Topological Phases in the
Plaquette Random-Cluster Model
and Potts Lattice Gauge Theory

The generalized random cluster model we study here was introduced by Hiraoka and Shirai [HS16]. Their model, which we call the $i$-dimensional plaquette random-cluster model, weights the probability of an $i$-complex $P$ in terms of the $(i-1)$-Betti number of the homology with coefficients in a field $\mathbb{F},$ denoted $\mathbf{b}_{i-1}\left(P;\,\mathbb{F}\right).$ We provide a definition of Betti numbers in Section 2 below.

As $\mathbf{b}_{0}$ is the number of connected components, this is indeed a generalization of the classical random-cluster model. In their paper introducing the plaquette random-cluster model, Hiraoka and Shirai’s main focus is proving an expression for the partition function in terms of a generalized Tutte polynomial. They also show positive association and construct a coupling between the model and a generalization of the Potts model. However, they do not mention that this generalized Potts model had been previously defined as Potts lattice gauge theory [HS16].

Generalized $i$-dimensional Potts lattice gauge theory assigns spins in an abelian group $\mathcal{G}$ to the $i$-dimensional faces of an oriented cell complex $X.$ Denote by $C^{i}\left(X;\,\mathcal{G}\right)$ the set of functions from the oriented $i$-cells of $X$ to $\mathcal{G}$ so that cells with opposite orientations are mapped to inverse group elements. We follow the language of algebraic topology and call elements of $C^{i}\left(X;\,\mathcal{G}\right)$ cochains. In other works on lattice gauge theory they are sometimes termed discrete differential forms, for reasons that we will describe in Section 2 below.

In this paper, we focus exclusively on the cases where $\mathcal{G}=\mathbb{F}_{q}$ is the additive group of integers modulo a prime number $q$ (which can be identified with the multiplicative group of $q$-th complex roots of unity $\mathbb{Z}\left(q\right)$). In keeping with conventions from algebraic topology, we use additive notation rather than the multiplicative notation of the definition of Potts lattice gauge theory. In particular, we will distinguish between the additive group of integers modulo $n$ $\mathbb{Z}_{n}$ and the multiplicative group of complex $n$-th roots of unity $\mathbb{Z}(n),$ though they are isomorphic. In the case where $\mathcal{G}=\mathbb{F}_{q},$ we will denote the Potts lattice gauge theory measure by $\nu_{\beta,q,i-1,d},$ or simply $\nu,$ and call this measure $q$-state Potts lattice gauge theory. We describe why the assumption that $q$ is prime is necessary in Section 1.2. We will be most interested in the Potts lattice gauge theory on $\mathbb{Z}^{d},$ and we will write $\nu^{\mathbf{f}}=\nu^{\mathbf{f}}_{\mathbb{Z}^{d},\beta,\mathcal{G},i-1,d}$ to denote the limiting Potts lattice gauge theory with free boundary conditions constructed in Section 5.2.

Potts lattice gauge theory is motivated by its similarities with Euclidean lattice gauge theory. Lattice gauge theory is in turn a discretized model of Euclidean Yang–Mills theory [Wil74, Wil04, Cha16]. Very briefly, Euclidean lattice gauge theory is defined so that $\mathbb{F}_{q}$ can be replaced with a complex matrix group $\mathcal{G}.$ In particular, $1$-dimensional lattice gauge theory on $\mathbb{Z}^{4}$ with $\mathcal{G}=U(1),$ $\mathcal{G}=SU(2),$ and $\mathcal{G}=SU(3)$, models the electromagnetic, weak nuclear, and strong nuclear forces, respectively. While Potts lattice gauge theory is not itself physical, it has been studied in the physics literature as it is more tractable and is thought to present some of the same behavior as more physically relevant cases [GGZ80, KS82, MO82, AF84, LMR89]. The special cases of $2$ and $3$-state Potts lattice gauge theory coincide with $\mathbb{Z}(2)$ (Ising) and $\mathbb{Z}(3)$ “clock” lattice gauge theory respectively after an appropriate rescaling of the coupling constant $\beta$ (where $\mathbb{Z}(n)$ denotes the multiplicative group of $n$-th complex roots of unity). These models have also been studied in the physical and the mathematical literatures as relatively approachable examples of lattice gauge theory [Alt78, Yon78, Frö79, MP79, MMS79, KPSS80, WHK⁺07, Weg17, Cha20].

The most important observables in lattice gauge theory are arguably Wilson loop variables, which we define for $q$-state Potts lattice gauge theory. These arise from evaluating a Potts state $f$ on a “loop” $\gamma$ made of $(i-1)$-cells. In topological language $\gamma\in C_{i-1}\left(X;\;\mathcal{G}\right)$ is called an $(i-1)$-cycle.

An important conjecture for the analogous quantities in Euclidean lattice gauge theory — called the Wilson area law — is that if $\gamma=\partial\rho$ is a $1$-boundary and $\rho$ is the minimal bounding chain then the expectation of $W_{\partial\rho}\left(\omega\right)$ should decay as $e^{-c\left|\rho\right|}$ in some cases. For $\mathcal{G}=SU(2)$ or $SU(3),$ this conjecture is thought to be related to the phenomenon of quark confinement: that charged particles for the weak or strong nuclear forces are not seen in isolation, unlike for for the electromagnetic force [Wil74, Cha21] where the corresponding $U\left(1\right)$ lattice gauge theory exhibits both “area law” and “perimeter law” phases [Gut80, FS82].

One-dimensional $q$-state Potts lattice gauge theory on $\mathbb{Z}^{4}$ is thought to undergo a sharp phase transition as the coupling constant $\beta$ increases, from a “perimeter law” to an“area law” regime. We state a more general form of this “sharpness” conjecture here.

Recall that $W_{\gamma}$ is defined as a complex number, so the expectation is complex as well. Here, the perimeter of an $(i-1)$-boundary $\gamma$ is the number of plaquettes in its support and its area is the number of plaquettes supported in the minimal bounding chain. Observe that when $i=1,$ $\gamma$ consists of two vertices $\left\{v,w\right\},$ its perimeter is $2$, and its area is the distance between $v$ and $w.$ As such, the $i=1$ case of the conjecture is the the sharpness of the phase transition for the Potts model, as proven by Duminil-Copin, Raoufi, and Tassion [DCRT19].

This conjecture is not known for any specific value of $q$ when $i>1$, but Laanait, Messager, Ruiz showed that such a transition occurs when $q$ is sufficiently large [LMR89]. In addition, classical series expansions have been used to prove the existence of perimeter law and area law regimes for sufficiently extreme values of $\beta.$ Specifically, Osterwalder and Seiler employed a “high–temperature expansion” to show existence of an area law regime for any lattice gauge theory [OS78, Sei82], and the argument should work for Potts lattice gauge theory as well. On the other hand, a perimeter law regime is demonstrated by a “low–temperature expansion,” a technique that has recently been used by Chatterjee [Cha20], Cao [Cao20], and Forsström, Lenells, and Viklund [FLV21] to prove precise estimates for the asymptotics of Wilson loop variables for lattice gauge theory when $\beta$ is large. We use the coupling of the plaquette random cluster model and Potts lattice gauge theory to give an alternate proof of the existence of perimeter law and area law regimes for $i$-dimensional $q$-state Potts lattice gauge theory in $\mathbb{Z}^{d},$ for prime values of $q.$ Towards that end, we prove an exact relationship between Wilson loop variables and a certain topological event.

An $i$-chain $\gamma$ is null-homologous in a cubical complex $P$ if $\gamma=\partial\rho$ for some $(i+1)$-chain $\rho\in C_{i+1}\left(P;\,\mathcal{G}\right)$ (in other words, $\gamma=0$ the $i$-dimensional homology group). Roughly speaking, when $i=1$ this means that $\gamma$ is “bounded by a surface of plaquettes,” with the surfaces under consideration being dependent on the coefficient group $\mathcal{G}.$ For $\mathcal{G}=\mathbb{Z},$ $\gamma$ is null-homologous if and only if it is bounded by an orientable surface of plaquettes; when $\mathcal{G}=\mathbb{F}_{2}$ any surface of plaquettes will suffice. We denote the event that $\gamma$ is null-homologous by $V_{\gamma},$ with the choice of coefficients being understood in context. We have the following result.

While we state this theorem for $\mathbb{Z}^{d},$ the same proof works for any finite cubical complex. A precise definition of the infinite volume $i$-dimensional plaquette random cluster model $\mu_{\mathbb{Z}^{d}}=\mu_{\mathbb{Z}^{d},p,q,i}$ is given in Section 4.2. As a corollary, we have that Conjecture 4 is equivalent to the special case of the following conjecture for the plaquette random cluster model when $q$ is a prime integer and $\mathbb{F}=\mathbb{Z}_{q}.$

When $i=1,$ this conjecture is the sharpness of the phase transition of the classical random cluster model, as proven in [DCRT19]. In addition, the special case of the conjecture for $2$-dimensional independent plaquette percolation in $\mathbb{Z}^{3}$ ($i=2,d=3,q=1$) is a theorem of Aizenman, Chayes, Chayes, Frölich, and Russo, who demonstrated that this phase transition is dual to the bond percolation transition³³3Their paper phrases this result in terms of possibly distinct critical probabilities, but these have since been shown to coincide [Men86, GM90].. Their proof works for general coefficient fields.

Another consequence of Theorem 5, together with a comparison with plaquette percolation, is a new proof of the existence of area law and perimeter law regimes for Potts lattice gauge theory. We state a more general result for the plaquette random cluster model.

Note that the constants could depend on the boundary conditions used to construct the infinite volume plaquette random cluster model.

The perimeter law phase can be interpreted as a “hypersurface–dominated regime” in a quantitative sense, where there are many large hypersurfaces of $i$-plaquettes in $\mathbb{Z}^{d}$ [ACC⁺83]. For plaquette percolation on the torus, Duncan, Kahle, and Schweinhart [DKS20] showed that this phase coincides with a “hypersurface–dominated regime” in the qualitative sense of homological percolation, that is, one marked by the appearance of $i$-dimensional cycles in the random cubical complex that are representatives of nonzero homology classes in the ambient torus (see Section 2 for more detail). Unlike the theorem of [ACC⁺83], this homological percolation result generalizes to to $i$-dimensional plaquette percolation in the $2i$ dimensional torus. Here, we further generalize it to the $i$-dimensional random-cluster model in the $2i$-dimensional torus.

When $q$ is a prime integer and $i=2$, the phase transition is marked by the emergence of “giant sheets” on which the Wilson loop variables are constant within homology classes. The critical point is alternatively characterized by a qualitative change in the generalized Swendsen–Wang dynamics from local to non-local behavior, as defined in Section 5.1 below.

We also generalize the other two main theorems of [DKS20] from plaquette percolation to the plaquette random-cluster model. The proofs of these results and the previous one rely heavily on the duality properties of the plaquette random-cluster model; in Theorem 18 we show that the dual of $\mu_{\mathbb{T}^{d}_{N},p,q,i}$ is distributed approximately as $\mu_{\mathbb{T}^{d}_{N},p^{*},q,d-i}$ in a precise sense, where

Next, we prove that there are dual sharp phase transitions for $i=1$ and $i=d-1$ that are consistent with the critical probability for the random-cluster model in $\mathbb{Z}^{d},$ assuming a conjecture about the continuity of the critical probability in slabs. Let

Fix $q\geq 1$ and let $p_{c}\left(S_{k}\right)$ be the critical probability for the $1$-dimensional random-cluster model with parameter $q$ on $S_{k}$ with free boundary conditions. This can be constructed by a limit of free random-cluster models on

Since $S_{k,l}\subset S_{k+1,l},$ it follows that $p_{c}\left(S_{k}\right)$ is decreasing in $k.$ Then let

Let $\hat{p}_{c}=\hat{p}_{c}(q,d)$ be the critical threshold for the random-cluster model with parameter $q$ on $\mathbb{Z}^{d}.$ These two critical values are conjectured to coincide.

In general dimensions we demonstrate the existence of a sharp threshold function defined as follows: Let $\lambda=\lambda\left(N,q,i,d\right)$ satisfy

and let $p_{l}=p_{l}\left(q,i,d\right)\coloneqq\liminf_{N\to\infty}\lambda\left(N,q,i,d\right)$ and $p_{u}=p_{u}\left(q,i,d\right)\coloneqq\limsup_{N\to\infty}\lambda\left(N,q,i,d\right).$

Our results for $q$-state Potts lattice gauge theory are for cases where the cohomology coefficients are a finite field $\mathbb{F}_{q}$ where $q$ is a prime integer, instead of the more general additive group $\mathbb{Z}_{n}.$ This is not just for convenience, though it is more convenient to work with homology and cohomology with field coefficients since the associated groups are then vector spaces. There is one key place where this assumption is necessary, as noticed by Aizenman and Frölich [AF84]. It arises trying to couple the plaquette random-cluster model with Potts lattice gauge theory. Recall that the definition of the random-cluster model involves the term $q^{\mathbf{b}_{i-1}\left(P\right)},$ where $\mathbf{b}_{i-1}\left(P\right)$ is dimension of the $\mathbb{F}$-vector space $H_{i-1}\left(P;\,\mathbb{F}\right)$. If we were to use $\mathbb{Z}_{n}$ coefficients, we could try to replace $\mathbf{b}_{i-1}\left(P\right)$ with the rank of the $\mathbb{Z}_{n}$-module $H_{i}\left(P;\,\mathbb{Z}_{n}\right)$ (that is, the size of the minimal generating set). However, this does not lead to the same model as the marginal of the generalized Edwards–Sokal coupling defined in Section 5. The argument in Proposition 20 breaks down because, in general,

This could be resolved by defining a dependent plaquette percolation model so that

We expect some of our results to carry over to this model, but we defer them to a later paper.

Aizenman and Frölich identified a second issue, where they constructed a plaquette system in which there exist $(i-1)$-cycles $\gamma$ and $\gamma^{\prime}$ so that $\left[\gamma\right]$ is homologous to $q\left[\gamma^{\prime}\right].$ Then an analogue of Theorem 5 fails when $V_{\gamma}$ is defined to be the event that $\gamma$ is bounded by a surface of plaquettes. However, we resolve this by redefining $V_{\gamma}$ to depend on the choice of $q$; clearly $\left[\gamma\right]=0$ in $H_{i-1}\left(P;\;\mathbb{Z}_{q}\right)$ in this example.

Finally, in the proof of the homological percolation phase transition, Lemma 41 requires that the homology groups are vector spaces.

We outline the remainder of the paper. First, in Section 2 reviews the definitions of homology and cohomology for cubical complexes. These are used in Section 3 to show useful topological duality results. Next, we define the plaquette random-cluster model in Section 4 and prove several of its basic properties. These include positive association (which was previously shown by Hiraoka and Shirai [HS16]), the existence of the infinite volume limit, and duality. Section 5 covers the relationship between the plaquette random-cluster model and Potts lattice gauge theory: the coupling between them, the definition of the plaquette Swendsen–Wang algorithm, and proofs of Theorems 5 and 7. Finally, we prove our results for homological percolation — Theorems 8, 10, and 11 — in Section 6.

In order to convert vague geometric questions about “the number of holes” of a topological space into concrete algebraic quantities, homology theory defines a space $C_{i}\left(X;\,\mathcal{G}\right)$ of linear combinations with coefficients in an abelian group $\mathcal{G}$ (for a concrete example, take $\mathcal{G}=\mathbb{Z}$) of $i$-plaquettes called chains, and boundary operators, which map an $i$-plaquette to a sum of its $(i-1)$-faces. The continuous analogues of these concepts are, roughly speaking “$i$-dimensional spaces on which one can integrate an $i$-form” (called an $i$-current) and the geometric boundary of a space.

For reasons that become apparent below, it is important to give each term in the boundary of a plaquette a sign corresponding to the orientation of a plaquette. The formula is relatively simple in low dimensions. A zero-plaquette is a vertex and its boundary is zero. A $1$-plaquette is an edge $\left(v_{1},v_{2}\right)$ and its boundary is the difference $v_{2}-v_{1}.$ A two-plaquette is an oriented unit square with vertices $\left(v_{1},v_{2},v_{3},v_{4}\right)$ and its boundary is $\left(v_{1},v_{2}\right)+\left(v_{2},v_{3}\right)+\left(v_{3},v_{4}\right)-\left(v_{1},v_{4}\right).$ More generally, let $1\leq k_{1}<k_{2}<\ldots<k_{i}\leq d$ and let $I_{j}=\left[0,1\right]$ for $j\in\left\{k_{1},k_{2},\ldots,k_{i}\right\}$ and $I_{j}=\left\{0\right\}$ for $j\in\left[d\right]\setminus{k_{1},k_{2},\ldots,k_{i}}.$ Then $\sigma=\prod_{1\leq j\leq d}I_{j}$ is an $i$-plaquette in $\mathbb{R}^{d},$ and its boundary is given by

The reason that the sum is alternating in sign is so that the boundary operator satisfies the equation

Of particular interest are the chains $\alpha\in C_{i}\left(X;\,\mathcal{G}\right)$ satisfying $\partial\alpha=0.$ Such chains are called cycles, the space of which is denoted $Z_{i}\left(X;\,\mathcal{G}\right).$ Equation 3 provides one source of $i$-cycles, namely the boundaries of $(i+1)$-plaquettes. We denote this space $B_{i}\left(X;\,\mathcal{G}\right).$ It turns out that the most interesting cycles are the ones that are not boundaries.

To illustrate why this is the case, consider the unit square $Q=\left[0,1\right]^{2}\subset\mathbb{R}^{2},$ seen in Figure 1. In the cubical complex structure we defined earlier, we have that $Z_{1}\left(Q;\,\mathcal{G}\right)\simeq\mathcal{G}$ consists of multiples of a single $1$-cycle, namely the boundary of $Q.$ But now imagine that we make our lattice spacing $1/2$ instead of $1.$ Now $Z_{1}\left(Q;\,\mathcal{G}\right)$ contains linear combinations of the boundaries of the four $2$-plaquettes contained in $Q.$ One such combination can be seen in Figure 2. Though $Q$ has not changed, our choice of lattice has made the cycle space significantly more complicated.

Therefore, in order to measure the shape of spaces in a way that does not depend on our choice of lattice, we define the $i$-dimensional homology group as the quotient

Returning to the unit square $Q,$ every $1$-cycle is a boundary regardless of the cubical complex structure on $Q$, so $H_{1}\left(Q;\,\mathcal{G}\right)=0.$ In general, it turns out that homology groups do not depend on the cubical complex structure of a space, and are invariant to continuous deformations of the space. In particular, one can show that $H_{1}\left(Q;\,\mathcal{G}\right)=0$ by continuously deforming $Q$ to a point, which has no nonzero $1$-chains let alone $1$-cycles.

For an example with nontrivial first homology, consider $H_{1}\left(\partial Q;\,\mathcal{G}\right),$ where $\partial Q$ is the topological boundary of $Q,$ i.e. the empty square formed by the union of the four sides of $Q.$ There is still the $1$-cycle from before but there are no $2$-plaquettes, so $H_{1}\left(\partial Q;\,\mathcal{G}\right)\neq 0.$ This example illustrates the often repeated informal description of $H_{i}\left(X\right)$ as a measurement of the number of $i$-dimensional holes of $X.$ In the case of $H_{1},$ the cycles that remain are the loops that cannot be filled in by $2$-plaquettes.

We can now define giant cycles in a subcomplex of the torus. If $X\subset\mathbb{T}^{d}_{N}$ consists of a subset of the cubical cells of $\mathbb{T}^{d}_{N},$ then the chain groups of $X$ are subgroups of the chain groups of $\mathbb{T}^{d}_{N}.$ The giant cycles of $X$ are the cycles that are not boundaries when considered as chains in $\mathbb{T}^{d}_{N}.$ In the case $i=1,$ a giant cycle is a loop that spans the torus $\mathbb{T}^{d}_{N}$ in some sense, and must therefore have length at least $N.$ We interpret this as a finite volume analogue of an infinite path in the lattice. See Figure 3 for illustrations of two giant cycles.

So far, the choice of coefficient group $\mathcal{G}$ has not been important to our computations in this section, though we have mentioned their importance to the random-cluster model in the introduction. In this paper, we will almost exclusively work with homology over the finite field of integers mod $q$ for prime $q,$ denoted $\mathbb{F}_{q}.$ We will see later that this is the correct choice to study Potts lattice gauge theory, and it has the advantage of simplifying the algebra required since a homology group with field coefficients is a vector space.

For an example where different coefficients produce different results, consider the identifications of the sides of the $2$-plaquette in Figure 4. In the first case we roll up the plaquette into a cylinder and then put the opposite ends of the cylinder together to form a torus $\mathbb{T}.$ In the second, we also form a cylinder but put the opposite ends of the cylinder together in the reverse orientation to form a Klein bottle $\mathbb{K}.$ Consider the second homology of both spaces. In both cases there are no $3$-plaquettes and only one $2$-plaquette $\sigma,$ so it suffices to check if the $2$-chains generated by $\sigma$ contain nontrivial cycles. In the torus,

so $H_{2}\left(\mathbb{T};\,\mathbb{F}\right)\simeq\mathbb{F}$ for any field $\mathbb{F}.$ In the Klein bottle,

which is nonzero over most fields, with a number of exceptions including $\mathbb{F}_{2}.$ Thus, $H_{2}\left(\mathbb{K};\,\mathbb{F}_{2}\right)\simeq\mathbb{F}_{2},$ and $H_{2}\left(\mathbb{K};\,\mathbb{F}_{q}\right)=0$ for any prime $q\neq 2.$ There is also a difference in first homology; one can compute $H_{1}\left(\mathbb{K};\,\mathbb{F}_{2}\right)\simeq\mathbb{F}_{2}\oplus\mathbb{F}_{2}$ and $H_{1}\left(\mathbb{K};\,\mathbb{F}_{q}\right)\simeq\mathbb{F}_{q}$ for $q\neq 2.$ Of course, there are no single plaquettes with sides identified this way in the integer lattice, but non-orientable surfaces such as the Klein bottle appear as unions of plaquettes in dimensions $4$ and higher. Homology that appears over some finite fields but not others is sometimes called torsion.

It is worth mentioning a special situation that arises for the $0$-th homology group. It is not hard to check that $H_{0}\left(X;\,\mathbb{F}\right)=\mathbb{F}^{{c_{X}}},$ where $c_{X}$ is the number of connected components of $X.$ Some standard results in algebraic topology have simpler statements when all of the homology groups of a space consisting of a single point are zero. This is not true in the setup that we have described so far, since there is one connected component. To fix this, define the reduced homology groups

The Euler–Poincaré formula is an important result that relates the Euler characteristic of a cubical complex to the ranks of its homology groups with coefficients in any field. Define the $i$-th Betti number of a cubical complex $X$ with coefficients in $\mathbb{F}$ by

Also, let $X^{i}$ be the set of $i$-plaquettes of $X$ and let $\left|X^{i}\right|$ be their number. The Euler characteristic of $X$ is then

The following theorem is a straightforward consequence of the rank–nullity theorem.

A small algebraic change in the definitions that give the homology groups leads to cohomology. The $i$-th cochain group $C^{i}\left(X;\,\mathbb{F}\right)$ is defined as the group of $\mathbb{F}$-linear functions from $C_{i}\left(X;\,\mathbb{F}\right)$ to $\mathbb{F}.$ Since we are using field coefficients, this is the dual vector space to $C_{i}\left(X;\,\mathbb{F}\right).$ Then the coboundary operator $\delta:C^{i}\left(X;\,\mathbb{F}\right)\to C^{i+1}\left(X;\,\mathbb{F}\right)$ is defined by

for $f\in C^{i}\left(X;\,\mathbb{F}\right),\alpha\in C_{i+1}\left(X;\,\mathbb{F}\right).$ The continuous analogues of an $i$-cochain and and its coboundary are a differential form and its exterior derivative. Note that if $f$ is an $i$-cochain and $\alpha$ is an $i$-chain, the evaluation $f\left(\alpha\right)$ corresponds to integration and the definition of the coboundary operator corresponds to Stokes’ Theorem.

Analogously to homology groups, we define the $i$-th cocycle group $Z^{i}\left(X;\,\mathbb{F}\right)$ to be the kernel of $\delta$ and the $i$-th coboundary group $B^{i}\left(X;\,\mathbb{F}\right)$ to be the image of $\delta$ applied to $C^{i+1}\left(X;\,F\right).$ Then $i$-th cohomology group is the quotient

In the continuous setting, the analogue of the $i$-dimensional cohomology group is the $i$-dimensional de Rham cohomology of closed forms modulo exact forms. This is more than just as analogy: de Rham’s theorem states that the de Rham cohomology of a smooth manifold is isomorphic to $H^{i}\left(M,\mathbb{R}\right)$ computed using a cubical complex structure on $M$ [Bre13].

In some ways cohomology does not add more information than homology already provided. The universal coefficient theorem for cohomology tells us that $H_{i}\left(X;\,\mathbb{F}\right)\simeq H^{i}\left(X;\,\mathbb{F}\right)$ when $\mathbb{F}$ is a field (see Corollary 3.3 of [Hat02]). However, this algebraic equivalence does not preclude problems lending themselves more naturally to one perspective or the other. For example, it is arguably more intuitive to think of giant cycles as homological in nature, due to their interpretation as giant “hypersurfaces” spanning the surface. On the other hand, we argue that lattice gauge theory should be thought of as a random cochain and Wilson loop variables as cohomological quantities. It is also worth noting that although the homology and cohomology groups themselves may be isomorphic, there is a relationship between the different dimensional cohomology groups in the form of a multiplication of cocycles called the cup product that does not always have a homological analogue (see Section 3.2 of [Hat02]).

Homology and cohomology come together in global duality theorems, of which there are many versions. We primarily use variations of Alexander duality, the original form of which says that for a sufficiently “nice” subspace $X\subset S^{d},$

for $1\leq i\leq d-1.$ In the simplest case, this says that the number of bounded components of a “nice” subset of the plane equals the number of “minimal contours” of its complement. Since the complement of a set of plaquettes is a thickening of the dual system (as shown in Proposition 13 below), we are able to use techniques related to the proof of Alexander duality to relate both the giant cycles and local cycles of the two sets (Theorem 14 below). This relationship is quantified in terms of the dimension of the relevant subspaces of the homology groups, and is a main ingredient in the proof that the dual complex is close to a random-cluster model.

We will now briefly discuss the relevant information about the topology of the torus specifically. Although in percolation theory the torus is most often thought of as a cube with periodic boundary conditions, it is useful topologically to think of it as the product of $d$ copies of the circle $S^{1}.$ In such a product space, the Künneth formula for homology (Section 3B of [Hat02]) tells us that there is an isomorphism of the form

Since $H_{1}\left(S^{1};\,\mathbb{F}\right)\simeq H_{0}\left(S^{1};\,\mathbb{F}\right)\simeq\mathbb{F}$ and $H_{i}\left(S^{1};\,\mathbb{F}\right)=0$ for all $i\geq 2,$ we see that

Furthermore, there is a set of generating $i$-cycles consisting of the products of $i$ of the $d$ possible $S^{1}$ factors.