On three-point connectivity in two-dimensional percolation
aa r X i v : . [ h e p - t h ] S e p SISSA 60/2010/EP
On three-point connectivity in two-dimensional percolation
Gesualdo Delfino a,b and Jacopo Viti a,ba
International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy b Istituto Nazionale di Fisica Nucleare, sezione di Trieste, Italy
Abstract
We argue the exact universal result for the three-point connectivity of critical percolationin two dimensions. Predictions for Potts clusters and for the scaling limit below p c are alsogiven. . The connectivity functions P n ( x , . . . , x n ), i.e. the probabilities that n points belongto the same finite cluster, play a fundamental role in percolation theory [1, 2]. In particular,their scaling limit determines the universal properties that clusters exhibit near the percolationthreshold p c . Although most quantitative studies focus on the two-point connectivity, whichdetermines observables as the mean cluster size S = P x P ( x, n > n = 3 for clusters in random percolation and in the q -state Potts model ( q ≤
4) in twodimensions, in the scaling limit on the infinite plane. In particular, we will determine exactlythe universal quantity R ( x , x , x ) = P ( x , x , x ) p P ( x , x ) P ( x , x ) P ( x , x ) (1)at p c , as well as some of its features below p c . The connectivity combination (1) was studied in[3, 4] for the case in which two of the three points are located on the boundary of the half-plane,it was shown to be a constant at p c and determined numerically and also analytically, exploitingthe fact that points on the boundary yield linear differential equations for the connectivitiesat criticality [5]. It is not known how to write differential equations for random percolationconnectivities if all the points are in bulk.It is well known that random percolation can be described as the limit q → q -statePotts model with lattice hamiltonian [6, 7] H = − J X
1, the formula X m,n = [( t + 1) m − tn ] − t ( t + 1) (15)determines, for m and n positive integers, the scaling dimensions of scalar “degenerate” primaryfields having the property that correlation functions containing them satisfy linear differentialequations of order mn . For some discrete values of c , and in particular for any integer t >
2, thereexist “minimal models” whose space of fields decomposes into a finite number of subspaces, eachone originating from a degenerate primary [12]; the number of primaries with a given scalingdimension is an integer in minimal models [14, 15]. In the Potts model, the fields σ α have [16, 13] X σ = X ( t +1) / , ( t +1) / (16)and multiplicity q − P α σ α = 0), allowing minimality only for q = 2 (Ising model, t = 3) and q = 3 ( t = 5); the leading S q -invariant field ε , with X ε = X , , is degenerate for generic valuesof q .In [17] the properties of the degenerate fields were exploited to compute the OPE coefficients C kij for the “diagonal” series of minimal models, in which all the primaries appear with multi-plicity one. The Potts model belongs to this class for q = 2, but even for this case the results forthe minimal OPE are not sufficient to determine R c . Indeed, it is clear from (11) that a finite R c requires C σ α σ α σ α = 0 at q = 2, and this is trivially ensured by the spin reversal symmetry ofthe Ising model. Hence, even the determination of R c at q = 2 requires the computation of theOPE coefficient for continuous values of q ; even forgetting the problem with the multiplicity ofthe fields, the formulae of [17] can be evaluated only for the discrete values of t correspondingto minimal models.Few years ago Al. Zamolodchikov [18] approached the problem of the derivation of the OPEcoefficients of c < X , or X , . The other three fields are simply required to appear with multiplicity one; themathematical treatment does not put any constraint on their scaling dimensions X i , i = 1 , , c ≥
25) [19]. The solutionfor c <
1, however, is not an analytic continuation of the Liouville solution, and reads [18] C X X ,X = C X ,X ,X = (17) A Υ( a + a − a + β )Υ( a + a − a + β )Υ( a + a − a + β )Υ(2 β − β − + a + a + a )[Υ(2 a + β )Υ(2 a + 2 β − β − )Υ(2 a + β )Υ(2 a + 2 β − β − )Υ(2 a + β )Υ(2 a + 2 β − β − )] , β = p t/ ( t + 1) , (18) X i = 2 a i ( a i + β − β − ) , (19) A = β β − − β − [ γ ( β ) γ ( β − − / Υ( β ) , γ ( x ) ≡ Γ( x )Γ(1 − x ) , (20)Υ( x ) = exp Z ∞ dtt (cid:18) Q − x (cid:19) e − t − sinh h(cid:16) Q − x (cid:17) t i sinh βt sinh t β , Q = β + β − . (21)The integral in (21) is convergent for 0 < x < Q ; outside this range Υ( x ) can be computed usingthe relations Υ( x + β ) = γ ( βx ) β − βx Υ( x ) , (22)Υ( x + 1 /β ) = γ ( x/β ) β x/β − Υ( x ) . (23)The function (17) can be evaluated for continuous values of t and of the X i ’s. Taken literally,it would give the OPE coefficients for a theory with arbitrary c < c corresponding to minimal models, (19) gives the scaling dimensions(15) for a i equal to a m,n = ( n − β − ( m − β − . (24)Although a general proof is not available, checks case by case show that (17) reproduces theOPE coefficients of minimal models obtained in [17], at least when these differ from zero. Insome cases for which the minimal OPE prescribes vanishing coefficients, (17) gives instead finitenumbers whose interpretation is considered “mysterious” in [18]. As an example, (17) evaluatedfor X = X = X = X , at t = 3 does not vanish, despite the fact that the spin three-pointfunction is zero in the Ising model.Recall now that the two-dimensional Potts model contains also disorder fields, dual to thespin fields and with the same scaling dimension X σ . They satisfy the OPE µ αβ µ βγ = δ αγ ( I + C ε ε + . . . ) + (1 − δ αγ )( C µ µ αγ + . . . ) , (25)where we omit the coordinate dependence for simplicity and the coefficient in front of the identityis fixed to 1 by (6) and (12). Use of (6-7) at the self-dual point then leads to the result R c = C µ . S q -invariance gives to (25) a two-channel structure ( α = γ or α = γ ) equivalent to that producedby two fields µ and ¯ µ satisfying µ ¯ µ = I + C ε ε + . . . , (26) µ µ + ¯ µ ¯ µ = C µ ( µ + ¯ µ ) + . . . . (27)4he field φ ≡ ( µ + ¯ µ ) / √ φ φ = I + C ε ε + C µ √ φ + . . . , (28)namely a “neutral” OPE for fields with multiplicity one, as the one assumed in the derivationof (17). Notice also that, due to the neutrality of the fields in (28), C µ has no reason to vanishfor any value of q ; in particular, this is not in conflict with the symmetries of the Ising model,because the absence for q = 2 of the vertex with three disorder fields in (25) is enforced by thefactor 1 − δ αγ .These observations may suggest the following interpretation for the function (17): it encodesthe “dynamical” information about the OPE of conformal field theory for c <
1, and knowsnothing about internal symmetries, which, on the other hand, are not uniquely determined bythe value of c ; symmetry considerations have to be developed separately and produce a dressingof (17) by factors which may vanish, suppressing the vertices not compatible with the givensymmetry. Letting aside the general validity of this interpretation, the OPE’s (25) and (28)make it plausible for the case we are discussing, and lead us to take R c = √ C X σ ,X σ ,X σ , (29)with X σ given in (16). The results for the integer values of q are given in Table 1; the valuefor q = 4 is obtained in the limit t → ∞ . In [20] the constant value ≈ .
022 was obtainednumerically for critical percolation on a cylinder and (1) evaluated for x on one edge, x on theother edge and x far away from both edges. Since in such a configuration the cylinder, seenfrom x , looks infinitely long and becomes conformally equivalent to the plane, we see in thisresult a confirmation of our analytic value for R c at q = 1. q R c . .. . .. . .. . .. ˜ R c − . .. . .. . .. Γ KKK . .. . .. . .. . .. Table 1: R c and Γ KKK are the values of the quantities (11) and (32) for KF clusters; q = 1corresponds to random percolation. ˜ R c is obtained from (29) with X σ replaced by X ˜ σ , thescaling dimension associated to spin clusters.The result (29) refers to KF clusters. Concerning the ordinary spin clusters, i.e. thoseobtained connecting nearest neighbors with the same value of the spin, they are also critical at J c in two dimensions [10], with connectivities related to the correlation functions of the fieldwith scaling dimension X ˜ σ = X t/ ,t/ [21, 22]. For the Ising case, the best understood fieldtheoretically [23, 24], the ratio (1) is expected to be given by (29) with X σ replaced by X ˜ σ . Wegive this value ˜ R c in Table 1; those obtained in the same way at q = 3 , π/3 2π/3 x rr rr r r Yx x
12 31312 23
Figure 1: Triangle identified by the points x , x , x . The lines joining the vertices of the triangleto the Fermat (or Steiner) point Y form 2 π/ C σ α σ α σ α and its relation with C µ . The re-lations C σ α σ β = ( qδ αβ − /q and C σ σ σ = (1 − q ) C σ σ σ = ( q − q − C σ σ σ are aconsequence of the symmetry. Eqs. (6) and (28) lead to C ε = − q ( q − − C εσ α σ α = C X σ ,X σ ,X ε ,where we also took into account that ε changes sign under duality . The OPE between σ α and µ βγ produces parafermionic fields. An analysis similar to that of [25] suggests that they havespin X , /
2; this is certainly the case for q = 2 , Away from criticality the Potts field theory is solved exactly in the framework of thefactorized S -matrix [27], from which large distance expansions can be obtained for the correlationfunctions [28, 29]. The S q symmetry is more transparently implemented working in the orderedphase ( J > J c ), where the elementary excitations (kinks) are interpolated by the disorder fields;the results for the disordered phase are obtained by duality. If x , x , x are the vertices of atriangle whose internal angles are all smaller than 2 π/
3, the asymptotic behavior of the correlator(7) when all the distances between the vertices become large reads [27] ([30] for a derivation) h µ αβ ( x ) µ βγ ( x ) µ γα ( x ) i J>J c = F µ π Γ KKK K ( r Y /ξ ) + O (e − ρ/ξ ) . (30)Here F µ is the one-kink form factor of the disorder field, K ( x ) = R ∞ d y e − x cosh y is a Bessellfunction, and r Y ≡ r + r + r is the sum of the distances of the vertices of the triangle fromthe Fermat (or Steiner) point Y , which has the property of minimizing such a sum (Fig. 1); r Y < ρ ≡ min { r + r , r + r , r + r } . ξ is the connectivity length (inverse of the kinkmass), also determined by the large distance decay of the two-point function h µ αβ ( x ) µ βα ( x ) i J>J c = F µ π K ( r /ξ ) + O (e − r /ξ ) , (31) For example (17) gives C ε | q =2 = − / KKK is the three-kink vertex given by [27]Γ
KKK = r λ sin 2 πλ g ( λ ) , (32) g ( λ ) = exp "Z ∞ d t sinh (cid:18) t (cid:19) sinh (cid:2) t (cid:0) − λ (cid:1)(cid:3) − sinh (cid:2) t (cid:0) λ − (cid:1)(cid:3) t sinh t λ cosh t , (33)with q = 2 sin πλ , λ ∈ (0 , / p c . It follows R ( x , x , x ) ≃ √ π Γ KKK K ( r Y /ξ ) p K ( r /ξ ) K ( r /ξ ) K ( r /ξ ) , r ij ≫ ξ, p → p − c . (34)The dynamical information is entirely contained in the three-kink vertex, whose values for q integer are given in Table 1. In the opposite limit, in which all the distances r ij are muchsmaller than ξ (always remaining much larger than the lattice spacing), the function R tends toits constant critical value R c .Finally, consider the ordinary spin clusters for the Ising model. It was argued in [23] that inthis case the scaling limit for p → p − c (in zero magnetic field) corresponds to a renormalizationgroup trajectory with infinite connectivity length ending into a random percolation fixed point atlarge distances. Then one expects (1) to interpolate from ˜ R c | q =2 , when all the distances betweenthe points are small, to R c | q =1 , when all of them are large. Although universal, this crossovercould be easier to observe on the triangular lattice, which has p c = 1 / H , the connectivity length is finite and the largeseparation behavior of (1) for clusters of positive spins and H → − has again the form (34),with Γ KKK = 5 . .. given by (32) evaluated at λ = 5 / λ ≥ /
2, one uses the analytic continuation g ( λ ) = Γ (cid:0) − λ (cid:1) Γ(1 + λ )Γ (cid:0) λ (cid:1) exp "Z ∞ d t sinh (cid:18) t (cid:19) sinh (cid:2) t (cid:0) − λ (cid:1)(cid:3) − e − t sinh (cid:2) t (cid:0) + λ (cid:1)(cid:3) t sinh t λ cosh t for 3 / ≤ λ < Acknowledgments.
We thank J. Cardy, P. Kleban, J. Simmons and R. Ziff for discussions.GD thanks NORDITA (Stockholm) for hospitality during the final stages of this work. Worksupported in part by ESF Grant INSTANS and by MIUR Grant 2007JHLPEZ.
References [1] D. Stauffer and A. Aharony, Introduction to percolation theory (2nd ed.), Taylor & Francis,London, 1992. The value Γ
KKK | q =1 quoted in [27] is affected by a typo in the formula corresponding to our (33).quoted in [27] is affected by a typo in the formula corresponding to our (33).