Nested Closed Paths in Two-Dimensional Percolation
Yu-Feng Song, Xiao-Jun Tan, Xin-Hang Zhang, Jesper Lykke Jacobsen, Bernard Nienhuis, Youjin Deng
NNested Closed Paths in Two-Dimensional Percolation
Yu-Feng Song,
1, 2
Xiao-Jun Tan,
1, 3
Xin-Hang Zhang, Jesper LykkeJacobsen,
4, 5, 6, 7, ∗ Bernard Nienhuis, † and Youjin Deng
1, 2, ‡ Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China MinJiang Collaborative Center for Theoretical Physics,Department of Physics and Electronic Information Engineering,Minjiang University, Fuzhou, Fujian 350108, China Cainiao Network, Hanzhou, Zhejiang 310013, China Laboratoire de Physique de l’ ´Ecole Normale Sup´erieure, ENS, Universit´e PSL,CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France Sorbonne Universit´e, ´Ecole Normale Sup´erieure, CNRS,Laboratoire de Physique (LPENS), F-75005 Paris, France Universit´e Paris Saclay, CNRS, CEA, Institut de Physique Th´eorique, F-91191 Gif-sur-Yvette, France Institut des Hautes ´Etudes Scientifiques, Universit´e Paris Saclay, CNRS,Le Bois-Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette, France Delta Institute of Theoretical Physics, Instituut Lorentz,P.O. Box 9506, NL-2300 RA Leiden, The Netherlands (Dated: February 16, 2021)For two-dimensional percolation on a domain with the topology of a disc, we introduce a nested-path operator (NP) and thus a continuous family of one-point functions W k ≡ (cid:104)R · k (cid:96) (cid:105) , where (cid:96) isthe number of independent nested closed paths surrounding the center, k is a path fugacity, and R projects on configurations having a cluster connecting the center to the boundary. At criticality,we observe a power-law scaling W k ∼ L − X np , with L the linear system size, and we determine theexponent X np as a function of k . On the basis of our numerical results, we conjecture an analyticalformula, X np ( k ) = φ − φ / ( φ − ) where k = 2 cos( πφ ), which reproduces the exact resultsfor k = 0 , X np for other k values. In addition, weobserve that W ( L ) = 1 for site percolation on the triangular lattice with any size L , and we provethis identity for all self-matching lattices. Introduction —
Percolation [1–4] is a paradigmaticmodel in the field of phase transitions and critical phe-nomena and a central topic in probability theory. Italso finds important applications in various fields suchas fluids in porous media [5], gelation [6] and epidemi-ology [7]. Bond percolation corresponds to the Q → Q -state Potts model [8, 9], and provides asimple illustration of many important concepts for thelatter [10]. In two dimensions (2D), the algebraic use ofsymmetries—lattice duality [11], Yang-Baxter integrabil-ity [12, 13] and conformal invariance [14, 15]—has led to ahost of exact results. Critical exponents are predicted byCoulomb-gas (CG) arguments [16], conformal field the-ory [17] and Stochastic Loewner Evolutions [18], and havebeen proven rigorously for e.g. triangular-lattice site per-colation [19].In site percolation, the sites of a lattice are occupiedwith probability p and empty otherwise. A sequence ofdistinct, occupied sites of which each is nearest neighborto its predecessor is called a path. Two occupied sitesare connected if there is a path from one to the other.Two paths are independent if they do not have a sitein common. A closed path has neighboring first and lastsites. A maximal set of connected sites is called a cluster.In bond percolation, the edges or bonds of the lat- (a) (b)FIG. 1. Examples of configurations. (a) Triangular-latticesite percolation (STr). The central site is neutral (white),the occupied (empty) sites are in red (green), and the threeindependent nested paths are specified by black lines. (b)Square-lattice bond percolation (BSq). The occupied bondsare shown red and the unoccupied as green bonds on thedual lattice. The neutral central bond is blank, and the threeindependent paths surrounding it are marked as white lines.They may visit the same site but not share a bond. tice are occupied with probability p , or vacant (empty).Paths, connectivity and clusters follow the same defini-tions as in site percolation, with sites replaced by bonds of the lattice, and nearest neighbor by having a site incommon . Two paths are independent if they do not havea bond in common, and do not cross, but they may sharea site.Clusters and paths can also be introduced for empty a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b elements. For bond percolation, these paths and clus-ters typically consist of bonds on the dual lattice. Adual bond is occupied if the (primal) bond it intersectsis empty, and vice versa. For this reason, the paths onempty elements are often referred to as dual.A path between two regions is typically not unique. Incontrast, cluster boundaries form trajectories which areuniquely determined by the configuration. Two differentcluster boundaries are by nature non-overlapping.Cluster boundaries define two families of exponents,which have been computed by CG methods. The so-called watermelon exponents [20, 21] govern the proba-bility that two (or more) distant regions are connected bya given number, say n , of cluster boundaries. The water-melon exponents have the value X wm ( n ) = ( n − / nested-loop oper-ator (see [22, 23]). This operator gives a weight, say k ,to each cluster boundary surrounding the insertion point(multiple such boundaries must be nested, whence thename). Its two-point function gives a weight to thosecluster boundaries that surround one, but not both ofthe insertion points. The corresponding exponent is X nl ( k ) = 34 φ − , where k = 2 cos( πφ ) . (1)Analogous to X wm ( n ) are the so-called monochromatic n -path exponents X mp ( n ), governing the decay of proba-bility that between two distant regions there are n inde-pendent paths, all on clusters (or, equivalently, all ondual clusters). The case n = 1 means that no clus-ter boundary separates the two regions, so X mp (1) = X nl (0) = . But the exponents for n = 2 (backbone ex-ponent) or higher n do not appear amenable to CG anal-ysis, and hence they are known only numerically [24–26].As a side-remark we mention that when one or more, butnot all, of the paths are on dual clusters, the exponentsare different and in fact identical to X wm ( n ) [25, 27].In this Letter we propose to similarly consider the pathanalogue of the nested-loop operator: the nested-path(NP) operator. It gives a weight k to each independentclosed path surrounding the insertion point. We inves-tigate here the exponent of this operator by numericalmeans. For simplicity, we simulate its one-point func-tion: 2D percolation with the NP operator placed at thecenter of a compact domain of linear dimension L . Wethus estimate the expectation value W k ≡ (cid:104)R · k (cid:96) (cid:105) , where R is the indicator function of a path from the center tothe boundary of the domain, and (cid:96) the number of inde-pendent closed paths surrounding the center. The factor R ensures that two consecutive surrounding paths areconnected, rather than separated by two cluster bound-aries. The configurations with R = 1 we call percolating. Results, summary —
We show that, at the perco-lation threshold, the scaling of W k obeys a power law W k ∼ L − X np with an exponent X np ( k ), that depends con- − − (a) STr (b) BSq W k L k = 57.7537.6523.1813.086.302.001.000.00-0.48-1.00-1.35 L k = 57.7537.6523.1813.086.302.001.000.00-0.48-1.00-1.35 FIG. 2. Log-log plot of W k versus linear size L , for STr (a)and BSq (b). The lines represent the fitting curves by Eq. (2)and strongly indicate the algebraic dependence of W k on L . tinuously on the weight k . A high-precision estimate of X np is obtained as a function of k . For k = 2, we observethat W ( L ) = 1 for site percolation on the triangularlattice with any L , and prove this to be true for any self-matching lattice [28]. We present an analytical formula(3), analogous to (1), which reproduces some exact val-ues and agrees so well with the numerical results, thatwe conjecture it to be exact. Results, details —
We study site percolation on a tri-angular lattice (STr) in a hexagonal domain with freeboundaries. The scale L is the length of the diagonal.Fig. 1a shows a sample configuration, with L = 17. Thecentral site is neutral, and the other sites are occupiedwith probability p . For each configuration we calculatethe number (cid:96) of independent closed paths that surroundthe center, and R which is 1 (0) if there is (not) a pathfrom the center to the boundary. While (cid:96) is well-defined,the paths are not unique. In Fig. 1a, R = 1 and (cid:96) = 3.Analogous procedures are applied to bond percolationon the square lattice (BSq), see Fig. 1b, with a neutralbond placed at the center, and the length of the diagonal, L = 15.We are interested in the scaling behavior of W k ( L )at the percolation threshold p c . For k = 1, W ≡ (cid:104)R(cid:105) represents the probability that the central site is con-nected to the boundary, which is seen from (1) to decay as (cid:104)R(cid:105) ∼ L − / . The contributions to W are those, whichhave a path from the center to the boundary, but no closed path surrounding the center. These configurationsmust have a path of occupied and one of empty elementsfrom the center to the boundary and consequently twocluster boundaries connecting the center to the bound-ary. These events are selected for the watermelon expo-nent X wm (2) = 1 /
4. Thus, proposing W k ∼ L − X np ( k ) , wealready know that X np (1) = 5 /
48 and X np (0) = 1 / p c = 1 /
2, with geometric . . (a) STr . (b) BSq W L FIG. 3. Observable W versus L , for STr and BSq. The STrvalues for L ≤ shapes as in Fig. 1. The linear size L is taken in range3 ≤ L ≤ L , the numberof samples is at least 5 × for L ≤ × for100 < L ≤ × for 100 < L ≤ × for L > W k versus L is shown in Fig. 2. Thedata clearly support asymptotic power-law dependenceof W k on L . We fit (by least squares) the W k data to W k = L − X np ( a + b L − ω + b L − ω ) . (2)We admit only data points with L ≥ L m for the fits, andsystematically study the effect on the residual χ value(weighted according to confidence level) when varying L m . In the best fits, ω ≈
1. The results with ω = 1 aregiven in Tab. I. The estimates of X np (1) and X np (0) agreewell with the exact values 5 /
48 and 1 /
4, respectively.Table I strongly indicates that X np (2) = 0 for k = 2. φ -1 -3/4 -1/2 -1/4 0 k φ k X np ( k ) as a function of k , withthe rows ‘conj’ from Eq. (3). The W data for STr and BSq are listed in Tab. II andplotted in Fig. 3 versus L . For STr we find for L ≤ L within statisticalerrors, that W = 1. For BSq, we find W = 1 for L = 3and 2097075 / for L = 5, shown in Tab. II togetherwith simulation data for larger L . As L increases, W converges to a value slightly smaller than, and clearly L L
13 29 61 125STr 0.999997(6) 0.999997(7) 1.000010(9) 0.99997(4)BSq 0.999088(5) 0.99764(1) 0.99649(1) 0.99567(5) L
253 509 1021 2045STr 1.00007(5) 0.99995(6) 0.9999(2) 1.0001(2)BSq 0.99538(6) 0.99503(8) 0.9949(3) 0.9948(4)TABLE II. Observable W . For L = 4093 and 8189, we haverespectively W = 0 . W =0 . . different from 1. A least-squares fit W ( L ) = W , ∞ + bL − for L >
20, gives the asymptotic value as W , ∞ =0 .
994 5(2). Thus, we conjecture X np (2) = 0 in general,and, for STr, W ( L ) = 1.Then, in an attempt to find a formula analogous to (1),we set k = 2 cos( πφ ) and plot X np as a function of φ , asshown in Fig. 4. Noting (i) X np (2) = 0, (ii) an apparentpole for some positive φ <
1, and (iii) an asymptoticslope of 3/4 for negative φ , just as X NL , (1), it is naturalto propose X np = 3 φ / aφ / ( φ − b ) as the simplestrational function that matches these observations. Thenthe exact results for X np (0) and X np (1) fix X np ( k ) = 34 φ − φ ( φ − / , (3)in which some well-known exponents of 2D percolationseem to appear. The excellent agreement with the nu-merical results shown in Fig. 4 and Table I, leads us toconjecture that (3) is exact. But, in spite of some sim-ilarity with (1) we have not found any theoretical basisfor (3). Method —
In simulations, each site (bond) is ran-domly occupied with probability p c = 1 /
2, a cluster isgrown from the center by standard breadth-first search. -1.0-0.5 − . -1.0 − . . φ = − X N P φ STrBSq
FIG. 4. Exponent X np ( k ) versus φ . Estimates of X np for STrand BSq agree well with Eq. (3), shown as the solid curve. If the cluster does not reach the boundary, R = 0 andthere is no contribution to W k . Otherwise, we computethe number (cid:96) of paths surrounding the center. This canbe achieved efficiently by the following algorithm.The m -th path surrounding the center acts as the seedof one or more dual (empty) clusters, linked together bythe ( m − m − m -th path as the chain of occupied sites (orbonds) that stops the dual-cluster growth. A caveat isthat for BSq the dual cluster consists of bonds of the dual lattice. Algorithm 1 sketches the corresponding proce-dure, in which Ω is the region that is encircled by thenext surrounding path. It is written for STr, but whensites are replaced by bonds, and adjacent sites by bondssharing a site, it works for BSq too. Algorithm 1
Calculate (cid:96) for site percolation (cid:96) = 0Ω ← the central siteGrow the empty cluster around Ω while boundary is not reached do (cid:96) = (cid:96) + 1Ω ← set of occupied sites encircling the empty clusterGrow the empty cluster around Ω end while Proof of the identity W ( L ) = 1 for STr — Take anarbitrary STr configuration on a simply connected pieceof the lattice, with a distinguished, ‘central’ site. Con-sider a maximal set of independent paths surrounding thecenter, with each path consisting either of only occupied(red) sites or of only empty (green) sites. The innermost such colored paths, given the interior ones, are uniquelydefined (and can be constructed by Algorithm 1).Let (cid:96) be the number of these nested paths. By P i wedenote the map that inverts (red ↔ green) the color ofpath i (counted from the center), and of all sites strictlybetween path i and path i − P i .Since p c = , the set { P i } (cid:96)i =1 generates 2 (cid:96) equiprobableconfigurations. Within this ensemble, R = 1, if and onlyif all the paths are occupied. The entire ensemble canbe generated by the P i from any of its members. There-fore all configurations are member of precisely one suchensemble, and as a consequence W = (cid:104)R · (cid:96) (cid:105) = 1. (cid:3) An essential property used in the proof is that invertinga path of occupied sites surrounding the center creates abarrier preventing a path of occupied sites to connect thecenter with the boundary. This is an obstacle against ap-plying the proof to BSq, where a path of occupied bondscan cross a path of empty bonds, see [29] for more expla-nation.The regularity of the lattice and of the domain are notused in the proof, but having p c = is crucial. However, FIG. 5. Example of a set of (cid:96) = 2 configurations, related bythe inversion map P i in the proof. Paths of occupied (empty)sites, denoted by solid (dotted) lines, are uniquely located byAlgorithm 1. The map P leads to (a) ↔ (b) and (c) ↔ (d),while P gives (a) ↔ (c) and (b) ↔ (d). p c = for any self-matching lattice, and in particular forlattices of which all faces are triangles. Hence, the resultis also valid for regular or irregular planar triangulationgraphs, of any shape and position of the center. Discussion —
We construct a new family of geome-tric quantities W k for critical percolation in two dimen-sions and determine a continuous set of critical exponents X np ( k ) with high precision. An identity W ( L ) = 1, in-dependent of the linear size L , is found for triangular-lattice site percolation and proven for any lattice withonly triangular faces. This implies an exact exponent X np (2) = 0. The universality of the critical exponent X np is well supported by simulations for both bond andsite percolation. Apart from the special cases k = 2 , , X np are unknown. We conjecture ananalytical function, Eq. (3), which reproduces the knownexact results and agrees excellently with numerical esti-mates of X np for other k values. We note that, thoughEq. (3) is somewhat similar to existing results, provingit remains elusive.Future work will involve the Q -state Potts model in theFortuin-Kasteleyn cluster representation, which includesbond percolation as a special case for Q → ∗ [email protected] † [email protected] ‡ [email protected] [1] S. R. Broadbent and J. M. Hammersley, Percolation Pro-cesses , Math. Proc. of the Cambridge Phil. Soc. , 629(1957).[2] D. Stauffer and A. Aharony, Introduction to percolationtheory (Taylor & Francis, London, 1994), 2nd ed.[3] G. R. Grimmett,
Percolation (Springer, Berlin, 1999),2nd ed.[4] B. Bollob´as and O. Riordan,
Percolation (CambridgeUniversity Press, 2006).[5] A. Hunt, R. Ewing and B. Ghanbarian,
Percolation the-ory for flow in porous media (Springer, 2014), 3rd ed.[6] D. Stauffer, A. Coniglio and M. Adam,
Gelation and crit-ical phenomena (Springer, 1982).[7] L. Meyers,
Contact network epidemiology: Bond percola-tion applied to infectious disease prediction and control ,Bulletin of the American Mathematical Society (1),63-86 (2007).[8] P.W. Kasteleyn and C.M. Fortuin, Phase transitions inlattice systems with random local properties , J. Phys. Soc.Jpn. Suppl., 11 (1969).[9] R. B. Potts,
Some generalized order-disorder transforma-tions , Proc. Cambridge Philos. Soc. , 106 (1952).[10] F. Y. Wu, The Potts model , Rev. Mod. Phys. , 235(1982).[11] H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. Part I , Phys. Rev. , 252(1941).[12] E. H. Lieb, Exact solution of the problem of the entropyof two-dimensional ice , Phys. Rev. Lett. , 692 (1967).[13] R. J. Baxter, Partition function of the eight-vertex latticemodel , Ann. Phys. (NY) , 193 (1972).[14] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quan-tum field theory , Nucl. Phys. B , 333 (1984).[15] D. Friedan, Z. Qiu and S. Shenker,
Conformal invariance,unitarity, and critical exponents in two dimensions , Phys.Rev. Lett. , 1575 (1984).[16] B. Nienhuis, in Phase transitions and critical phenom-ena , edited by C. Domb, M. Green and J. L. Lebowitz(Academic Press, London, 1987), Vol. 11.[17] J. L. Cardy, in
Phase transitions and critical phenom-ena , edited by C. Domb, M. Green and J. L. Lebowitz(Academic Press, London, 1987), Vol. 11.[18] G. F. Lawler, O. Schramm and W. Werner,
The dimen-sion of the planar Brownian frontier is 4/3 , Math. Res.Lett. , 401 (2001).[19] S. Smirnov and W. Werner, Critical exponents for two-dimensional percolation , Math. Res. Lett. , 729 (2001).[20] B. Duplantier and H. Saleur, Exact determination of thepercolation hull exponent in two dimensions , Phys. Rev.Lett. , 2325 (1987)[21] B. Nienhuis, Critical behavior in two dimensions andcharge asymmetry of the Coulomb gas , J. Stat. Phys. ,731 (1984).[22] S. Mitra and B. Nienhuis, Exact conjectured expressionsfor correlations in the dense O(1) loop model on cylin-ders , J. Stat. Mech.: Theory Exp., P10006 (2004)[23] M. den Nijs,
Extended scaling relations for the magneticcritical exponents of the Potts model , Phys. Rev. B ,1674 (1983).[24] J. L. Jacobsen and P. Zinn-Justin, Monochromatic pathcrossing exponents and graph connectivity in 2D percola-tion , Phys. Rev. E , 055102(R) (2002).[25] V. Beffara and P. Nolin, On monochromatic arm expo- nents for 2D critical percolation , The Annals of Proba-bility , 1286–1304 (2011).[26] X. Xu, J. F. Wang, Z. Z. Zhou, T. M. Garoni and Y. J.Deng, Geometric structure of percolation clusters , Phys.Rev. E , 012120 (2014).[27] M. Aizenman, B. Duplantier and A. Aharony, Path-crossing exponents and the external perimeter in 2D per-colation , Phys. Rev. Lett. , 1359 (1999).[28] M. F. Sykes and J. W. Essam, Exact critical percolationprobabilities for site and bond problems in two dimen-sions , J. Math. Phys. , 1117 (1964).[29] See Supplemental Material at [URL] for an explanationwhy the proof for STr does not work for BSq, and anattempt to remedy that situation. Supplementary material withNested Closed Paths in Two-Dimensional Percolation
Y.-F. Song, X.-J. Tan, X.-H. Zhang, J.L. Jacobsen, B. Nienhuis, Y. Deng
APPLICABILITY OF THE PROOF OF W = 1 The proof that W = 1 for site-percolation on latticeswith only triangular faces fails for bond-percolation one.g. the square lattice. This note explains why, and dis-cusses some unsuccesful attempts to remedy this.The first issue is that in the proof we consider pathsboth on occupied and on empty sites, occupied paths andempty paths, for short. For bond percolation there is achoice how to construct the empty paths; this is between(i) the bonds of the lattice that happen to be empty, or(ii) the bonds of the dual lattice at the positions wherethe primal bond is empty. The proof uses two propertiesof the relation between occupied and empty paths: (a) anoccupied path becomes an empty path (and v.v.) underinversion, and (b) an empty path and an occupied pathcannot cross. With choice (i) we have property (a) butnot (b), and with choice (ii) we property (b) but not (a).Fig. 6 below illustrates these facts. As a consequence,a straightforward translation of the proof for BSq is notpossible. cba d e FIG. 6. Properties of the inversion map. Occupied bondsare marked red, and empty bonds green. A path of occupiedbonds (b), converted by an inversion transformation, to (a)an empty path on the same lattice, under choice (i), or underchoice (ii) to (b) a sequence of dual bonds, not forming a path.Part (d) shows how an occupied path and an empty path cancross, but (e) an occupied path cannot cross an empty pathon the dual lattice.
One may consider an alternative definition of W k thatwould allow to reconstruct the proof. A central conceptin the proof is the inversion transformation P i of a pathand (a part of) its interior. Under such transformationa path remains a path but changes color. This seems tomake choice (ii) hopeless.So we concentrate on choice (i), and consider bondsand paths on the primal lattice only. Then we have todeal with the difficulty that paths can cross each other. We rule out crossing paths of the same color by construc-tion, so that it is still possible to define innermost paths.But crossing paths of different colors are difficult to ruleout, as in this case one must give precedence to one orthe other.Thus the restriction that the center is connected to theboundary over an occupied path, does not rule out theexistence of an empty path surrounding the center. It istempting to redefine W k by giving the weight k to eachoccupied path surrounding the center, while demandingthere are no empty paths surrounding the center, thiscondition replacing the one that an occupied path con-nects the center to the boundary. But for a configura-tion with occupied and empty paths, both surroundingthe center, and crossing each other, we did not succeedto define the transformations P i appropriately. The dif-ficulties are illustrated in Fig. 7a showing a configura-tion in which the center (black circle) is surrounded byboth an occupied and an empty path. If the empty pathis inverted, the resulting configuration has two occupiedpaths, but with a different structure (b). One may arguethat in the original definition, not only the path itself,but also (part of) its interior was inverted. But invertingthe empty path with its interior results in an even lesshopeful configuration, with only one surrounding pathleft (Fig. 7c).In conclusion, we have not found a variant definitionof W k such that we can prove that W = 1 also for BSq. a b ca b c