Exact factorization of correlation functions in 2-D critical percolation
aa r X i v : . [ m a t h - ph ] S e p Exact factorization of correlation functions in 2-D critical percolation
Jacob J. H. Simmons + ∗ and Peter Kleban † LASST and Department of Physics & Astronomy, University of Maine, Orono, ME 04469, USA + current address: Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK Robert M. Ziff ‡ MCTP and Department of Chemical Engineering,University of Michigan, Ann Arbor, MI 48109-2136 USA (Dated: November 1, 2018)By use of conformal field theory, we discover several exact factorizations of higher-order densitycorrelation functions in critical two-dimensional percolation. Our formulas are valid in the upperhalf-plane, or any conformally equivalent region. We find excellent agreement of our results withhigh-precision computer simulations. There are indications that our formulas hold more generally.
Keywords: correlation functions, factorization, percolation
I. INTRODUCTION
Correlation functions play an important role in the theory of fluids at thermal equilibrium, appearing in expressionsfor many experimental as well as theoretical quantities. Higher-order correlations, i.e. correlations of quantities suchas the density ρ ( x ) at several points, e.g. h ρ ( x ) ρ ( x ) ρ ( x ) ... i , where the brackets denote a thermal average, occurin many contexts. Calculating such quantities is therefore a central goal of the theory of fluids. This is, however,an especially challenging problem, and many approaches have been proposed (for a review, see [1]). One idea isto factorize the higher-order correlations in terms of lower-order correlations (with fewer points). In this article,we consider percolation in two dimensions at the percolation point in the upper half-plane (or any simply connectedregion). In this case, by use of conformal field theory, we are able to exhibit several exact formulas in which three-pointcorrelations factorize in terms of two-point correlations or correlations involving one point and an interval. There areno similar exact results in the theory of fluids, to our knowledge.Percolation in two-dimensional systems has a long history, and has been examined by a very wide variety of methods,especially at the percolation point (see [2] for some representative references). In this paper we report the results ofcalculations of correlation functions of the density in the upper half-plane. Our formulas follow from conformal fieldtheory [3, 4], which is applicable to critical 2-D percolation in the continuum limit.We focus on the density, defined as the number of samples for which a site belongs to clusters satisfying somespecified boundary condition (such as clusters touching certain parts of the boundary) divided by the total numberof samples N , in the limit N → ∞ . The density is an interesting and also practically important universal featureof percolation at the critical point. Note that the density at a point z of clusters which touch specified parts of theboundary is proportional to the probability of finding a cluster that connects those parts of the boundary with a smallregion around the point z . Thus density correlation functions may also be viewed as probabilities of configurationswith certain specified connections.In conformal field theory, operators in the bulk (except the unit operator) are defined so that their expectationvalued vanishes, e.g. h ψ ( z ) i = 0. Hence the density at z calculated below is subtracted . In particular, it will vanishwhen z is sufficiently far from the other points or intervals with which it connects.In a recent Letter [2], we considered the problem of clusters simultaneously touching one or two intervals onthe boundary of a system, and also considered cases where those intervals shrink to points (anchor points). In[2] we exhibited factorization in one particular case, demonstrated a relation to two-dimensional electrostatics, andhighlighted the universality of percolation densities. (In particular, we pointed out that by conformal covariance,the factorization is valid in any simply connected domain.) In this paper, we find more factorizations, and also giveexplicit expressions for the coefficients in the factorization formulas.In addition, we confirm our theoretical results via numerical simulations to a high degree of accuracy. One case ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: rziff@umich.edu confirms numerical results in [2] using a different realization of percolation; simulations of our new predictions arealso included.In section II, we first give the derivation of the factorization results, and then compare them with computersimulations. Section III includes a few concluding remarks and discussion.
II. PREDICTIONS AND SIMULATIONS
This section begins by presenting the derivation of our formulas, then compares them with computer simulations.First we recapitulate the derivation of the factorization presented in [2]. Employing conformal field theory, appliedto percolation, one may identify three operators of interest. The boundary operator φ , ( x ), with conformal dimension h , = 0, changes the boundary conditions from fixed to free at a point x on the boundary (here, the real axis). Thisoperator appears in the field theory limit of a percolation system at a boundary point between a segment where allthe sites are occupied (or all empty) and one where the sites are unconstrained. Similarly, the boundary operator φ ( x ) := φ , ( x ), with conformal dimension h φ = 1 /
3, anchors a cluster at a point x on a free boundary segment; thisoperator appears along a free boundary at a point where a cluster touches the boundary. Finally, the “magnetization”operator ψ ( z ) := φ / , / ( z ), with conformal dimension h ψ = 5 /
96, measures the density of clusters at a point z inthe upper-half plane. It is the field theory limit of the corresponding lattice quantity. Correlations involving theseoperators are proportional to the probability of finding a cluster connecting the various points, or intervals betweenboundary points, that they define.The notation in the following formulas omits, for brevity and clarity, various constants of proportionality. Some ofthese are universal (independent of the particular realization of percolation), while others are not. The latter typeincludes two kinds: constants multiplying the conformal operators, which are associated with the particular realizationof the operator for the system of interest, and constants specifying the dimension of the small regions with which theclusters are conditioned to connect. Our final results are homogeneous in operators and dimensional constants, soignoring these constants makes no difference. The remaining (universal) constants are evaluated directly by takingan appropriate limit.Consider the probability P ( x , x ) of a cluster in the upper half-plane (as are all the clusters considered herein)that connects the points x and x on the real axis. This (for x < x ) is P ( x , x ) ∝ h φ ( x ) φ ( x ) i ∝ (cid:16) x − x (cid:17) / . (1)This result follows from Cardy’s formula for the crossing probability [5] in the appropriate limit.The probability P ( z ) of a cluster connecting any point on the boundary with a point z = x + iy in the upperhalf-plane is simply P ( z ) ∝ h ψ ( z ) ψ (¯ z ) i ∝ (cid:16) y (cid:17) / , (2)consistent with the considerations in [6]. Here, the point ¯ z appears because because half-plane correlation functionsare given by full-plane correlators (which the expectation value denotes) with “image” operators [4].The probability P ( x , z ) of a cluster constrained to touch the (boundary) point x and a point z = x + iy in theupper-half plane is similarly given by a three-point correlation function, P ( x , z ) ∝ h φ ( x ) ψ ( z ) ψ (¯ z ) i ∝ y / | z − x | / . (3)We now consider two more complicated objects. The first is the probability P ( x , x , z ) of a cluster touching twoboundary points x and x as well as a point z in the upper-half plane. This is given by a four-point correlationfunction, P ( x , x , z ) ∝ h φ ( x ) φ ( x ) ψ ( z ) ψ (¯ z ) i = ( x − x ) − / y − / F ( η ) , (4)where the cross-ratio η = ( z − x )(¯ z − x )(¯ z − x )( z − x ) , (5)(this form is slightly different from, but equivalent to, the expression in [2]). It is convenient, in what follows, toexpress the cross-ratio in terms of the angle ζ (see Fig. 1) η = e − i ( θ + θ ) = e iζ . (6) FIG. 1: (Color online).The geometry used to define ζ . Since φ is a level-three operator, F ( η ) satisfies a third-order differential equation. The appropriate solution may beidentified via physical arguments in the limit x → x [2]. In terms of ζ , it is F ( ζ ) = sin / ( ζ ) . (7)Next, combining (1) and (3-7) one finds by simple algebra that P ( x , x , z ) = C p P ( x , x ) P ( x , z ) P ( x , z ) . (8)Here, the constant C may be evaluated by taking x → x , so that the lhs of (8) becomes a three-point function,with C = C a (boundary) operator product expansion coefficient [2]. This is evaluated in [7], giving C = 2 / π / / Γ(1 / / . (9)One finds C = 1 . . . . . In [2] we report simulation results verifying (8). For bond percolation on the squarelattice we find C = 1 . ± . C , when one or both of the points x , x is moved off the boundary [2]. However in this case thefactorization only holds asymptotically, when the points are far apart compared to the distance to the edge.We emphasize that (8), along with (14), (16) and (18) below, are both exact and universal. Furthermore, they (by aconformal mapping) also hold in any simply-connected region, with the same proportionality constants. In a differentgeometry, the functions in (8), (14), (16) or (18) will change, but the relation remains, with the same proportionalityconstant.Note that (8) resembles the Kirkwood superposition approximation [8], with the difference that here there is asquare root and a coefficient on the rhs. Further, the Kirkwood formula is apparently only exact asymptotically, bycontrast to (8), which is both exact and universal.Now we examine the probability P (( x , x ) , z ) (cid:16) P (cid:0) ( x , x ) , z (cid:1)(cid:17) of a cluster touching the boundary on (outside) the interval ( x , x ) as well as a point z in the upper-half plane. Both these quantities are given by the correlator h φ , ( x ) φ , ( x ) ψ ( z ) ψ (¯ z ) i = y − / G ( ζ ) . (10)Since φ , is a level-two operator, G satisfies a second-order differential equation. The solution corresponding to P (( x , x ) , z ) [2] may be written as P (( x , x ) , z ) ∝ y − / sin / ( ζ/
2) ; (11)and it is straightforward to verify that P (cid:16) ( x , x ) , z (cid:17) ∝ y − / cos / ( ζ/ . (12)The identity sin( ζ ) = 2 sin( ζ/ ζ/
2) then immediately implies P ( x , x , z ) P ( z ) ∝ P ( x , x ) P (( x , x ) , z ) P (cid:16) ( x , x ) , z (cid:17) . (13) FIG. 2: (Color online).Diagrammatic illustrations of (8) and (14).
We can evaluate the constant in (13) by taking the limit x → x (the same procedure used to evaluate C in (8)).The leading term gives P ( x , x , z ) P ( z ) = C P ( x , x ) P (( x , x ) , z ) P (cid:16) ( x , x ) , z (cid:17) , (14)with the universal constant C equal to the ratio of the boundary operator product expansion coefficients C , givenabove, to C , given below. Specifically, C = C /C , so that C = 8 π / , (15)with C = 1 . . . . . Note that although (14) includes correlation functions involving specified intervals, which areperhaps more complicated than the correlation functions in (8), there is no square root.Next, for completeness, we present two factorized expressions that follow from the above, but have different formsand certain new features. First, one can eliminate P ( x , x , z ) between (8) and (14). This gives P (( x , x ) , z ) P (cid:16) ( x , x ) , z (cid:17) p P ( x , x ) = C P ( z ) p P ( x , z ) P ( x , z ) , (16)with C = C . Thus (see [7]) C = 2 / / π / Γ(1 / / , (17)and C = 0 . . . . . Finally, multiplying (8) by (16) (or dividing the square of (8) by (14)) gives P (( x , x ) , z ) P (cid:16) ( x , x ) , z (cid:17) P ( x , x , z ) = C P ( z ) P ( x , z ) P ( x , z ) , (18)with C = C C , so that C = 2 π / Γ(1 / , (19)and thus C = 0 . . . . . Equation (18) is “homogeneous in averages”, as discussed below.Figure 2 illustrates (8) and (14) diagrammatically.We have simulated (8) and (14) (note that [2] includes other numerical results for (8)). We used site percolationon a square lattice of size 510 × p c = 0 . × samples. The boundary sites, chosen as( x, y ) = ( x = 192 ,
1) and ( x = 320 ,
1) (i.e. 3 / / x , x ) (and its complement ( x , x )). We considered each site z in the entire lattice, and determinedwhich of the various boundary points and intervals that it connects with. The fraction of samples in which thetwo interval boundary points were connected together was found to be P ( x , x ) = 0 . P ( x , x , z ) / p P ( x , x ) P ( x , z ) P ( x , z ), which is predicted by (8) to equal C = C , and also theratio P (( x , x ) , z ) P (cid:16) ( x , x ) , z (cid:17) p P ( x , x ) / P ( z ) p P ( x , z ) P ( x , z ) predicted to be C = C by (16). These y c o rr e l a t i o n r a t i o s FIG. 3: (Color online).Numerical results for correlation ratios predicted to be C = 1 . ... (upper two curves) and C = 0 . ... (lower twocurves), for a system of 510 ×
510 sites, for anchors or endpoints at x = 192, y = 1, and x = 320, y = 1. In each pair ofcurves, the upper (blue online) is along x = 256 (the centerline), shifted up by 0 .
1, while the lower (red online) represent thedata for x = 192, touching the point x . quantities are shown along the line x = 192, 0 < y < x , x = 256, 0 < y < x , P ( x , x , z ), which is a rarer event and more subject tofluctuations than the other quantities.For the first ratio, we find C = C = 1 . ± . C = 1 . . . . ,and identical to the value found in [2] using bond percolation on the square lattice. For the second ratio, we find C = C = 0 . ± . C = 0 . . . . These valueswere determined by averaging the point z over all points on the lattice. III. DISCUSSION
Any of our factorization results, namely (8), (14), (16) or (18), when written as expectation values of conformaloperators, as mentioned, is necessarily “homogeneous in operators”, i.e. each distinct operator either appears to thesame power on each side of the equation, or pairs of operators are replaced by a single operator and the appropriateoperator product expansion coefficient. If this were not so, a universal expression would not be possible.Now (8) may be regarded as a generalization of the three-point function of conformal field theory [9], in the casewhen all three operators are the same. (It reduces to this when the point z is on the boundary.) The three-pointresult only requires covariance under the special conformal group, not the full machinery of conformal field theory.This may indicate that (8) is more generally valid.Equation (18) is “homogeneous in averages”, i.e. the same number of brackets appears on each side. This meansthat it may be verified numerically without overall normalization–one can use the raw data for the number of samplesfor each specified probability P without dividing by the total number of samples.As remarked in [2], preliminary calculations and numerical data show that factorization generalizing (8) holds forFortuin-Kastleyn clusters in the critical Potts models as well [10].It is interesting to understand the factorization in terms of two-dimensional electrostatics. There are several waysto do this. Defining ϕ ( x ; z ) = 1 / ( z − x ) as the generalized (complex) potential at z of a unit dipole at x expressesthe potential of a dipole of strength | p | in the direction Arg( p ) as p ϕ ( x ; z ). Now, establishing our factorizationresults involves writing P ( x , x , z ), P (( x , x ) , z ), or P (cid:0) ( x , x ) , z (cid:1) in terms of simpler correlation functions. Thekey algebraic step needed is expressible assin ( ζ ) ∝ | ϕ ( x ; z ) ϕ ( x ; z ) | y ( x − x ) ∝ | ϕ ( x ; z ) ϕ ( x ; z ) | | ϕ ( z ; ¯ z ) ϕ ( x ; x ) | . (20)This may be used, for example, in conjunction with (2-7), (11), and (12) to establish (18). (These manipulations donot give an expression for C , of course.)Finally, one might wonder why the numerical accuracy of the conformal prediction is so high, especially at shortdistances. In general, the field theory limit of an appropriate lattice quantity, for instance the order parameter, is givenby a conformal primary field plus correction terms proportional to its descendant fields. These descendants necessarilyhave dimensions larger than that of the primary, and hence give rise to terms in a given correlation function that dieaway more rapidly with distance than those due to the primary, but which may be substantial at short distances,even for very large lattices. However, in critical percolation in two dimensions, previous numerical work on closelyrelated quantities (see [2, 11]) has shown that such effects are very small. The reason for this is, to our knowledge,not known, but it follows that the accuracy which we observe herein is not surprising.In conclusion, we have presented several new formulas for higher-order correlation functions applicable to criticalpercolation in two dimensions. These have the property of exact factorization in terms of lower-order correlations orcorrelations involving intervals. Our predictions agree with the results of high-precision simulations.For the future, it might be possible, using perturbation theory, to find the corrections to our factorization resultsfor p = p c . However the calculations required do not appear to be simple. IV. ACKNOWLEDGMENTS
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