) 63A-C (1998) 664-666. [46] J. Cardy, The Number of Incipient Spanning Clusters in Two-Dimensional Percolation, J. Phys. A 31 (1998) L105-L110. 16

[47] M. Aizenman, K. Kesten and C.M. Newman, Uniqueness of the Infinite Cluster and Continuity of Connectivity Functions for Short and Long Range Percolation, Commun. Math. Phys. 111 (1987) 505-531. [48] H. Kesten, The incipient infinite cluster in two-dimensional percolation, Probab. Theory Related Fields 73 (1986) 369-394. [49] M. Aizenman, The Geometry of Critical Percolation and Conformal Invariance, in “Proceedings of the 19th IUPAP Intern. Conf. on Statistical Physics (Xiamen, China 1995)”, H. Bai-lin (Ed.), World Scientific 1995. [50] M. Aizenman, Scaling Limit for the Incipient Spanning Cluster , in “Mathematics of Materials: Percolation and Composites”, by K.M. Golden, G.R. Grimmett, R.D. James, G.W. Milton, and P.N. Sen. (Eds.), The IMA Volumes in Mathematics and its Applications (Springer Verlag, Berlin, 1997). [51] P. Erd˝os and A. R´enyi, On the Evolution of Random Graphs, Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl. 5 (1960) 17-61. [52] B. Bollob´as, The Evolution of Random Graphs, Trans. Amer. Math. Soc. 286 (1984) 257-274. [53] T. Luczak Component Behavior near the Critical Point of the Random Graph Process, Random Structures Algorithms 1 (1990) 287-310. [54] D. Stauffer, Finite-Size Effect in Seven-Dimensional Site Percolation, Physica A 210 (1994) 317-319. [55] G. Kozma and A. Nachmias, A note about critical percolation on finite graphs, J. Theor. Prob. 24 (2011) 1087-1096. [56] J. Rudnick, G. Gaspari and V. Privman, Effect of Boundary Conditions on Critical Behavior of a Finite High-Dimensional Ising Model , Phys. Rev. B 32 (1985) 7594-7596. [57] P. H. Lundow and K. Markstr¨om, Non-Vanishing Boundary Effects and Quasi-First Order Phase Transitions in High Dimensional Ising Models, Nucl. Phys. B 845, 120 (2011). [58] P. H. Lundow and K. Markstr¨om, Finite Size Scaling of the 5D Ising Model with Free Boundary Conditions, Nucl. Phys. B 889, 249 (2014). [59] E. Br´ezin, An Investigation of Finite Size Scaling, J. Physique 43 (1982) 15-22. [60] W. Janke and R. Kenna, Finite-Size Scaling and Corrections in the Ising Model with Brascamp-Kunz Boundary Conditions, Phys. Rev. B 65 (2002) 064110. [61] M. Wittmann and A.P. Young, Finite-Size Scaling above the Upper Critical Dimension, Phys. Rev. E 90 (2014) 062137. 17

[62] G. Andronico, A. Coniglio and S. Fortunato, A Geometrical Interpretation of Hyperscaling Breaking in the Ising Model , Nucl. Phys. B (Proc. Suppl.) 119 (2003) 876-878.

18