Topological effects and conformal invariance in long-range correlated random surfaces

We consider discrete random fractal surfaces with negative Hurst exponent $H<0$. A random colouring of the lattice is provided by activating the sites at which the surface height is greater than a given level $h$. The set of activated sites is usually denoted as the excursion set. The connected components of this set, the level clusters, define a one-parameter ($H$) family of percolation models with long-range correlation in the site occupation. The level clusters percolate at a finite value $h=h_c$ and for $H\leq-\frac{3}{4}$ the phase transition is expected to remain in the same universality class of the pure (i.e. uncorrelated) percolation. For $-\frac{3}{4}<H<0$ instead, there is a line of critical points with continously varying exponents. The universality class of these points, in particular concerning the conformal invariance of the level clusters, is poorly understood. By combining the Conformal Field Theory and the numerical approach, we provide new insights on these phases. We focus on the connectivity function, defined as the probability that two sites belong to the same level cluster. In our simulations, the surfaces are defined on a lattice torus of size $M\times N$. We show that the topological effects on the connectivity function make manifest the conformal invariance for all the critical line $H<0$. In particular, exploiting the anisotropy of the rectangular torus ($M\neq N$), we directly test the presence of the two components of the traceless stress-energy tensor. Moreover, we probe the spectrum and the structure constants of the underlying Conformal Field Theory. Finally, we observed that the corrections to the scaling clearly point out a breaking of integrability moving from the pure percolation point to the long-range correlated one.


Introduction
The percolative properties of random fractal surfaces have been studied for a long time [1][2][3][4]. The universality class of their critical points remains a very active subject of research in the mathematical [5][6][7] and in the theoretical physics [8] communities, mainly because they challenge our understanding of both the emergence of conformal symmetry and of the way this symmetry is implemented. Let us consider a random stationary function u(x) on a lattice u(x) : Z 2 → R which satisfies: E [u(x)] = 0, E (u(x) − u(y)) 2 ∼ C(H) |x − y| 2 H (|x − y| >> 1) (1.1) where E [· · · ] is the average over the instances of u(x), the symbol ∼ stands for asymptotically equivalent and C(H) is some constant depending on H. The number H, H ∈ R, is the surface roughness exponent [9], also known as Hurst exponent. The fractional Gaussian surfaces [7] that we consider here, see (1.4) below, is a class of random surfaces which satisfy the above properties. A site is therefore activated with probability p(h): where we use the translational invariance in law. The set of activated points is usually known as the excursion set [10]. The study of the connected components of the excursion set, hereafter referred to as level clusters, defines a site percolation model [8,11]. For general values of H there is a finite value of h = h c > −∞ below which a level cluster of infinite size is found with probability one [12]. This is the percolation critical point. Note that the characterisation of the class of random fields which permit percolation has been given in [2,13,14]. Close to the critical point, the main scaling behaviours are described by two critical exponents, the correlation length ν and the order parameter β exponents [11].
In particular, they determine the scaling of the h c width distribution with the size of the system, see (B.1), and the Hausdorff dimension D f of the level cluster, D f = 2 − β/ν. For H > 0, due to unbounded fluctuations of u(x) and to the strong correlations, the level clusters are compact (i.e. without holes) regions with fractal dimension D f = 2. The exponent ν is infinite ν = ∞, as one can see from the fact that the h c width distribution remains finite in the thermodynamic limit (self-averaging is broken) [12]. At H > 0 the transition is not critical. At the point H = 0, the fluctuations of the surface remain unbounded and the fractal dimension remains D f = 2, as argued in [15] and recently proven in [16,17] for the Gaussian free field. For negative roughness exponent instead, the surface fluctuations are bounded, the correlation length exponent ν is finite (ν < ∞) and a genuine continous transition of percolation type occurs. Correspondingly, the level clusters have a richer fractal structure with D f < 2.
In this paper we consider random surfaces with negative roughness exponent. If not stated otherwise, we take H < 0 henceforth. We generate a fractional Gaussian process on a flat torus of dimension M × N . The surface u(x) takes the form (1. 4) In the above equation λ k and e i k x are respectively the eigenvalues and the eigenvectors of the discrete Laplacian operator ∆ x u(x) = y,|y−x|=1 (u(y) − u(x)) on the flat torus, and theŵ(k) are independent normally distributed random variables. For H = 0 the function u(x) is the discrete two-dimensional Gaussian free field on the torus. The role of open boundary conditions in one-dimensional fractional Gaussian processes is discussed in [18][19][20]. We generate also a second type of long range correlated random surface where thê w(k) are drawn by a different distribution. Full details on how we generate the surfaces are given in Appendix A. The probability of activating two distant sites inherits the long-range correlation of the random surfaces:  Table B.2 are shown in the second column, while the third column shows the level clusters. The yellow points in the third column are the points belonging to the percolating level cluster. Note that by increasing H, i.e. the correlation, the level clusters have less holes. This is consistent with the prediction that the fractal dimension D f → 2 for H → 0 − .
The common understanding is that the percolating universal properties only depend on the asympotic behaviour of the covariance (1.1) and therefore on H. In [21] an extended Harris criterion was proposed, according to which the universality class remains the one of pure percolation for H < −3/4. Recent new arguments, based on the fractal dimension of the pivotal points support this prediction [22,23]. The exponents ν and D f are expected to be where ν pure and D pure f are the pure percolation (H = −1) exponents. The fact that the system behaves as pure percolation for H < −2 was put on more rigorous grounds by [5,24]. For −3/4 < H < 0 instead, the slower decay allows the correlation to change the large distance behaviour of the system, as was also argued in [3]. In particular, it was shown in [21] that there is a new line of critical points with an exponent ν = ν long which varies continuously with H: The above prediction was supported by many numerical works, see for instance [3,4,12,25,26]. There are no theoretical predition for D f in the range −3/4 < H < 0. In Figure  1.1 the level clusters become visibly more compact by increasing the value of H. One can expect then D f to increase when H → 0 − . Even if the numerical results are not conclusive about the value of D long f , there are strong evidences that [12,[25][26][27]: In Appendix B, we numerically compute D f . The results, summarised in Table B.6, support the above scenario. The following diagram summarises the actual state of the art: Pure percolation, RG arguments [21] Line of new critical points [21] −2 −1 − We stress the fact that the results mentioned above are based on the assumption that the kernel has a definite sign at large distances. For other important classes of random functions, this is not true anymore. This is the case for instance of the random plane wave [28]: this random function has an oscillating kernel which decays with an exponent H = −1/4, and the universality class of its percolation transition is conjectured to be the one of pure percolation [29]. Most of the results on pure critical percolation have been discovered by using conformal invariance [30]. The values (1.6) have been predicted by the conformal field theory approach [31,32], which allowed also the computation of the full partition function [33] and the derivation of exact formulas for cluster crossing probabilities [34]. Conformal invariance has been rigourously proven in [35]. Even if a complete bootstrap solution is still missing, recent important insights come from the study of many-point cluster connectivities [36][37][38][39][40]. On the other hand, the universality classes of the new critical points of Figure  1.2 remain by far less understood. Recent numerical results have shown the emergence of conformal invariance [41], while in [42], where a random surface with H = −2/3 was considered, conformal symmetry has been ruled out. These papers check if the boundary of the percolation level cluster is described by a Stochastic Loewner Evolution (SLE) process [43]. The SLE numerical tests are in general very subtle and, in some cases not conclusive, as argued for instance in [44]. Moreover we observe that, in case of a positive SLE test as in [41], one expects the boundaries of the level clusters to be described by the loops of the O(n) models either in their dense or critical phases [45]. In these models, the fractal dimensions of the loops D b and of their interior D f vary with n [46]. For instance, in the O(n) dense phase, they are related by ). This scenario is not consistent with the numerical findings for the level clusters of long-range correlated random surfaces, as can be directly seen from the fact that D f does not vary for −3/4 < H < −1/2 while D b does [41]. Moreover, we provide further evidences that the line −3/4 < H < 0 is not the one of the O(n) models. This point illustrates the fact that many fundamental questions remain open.
Our objective is to test conformal invariance and to extract new information about these critical points. We use a completely different protocol based on the study of the level clusters and their connectivity function. This is the probability that two sites belong to the same level cluster, see (2.2). Because the random surfaces have double periodicity, the level clusters live on a torus. For pure percolation, signatures of conformal invariance were shown to be encoded in toric boundary conditions effects in the connectivity function [47]. These effects depend on a non trivial combination of the two exponents ν and D f , fixed by conformal invariance. Moreover, when the lattice is rectangular, M = N , a soft breaking of rotational symmetry is introduced. Using this anisotropy, we show that the connectivity function directly probes the existence of the two components of a traceless stress-energy tensor. The existence of this pair of fields is the most basic manifestation of conformal symmetry. Finally, we provide the first numerical measurements of quantities related to the conformal spectrum and structure constants of this new conformal critical points.
In Section 2 we define the connectivity function and we give the theoretical predictions for the toric effects. We discuss the main ideas behind the CFT approach on which these predictions are based. In Section 3 we provide the numerical evidences on the connectivity function. In Appendix A we provide full details on how we generate the random surfaces and in Appendix B,on how we locate the critical percolation point and compute the exponents ν and D f .

Critical two-point connectivity of level clusters
In this section we consider the two-point connectivity p 12 (x 1 , x 2 ), referred to as simply correlation function in [11]. Defining the event: Conn(x 1 , x 2 ) = x 1 and x 2 belong to the same level cluster, (2.1) we define: where translational invariance in law has been taken into account. A study of two-point connectivity for general Gaussian random surfaces can be found in [48] where the large h asymptotic behaviour of (2.2) has been considered. Here we are interested in the behaviour of this probability at the critical point h = h c .

Scaling limit in the infinite plane M, N = ∞
Let us consider first the regime in which toric size effects are negligeable. It corresponds to M, N = ∞, i.e. the infinite plane limit.
At the critical point, h = h c , we have p 12 (x) ∼ |x| −η , where η is the standard notation for the anomalous dimension of the two-point function [11]. Percolation theory tells us that η is directly related to the level cluster dimension D f via the scaling relation η = 4 − 2D f [49]. One has therefore: where d 0 is a non-universal constant which we evaluate numerically, see Table 3.4. We can use (2.3) to determine D f . The corresponding values are denoted as D (2) f in Table B.6.
The good agreement with the values D   Table B.6. For distances 6 < |x 1 − x 2 | < 100 the data matches very well with the infinite plane prediction. For larger distances, the effect of the toric boundary conditions becomes visible. As can be seen in Figure 2.1, when the distance between points approaches N/2, the data points start to deviate from the power-law behaviour: the contributions of the paths connecting the two points on the other side of the torus become non negligeable. We say that the topological corrections become visible. We expect these corrections to provide sub-leading |x|/N terms in (2.3) of universal nature. These effects have been studied in [47] for pure percolation (H = −1). In the scaling limit, our system lives on a flat torus T q of periods M and N and nome q: As the connectivity between two points always depend on the vector connecting them, it is convenient to introduce the vector x, x ⊥ ∈ T q that have polar coordinates |x| and θ: We propose the following form for the scaling limit of p 12 on a torus: The coefficients c ν (q), and c T (q), given in (2.11), are universal coefficients which depend only on the geometry of the torus. To explain the origin of (2.6) and the information we can extract from this formula, we need to recall some basic definitions and notions on CFT.

Basic notions of CFT
A CFT is a massless quantum field theory in which each (quantum) field V ∆,∆ (x) is characterised by a pair of numbers (∆,∆), called left and right conformal dimensions, which give the scaling dimension (∆ phys = ∆ +∆) and the spin (s = ∆ −∆) of the field. The set of fields entering a CFT is called the spectrum S of the theory, S = ⊕ (∆,∆) V (∆,∆) . The most important landmark of conformal invariance is the existence of two fields, commonly denoted as T andT , with left-right dimensions (∆,∆) = (2, 0) and (∆,∆) = (0, 2). These fields are the conserved (chiral) Noether current associated to the conformal symmetry, and they correspond to the components of the traceless stress-energy tensor field. In the CFT approach to statistical models, there is a correspondance between lattice operators and fields V ∆,∆ (x). In particular, the long distance behaviour of lattice observables is described by the correlation functions of the fields V ∆,∆ (x). Scale invariance fixes the infinite plane limit of the two-point functions. For a spinless field V ∆,∆ we have: where · · · q denotes the torus CFT correlation function on T q . A quantum field theory is completely solved if we can compute all its correlation functions. For a CFT, one needs two basic inputs: the spectrum S and the structure constants The latter are real constants associated to the amplitude with which two fields V 1 and V 2 fuse into a third one V 3 . Said in other words, the constants C V 3 V 1 ,V 2 determine the short-distance behaviour of the CFT correlation functions which is encoded, in the CFT jargon, in the Operator Product Expansion (OPE). Among all the fields in a CFT, a major role is played by the density energy field ε = V ∆ε,∆ε and the magnetic (order parameter) field σ = V ∆σ,∆σ , which are the (spinless) fields with the lowest scaling dimension in the thermal and magnetic sector. Their names come from the fact that, in a ferromagnetic/paramagnetic type transition, these are the fields which couple respectively to the temperature and to the magnetic field. Their dimension ∆ ε and ∆ σ give the exponents ν and β of a critical point. In terms of ν and D f we have:

Three main assumptions
Our prediction (2.6) is based on three assumptions which have been verified for pure percolation [47,50]. The first two assumptions are more general and concern the fact that the connectivity, which is non-local in nature, can be studied by correlations of local fields in a CFT.
• 1: The system is conformally invariant in the scaling limit.
• 2: The scaling limit of the connectivity (2.2) is described by the two spin field torus correlator: The two-point function σσ q can be expressed as an (s-channel) expansion: with x = |x|(cos(θ), sin(θ)), see (2.5). We refer the reader to [47] for a detailed derivation of the above formula which is a direct consequence of the existence of an operator algebra and of the symmetry between the holomorphic and anti-holomorphic sectors. This latter symmetry is very natural for CFTs without boundaries and implies that if a field with spin s > 0 enters in the spectrum, then also its anti-holomorphic partner does, with the same physical dimension and with spin with opposite sign −s. The expansion (2.10) is then valid for almost all the CFTs. The information which characterise a specific CFT is encoded in the spectrum S, i.e. on which fields the above sum runs over, and in the structure constants C V ∆,∆ σ,σ . In the case of pure percolation, for instance, the spectrum is known but not the structure constants, even if very recent progresses have paved the way to their determination [40]. The plane limit M, N = ∞ is recovered by noting that all the one-point functions V ∆,∆ q vanish but the identity one Id q = V 0,0 q = 1. One . Note that, in the infinite plane limit, one can prove for pure percolation (or more generally for the O(n) models in their dense or critical phases) that p 12 is given by the correlator of two spin fields σ, see for instance [46,51]. The exponent η is therefore η = 4∆ σ which, by (2.8) gives equation (2.3). It has been shown in [47] that the first dominant terms in the above series can be computed for pure percolation. Our third assumption is motivated by a generalisation of these results to the case of long-range percolation: • 3: The identity field (∆ =∆ = 0), the density energy density field ε and the stress-energy tensor fields T (∆ = 2,∆ = 0),T (∆ = 0,∆ = 2) are the fields with the lowest conformal dimension that appear in the fusion of two fields σ and whose torus one-point function does not vanish.
Using the above assumption in the expansion (2.10), one obtains expression (2.6) with the coefficients c ν (q) and c T (q) given by: where c is the CFT central charge (which provides for instance the universal Casimir amplitude [52]). We refer the reader to [47,50] for a detailed explication of the CFT techniques used to study the topological effects.
Let us detail further the information one can extract from c ν (q) and c T (q). The spectrum S and some structure constants C V 3 V 1 ,V 2 enter in the determination of these coefficients. For a general CFT, the spectrum defines the torus partition function [53]: where n V ∆,∆ is the multiplicity of the field V ∆,∆ . For small values of q, the leading contributions to the partition function are given by the representations with the smallest physical dimensions. The Identity field V 0,0 has the lowest physical dimension 0, with n Id = 1. We will assume that the sub-leading contribution to the partition function is given by a spinless field V ∆,∆ with multiplicity n V ∆,∆ . For non unitary CFTs, the number n V ∆,∆ can take general real values. This is the case of the Q− state Potts model [33], in which the sub-dominant contribution is given by the spin field σ with multiplicity n σ = Q − 1.
In a general CFT, one-point torus functions can be expressed in the variable q, in a way similar to the partition function (2.12). As detailed in [47], the three assumptions of Section 2.4 lead to the following form for the energy density one-point torus function: The coefficient c ν (q), given by (2.11), can therefore be expanded in q as: In a similar way, using the formula [53]: and expression (2.12) of the partition function, the coefficient c T (q) (given by (2.11)) admits the following small q expansion: (2.16) The above three assumptions do not put any constraint on the dimension ∆ and multiplicity n V ∆,∆ of the field giving the leading contribution to (2.16). For pure percolation, for which the partition function (2.12) is known exactly, this leading contribution is given by the spin field σ: In that case the ratio n σ /c can be obtained as the limit Q → 1 of the analogous expression for the Q− Potts model. Using the fact that in this limit the central charge c Q ∼ Q−1 (|Q−1| 1), the limit c → 0 of n σ /c yields a finite non-zero limit, n σ /c = 4π/(5 √ 3).

Numerical protocols for testing CFT predictions
We have seen that, by using a CFT approach, the topological effects on p 12 encode in principle highly non-trivial information about the critical point. We discuss now how to efficiently extract this information from a numerical study of p 12 and how to interpret these results. The torus shape can be exploited to disentangle the contributions of sub-leading and subsub leading terms in (2.6). This can be done by comparing the connectivities p 12 (x 2 − x 1 ) and p(x 3 − x 1 ) between pairs of points x 2 and x 1 and x 3 and x 1 that are aligned on orthogonal axes, as illustrated in Figure 2.2. Note that similar ideas were used in [47].
Let us consider first the square torus, M = N or q = e −2π and the case where ) and x v = |x|(0, 1). As the two cycles are equivalent, one has p 12 (x h ) = p 12 (x v ). From (2.6) and (2.10), In particular T q=e −2π = 0 and therefore: The connectivity (2.6) therefore reduces to: Let us consider now the rectangular torus M > N with again Figure 2.3 we show the corresponding measurements of p 12 (x h ) and p 12 (x v ) when M = 2N . The two connectivities are now different, which is explained by the simple fact that the paths closing on the other side of the small cycle (N ) start to contribute for smaller distances than the ones closing on the largest one (M ). From (2.6) and for general x we have: where x and x ⊥ are parametrised as in (2.5), and c T (q) has been replaced by its expression (2.11).
x 2 x  Equation (2.20) is a clear consequence of the fact that, whenever an anisotropy is introduced, the response of the system is bound to be determined by the stress-energy tensor components T andT (see for instance Section 11.3 in [54]). It is interesting to note that Monte Carlo algorithms, based on the properties of rectangular torii [55,56], have been proposed to measure the central charge and the leading fields in the partition function [57]. However, these methods can be applied to statistical models for which a direct lattice representation of the stress-energy tensor is available, such as the Ising model. In our case we do not know the stress-energy lattice representation. Actually, away from the pure percolation point H = −1, we do not even know the energy density lattice representation. This is also the reason why the connectivity functions are the most natural observables to study universal critical amplitudes of non-local models.
From the expansion (2.10) of the connectivity, the difference (2.20) gets in general contributions only from fields with a non-zero spin. By lattice symmetry arguments, this difference vanishes for θ = π/4, as shown in Figure 2 In pure percolation there are no fields in the spectrum with spin greater than 2 and physical dimension ∆ +∆ < 6. If we assume this is true also for correlated percolation H > −1, then we have: (2.21)

Numerical evidences
We summarise here the main numerical results for p 12 and the conclusions we can draw by comparing these results with the CFT predictions.

Conformal invariance
The quantity (2.6) is, first of all, a powerful test of conformal invariance. Via the numerical simulation of the connectivity we test two predictions: • The dominant topological correction shows a precise interplay between the exponents ν and D f . In particular the leading correction behaves as |x| 2(2D f −2) (|x|/N ) 2−1/ν . This effect is more clearly seen on the square torus, see (2.19). Figure 3.5 shows that the numerical results for the values H < −1/2 agree with this prediction.
• The sub-leading term is ∝ |x| 2(2D f −2) cos(2 θ)(|x|/N ) 2 . As explained above, the presence of such term implies the existence of a pair of fields with scaling dimension ∆ +∆ = 2, which corresponds to the power 2 in the (|x|/N ) 2 decay, and with spin ∆ −∆ = ±2, which fixes the θ dependence. If such fields exist, they correspond by definition to the stress-energy tensor components T andT . The presence of T and T is the most basic and direct consequence of conformal invariance. In numerical simulations, the sub-leading term is seen by considering a rectangular torus. Figures 3.6 and 3.7 show clearly the (|x|/N ) 2 decay and the cos(2θ) dependence. Figure 3.8 shows further that the data is well described by the form (2.21).

Spectrum and structure constants
• The values of c ν (q) for different values of q have been measured for −1 < H < −1/2 and reported in Table 3.9. The results support the fact that for H −3/4 the universality class is the one of pure percolation. Note that this a highly non-trivial verification, as it not only based on the values of critical exponents, but on the values of constants which depend on the spectrum and fusion coefficients of the theory. For H > −3/4, the data are quite well consistent with the CFT prediction (2.13), as shown in Figure 3.10. This is also consistent with the fact that the fusion between two spin field produces an energy field.
• We could measure with good precision the dependence of the coefficient c T (q) with q. Figure 3.16 shows that (2.17) is satisfied, and that the dimension of the most dominant field coincides with the dimension of the spin field.
Exploiting the translational invariance of the surface distribution, we average over the position x 1 for each instance of u(x), and then over 10 5 instances. In the scaling limit, the dependence of p 12 (x) with respect to the lattice size N is expected to be of the form |x|/N . Plotting the connectivity as a function of |x|/N , we observe that the corrections to the scaling are still visible as the data points for different sizes do not collapse at large distances. In Figure 3.1a we show the data for H = −5/8 and for lattice sizes M = N = 2 9 − 2 12 . One can see that the scaling limit is still not attained. These non-universal effects become even more important for larger H. As shown in Figure 3.1b for H = −3/8, even the infinite plane scaling limit is not clearly attained at the sizes of our simulations. Of course these non-universal effects make the analysis of the universal topological effects less precise, in particular for studying the contributions of the spinless fields. On the other hand, we observed that the non-universal effects are less important for the surface (A.14) generated by the kernelŜ 2 (k), at least for values of H < −1/2. This is shown in 3.2b. For values of H < −1/2 and for the two surfaces (A.10) and (A.14) we could determine the non-universal constant d 0 , as well as the dimension of the leading spinless contribution. For this latter, the consistency of the results obtained from the two surfaces makes the verification of the CFT predictions more solid. The coefficient c ν and its dependence on the aspect ratio, on the other hand, could only be determined with sufficient precision for the surface (A.14).   A very remarkable fact is that, for both surfaces, these corrections to the scaling terms cancel when one takes the differences between connectivities. This is shown in Figure 3.3 for the same values of H. The corrections may originate, for instance, from the fact that we are not sufficiently close to the critical point. More generally, any perturbation that drives the system out of the critical point and that does not break rotational invariance is related to a spinless field, whose contributions to the connectivity are isotropic. This explains why they disappear by taking the difference p 12 (x) − p 12 (x ⊥ ). This mechanism allows to test the contribution of the fields with spin, and therefore of the stress-energy tensor, with a very good precision. For H < −1/2, our determination of the constants d 0 allowed moreover to acces the value of the universal coefficient c T (q). For H > −1/2, we could only determine the behaviour of d 0 c T (q) with q.

Plane limit
For N = M = 2 12 , we fit the data points for |x| ∈ [4,128], expected to be well described by the infinite plane limit (2.3) (see Figure 2.1), to the form The values D (2) f of the fractal dimension are given in Table B.6. To extract the topological corrections (2.19), we fit our numerical data to the form: 3) The first factor takes into account the non-universal, small distance effects due to the lattice. We refer the reader to [36,38] for a more detailed discussion of these ultraviolet corrections. The values of d 0 are reported in Table 3.4. The numerical values for the universal coefficient c ν are given in Table 3.9. They were obtained from the data generated using kernel (A.12), which converge faster to the scaling limit, and for which the agreement with (3.3) is excellent. This is shown in Figure 3.5.

Evidences of conformal invariance
With M = N , and following prediction (2.20), the quantity log |x| 2(2−D (2) f ) p 12 (x) − p 12 (x ⊥ ) should follow a line of slope 2. This is very clear for H < −1/2, as shown in Figure 3.6. When H > −1/2, the slope increases significantly: either there is no order 2 term (conformal invariance is broken), or this term is still present, with higher-order corrections making the effective slope significantly greater than 2. Assuming the latter and that the difference of connectivities is described by (2.21) on the whole line H < 0, we fit our data for different angles θ to the form: This fit shows good consistency with the data for all values of H, and allows to determine c 2 (θ) with good precision. In Figure 3.7 we show that c 2 (θ) has the expected behaviour (2.10): c 2 (θ) ∝ cos(2θ). This makes manifest the presence of a field with conformal dimension 2 and spin 2, and therefore of conformal invariance for all H < 0.

Spectrum and structure constants
Setting θ to zero, we varied the aspect ratio and obtained c ν and c T as functions of M/N , given in Tables 3.9 with the slope x given by the dimension of the spin field x = 2 − D f , see Table 3.11.  Setting x to 2−D f , a fit of c ν (q) as a function of q 2−D f gives an estimation of the quantity C ε σ,σ 2 n σ (see 2.14), given in Table 3.12.   [47].
Conversely, to obtain c T (q) we fit the difference |x| 2(2−D (2) f ) p 12 (x) − p 12 (x ⊥ ) to the form: where c 2 (q) = 4d 0 c T (q). (3.7) The values we obtained for c T (q), for both types of surfaces, are given in Tables 3.13, 3.14.  Following prediction (2.16), we fit the quantity log 2 as a function of log q to a line. This is shown in Figure 3.16, and we obtain values for the dominant dimension ∆ close to the dimension of the spin field, see Table 3.17. Assuming that this dimension is indeed the one of the spin field, 2∆ = 2∆ σ = 2 − D f , we fit c 2 (q)/d 0 as a function of q 2−D f : see Figure 3.16. In particular, from (2.16): The values of the cylinder (q → 0) limit and of the ratio n σ /c obtained are given in Tables  3.18 (2) 1.51 (7) Table 3.19: Cylinder limit c 2 (0)/d 0 and ratio of the spin field multiplicity n σ to the central charge c, obtained from fit (3.8), for surfaces (A.14).
When H > −1/2, we could not determine the value of the plateau d 0 , so we cannot determine the leading dimension in the expansion (2.16) as above. In

Conclusion
In this paper we have studied the percolative properties of fractional random surfaces with negative Hurst exponent H. Via the connected components of their excursion sets, the level clusters, this problem is reformulated in terms of a long-range correlated twodimensional site percolation model. The main motivation here was to better understand the universality of their percolation critical points, in particular in the region −3/4 < H < 0 where the correlation effects drive the system into universality classes different from the one of pure percolation. When the problem is defined on a rectangular domain of size M × N with toric boundary conditions, we argued that the two-point connectivity (2.2) represents an excellent observable to test conformal invariance. On the basis of three main assumptions, explained in Section 2.4, we predicted the leading contributions to the toric corrections, see (2.6) and (2.21). We tested these predictions by generating two types of fractional random surfaces (A.10) and (A.14), expected to have the same long distances behaviour. The comparison between the theory and the numerical simulations is summarised in Section 2.6. The main result is shown in Figure 3.6 and in Figure 3.7 and points out, for the first time, the existence of the two components of a traceles stressenergy tensor for all H < 0. Furthermore, the two point connectivity on rectangular torus lattices gives access to the spectrum and to some fundamental structure constants of the underlying CFT, still unknown for any H < 0. In particular we conclude that the leading contribution to the conformal partition function is the magnetic field σ with scaling dimension 2 − D f , as shown in Figure 3.16 and in Table 3.17. The ratio n σ /c of the multiplicity of the magnetic field to the central charge has also been determined numerically with quite good precision, and it is reported in Table 3.18. Finally, we succeeded in evaluating the product C ε σ,σ 2 n σ , directly proportional to the fusion between the thermal and magnetic field. The results are given in Table 3.12. We conclude by noting that the fact that, for H < −3/4, the long-range correlation is irrelevant is a very established one. Nevertheless, the results in Table 3.12 verify this conjecture at the level of the structure constants of the theory, which encode much more information than the critical exponents. At the best of our knowledge, this is the first time such verification has been done. A last noteworthy observation concerns the corrections to the scaling of the critical level, when using the Binder method to locate the critical point (see Appendix B). From the values of the corresponding exponent ω given in Table B.3, we argue that the long-range correlations break the integrability of the model.

A Fractional Gaussian surfaces
To generate a random function u(x) satisfying the properties (1.1), we use a method based on the Fourier Filtering Method [9]. The principle is to create correlated random variables by linearly combining uncorrelated ones. Let us first briefly sketch the method. Given a set of uncorrelated random variable w(x), E [w(x)w(y)] = δ x,y , one can define, via a convolution, a new set of random variables u(x): The convolution kernel S(x) is a non-random function which determines the u(x) covariance function: By Fourier transforming both sides of the above equation, one can see that the large distance asymptotics (1.1) is determined by the small k asymptotics ofŜ(k) 2 , whereŜ(k) is the Fourier transform of S(x). In particular,Ŝ(k) ∼ |k| −H−1 (for |k| << 1). We apply this procedure to generate random long-range correlated surfaces. We consider a domain [0, · · · , N − 1] × [0, · · · , M − 1] ⊂ Z 2 where x = (x 1 , x 2 ) denotes a lattice site: x = (x 1 , x 2 ), x 1 = 0, · · · N − 1 and x 2 = 0, · · · , M − 1. (A.3) A random function w(x) is generated by drawing its values independently at each point by an initial Gaussian distribution P (w) = N (0, 1). The probability distribution function P [w(x)] is therefore: The discrete Fourier transform of w(x) is defined as: From (A.4) one has: We use the convolution kernel: where: We generate the random surface u(x) by doing the following inverse Fourier transform: The universal properties do not depend on the initial distribution P [w(x)] distribution nor on the precise form of the kernel as long asŜ(k) has the same small k asymptotic behaviour [58]. As we explain in Section 3, we find useful to generate long-range correlated random surfaces by using another distribution P 2 [w(x)] for w(x) and a different kernel. In particular, the P 2 [w(x)] is determined by the uniform distribution: and the kernel: where: The second kind of surfaces we generate are where we indicated asŵ 2 (k) the Fourier transforms of the random function w(x) of law (A.11). In the above equations we assumed M = N , but the generalization to M = N is straightforward. Note that, due to the (Lyupanov) central limit theorem,ŵ 2 (k) is described in the large N limit by a Gaussian distribution and the function u(x) can be considered an instance of a fractional Gaussian surface. For H < 0, the surface u(x), generated by (A.10) or by (A.14): • is real, u(x) ∈ R, from the property (A.7) and the symmetry of the kernel (A.8) • satisfies (1.1). In Figure  • has a zero mode which vanishes in law: • is normalised such that: We study here the critical percolative properties of the level clusters of the surface (A.10) and (A.14). In particular we determine numerically the critical level h c and the exponents ν and D f .

B.1 Critical level and correlation length exponent ν
For a sign-symmetric random function u(x) on the Euclidean space, x ∈ R 2 , the critical level is h c = 0 by symmetry argument [13]. Our function u(x) is defined on a lattice and h c is expected to be negative. We determine the critical level h c by the standard procedure of percolation theory [11]. We consider square domains of different sizes N × N . We determine the average E [h c (N )] of the level h c (N ) at which a level cluster connecting the top and the bottom of the lattice appears. This quantity scales with the size of the lattice as: The  Another way to determine the critical point is based on the Binder method. We apply this method to study the surface (A.14). Defining the moments M m as: with n i the number of level clusters composed of i sites, one computes the ratio r Bin where the average E[· · · ] is weighted by the distribution (A.11). The ratio r Bin N (h) depends on the level h and on the system size N through a scaling relation of the type: where the function f is some scaling function, and the term a N −ω , with a a non-universal prefactor, is a correction to the scaling term. The interpretation of ω is discussed below. From (B.4), one can find the point h c (N ) where the curves r Bin N (h) and r Bin 2N (h) intersect [59] and use the fitting form: to determine h c , with x a free parameter. For each value in (3.1), we compute (B.3) for sizes N = 2 s , s = 4, · · · , 9 and N = 3 × 2 s , s = 3, · · · , 7 averaged over 10 5 instances. We interpolate the curves and find their intersections. The Binder method shows less precision for H approaching 0. Indeed the correlation length exponent ν = −1/H increases fast, making the size effects much smaller. The curves r Bin N (h) and r Bin N (h) tend to be parallel, and localising their crossing point becomes difficult. In Figure B.4 we show the scaling of the crossing points h c (N ) for some values of H. Once the critical point is located, the thermal exponent ν can be estimated by using that:  It is quite interesting to comment on the exponent ω, which determines the correction to the scaling. The exponent ω is expected to be the conformal dimension of the first irrelevant thermal field. As observed in [60], when the system is integrable the value of ω is universal, and should match with one of the Kac table of the underlying conformal field theory. In particular, using the minimal model notation, ω = 2∆ (3,1) where ∆ (3,1) is the dimension of the thermal spinless field with the lowest dimension after the energy density ε (note that, in this notation, ∆ ε = ∆ (2,1) ). A discussion of this exponent can be found for instance in Appendix D of [61]. In the case of pure percolation, which is an integrable model, we find indeed ω = 2∆ (3,1) = 2. We observe in Table B.3 that, when H = −1, a non-universal correction with ω ∼ 1 to the scaling dominates.

B.2 Fractal dimension D f
At the critical point h = h c , the level clusters have fractal dimension D f . This dimension determines the scaling of the average mass (i.e. number of points) A l of a level cluster with respect to its length l, A l ∼ l D f . The length of a level cluster can be defined as its radius of gyration. One effective way to measure D f is to consider the percolating level cluster whose size is of the same order of the system size, l ∼ N . To determine D f , we use then the following relation: E [# sites of the p.l.c.] ∼ N D f , p.l.c.=percolating level cluster. (B.7) A representative example of a numerical measurement of the above average is shown in Figure B.5a, for H = −2/3. To remove the small sizes effects, we perform fits with the successive lower sizes removed, and expect the best fit parameter to converge to the fractal dimension, as in Figure B.5b. The values D (1) f obtained are given in Table B.6.    and (2) from the power-law decay of the two-point connectivity (2.3), and comparison with previous numerical work [25].