A solution space for a system of null-state partial differential equations 1
This article is the first of four that completely and rigorously characterize a solution space for a homogeneous system of linear partial differential equations (PDEs) in variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE). In CFT, these are null-state equations and conformal Ward identities. They govern partition functions for the continuum limit of a statistical cluster or loop-gas model, such as percolation, or more generally the Potts models and O models, at the statistical mechanical critical point. (SLE partition functions also satisfy these equations.) For such a lattice model in a polygon with its sides exhibiting a free/fixed side-alternating boundary condition , this partition function is proportional to the CFT correlation function
where the are the vertices of and where is a one-leg corner operator. (Partition functions for “crossing events” in which clusters join the fixed sides of in some specified connectivity are linear combinations of such correlation functions.) When conformally mapped onto the upper half-plane, methods of CFT show that this correlation function satisfies the system of PDEs that we consider.
In this first article, we use methods of analysis to prove that the dimension of this solution space is no more than , the th Catalan number. While our motivations are based in CFT, our proofs are completely rigorous. This proof is contained entirely within this article, except for the proof of lemma 14, which constitutes the second article . In the third article , we use the results of this article to prove that the solution space of this system of PDEs has dimension and is spanned by solutions constructed with the CFT Coulomb gas (contour integral) formalism. In the fourth article , we prove further CFT-related properties about these solutions, some useful for calculating cluster-crossing probabilities of critical lattice models in polygons.
We consider critical bond percolation on a very fine square lattice inside a rectangle with wired (or fixed) left and right sides (i.e., all bonds are activated on these sides) and free top and bottom sides (i.e., we do not condition the state of any of the bonds on these sides). In , J. Cardy used conformal field theory (CFT) ; ;  methods to argue that, at the critical point and in the continuum limit, the partition function for this system is proportional to the CFT correlation function
where is the th vertex of and is a CFT one-leg corner operator ; ; ; ; ;  that implements the boundary condition change (BCC) from free to fixed ; ;  at this vertex. The essence of this argument supposed the emergence of conformal invariance at the critical point for bond percolation in the continuum limit, a feature that was previously observed in computer simulations . By considering the limit of a particular combination of certain -state random cluster model partition functions given by (1), Cardy then predicted a formula for the probability that the wired sides of are joined by a cluster of activated bonds (figure 1). His prediction, called Cardy’s formula, is 
Here, corresponds one-to-one with the aspect ratio of the rectangle via the second equation in (2), with the complete elliptic function of the first kind . Computer simulations ;  have numerically verified this prediction (2), thus giving very strong evidence for the presence of conformal symmetry in the continuum limit of critical percolation. Other simulations  consistently suggest that many observables, such as the probability of the left-right cluster-crossing event, common to different homogeneous models of critical percolation (e.g., site vs. bond percolation and percolation on different regular lattices) converge to the same value in the continuum limit, a phenomenon called universality. Later, S. Smirnov rigorously proved Cardy’s formula for site percolation on the triangular lattice . Also after Cardy’s result (2), researchers have used CFT to predict other formulas involving critical percolation cluster crossings ; , densities ; ; ; , and pinch points .
The setup for Cardy’s formula has interesting generalizations that motivate the analysis presented in this article. Looking beyond rectangles, we may consider system domains that are even-sided polygons , with the boundary condition (BC) alternating from wired to free to wired, etc., as we trace the boundary of from side to side. We call this a free/fixed side-alternating boundary condition (FFBC) (figure 2). And looking beyond percolation, we may consider other lattice models with critical points that have CFT descriptions in the continuum limit. These include the Potts model  and its close relative, the random cluster model . If we enumerate the FFBC events and condition one of these systems to exhibit the th FFBC event on the boundary of , then we may adapt Cardy’s argument to predict that the conditioned partition function is proportional to the -point CFT correlation function
Here, is the th vertex of , and is a CFT one-leg corner operator ; ; ; ; ;  that implements the BCC from free to fixed ; ;  at this vertex (appendix A). If we condition the system to exhibit the th FFBC event, then Potts model spin clusters or FK clusters, anchor to the wired sides of and join these sides in some topological crossing configuration with some non-trivial probability. We call these boundary clusters. An induction argument  shows that there are such configurations (figure 3), with the th Catalan number given by
Formulas for crossing probabilities, or probabilities of these crossing events, as functions of the shape of generalize Cardy’s formula (2), which corresponds to the case of critical percolation with . In , we use results from this article and its sequels ; ;  to predict some of these formulas, extending recent results on crossing probabilities for hexagons ; .
To calculate the -point function (3), we conformally map onto the upper half-plane (figure 2). After we continuously extend it to the boundary of , this map also sends the vertices , onto real numbers , and it sends the one-leg corner operator hosted by the th vertex of to a one-leg boundary operator at ; . In the Potts model (resp. random cluster model) ; , the one-leg boundary operator is a primary operator that belongs to the (resp. ) position of the Kac table ; ; ; ; . (We discuss one-leg boundary operators further in appendix A.) Thus, the CFT null-state condition implies that this half-plane version of (3) satisfies the system of null-state partial differential equations (PDEs) ; 
Here, is the central charge of the CFT, and it corresponds to the model under consideration. For example, corresponds to percolation , and corresponds to the Ising model (i.e., the two-state Potts model) and the two-state random cluster model . Also, the sign to be used in (6) depends on the model. For example, we use the sign for the Potts model and the sign for the random cluster model.
Aside from the null-state PDEs (5), any CFT correlation function must satisfy three conformal Ward identities ; ; . For the half-plane version of the particular -point function (3) above, these are
If a function satisfies these PDEs (7), then it is covariant with respect to conformal bijections of the upper half-plane onto itself, with each coordinate having conformal weight . We elaborate on this in section I.1 below.
If , then it is easy to show that any solution to the system (5, 7) is of the form for some arbitrary constant . Thus, the rank (i.e., the dimension of the solution space) equals the first Catalan number, . If , then we may use the conformal Ward identities (7) to convert the system of four null-state PDEs (5) into a single hypergeometric differential equation . The general solution of this differential equation completely determines the solution space, so the rank of the system equals the second Catalan number, . (After setting , an appropriate boundary condition argument gives Cardy’s formula (2).) If , then a similar but more complicated argument  shows that the rank of the system equals the third Catalan number, , at least when . Beyond this, the need for linearly independent crossing-probability formulas suggests that the rank of the system is at least , but it does not apparently suggest that the rank is exactly. In spite of this, the Coulomb gas formalism ; ;  allows us to construct many explicit classical (in the sense of ) solutions of the system for any , a remarkable feat! Neither this article nor its first sequel  uses these solutions, but the latter sequels ;  do.
In addition to CFT, we may use multiple SLE ; ; ; ; , a generalization of SLE (Schramm-Löwner evolution) ; ; , to study the continuum limit of a critical lattice model inside a polygon with an FFBC. As we use this approach, we forsake the boundary clusters and study their perimeters instead. These perimeters, called boundary arcs, are random fractal curves that fluctuate inside . Their law is conjectured, (and proven for some models in the case of (ordinary) SLE, see table 1), to be that of multiple SLE, a stochastic process that simultaneously grows fractal curves, one from each vertex, inside . These curves explore the interior of without crossing themselves or each other until they join to form distinct, non-crossing boundary arcs that connect the vertices of pairwise (figure 4) in a specified connectivity ; . An induction argument  shows that these curves join in one of possible connectivities, called boundary arc connectivities  (figure 3). Furthermore, we identify each boundary arc connectivity with the particular cluster-crossing event whose boundary arcs join in that connectivity.
|Random walk or critical lattice model||Current status|
|The loop-erased random walk ||2||proven |
|The self-avoiding random walk ||8/3||0||conjectured |
|Potts spin cluster perimeters ||3||1/2||proven |
|Potts spin cluster perimeters ||10/3||4/5||conjectured |
|Potts spin/FK cluster perimeters ; ||4||1||conjectured |
|The level line of a Gaussian free field ||4||1||proven |
|The harmonic explorer ||4||1||proven |
|Potts FK cluster perimeters ||24/5||4/5||conjectured |
|Potts FK cluster perimeters ||16/3||1/2||proven |
|Percolation and smart-kinetic walks ; ||6||0||proven |
|Uniform spanning trees ||8||proven |
Multiple SLE provides a different, rigorous approach to calculating some observables that may be predicted via CFT, and although these two approaches are fundamentally different, they are closely related . Two entities determine the multiple-SLE process ; :
The first is stochastic: a collection of absolutely continuous martingales with zero cross-variation and total quadratic variation , with the SLE speed or parameter and the evolution time. The equation 
relates a CFT of central charge (resp. ) to a multiple SLE with one of two possible speeds, one in the dilute phase , and one in the dense phase (resp. with one speed in the dilute phase)  of SLE. Further arguments provided in appendix A show that if we substitute (8) into (6), then we must use the (resp. ) sign in the dilute (resp. dense) phase, so 
The second is deterministic: a nonzero function , which we call an partition function. (This is similar to but slightly different from the actual partition function of the critical system under consideration. See appendix A and .) The only condition imposed on is that it satisfies the system of null-state PDEs (5) and the three conformal Ward identities (7) (in the classical sense of ) with given by (9), and that it never equals zero.
Because all boundary arcs have the statistics of SLE curves in the small regardless of our choice of SLE partition function, the partition function that we do use may only influence a large-scale property of multiple SLE, such as the eventual pairwise connectivity of its curves in the long-time limit. Indeed, two multiple-SLE processes whose curves are conditioned to join in different connectivities obey the same stochastic PDEs driven by the martingales of condition I. Because which SLE partition function to use for item II above is the only unspecified feature of these equations, we expect that this choice influences the eventual boundary arc connectivity.
This supposition naturally leads us to conjecture the rank of the system (5, 7) previously considered in our CFT approach above. As mentioned, there are possible boundary arc connectivities, which we enumerate one through . Thus, there must be at least one SLE partition function per connectivity that conditions the boundary arcs to join pairwise in that connectivity almost surely. But furthermore, if our SLE partition function does not influence any of the boundary arcs’ other large-scale properties, then there may be at most one SLE partition function , called the th connectivity weight, that conditions the boundary arcs to join in, say, the th connectivity. If this is true, then we anticipate that the set is a basis for the solution space of the system (5, 7), and the rank of the system is therefore . Proving this last statement is one of the principal goals of this article and its sequels ; ; .
i.1 Objectives and organization
So far, we have used the application of the system (5, 7) to critical lattice models and multiple SLE to anticipate some of the properties of its solution space. In this section, we set the stage for our proof of some of these properties by declaring the goals, describing the organization, and establishing some notation conventions for this article and its sequels ; ; . Inserting (9) in the system (5, 7) gives the null-state PDEs
with and (however we consider only in this article), and the three conformal Ward identities
We call the th null-state PDE among (5) the null-state PDE centered on . Although this system (10, 11) arises in CFT in a way that is typically non-rigorous, our treatment of this system here and in ; ;  is completely rigorous. Before declaring what we intend to prove about this system of PDEs, we observe some important facts about it.
The subsystem of null-state PDEs (10) is undefined on the locus of diagonal points in , or points with at least two of its coordinates equal. We let be the complement of the locus of diagonal points in . Then the diagonal points make up the boundary , and all together, they divide into connected components, each of the form
for some permutation . By symmetry, it suffices to restrict the domain of our solutions to the component corresponding to the identity permutation . That is, we take whenever without loss of generality. In this article and its sequel , we refer to as a point in a component of and as the th coordinate of that point, but in the sequels ; , we refer to as a point.
The subsystem (10) is elliptic, so all of its solutions exhibit strong regularity. Indeed, after summing over all null-state PDEs, we find that any solution satisfies a linear homogeneous strictly elliptic PDE whose coefficients are analytic in any connected component of . (In fact, the principal part of this PDE is simply the Laplacian.) It follows from the theorem of Hans Lewy  that all of its solutions are (real) analytic in any connected component of . We use this fact to exchange the order of integration and differentiation in many of the integral equations that we encounter here and in .
We may explicitly solve the conformal Ward identities (11) via the method of characteristics. It follows that any function that satisfies these identities must have the form
where is any set of independent cross-ratios that we may form from , where is a (real) analytic function of , and where is any pairing (i.e., a permutation other than the identity with ) of the indices .
We suppose that is a Möbius transformation sending the upper half-plane onto itself, and we define and . Then the mapping defined by sends onto a possibly different connected component of . Because the cross-ratios , are invariant under , the right side of (13) evaluated at any is well-defined. If we enumerate the permutations in so , , are all of the components of that may be reached from by such a transformation , then we use (13) to extend to the function
It is evident that because in (13) satisfies the system of PDEs (10, 11), must satisfy this system too on each component of in its domain. Now, it is easy to show that (14) transforms covariantly with respect to conformal bijections of the upper half-plane onto itself, with each of the independent variables having conformal weight (9). In other words, the functional equation (where )
holds whenever is a Möbius transformation taking the upper half-plane onto itself. Such transformations are compositions of translation by , dilation by , and the inversion (all of which have positive-valued derivatives). Hence, is invariant as we translate the coordinates of by the same amount and is covariant with conformal weight (9) as we dilate all of them by the same factor or invert all of them. The first, second, and third Ward identities (11) (counting from the left) respectively induce these three properties.
In the case, we use the first conformal Ward identity of (11) (counting from the left) to reduce either PDE in (10) to a second order Euler differential equation in the one variable . The Euler equation has two characteristic powers (given in (29) below), and the second conformal Ward identity of (11) permits only the power . Thus, the solution space is
It is easy to show that the elements of satisfy the third conformal Ward identity of (11).
In the case, the conformal Ward identities demand that our solutions have the form (13), which we write as
with an unspecified function. By substituting (17) into any one of the null-state PDEs, we find that satisfies a second order hypergeometric differential equation. This restricts to a linear combination of two possible functions and given by
with the Gauss hypergeometric function . Thus, with , the solution space is
Sometimes, we write to specify the particular solution space with and use a similar notation for subsets of too. But usually, we suppress reference to the parameter and simply write .
One may explicitly construct many putative elements of by using the Coulomb gas formalism first proposed by V.S. Dotsenko and V.A. Fateev ; . This method is non-rigorous, but in , J. Dubédat gave a proof that these “candidate solutions” indeed satisfy the system of PDEs (10, 11). We call these solutions Coulomb gas solutions.
Rigorously prove that is spanned by real-valued Coulomb gas solution.
Rigorously prove that .
Argue that has a basis of connectivity weights and find formulas for all of the connectivity weights.
(For all , the multiple-SLE curves are space-filling almost surely ; . Although we suspect that the findings of this article and its sequels ; ;  are true for all , our proofs do not carry over to this range.) Goals 1 and 2 determine the size and content of . In this article, we prove the upper bound . To obtain that upper bound in this article, we construct a basis for the dual space of linear functionals acting on . The derivation of this bound is contained entirely within this article, except for the proof of lemma 14 below, which we defer to the second article  of this series. In the third article , we use the results of this article and  to achieve goals 1 and 2 above. In the fourth article , we investigate connectivity weights among other topics, and we use them to predict a formula for the probability of a particular multiple-SLE boundary arc connectivity. A heuristic, though non-rigorous, argument shows that the basis for that is dual to comprises all of the connectivity weights. This realization gives a direct method for their computation, as desired in goal 3.
In a future article , we use the connectivity weights to derive continuum-limit crossing-probability formulas for critical lattice models (such as percolation, Potts models, and random cluster models) in a polygon with an FFBC. We verify our predictions with high-precision computer simulations of the critical random cluster model in a hexagon, finding excellent agreement.
In appendix A, we survey some of the CFT methodologies used to study critical lattice models. This formalism (non-rigorously) anticipates many of our results, so we often interpret our findings in context with CFT throughout this article. The reader who is not familiar with this approach but wishes to understand our asides to it may consult this appendix. (We emphasize that, in spite of our occasional references to CFT, all of our proofs are rigorous, and none of them use assumptions from CFT. Rather, CFT tells us what ought to be true, and we prove those facts using rigorous methods.)
i.2 A survey of our approach
In this section, we motivate our method to achieve goals 1–3 (stated in the previous section), momentarily restricting our attention to percolation () ; ;  for simplicity. To begin, we choose one of the available boundary arc connectivities in a -sided polygon with vertices . Topological considerations show that there are at least two sides of whose two adjacent vertices are endpoints of a common boundary arc (i.e., multiple-SLE curve), and we let be such a side.
Next, we investigate what happens as the vertices and approach each other. What we find depends on the boundary condition for . If this side is wired, then the adjacent free sides and fuse into one contiguous free side of a -sided polygon , and the isolated boundary cluster previously anchored to contracts away. Or if is free, then the adjacent wired sides fuse into one contiguous wired side of , and the boundary clusters previously anchored to these sides fuse into one boundary cluster anchored to this new wired side. In either situation, the original crossing configuration for the -sided polygon goes to a crossing configuration for a -sided polygon, and the connectivity weight for the former configuration goes to the connectivity weight for the latter configuration (figure 5). (In percolation, “connectivity weight” and “crossing-probability formula” are synonymous.) If we repeat this process more times, then we end with a zero-sided polygon, or disk, whose boundary is either all wired or all free. The disk trivially exhibits just one “boundary arc connectivity” with zero curves, so this cumulative process sends the original connectivity weight to one.
Now, we might bring together in a different order the same pairs of vertices that approach each other in the previous paragraph. But because all of these variations send the same connectivity weight to one, we anticipate that all of them are different realizations of the same map.
Next, we investigate what happens if two adjacent vertices and that are not endpoints of a common boundary arc approach each other. Using the same boundary arc connectivity as before, we observe one of two outcomes. If the side is wired, then the adjacent free sides and of do not fuse into one contiguous free segment. Instead, they remain separated by an infinitesimal wired segment centered on the point within the side of , and the boundary cluster previously anchored to now anchors to this infinitesimal segment. Or if is free, then the adjacent wired sides do not fuse into one contiguous wired segment. Instead, they remain separated by an infinitesimal free segment centered on , and the boundary clusters that originally anchored to the adjacent wired sides remain separated by this segment. The likelihood of witnessing either of these two configurations in , with respect to the point on its boundary, is zero. Hence, pulling together two vertices not connected by a common boundary arc sends the connectivity weight for the original configuration in to zero. In CFT, this corresponds to the appearance of only the two-leg fusion channel in the OPE of the one-leg corner operators at and . (See appendix A.)
All of these mappings that pull pairs of adjacent vertices of together are subject to one constraint: the vertices and of the -sided polygon to be brought together at the th step of this mapping cannot be separated from each other by any other vertices within the boundary of . If we imagine connecting and with an arc crossing the interior of for each , then this condition is satisfied if and only if we may draw these arcs in so they do not intersect. Furthermore, two mappings that bring the same pairs of vertices together in a different order have the same such arc connectivities, of which there are (4) (figure 6). Now, if changing the order in which we collapse the sides of does not change the image of any of these mappings, then there are effectively only distinct mappings. Assuming that the order indeed does not matter, we enumerate these mappings as we enumerated their corresponding boundary arc connectivities, we denote the th of them by , and we let
If the arc connectivity for some specified mapping in matches (resp. does not match) the boundary arc connectivity for some specified connectivity weight, then our arguments imply that this mapping sends that connectivity weight to one (resp. zero). Hence, we anticipate that for all and, assuming that is finite-dimensional with basis , the set of connectivity weights is the basis of dual to . (Therefore, in the sequel , we define the th connectivity weight to be the element of dual to . Both here and in that article, we assume that this precise definition agrees with the multiple-SLE definition given above.)
If is a basis for , then this duality relation implies that is the coefficient of the th connectivity weight in the decomposition of over . Furthermore, the linear mapping with the th coordinate of equaling is a bijection. That is, although destroys the pointwise information contained in each element of , it preserves the linear relations between these elements.
This reasoning motivates our strategy for proving goals 1–3 in section I.1 but in what follows, we order our steps differently because we do not know how to prove that is a basis for a priori. So after constructing the elements of in sections II and III, we prove that the linear mapping is injective in section IV (deferring there the proof of lemma 14 to ). Then the dimension theorem of linear algebra bounds the dimension of by . Finally, to prove that the dimension of is indeed in , we use the Coulomb gas formalism to construct explicit elements of and then use results of this article to prove that they are linearly independent. That is a basis for follows from these results.
Ii Boundary behavior of solutions
Motivated by the observations of section I.2, we investigate the behavior of elements of near certain points in the boundary of . (See the discussion surrounding (12).) If we conformally map the -sided polygon onto the upper half-plane, with its th vertex sent to the th coordinate of , then the action of bringing together the vertices and sends to the boundary point . This point is in the hyperplane within , whose points have only the th and th coordinates equal. Hence, to implement the mappings described in the previous section for any , we must study the limit of as for any first.
Interpreting as a half-plane correlation function of one-leg boundary operators (3), we anticipate this limit using CFT. (These correlation functions appear, e.g., on the right side of (132). See appendix A for further details and a review of the CFT nomenclature that we refer to here.) We envisage the th coordinate of as hosting a one-leg boundary operator . If we send for some , then the operators and fuse into some combination of an identity operator (which is actually independent of ) and a two-leg boundary operator at . After inserting their OPE into the -point function , we find the Frobenius series expansion
Here, and are arbitrary real constants (for our present purposes), is the conformal weight of the -leg boundary operator (140)
Below, (29) gives the powers appearing in (22, 23). Loosely speaking, if appropriately normalized, we refer to either (22) or (23) in CFT as a “conformal block,” and (22) and (23) correspond to the identity and two-leg fusion channels of the constituent one-leg boundary operators respectively
Motivated by this interpretation of , we suppose that any has a Frobenius series expansion in centered on :
Using the null-state PDEs (10) centered on and (i.e., with and ) and collecting the leading order contributions, we find from either equation that
Solving this equation for , we find the two powers
that appear in (22) and (23) respectively. In this article, we restrict our attention to the range over which is the smaller power. At the next order, the null-state PDEs centered on and respectively give
Taking their difference when gives , so immediately follows if . (We re-examine the case with more closely in section II of .) In the CFT language, the condition that is equivalent to the vanishing of the level-one descendant of the identity operator, and the condition implies that the identity operator is nonlocal. Comparing (25) with (22) suggest that we interpret as a -point function of one-leg boundary operators. If this supposition is true, then must satisfy the system of PDEs (10, 11) in the coordinates of with replaced by . This observation echoes our previous claim that the limit (previously ) sends a connectivity weight for a -sided polygon to that of a -sided polygon. If instead, then (30) and (31) are identical, so is typically not zero. In the CFT language, this implies that the two-leg operator is local. We focus our attention on the case of (29) for now and postpone consideration of the case to ; .
These heuristic calculations suggest that for all , if we let approach with the values of and the other coordinates of fixed, then either grows or decays with power or greater. Lemma 3 below establishes this fact, but before we prove it, we introduce some convenient notation.
We define and to be the projection maps removing the th coordinate and both the th and th coordinates respectively from :
More generally, we define to be the projection map removing the coordinates with indices from .
In this article, we often identify with the subset of the boundary of whose points have only two coordinates, the th and the th, equal. Furthermore, we sometimes identify , explicitly given by
with the subset of the boundary of whose points have only three coordinates, the th, the th and the th, equal.
Suppose that and , and for some , let
Then for all and any compact subset of , the supremums
are as .
For each point , we let , we relabel the coordinates in as in increasing order, and we let . With this new notation, we define
Finally, we choose an arbitrary compact subset , and with determined by (20), we choose bounded open sets , with and , such that they are sequentially compactly embedded:
(We choose large enough so .) Our goal is to prove that the quantities in (35) are as .
To begin, we write the null-state PDE (10) centered on as , where contains all of the terms that are seemingly largest when . With referring to a derivative with respect to , we have