Universality for bond percolation
in two dimensions
Abstract.
All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star–triangle transformation and the boxcrossing property. The exponents in question are the onearm exponent , the alternatingarms exponents for , the volume exponent , and the connectivity exponent . By earlier results of Kesten, this implies universality also for the nearcritical exponents , , , (assuming these exist) for any of these models that satisfy a certain additional hypothesis, such as the homogeneous bond percolation models on these three lattices.
Key words and phrases:
Bond percolation, inhomogeneous percolation, universality, critical exponent, arm exponent, scaling relations, boxcrossing, star–triangle transformation, Yang–Baxter equation2010 Mathematics Subject Classification:
60K35, 82B431. Introduction and results
1.1. Overview
Twodimensional percolation has enjoyed an extraordinary renaissance since Smirnov’s proof in 2001 of Cardy’s formula (see [15]). Remarkable progress has been made towards a full understanding of site percolation on the triangular lattice, at and near its critical point. Other critical twodimensional models have, however, resisted solution. The purpose of the current work is to continue our study (beyond [6]) of the phase transition for inhomogeneous bond percolation on the square, triangular, and hexagonal lattices. Our specific target is to show that such models belong to the same universality class. We prove that critical exponents at the critical point are constant within this class of models (assuming that such exponents exist). We indicate a hypothesis under which exponents near criticality are constant also, and note that this is satisfied by the homogenous models.
We focus here on the onearm exponent , and the alternatingarms exponents for . By transporting open primal paths and open dual paths, we shall show that these exponents are constant across (and beyond) the above class of bond percolation models. More precisely, if any one of these exponents, say, exists for one of these models, then exists and is equal for every such model. No progress is made here on the problem of existence of exponents.
Kesten [10] showed that the exponents and are specified by knowledge of , under the hypothesis that exists. Therefore, and are universal across this class of models. Results related to those of [10] were obtained in [11] for the ‘nearcritical’ exponents , , , . This last work required a condition of rotationinvariance not possessed by the strictly inhomogeneous models. This is discussed further in Section 1.4.
It was shown in [6] that critical inhomogeneous models on the above three lattices have the boxcrossing property; this was proved by transportations of open boxcrossings from the homogeneous squarelattice model. This boxcrossing property, and the star–triangle transformation employed to prove it, are the basic ingredients that permit the proof of universality presented here.
A different extension of the star–triangle method has been the subject of work described in [2, 18, 19]. That work is, in a sense, combinatorial in nature, and it provides connections between percolation on a graph embedded in and on a type of dual graph obtained via a generalized star–triangle transformation. In contrast, the work reported here is closely connected to the property of isoradiality (see [3, 8]), and is thus more geometric in nature. It permits the proof of relations between a variety of twodimensional graphs. The connection to isoradiality will be the subject of a later paper [5].
The paper is organized as follows. The relevant critical exponents are summarized in Section 1.3, and the main theorems stated in Section 1.4. Extensive reference will be made to [6], but the current work is fairly selfcontained. Section 2 contains a short account of the star–triangle transformation, for more details of which the reader is referred to [6]. The proofs are to be found in Section 3.
1.2. The models
Let be a countable connected planar graph, embedded in . The bond percolation model on is defined as follows. A configuration on is an element of the set . An edge with endpoints , is denoted . The edge is called open, or open, in (respectively, closed) if (respectively, ).
For and , we say is connected to (in ), written (or ), if contains a path of open edges from some to some . An open cluster of is a maximal set of pairwiseconnected vertices, and the open cluster containing the vertex is denoted . We write if is the endpoint of an infinite open selfavoiding path.
The homogeneous bond percolation model on is that associated with the product measure on with constant intensity . Let denote a designated vertex of called the origin. The percolation probability and critical probability are given by
We consider the square, triangular, and hexagonal (or honeycomb) lattices of Figure 1.1, denoted respectively as , , and . It is standard that , and is the root in the interval of the cubic equation . See the references in [4, 6] for these and other known facts quoted in this paper.
We turn now to inhomogeneous percolation on the above three lattices. The edges of the square lattice are partitioned into two classes (horizontal and vertical) of parallel edges, while those of the triangular and hexagonal lattices may be split into three such classes. We allow the product measure on to have different intensities on different edges, while requiring that any two parallel edges have the same intensity. Thus, inhomogeneous percolation on the square lattice has two parameters, for horizontal edges and for vertical edges, and we denote the corresponding measure where . On the triangular and hexagonal lattices, the measure is defined by a triplet of parameters , and we denote these measures and , respectively. Let denote the set of all such inhomogeneous bond percolation models on the square, triangular, and hexagonal lattices, with edgeparameters belonging to the halfopen interval .
These models have percolation probabilities and critical surfaces, and the latter were given explicitly in [4, 6, 9]. Let
It is well known that the critical surface of the lattice (respectively, , ) is given by (respectively, , ). Bond percolation on may be obtained from that on by setting one parameter to zero.
The triplet is called selfdual if . We write for the triplet , and also for the natural numbers, and for the integers.
1.3. Critical exponents
The percolation singularity is of powerlaw type, and is described by a number of socalled ‘critical exponents’. These may be divided into two groups of exponents: at criticality, and near criticality.
First, some notation: we write as if there exist strictly positive constants , such that
in some neighbourhood of (or for all large in the case ). We write if . Two vectors , satisfy if for all , and .
For simplicity we restrict ourselves to the percolation models of the last section. Let be one of the square, triangular, and hexagonal lattices. Let be invariant under translations of as above, and let . The lattice has a dual lattice , each edge of which is called open if it crosses a closed edge of . Open paths of are said to have colour , and open paths of colour . We shall make use of duality as described in [4, Sect. 11.2].
Let be the set of all vertices within graphtheoretic distance of the origin , with boundary . Let be the annulus centred at , with interior radius and exterior radius . We call (respectively, ) its exterior (respectively, interior) boundary. We shall soon consider embeddings of planar lattices in , and it will then be natural to use the metric rather than graphdistance. The choice of metric is in fact of no fundamental important for what follows. For , we write
Let be a vector lying on the critical surface. Thus, is critical in that
Let , and let ; we call a colour sequence. The sequence is called monochromatic if either or , and bichromatic otherwise. If is even, is called alternating if either or . For , the arm event is the event that the inner boundary of is connected to its outer boundary by vertexdisjoint paths with colours , taken in anticlockwise order. [Here and later, we require arms to be vertexdisjoint rather than edgedisjoint. This is an innocuous assumption since we work in this paper with alternating colour sequences only.]
The choice of is in part immaterial to the study of the asymptotics of as , and we shall assume henceforth that is sufficiently large that, for , there exists a configuration with the required coloured paths. It is believed that there exist constants such that
and these are the armexponents of the model. [Such asymptotics are to be understood in the limit as .]
We concentrate here on the following exponents given in terms of , with limits as :

volume exponent: ,

connectivity exponent: ,

onearm exponent: ,

alternatingarms exponents: , for each alternating colour sequence of length .
It is believed that the above asymptotic relations hold for suitable exponentvalues, and indeed with replaced by the stronger relation . Essentially the only twodimensional percolation process for which these limits are proved (and, furthermore, the exponents calculated explicitly) is site percolation on the triangular lattice (see [15, 16]).
The arm events are defined above in terms of open primal and open dual paths. When considering site percolation, one considers instead open paths in the primal and matching lattices. This is especially simple for the triangular lattice since is selfmatching. It is known for site percolation on the triangular lattice, [1], that for given , the exponent for is constant for any bichromatic colour sequence of given length . This is believed to hold for other twodimensional models also, but no proof is known. In particular, it is believed for any model in that
for any bichromatic colour sequence of length , and any .
We turn now to the nearcritical exponents which, for definiteness we define as follows. Let and , and write for the product measure on in which edge is open with probability
By subcritical exponentialdecay (see [4, Sect. 5.2]), for , there exists such that
The function is termed the correlation length.
Here are the further exponents considered here:

percolation probability: as ,

correlation length: as ,

mean clustersize: as ,

gap exponent: for , as ,
We have written for the mean of under the probability measure , and where is the indicator function of the event .
As above, the nearcritical exponents are known to exist essentially only for site percolation on the triangular lattice. See [4, Chap. 9] for a general account of critical exponents and scaling theory.
1.4. Principal results
A critical exponent is said to exist for a model if the appropriate asymptotic relation (above) holds, and is called invariant if it exists for all and its value is independent of the choice of such .
Theorem 1.1.
For every , if exists for some model , then it is invariant.
By the boxcrossing property of [6, Thm 1.3], we may apply the theorem of Kesten [10] to deduce the following. If either or exists for some , then:

both and exist for ,

exists for ,

the scaling relations and are valid.
Taken in conjunction with Theorem 1.1, this implies in particular that and are invariant whenever either or exist for some .
We note in passing that Theorem 1.1 may be extended to certain other graphs derived from the three main lattices of this paper by sequences of star–triangle transformations, as well as to their dual graphs. This includes a number of uniform tessellations (see [7]) and, in particular, two further Archimedean lattices, namely those denoted and and illustrated in Figure 1.2. The measures on these two lattices are as follows. Let be selfdual. Edge is open with probability where:

if is horizontal,

if is parallel to the right edge of an upwards pointing triangle,

if is parallel to the left edge of an upwards pointing triangle,

the two parameters of any rectangle have sum .
Theorem 1.1 holds with augmented by all such bond models on these two lattices. The proofs are essentially the same. The methods used here do not appear to extend to homogeneous percolation on these two lattices, and neither may they be applied to the other six Archimedean lattices. Drawings of the eleven Archimedean lattices and their duals may be found in [13].
There is a simple reason for the fact that Theorem 1.1 concerns the alternatingarm exponents rather than all arm exponents. We shall see in Section 2 that the star–triangle transformation conserves open primal and open dual paths, but that, in certain circumstances, it allows distinct paths of the same colour to coalesce.
The boxcrossing property of [6] implies a type of affine isotropy of these models at criticality, yielding in particular that certain directional exponents are independent of the choice of direction, For example, if one insists on onearm connections in a specific direction, the ensuing exponent equals the undirected exponent . A similar statement holds for armdirections in the alternatingarm exponents.
Kesten has shown in [11] (see also [12]) that the above nearcritical exponents may be given explicitly in terms of exponents at criticality, for twodimensional models satisfying certain hypotheses. Homogeneous percolation on our three lattices satisfy these hypotheses, but it is not known whether the strictly inhomogeneous models have sufficient regularity for the conclusions to hold for them. The basic problem is that, while the boxcrossing property of [6] implies an isotropy for these models at criticality, the corresponding isotropy away from criticality is unknown. For this reason we restrict the statement of the next theorem to homogeneous models.
Theorem 1.2.
Assume that and exist for some . Then , , , and exist for homogeneous percolation on the square, triangular and hexagonal lattices, and they are invariant across these three models. Furthermore, they satisfy the scaling relations
Other authors have observed hints of universality, and we mention for example [14], where it is proved that certain dual pairs of lattices have equal exponents (whenever these exist).
2. Star–triangle transformation
Consider the triangle and the star , as drawn in Figure 2.1. Let . Write with associated product probability measure , and with associated measure , as illustrated in the figure. Let and . For each graph we may consider open connections between its vertices, and we abuse notation by writing, for example, for the indicator function of the event that and are connected by an open path of . Thus connections in are described by the family of random variables, and similarly for .
It may be shown that the two families
of random variables have the same joint law whenever . That is to say, if is selfdual, the existence (or not) of open connections is preserved (in law) under the star–triangle transformation. See [4, Sect. 11.9].
The two measures and may be coupled in a natural way. Let be selfdual, and let (respectively, ) have associated measure (respectively, ) as above. The random mappings and of Figure 2.2 are such that: has law , and has law . Under this coupling, the presence or absence of connections between the corners , , is preserved.
The maps and act on configurations on stars and triangles. They act simultaneously on the duals of these graph elements, illustrated in Figure 2.3. Let , and define for each primal/dual pair / of the left side of the figure. The action of on induces an action on the dual space , and it is easily checked that this action preserves connections. The map behaves similarly. This property of the star–triangle transformation has been generalized and studied in [2] and the references therein.
Socalled mixed lattices were introduced in [6]. These are hybrid embeddings of the square lattice with either the triangular or hexagonal lattice, the two parts being separated by a horizontal interface. By means of appropriate star–triangle transformations, the interface may be moved up or down, and this operation permits the transportation of open boxcrossings between the square lattice and the other lattice. Whereas this was suited for proving the boxcrossing property, a slightly altered hybrid is useful for studying arm exponents.
Let , and consider the mixed lattice drawn on the left of Figure 2.4, formed of a horizontal strip of the square lattice centred on the axis of height , with the triangular lattice above and beneath it. The embedding of each lattice is otherwise as in [6]: the triangular lattice comprises equilateral triangles of side length , and the square lattice comprises rectangles with horizontal (respectively, vertical) dimension (respectively, ). We require also that the origin of be a vertex of the mixed lattice.
Let , and let be the product measure on for which edge is open with probability given by:

if is horizontal,

if is vertical,

if is the right edge of an upwards pointing triangle,

if is the left edge of an upwards pointing triangle.
Suppose further that is selfdual, in that , and let . We denote by (respectively, ) the transformation of Figure 2.2 applied to an upwards (respectively, downwards) pointing triangle. Write for the transformation of obtained by applying to every upwards pointing triangle in the upper half plane, and similarly in the lower half plane; sequential applications of star–triangle transformations are required to be independent of one another.
Similarly, we denote by (respectively, ) the transformation of Figure 2.2 applied to an upwards (respectively, downwards) pointing star. Write for the transformation of obtained by applying to all upwards pointing stars in the upper halfplane and similarly in the lower halfplane. It may be checked that lies in and has law . That is, viewed as a transformation acting on measures, we have .
The transformations and are defined similarly, and illustrated in Figure 2.4. As in that figure, for ,
We turn to the operation of these two transformations on open paths, and will concentrate on ; similar statements are valid for . Let , and let be an open path of . It is not difficult to see (and is explained fully in [6]) that the image of under contains some open path . Furthermore, lies within the neighbourhood of viewed as a subset of , and has endpoints within unit Euclidean distance of those of . Any vertex of in the square part of is unchanged by the transformation. The corresponding statements hold also for open paths of the dual of . These facts will be useful in observing the effect of on the arm events.
Arm exponents are defined in Section 1.3 in terms of boxes that are adapted to the lattice viewed as a graph. It will be convenient to work also with boxes of . Let be a mixed lattice duly embedded in , and write for the subset of lying on the axis. Let . For and , we write (with negation written ) if there exists an open path joining some and some using only edges that intersect . Let be the unit (Euclidean) disk of and write for the direct sum .
Proposition 2.1.
Let , , , and . For ,

if then ,

if then .
Proof.
(a) Let ; the case is similar (we assume where necessary). If , there exists an open path of from to using edges that intersect . Since , are not moved by , the image contains a open path of from to . It is elementary that transports paths through a distance not exceeding (see [6, Prop. 2.4]). Therefore, every edge of intersects .
(b) Suppose . By considering the star–triangle transformations that constitute the mapping (as in part (a)), we have that . ∎
3. Universality of arm exponents
This section contains the proof of Theorem 1.1. The reader is reminded that we work with translationinvariant measures associated with the square, triangular, and hexagonal lattices.
3.1. The arm exponents
Let and . The arm event is empty if is too small to support the existence of the required disjoint paths to the exterior boundary of the annulus . As explained in [12] for example, for each , there exists such that the arm exponent (assuming existence) is independent of the choice of . We assume henceforth that is chosen sufficiently large for this to be the case.
It is a significant open problem of probability theory to prove the existence and invariance of arm exponents for general lattices. This amounts to the following in the present situation.
Conjecture 3.1.
Let be selfdual. For and a colour sequence , there exists such that
Furthermore, is constant for all selfdual .
This is phrased for the triangular lattice, but it embraces also the square and hexagonal lattices, the first by setting a component of to , and the second by a single application of the star–triangle transformation. (See also [14].) We make no contribution towards a proof of the first part of this conjecture. Theorem 1.1 may be viewed as the resolution of the second part when .
Hereafter, we consider only the onearm event with , and the alternatingarms events with , with associated exponents denoted respectively as and . Thus with as in Section 1.3.
3.2. Main proposition
Let be one of the square, triangular, and hexagonal lattices, or a hybrid thereof as in Section 2. We embed in in the manner described in that section. Let , , denote the vertices common to these lattices to the right of the origin, and , the vertices of the dual lattice corresponding to the faces of lying immediately above the edge . For , let , with boundary . We recall that (respectively, ) denotes the open cluster of containing (respectively, the open cluster of containing ). For and any connected subgraph of either or , we write if contains vertices in both and . Note that we may have even when there are no vertices of belonging to . ,
For with , let
We write when the role of is to be stressed. Note the condition of disconnection in the definition of : it is required that the are not connected by open paths of edges all of which intersect .
A proof of the following elementary lemma is sketched at the end of this subsection. An alternative proof of the second inequality of the lemma may be obtained with the help of the forthcoming separation theorem, Theorem 3.5, as in the final part of the proof of Proposition 3.7. The latter route is more general since it assumes less about the underlying lattice, but it is also more complex since it relies on a version of the separation theorem of [11] whose somewhat complicated proof is omitted from the current work.
Lemma 3.2.
Let be selfdual. Let , and let be an alternating colour sequence of length (when we set ). There exists and such that
for .
Let be selfdual with , and consider the two measures (respectively, ) on the square (respectively, triangular) lattice. The proof of the universality of the boxcrossing property was based on a technique that transforms one of these lattices into the other while preserving primal and dual connections. The same technique will be used here to prove the following result, the proof of which is deferred to Section 3.3.
Proposition 3.3.
For any and any selfdual triplet with , there exist such that, for all ,
Proof of Theorem 1.1.
Suppose there exist , a selfdual , and , such that
(3.1) 
with the alternating colour sequence of length (when , we take ). By Lemma 3.2, (3.1) is equivalent to
(3.2) 
We say that ‘ satisfies (3.2)’ if (3.2) holds with replaced by . By selfduality, there exists such that , and we assume without loss of generality that . By Proposition 3.3, satisfies (3.2). Similarly, satisfies (3.2) for any selfdual of the form . The claim is proved after one further application of the proposition. ∎
Outline proof of Lemma 3.2.
First, a note concerning the event with . If , vertices , , are connected to by open paths. We claim that such open paths may be found that are vertexdisjoint and interspersed by open paths joining the to . This will imply the existence of arms of alternating types joining to , such that the open primal paths are vertexdisjoint, and the open dual paths are vertexdisjoint except at the . The claim may be seen as follows (see also Figure 3.1). The dual edge with endpoints is necessarily open. By exploring the boundary of at , one may find two open paths denoted , , joining to , and vertexdisjoint except at . Let . Since and , we may similarly explore the boundary of to find an open path of that joins to , and is vertexdisjoint from either or , and in addition from , . The dual paths are the required open arms.
The set induces a subgraph of whose boundary is denoted . We denote the inside of (that is, the closure of the bounded component of ) by also. It is easily seen that , and the first inequality follows immediately.
For the second inequality, we shall use the fact that , together with a suitable construction of open and open paths within . Let and suppose occurs. On an anticlockwise traverse of , we find points such that the (respectively, ) are endpoints of open (respectively, open) paths crossing the annulus . Note that the are not vertices of , but instead lie in open edges. Write , .
As illustrated in Figure 3.1, for sufficiently large and all vectors , of length , there exists a configuration of primal edges of such that for , and
satisfies . The details of the construction of the are slightly complex but follow standard lines and are omitted (similar arguments are used in [4, Sect. 8.2] and [17, Chap. 2]). It follows as required that
Whereas a naive construction of the succeeds when for all , a minor variant of the argument is needed if for some . The details are elementary and are omitted.
The case is similar but simpler. ∎
3.3. Proof of Proposition 3.3
A significant step in the arguments of [11] is called the ‘separation theorem’ (see also [12, Thm 11]). This states roughly that, conditional on the arm event , there is probability bounded away from that arms with the required colours can be found whose endpoints on the exterior boundary of the annulus are separated from one another by a given distance or more. A formal statement of this appears at Theorem 3.5; the proof is rather technical and very similar to those of [11, 12] and is therefore omitted. It is followed in Section 3.4 by an application (Proposition 3.7) to the lattices of which we make use here.
The proof of Proposition 3.3 relies on the following lemma, in which the measure is utilized within the star–triangle transformations comprising the map . Let .
Lemma 3.4.
Let be a mixed lattice, and let be a selfdual measure on . For and ,
The proof of the lemma is deferred to the end of this section. Let and be as in Proposition 3.7. By making applications of to , we deduce that . Therefore, for ,
by Lemma 3.4  
This proves the first inequality of Proposition 3.3.
Fix , and consider the event on the lattice . If we apply times the transformation to , we obtain via Lemma 3.4 applied to the event that:
Proposition 3.3 is proved.
Proof of Lemma 3.4.
Let , we shall consider the case separately. Let and . Note that the points , , are invariant under .
It is explained in Section 2 (see also [6, Sect. 2]) that the image of an open path contains a open path of lying within distance of . Therefore, for , if , then . The proof when is complete, and we assume now that . Let and . By Proposition 2.1, for , whence .
Finally, let . Let and . Let (respectively, ) be an open primal (respectively open dual) path starting at (respectively ), that intersects . Since and are unchanged under , they are contained, respectively, in and . By the remarks in Section 2 concerning the operation of on open dual paths, we conclude that in , and similarly in . The proof is complete. ∎
3.4. Separation theorem
The socalled ‘separation theorem’ is a basic element in Kesten’s work on scaling relations in two dimensions. It asserts roughly that, conditional on the occurrence of a given arm event, there is probability bounded from that such arms may be found whose endpoints on the interior and exterior boundaries of the annulus are distant from one another. The separation theorem is useful since it permits the extensions of the arms using boxcrossings.
Kesten proved his lemma in [11] for homogeneous site percolation models, while noting that it is valid more generally. The proof has been reworked in [12], also in the context of site percolation. The principal tool is the boxcrossing property of the critical model. In this section, we state a general form of the separation theorem, for use in both the current paper and the forthcoming [5]. The proof follows closely that found in [11, 12], and is omitted.
Let be a connected planar graph, embedded in the plane in such a way that each edge is a straight line segment, and let be a product measure on . As usual we denote by the dual graph of , and more generally the superscript indicates quantities defined on the dual. We shall use the usual notation from percolation theory, [4], and we assume there exists a uniform upper bound on the lengths of edges of and , viewed as straight line segments of .
The hypothesis required for the separation theorem concerns a lower bound on the probabilities of open and open boxcrossings. Let and let be a (nonsquare) rectangle of . A latticepath is said to cross if contains an arc (termed a boxcrossing) that lies in the interior of except for its two endpoints, which are required to lie, respectively, on the two shorter sides of . Note that boxcrossings lie in the longer direction. The rectangle is said to possess an open crossing (respectively, open dual crossing) if there exists an open path of (respectively, open path of ) crossing , and we write (respectively, ) for the event that this occurs. Let be the set of translations of , and . Let and , and let be minimal with the property that, for all and all , and possess crossings in both and . Let
(3.3)  
(3.4) 
and