Ising (Conformal) Fields
and Cluster Area Measures
We provide a representation for the scaling limit of the critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to and to Potts models are also considered.
Keywords: continuum scaling limit, critical Ising model, Euclidean field theory, conformal field theory, FK clusters, SLE, CLE.
AMS 2000 Subject Classification: 82B27, 60K35, 82B43, 60D05.
The Ising model in dimensions is perhaps the most studied statistical mechanical model and has a special place in the theory of critical phenomena since the groundbreaking work of Onsager . Its scaling limit at or near the critical point is recognized to give rise to Euclidean (quantum) field theories. In particular, the scaling limit of the lattice magnetization field should be a Euclidean random field and, at the critical point, the simplest reflection-positive conformal field theory [4, 13]. As such, there have been a variety of representations in terms of free fermion fields  and explicit formulas for correlation functions (see, e.g., [25, 28] and references therein). In this paper, we provide a construction of in terms of random geometric objects associated with Schramm-Loewner Evolutions (SLEs)  (see also [12, 19, 22, 44]) and Conformal Loop Ensembles (CLEs) [35, 36, 43] — namely, a gas (or random process) of continuum loops and associated clusters and (renormalized) area measures.
Two such loop processes arise in the results announced by Smirnov [37, 38, 39, 40, 41] (see also the work of Riva and Cardy in  — in particular Sections 6 and 7 there) that the full scaling limit of critical Ising spin cluster boundaries (respectively, FK random cluster boundaries) is given by the (nested version of) CLE with parameter (resp., ). One can try to associate with each continuum cluster or external boundary loop in the scaling limit a finite area measure representing the rescaled number of sites in the corresponding lattice cluster (where is SP for the spin case and FK for the random cluster case). We can in fact do this for the FK case and expect it to also be valid for the spin case.
Although one might try to represent the Euclidean field using spin clusters by a sum , where the ’s are or depending on whether corresponds to a or spin cluster, this does not seem to work. Instead, we use the FK clusters, which leads to , where the ’s are independent random signs. The (countable) family is a “point” process with each a “point” and where distinct “points” should be orthogonal measures.
For a bounded with nonempty interior, one expects that . This would follow from the scaling covariance expected for and described at the end of this section. The same happens for the corresponding measures in independent percolation that count so-called “one-arm” sites, as follows from work of Garban, Pete and Schramm [17, 16]. Nevertheless, for any only finitely many ’s will have support that intersects and has diameter greater than . Furthermore, with probability one, which leads to convergence (at least in ) of the sum with random signs . We note that divergence of means that is not a signed measure; i.e., even restricted to a bounded , it is not the difference of two positive finite measures. For negative results of a similar sort, but in the context of Gaussian random fields, see .
In the next section, we set up notation for the Ising model on the square lattice and its FK representation and review how the scaling limit of FK cluster boundaries may be viewed as a process of noncrossing continuum loops and associated continuum clusters . We then show why the natural scaling for the Ising spin variables at criticality to obtain a Euclidean (random) field leads to natural rescaled area measures supported on and to the representation of in terms of those measures. We also discuss why area measures for spin clusters are not appropriate for representing , by using an example taken from the infinite temperature Ising model on the triangular lattice, .
In Section 3, we use (see Proposition 3.1) and then discuss how to verify a decay property of the critical Ising two-point correlation or equivalently the FK connectivity function. Another essential ingredient in our analysis is a bound (see Prop. 3.2) on the number of macroscopic FK clusters. Although we focus on critical Ising-FK percolation on , similar arguments can be applied to other lattices and to independent (and, in principle, Ising spin) percolation. The case of independent percolation is discussed at the end of Section 3. In Section 4, we review the general conclusions of our work and discuss extensions to off-critical (or as they are sometimes called, near-critical) scaling limits, either as temperature , the critical temperature, with magnetic field , or else as with . Finally, we propose there that a cluster area measure representation should also be valid for the Ising model and for the -state Potts model with or .
Before concluding this section, we wish to emphasize that this paper is meant to serve as an introduction, readable by both mathematicians and physicists, to a representation for the Ising scaling limit field in terms of the limit rescaled area measures . We hope this will prove useful in providing a general conceptual framework for field-based scaling limits like Aizenman and Burchard  did for connectivity-based ones. Although detailed explanations and proofs are provided in this paper for certain issues, others are avoided. In particular, although the next two sections of the paper provide arguments for the existence of both and as (subsequence) limits of the corresponding lattice quantities, they do not provide the tools needed to prove that the limits are unique. This will be done in a future paper in collaboration with C. Garban, along with a proof of related properties such as that and have the expected conformal covariance including that for , and are equidistributed with and .
2 Ising (Euclidean) Field
We consider the standard Ising model on the square lattice with Hamiltonian
where the first sum is over nearest neighbor pairs in (or bonds ), the spin variables are -valued and the external field is in .
When there is a unique infinite volume Gibbs distribution for some value of and inverse temperature , we denote by its expectations. There is a critical such that nonuniqueness occurs only for and . In particular, the critical Gibbs measure is unique and in that case we use the notation . By translation invariance, the two-point correlation is a function only of , which in the critical case we denote by .
We want to study the random field associated with the spins on the rescaled lattice in the scaling limit . More precisely, for test functions of bounded support on , we can define for the critical model
with an appropriate choice of the scale factor . Since is a random variable with zero mean, it is natural to choose so that is bounded away from 0 and as . Choosing so that this second moment is exactly one for the indicator function of the unit square yields
where and .
One way to formulate the FK representation of the Ising model (for and ) is that coexisting with the -valued spin variables on the sites of are -valued occupation variables on the bonds of . The occupied or open () FK bonds determine FK clusters, , which are the sets of sites in connected to each other by paths of open FK bonds. One can generate the ’s from the ’s by assigning independent symmetric random signs to the ’s and then setting for every in . If we write to denote that and are in the same FK cluster, it is immediate that the FK connectivity function at criticality is simply given by
Denoting by expectation in the critical system, by the restriction of the cluster in to , and by the number of ()-sites in , we have
By the definition of we see that the rescaled areas are uniformly square summable in the sense that for all . We would like to argue that, at least along subsequences of ’s tending to zero, has a nontrivial limit in distribution. This is already partly clear — i.e., no can diverge to . But what prevents them all from tending to zero as ? It turns out that this uses the following hypothesis about (where is the appropriate constant for the lattice ) — roughly speaking, that it decays like with , where denotes Euclidean norm or that diverges as a power when . (It also uses that the crossing probability of an annulus is bounded away from one as — see (10) and Prop. 3.2.)
For some fixed , there are constants and such that for any small and then for any with large ,
for any with .
As we will discuss, the clusters on the rescaled lattice will converge in the scaling limit to full plane continuum clusters in . In that limit most of the lattice clusters disappear because they are not of macroscopic size. The importance of the lower bound on in (4) is that it guarantees (see Prop. 3.1) that the rescaled areas of the microscopic clusters are negligible (at least in a square summable sense). That is, the contribution to coming from clusters whose intersection with the unit square has small macroscopic diameter is negligible. A corresponding statement is true for the clusters that contribute to the field for more general test functions of bounded support. The significance of the upper bound on in (4) is that it easily implies that is bounded away from and as .
In a series of papers, the authors constructed  a certain process of loops in the plane and proved  (see also ) convergence to it in the scaling limit of the collection of boundaries of all (macroscopic) clusters for critical independent site percolation on the triangular lattice. In the limit there is no self-crossing or crossing of different loops but there is self-touching and touching between different loops. Moreover, the loops are locally SLE curves.
Similar results for the 2D critical Ising model on the square lattice have been announced by Smirnov [37, 38, 39, 40, 41]. There one considers either the boundaries between plus and minus spin clusters , or the loops in the medial lattice that separate FK from dual FK clusters  (see Figure 1). We will focus on those loops which separate FK clusters in the original lattice on their inside from dual FK clusters in the dual lattice on their outside. In the scaling limit of spin cluster boundaries one would obtain simple loops that do not touch each other and locally are SLE type curves. In the case of FK cluster boundaries, there would instead be self-touching and touching between different loops (but no crossing), like in the percolation case. Now, however, the loops would locally be SLE type curves.
In the FK case, each loop that we consider on the medial lattice of is the outer boundary of a rescaled FK cluster . The inner boundary of is made of “daughter” loops corresponding to the “holes” in . In the scaling limit , one can analogously identify a continuum cluster as the closed set left after removing from the (open) exterior of the loop and the (open) interiors of its daughter loops (with interiors and exteriors defined using winding numbers). We remark that because the scaling limit is only a limit in distribution and no special effort was made to coordinate indexing for clusters in the lattice and in the continuum, we use different letters, and , for the two indices. We denote by the finite measures supported on corresponding to the limit of the rescaled areas as , in the sense, e.g., that is the scaling limit of the rescaled areas . The existence and nontriviality of (or of for more general test functions of bounded support) will follow from Hypothesis 2.1 (see Prop. 3.1) and Prop. 3.2, as noted above. The collection ought to be a functional of as has recently been proved in the independent percolation context by Garban, Pete and Schramm [17, 16].
Letting denote i.i.d. symmetric -valued variables, one obtains the following representation of the Euclidean field : for test functions of bounded support,
To be more precise, the sums in (5) should first be restricted to clusters with diameter greater than and then convergence (in ) as the cutoff will follow from the square summability discussed earlier.
As noted in the Introduction, one might be tempted to represent the Euclidean field using spin clusters and hence SLE type loops. If we use and to denote the limits of appropriately rescaled areas of plus and minus spin clusters, respectively, then on a formal level, by decomposing the righthand side of (1) into the contribution from plus and minus clusters, one might expect that of (5) would also be given by (with some resummation needed to handle the difference of two presumably divergent series) as an alternative to . This appears not to be so, as can be understood by considering the simple situation of the Ising model on the triangular lattice at .
The latter is noncritical as an Ising model and the correct Euclidean field obtained by using the noncritical FK clusters (which are just isolated sites since ) and the version of (2), is two-dimensional Gaussian white noise. But if one considers the Ising spin clusters, this is critical independent site percolation on and the formal expression , besides the resummation issue, seems unrelated to white noise. Indeed, applying and then removing a cutoff, as explained after (5), in this case would probably not lead to the physically correct limit.
3 Area Measure
In the previous section we gave a representation of the Ising Euclidean spin field in terms of rescaled counting measures that give the “areas” of macroscopic Ising-FK clusters. In this section we first explain how to use Hypothesis 2.1 to get the existence of nontrivial limits in distribution of these area measures, at least along subsequences of ’s tending to zero. We then explain how to verify Hypothesis 2.1 first for critical Ising-FK percolation on and then for critical independent site or bond percolation on or . Using the notation introduced in Section 2 and denoting by the Euclidean diameter of , we have the following proposition.
Hypothesis 2.1 implies that
The usefulness of Prop. 3.1 is based on the
additional result that for every fixed , in the scaling limit there will
only be finitely many FK clusters with diameter larger than
that intersect ; this important feature of the scaling
limit will be discussed below — see Prop. 3.2.
Once one has Prop. 3.2, it then follows from
Prop. 3.1 that the collection
has nontrivial subsequential limits;
i.e., it is not possible that all
’s scale to zero as .
Said more physically, Prop. 3.1
implies that there is a negligible contribution to the magnetization
from FK clusters whose linear size is small on a macroscopic lengthscale
while Prop. 3.2 says that there are only finitely
many larger clusters touching any bounded region.
Together, they lead to the representation (5).
Proof of Proposition 3.1. Using Hypothesis 2.1, we can compare for small as to by using the second inequality of (4) to compare each to the ’s with in the unit length square centered on (so that we may take as ). Since there are approximately such sites, we have that
Using this lower bound (with and ) and (3), we have that
The proposition follows from the observation that
the last expression in (6)
tends to zero as since .
The next two lemmas will be used to verify Hypothesis 2.1. Let denote , and denote its -boundary by . If the subscript is omitted, we refer to the disc centered at the origin . We denote by ( for wired) the critical FK measure inside with wired (i.e., everything connected) boundary condition on . The next lemma is based on the FKG inequalities.
Proof of Lemma 3.1.
where in the last step we have used FKG.
The next lemma uses RSW bounds [30, 34], namely, that the probability that there is an open FK -circuit in an -annulus centered at is bounded away from zero and one as by constants that depend only on . In fact, we only need a lower bound; i.e.,
This is not immediate in the Ising case, since there is not currently a direct proof of RSW for critical FK percolation (as opposed to the independent percolation case). However, as we explain after the proof of the lemma, RSW follows from announced results about the scaling limit of spin cluster boundaries , combined with the Brownian loop soup representation of CLE [24, 36, 43]; also, the lower bound (7) for some implies both upper and lower bounds for all .
Assuming (7), there exists a constant such that
Before giving the proof, we state an immediate consequence of this lemma, the preceding one and the fact that .
Assuming (7), and are comparable (up to constants) as .
Proof of Lemma 3.2. Let denote the intersection of and the annulus with outer radius and inner radius centered at and let denote the event that there is an open FK circuit in surrounding . Let be the event that occurs and the outermost open FK circuit contained in and surrounding is connected to by an open FK path. We have
where the second inequality follows from FKG and the constant follows from RSW. We then note that
Inserting this bound into (8) concludes the proof.
RSW for FK percolation on . In order to get RSW, we assume (from ) that the “full” scaling limit of the Ising model converges to (the nested version of) CLE. We can then use the representation of CLE in terms of the Brownian loop soup [24, 36, 43], assuming that corresponds to a density of the Brownian loop soup below its critical density (which should correspond to ). A single Brownian loop has positive probability of “surrounding” a disc of fixed radius centered at the origin. Let be such a loop and consider its loop-cluster, built recursively from the (countably many) Brownian loops by saying that any two loops which touch (and thus cross, with probability one) are in the same loop-cluster. Given that the density of the Brownian loop soup is assumed to be below the critical density, the loop-cluster of is contained with probability one inside a sufficiently large disc. Thus, for some , there is strictly positive probability that the -annulus centered at the origin contains a CLE circuit. Back on the lattice , this gives a positive probability, bounded away from zero as , that the external boundary of an Ising spin cluster provides such a circuit. But this in turn implies the same for a closed (dual) FK circuit, and hence by self-duality at the critical point, the same for an open FK circuit.
To conclude the discussion of RSW, we note that it is not difficult to show that one can use FKG to obtain from open circuits contained in overlapping -annuli a “necklace” structure that provides open crossings of rectangles of arbitrary aspect ratio (see  for more details about such arguments). The rectangle crossings can then be used, once again with the help of FKG, to obtain circuits inside arbitrary annuli with probability bounded away from zero. By self-duality one also has closed (dual) crossings of rectangles and these can be used to bound the probability of open circuits away from one.
For , let denote the number of distinct clusters that include sites in both and . Assuming (7), for any , there exists such that for all and all small and any ,
It follows that for any bounded and , the number of distinct clusters of diameter touching is bounded in probability as .
Proof of Proposition 3.2. The proof is by induction on . For , the result follows from RSW since is equivalent to the absence of a closed (dual) circuit in the -annulus about , which by self-duality at the critical point has the same probability as absence of an open circuit, which in turn is bounded away from one as . Now suppose . Then one may do an exploration of the ’s that touch until are found that reach , making sure that all cluster explorations have been fully completed without obtaining information about the outside of the clusters. At that point, the complement of some random finite remains to be explored and the (conditional) FK distribution in is with a free boundary condition on the boundary (or boundaries) between and . By RSW, the -probability of an open crossing in of the -annulus is bounded above by the original . Thus we have
The last claim of the proposition follows from (9) because
one may choose points in
so that any of diameter
touching will be counted in
for at least one .
We next explain how to verify Hypothesis 2.1 for critical
Ising-FK percolation and for independent percolation; we do not
have a verification for Ising spin percolation, although we expect
it to be true in that case also. It may be of interest to note that
for critical independent percolation,
one can obtain a representation like (5), but with
the SLE-based measures replaced by
SLE-based ones , for the scaling limit of the
lattice “divide and color” model . (Here
and below we use the letters IN to distinguish independent from FK
percolation.) The original divide and color model, and
the one most analogous to the FK representation of the () Ising
model, takes the open clusters of independent bond
percolation, e.g., on , and colors them with random
signs to define the divide and color spin variables.
In Sec. 4, we will consider this model as the
density of open bonds approaches its critical value (we note
that in  a different phase transition is studied). For
independent site percolation, e.g., on , with
say probability and for white and black sites, the option
for defining the divide and color spin variables that we will use is
to “color” both the white and black clusters with random
signs. It is unclear whether the scaling limit of the critical
( on ) divide and color model corresponds to
some known conformal field theory. Note that the limit, in terms of
boundaries between clusters of different colors, is a sort of
“dilute CLE” and is conformally invariant, but is not itself
described by a CLE since the divide and color model lacks the
“domain Markov property.”
Hypothesis 2.1 for FK percolation on . In this case, the behavior of the two-point function is
known exactly along the direction from Ising
calculations, which yield
(e.g., , referred to in ,
and Chap. XI of ).
Using Lemmas 3.1 and
3.2 one obtains that, up to constants, the
two-point function has the same behavior in all directions.
Hypothesis 2.1 is then satisfied with .
Hypothesis 2.1 for independent percolation. From the analogues of Lemmas 3.1 and 3.2 for independent percolation, we know that is comparable (up to constants) with . This immediately gives the desired upper bound for . For the lower bound, it suffices to show that for some constant . Using FKG, we have
where from RSW, and with for any , and denoting the one-arm exponent.
For site percolation on the triangular lattice, has been proved to be using SLE computations . For other percolation models (e.g., bond percolation on the square lattice), the five-arm exponent is known to be equal to (see Lemma 2 of , Corollary A.8 of  and Section 5.2 of ), since it can be derived via a general argument that does not use SLE. Using this and the BK inequality , we obtain an upper bound of for . We note, as pointed out to us by P. Nolin, that a more elementary argument from  is available showing that without use of the five-arm exponent — see Eqn. (2.5) of .
In this paper, we provided a representation (see (5)) for the scaling limit Euclidean random field associated with the Ising model at its critical point (, ). This field, one of the basic objects of conformal field theory, is the scaling limit of the magnetization field (see (1)) on as . is represented as a sum with random signs and finite measures that are the limits of rescaled area measures associated with the macroscopic Ising-FK clusters. These measures are supported on continuum clusters whose outer boundaries are described by CLE loops with . A key to the representation is that natural field strength rescaling (see (2), (3)) insures that for bounded , and hence is convergent (in ).
We explained, at the end of Section 2, why the limits of area measures for Ising spin clusters do not appear useful for representing . We also noted, towards the end of Section 3, that a field can be constructed using critical clusters from independent in place of FK percolation, but that its physical significance is unclear. We next discuss how the representation (5) could be extended to off-critical models.
Independent percolation, . Here the percolation density (say of the white sites on ) converges to the critical density, (), appropriately as . In this case, the representation of the near-critical field should involve area measures from the near-critical modification of CLE obtained by the approach of [7, 8] and the results of Garban, Pete and Schramm [17, 16]. A feature of that work, which might also be valid in the FK-Ising context, is a natural probabilistic coupling, based on a Poissonian marking of certain pivotal locations, so that the one-parameter family of near-critical models parametrized by the strength of the off-critical perturbation lives on a single probability space. The appropriate speed at which is such that the correlation length is bounded away from zero and infinity. Since it is proved in  (and  — see Theorem there) that crossing probabilities and multi-arm probabilities are comparable (up to constants and up to distances of the order of the correlation length) to those of the critical system, RSW still holds and both Hypothesis 2.1 and Prop. 3.2 can be verified. One of the area measures , corresponding to the unique infinite cluster (white for and black for ), will now have unbounded support and infinite mass over , but its mass in any bounded region will be finite and will be convergent (in , at least for a bounded not chosen in a way that depends on ). We note that for bond percolation on with the density of open bonds, it will only be in the near-critical model that some has infinite mass.
FK percolation, . Here one keeps in the Ising model but lets appropriately as , which is the analogue of in independent percolation. It is natural to expect that the claims made above for independent percolation still hold in this case. Since one is now considering bond (FK) percolation, only for (the analogue of in independent percolation) will there be an infinite mass with unbounded support in . Including a random sign for the infinite mass means one is taking the scaling limit of the symmetric mixture of the plus and minus Gibbs measures.
FK percolation, . Here one sets with and then lets appropriately as . Intuitively, in the scaling limit, this should involve formally multiplying the measure describing the critical continuum system by a factor proportional to . According to (5) the critical Euclidean field is given by the sum of all the . Let denote the marginal distribution of the process of finite signed measures in the plane, denote the indicator function of the square in , , and finally . We ask: does converge to some as and is , obtained from as the sum of its individual signed measures, the physically correct near-critical Euclidean field? Heuristically, the correct normalization to obtain a nontrivial near-critical scaling limit is such that the correlation length remains bounded away from zero and infinity. Since for small , this gives , which coincides with the normalization needed to obtain a nontrivial Euclidean field, as can be seen from (2) and the asymptotic behavior of . Using this observation and the Ising critical exponent for the magnetization (i.e., ), the rough computation (where denotes the sum over in ),
suggests a positive answer to the previous questions.
We conclude this section with brief discussions of the applicability of our approach to higher dimensions, , and to -state Potts models with . Although the scaling limit Ising magnetization field should be conformal with close connections to CLE, as we have indicated, very little conformal or SLE machinery was actually used in our analysis. Basically, the two main ingredients were (see Hypothesis 2.1) that behaves at long distance like with and (see Prop. 3.2) that as ,
Although such decay of should be valid for all , the crossing probability bound (10) is a different matter and presumably fails above the upper critical dimension (see Appendix A of ). When it fails, there can be infinitely many FK clusters with diameter greater than in a bounded region and so Prop. 3.1 would not preclude from being a Gaussian (free) field. But it appears that at least for , both (10) and a representation of as a sum of finite measures with random signs ought to be valid.
As pointed out to us by J. Cardy, an analogous representation for the scaling limit magnetization fields of -state Potts models also ought to be valid, at least for values of such that for a given , the phase transition at is second order. The phase transition is believed to be first order for integer when and for when — see ; this leaves, besides the Ising case, and and . We denote the states or colors by and recall that in the FK representation on the lattice, all sites in an FK cluster have the same color while the different clusters are colored independently with each color equally likely. In the scaling limit, there would be finite measures and the magnetization field in the color- direction would be with the ’s taking the value with probability