1.1 Motivation

Understanding the behaviour of physical systems undergoing a continuous phase transition at and near their critical point is one of the major challenges of modern statistical physics on both the physics and mathematical sides. In the first half of the twentieth century, the understanding relied essentially on exact computations, as exemplified by the analysis of mean-field systems and Onsager’s revolutionary solution of the 2D Ising model [Reference OnsagerOns44]. In the 1960s, a major advance was achieved by American chemist Benjamin Widom, who proposed in [Reference WidomWid65] that many quantities at and near criticality follow power laws, translating the question of understanding the phase transition into the question of computing so-called critical exponents. Also, the arrival of the renormalisation group (RG) formalism (see [Reference FisherFis98] for a historical exposition) led to a (nonrigorous) deep physical and geometrical understanding of continuous phase transitions. The RG formalism suggests that ‘coarse-graining’ renormalisation transformations correspond to appropriately changing the scale and parameters of the model under study. The large-scale limit of the critical regime then arises as the fixed point of the renormalisation transformations.

A striking consequence of the RG formalism is that because the critical fixed point is usually unique, the scaling limit at the critical point must satisfy translation, rotation, scale and even conformal invariance [Reference PolyakovPol70]. In two dimensions, this prediction allowed for the computation of the critical exponents ruling the behaviour of thermodynamical quantities and the classification of models into universality classes, meaning classes of models undergoing the same critical behaviour.

Another observation related to the previous developments is that the critical exponents are related to each other: if the behaviours of the specific heat, order parameter, susceptibility, source-field, two-point function and correlation length are governed by the exponents $\alpha $ , $\beta $ , $\gamma $ , $\delta $ , $\eta $ and $\nu $ , respectively, then the following scaling relations were predicted by physicists; see, for example, [Reference Essam and FisherEF63, Reference FisherFis64, Reference WidomWid65] for some early works on the subject (below, the dimension d of the lattice is assumed to be equal to 2, but we state the relations in this generality as they are predicted to extend to any dimension below the so-called upper critical dimension of the system):

(1.1) $$ \begin{align} &\frac{2-\alpha}d=\nu=\frac{2\beta}{d-2+\eta}, \end{align} $$

(1.2) $$ \begin{align} &2-\eta=d\frac{\delta-1}{\delta+1}=\frac\gamma\nu. \end{align} $$

A striking feature of these relations is that they hold for different universality classes, meaning the critical exponents may vary for different models, but they are always related via equations (1.1) and (1.2).

The aim of this paper is to provide rigorous proofs of these scaling relations for a family of planar percolation models. Percolation models are models of random subgraphs of a given lattice. Bernoulli percolation is perhaps the most-studied such model, and breakthroughs in the understanding of its phase transition have often served as milestones in the exciting history of statistical physics. The random-cluster model (also called Fortuin-Kasteleyn percolation) is another example of a percolation model. It was introduced by Fortuin and Kasteleyn around 1970 [Reference FortuinFor71, Reference Fortuin and KasteleynFK72] as a generalisation of Bernoulli percolation. It was found to be related to many other models of statistical mechanics, including the Ising, Potts and Ashkin-Teller models, and to exhibit a very rich critical behaviour. Of particular importance from the point of view of physics and relevant to our paper is the fact that the scaling limits of the random-cluster models at criticality are expected to fall into different universality classes and to be related to various 2D conformal field theories.

Let us conclude this section by reminding the reader that the theory of Bernoulli percolation is now well developed, with a deep qualitative understanding of the properties of the scaling limit [Reference Aizenman and BurchardAB99] and crossing probabilities [Reference RussoRus78, Reference Seymour and WelshSW78], universal critical exponents [Reference Kesten, Sidoravicius and ZhangKSZ98], scaling relations [Reference KestenKes87, Reference NolinNol08], noise sensitivity, near-critical window [Reference Garban, Pete and SchrammGPS10, Reference Garban, Pete and SchrammGPS18] and so on. For a variant of the model (site percolation on the triangular lattice), the existence of the scaling limit and its conformal invariance was proved [Reference SmirnovSmi01] and critical exponents have been computed [Reference Smirnov and WernerSW01]; see [Reference Beffara and Duminil-CopinBD13] and references therein for an overview of two-dimensional Bernoulli percolation. Deriving all these properties for Bernoulli percolation relies on specific features, such as independence of the states of different edges and geometric interpretations of differential formulae using so-called pivotal events. These features are not satisfied for more general random-cluster models. Another more prosaic goal of this paper is therefore to develop robust tools enabling one to bypass these special characteristics of Bernoulli percolation to extend the results mentioned in the abstract to the whole regime of critical random-cluster models undergoing a continuous phase transition. As such, these tools may have a number of implications that are not mentioned in the present paper, in particular for the study of other planar-dependent percolation models.

1.2 Definition of the random-cluster model

As mentioned in the previous section, the model of interest in this paper is the random-cluster model, which we now define. For background, we direct the reader to the monograph [Reference GrimmettGri06] and the lecture notes [Reference Duminil-CopinDum17] for an exposition of recent results.

Consider the square lattice $(\mathbb Z^2,\mathbb E)$ that is the graph with vertex-set $\mathbb Z^2=\{(n,m):n,m\in \mathbb Z\}$ and edges between nearest neighbours. In a slight abuse of notation, we will write $\mathbb {Z}^2$ for the graph itself. Consider a finite subgraph $G = (V,E)$ of the square lattice (V denotes the vertex-set and E the edge-set), and let $\partial G$ be the set of vertices in V incident to at most three edges in E. Write $\Lambda _n$ for the subgraph of $\mathbb {Z}^2$ spanned by the vertex-set $\{-n,\dots , n\}^2$ . For $1 \le r < R$ , write $\mathrm {Ann}(r,R)$ for the annulus $\Lambda _R\smallsetminus \Lambda _{r-1}$ . We also write $\Lambda _n(x)$ and $\Lambda _n(e)$ for the boxes of size n recentred around x and the bottom left endpoint of the edge e, respectively.

To define the model, consider first a finite graph G. A percolation configuration $\omega $ on G is an element of $\{0,1\}^{E}$ . An edge e is said to be open (in $\omega $ ) if $\omega _e=1$ ; otherwise it is closed. A configuration $\omega $ can be seen as a subgraph of G with vertex-set V and edge-set $\{e\in E:\omega _e=1\}$ . When speaking of connections in $\omega $ , we view $\omega $ as a graph. For sets of vertices A and B, we say that A is connected to B if there exists a path of edges of $\omega $ that connect a vertex of A to a vertex of B. This event is denoted by $A{\longleftrightarrow }B$ . We also speak of connections in a set of vertices C if the endpoints of the edges of the path are all in C.

A cluster is a connected component of $\omega $ . The boundary conditions $\xi $ on G are given by a partition of $\partial G$ . We say that two vertices of G are wired together if they belong to the same element of the partition $\xi $ .

Definition 1.1. The random-cluster measure on G with edge-weight p, cluster-weight $q>0$ and boundary conditions $\xi $ is given by

$$ \begin{align*} \phi_{G,p,q}^{\xi}[\omega]:=\frac{1}{Z^{\xi}(G,p,q)} (\tfrac{p}{1-p})^{|\omega|}q^{k(\omega^{\xi})}, \end{align*} $$

where $|\omega |:=\sum _{e\in E}\omega _e$ , $k(\omega )$ is the number of connected components of the graph, $\omega ^{\xi }$ is the graph obtained from $\omega $ by identifying wired vertices together, and $Z^{\xi }(G,p,q)$ is a normalising constant called the partition function chosen in such a way that $\phi _{G,p,q}^{\xi }$ is a probability measure.

Two specific families of boundary conditions will be of special interest to us. On the one hand, the free boundary conditions, denoted 0, correspond to no wirings between boundary vertices. On the other hand, the wired boundary conditions, denoted 1, correspond to all boundary vertices being wired together.

The random-cluster model may be modified to accommodate an external magnetic field as follows. Add to the lattice $\mathbb {Z}^2$ a vertex $\mathfrak g$ called the ghost vertex, and connect it to each vertex v of $\mathbb {Z}^2$ with an edge $v\mathfrak g$ . The random-cluster measure $\phi _{G,p,q,h}^i$ (for $i=0$ or $1$ and $h \geq 0$ ) is defined exactly as the random-cluster model on G, except that the boundary is now $\partial G\cup \{\mathfrak g\}$ , and the edge-weight is p for the edges of G and $1-e^{-h}$ for edges having $\mathfrak g$ as an endpoint: that is,

$$ \begin{align*} \phi_{G,p,q,h}^i[\omega]=\frac{1}{Z^{\xi}(G,p,q,h)} (\tfrac p{1-p})^{|\omega|}(e^h-1)^{\Delta(\omega)}q^{k(\omega^i)}, \end{align*} $$

where $\Delta (\omega ):=\sum _{v\in V}\omega _{v{\mathfrak g}}$ . The probability that $\mathfrak g$ is in the cluster of 0 has an interpretation in terms of spin models with a magnetic field: for $q=2$ , this probability is equal to the spontaneous magnetisation with an external field h for the Ising model on the square lattice. A similar interpretation holds for the 3-state and 4-state Potts models. For more details on this topic, see [Reference Biskup, Borgs, Chayes and KoteckýBBCK00].

For $q\ge 1$ and $i=0,1$ , the family of measures $\phi _{G,p,q,h}^i$ converges weakly as G tends to the whole square lattice [Reference GrimmettGri06, Theorem 4.19]. The limiting measures on $\{0,1\}^{\mathbb {E}}$ are denoted by $\phi _{p,q,h}^i$ and are called infinite-volume random-cluster measures with free and wired boundary conditions. They are invariant under translations and ergodic. When $h=0$ , we simply drop it from the notation.

The random-cluster model undergoes a phase transition at $h=0$ and a critical parameter $p_c=p_c(q)$ in the following sense: if $p>p_c(q)$ , the probability

$$ \begin{align*} \theta(p):=\phi_{p,q}^1[0\text{ is in an infinite cluster}] \end{align*} $$

is strictly positive, while for $p<p_c(q)$ , it is equal to 0. In the past 10 years, considerable progress has been made in the understanding of this phase transition: the critical point was proved in [Reference Beffara and Duminil-CopinBD12] (see also [Reference Duminil-Copin and ManolescuDM16, Reference Duminil-Copin, Raoufi and TassionDRT18]) to be equal to

$$ \begin{align*} p_c(q)=\frac{\sqrt q}{1+\sqrt q}. \end{align*} $$

It was also shown in these papers that the correlation length

(1.3) $$ \begin{align} \xi(p)&:=\lim_{n\rightarrow\infty}-n/\log [\pi_1(p;n)-\theta(p)]\in[0,\infty] \end{align} $$

is finite as soon as $p\ne p_c$ , where

(1.4) $$ \begin{align} \pi_1(p;n):=\phi_{p,q}^1[0\longleftrightarrow\partial\Lambda_n] \end{align} $$

(when $p=p_c$ , we drop $p_c$ from this notation).

For $q\ge 1$ , it was proved in [Reference Duminil-Copin, Sidoravicius and TassionDST17, Reference Duminil-Copin, Gagnebin, Harel, Manolescu and TassionDGHMT16] (see also [Reference Ray and SpinkaRS20] when $q>4$ ) that the correlation length at $p_c$ is infinite if and only if $q\le 4$ . As the divergence of the correlation length is one of the characterisations of a continuous phase transition, and as we are interested in this type of phase transition only, in the whole paper we will restrict our attention to the range $q\in [1,4]$ . Also, since the $q=1$ case was already treated by Kesten in [Reference KestenKes87] and later solved in numerous other places (see references below), we will often assume that $q>1$ .

1.3 Stability below the characteristic length

When studying a noncritical system, a natural length-scale is provided by the characteristic length, which appeared in a simplified context of Bernoulli percolation in the work of Kesten [Reference KestenKes87] (see also [Reference Borgs, Chayes, Kesten and SpencerBCKS99]) and was explicit for the random-cluster model in [Reference Duminil-Copin, Garban and PeteDGP14]. To define the characteristic length, we first introduce the notion of crossing probability.

A quad $(\mathcal {D}; a,b,c,d)$ is a finite subgraph of $\mathbb {Z}^2$ whose boundary $\partial \mathcal {D}$ is a simple path of edges of $\mathbb Z^2$ , along with four points $a,b,c,d$ found on $\partial \mathcal {D}$ in counterclockwise order. These points define four arcs $(ab)$ , $(bc)$ , $(cd)$ and $(da)$ corresponding to the parts of the boundary between them. We also see the quad as a domain of $\mathbb R^2$ with marked points on its boundary by taking the union of the faces enclosed by $\partial \mathcal {D}$ . The typical example is the case of rectangles $[0,n]\times [0,m]$ or $\Lambda _n$ with $a,b,c,d$ being the corners of the rectangle, oriented in counterclockwise order, starting from the bottom-right corner. In this case, we omit $a,b,c,d$ from the notation. We say that the quad $(\mathcal {D},a,b,c,d)$ is crossed if $(ab)$ is connected to $(cd)$ in $\mathcal {D}$ . The event is denoted by $\mathcal {C}(\mathcal {D})$ .

We say that a quad $(\mathcal {D}; a,b,c,d)$ is $\eta $ -regular at scale R for some $\eta> 0$ if $\mathcal {D}$ is contained in $\Lambda _R$ and is the union of a finite number of translates of $\Lambda _{\eta R}$ by points of $\eta R\mathbb Z^2$ , and $a,b,c,d\in \eta R\mathbb {Z}^2$ .

Now, consider $\delta _L>0$ small enough. How small $\delta _L$ should be is dictated by the proof of equation (RSW), and we simply wish to mention here that $\delta _L$ can be taken independent of $q\in [1,4]$ and can easily be estimated (even though the value is irrelevant for our study); see also Remark 2.14.

Definition 1.2 (Characteristic length)

For each $q\ge 1$ and $p\in (0,1)$ , let

(1.5) $$ \begin{align} L(p) =L(p,q):= \inf\{ R \geq 1 : \phi_{p} [\mathcal{C}(\Lambda_R)] \notin[\delta_L, 1-\delta_L]\}\in[1,\infty]. \end{align} $$

We insist on the fact that we consider the unique infinite-volume measure $\phi _p$ for the definition.

Note that $L(p)<+\infty $ for every $p\ne p_c$ by [Reference Beffara and Duminil-CopinBD12]; by duality, $L(p_c)=+\infty $ as long as $\delta _L < 1/2$ , which we will always assume. The interest of the characteristic length lies in its connection with the scaling window: that is, the regime of parameters $(R,p)$ for which one expects typical properties of the random-cluster model in $\Lambda _R$ with parameters p to be similar to the critical ones. In physics, the statement that the system looks critical is usually related to another length-scale: namely, the correlation length $\xi (p)$ defined in equation (1.3). The correlation length encodes the rate of exponential decay of the probability of being connected to distance n but not to infinity as n tends to infinity; it is not a priori directly related to $L(p)$ . Nevertheless, the following result reunites the two notions of correlation and characteristic lengths, thus affirming that the characteristic length is simply the correlation length in disguise.

Theorem 1.3 (Equivalence correlation/characteristic lengths)

Fix $1\le q\le 4$ . We have that for $p\in (0,1)$ ,

(1.6) $$ \begin{align} L(p)\asymp \xi(p). \end{align} $$

The proof is based on a coarse-grained procedure. We wish to highlight that the result is new for every $1<q\le 4$ – even for $q=2$ , for which [Reference Duminil-Copin, Garban and PeteDGP14, Theorem 1.2] proves almost the same statement, but with a logarithmic control over the ratio of $L(p)/\xi (p)$ rather than a constant one. Combined with [Reference Beffara and Duminil-CopinBD12b], for instance, this enables us to estimate Ising quantities; see Theorem 1.13 below.

One of the main results of [Reference KestenKes87] is that the scaling window is simply the set of $(p;R)$ such that $R=O(L(p))$ . This result is sometimes referred to as stability below the characteristic length; it is the subject of the following theorem in the context of the random-cluster model. Together with Theorem 1.3, the stability result legitimates the two physical interpretations of the correlation length: in terms of rate of decay and in terms of scaling window. We state the result for $q\ne 1$ as the case $q=1$ was already treated in [Reference KestenKes87]. Recall the definition of $\pi _1(p;R)$ and the fact that we omit p when $p=p_c$ .

Theorem 1.4 (Stability below the characteristic length)

Fix $1< q \le 4$ .

(Stability of crossing probabilities) There exists $\varepsilon> 0$ such that for every $\eta $ -regular discrete quad $(\mathcal {D},a,b,c,d)$ at scale $R\ge 1$ and every $p \in (0,1)$ (the constant in $\stackrel{<}{\scriptstyle{\frown}}$ depends on $\eta $ but $\varepsilon $ does not),

(1.7) $$ \begin{align} |\phi_{p} [\mathcal{C}(\mathcal{D})] - \phi_{p_c} [\mathcal{C}(\mathcal{D})]| \stackrel{<}{\scriptstyle{\frown}}(\tfrac R{L(p)})^{\varepsilon}. \end{align} $$

(Stability of the one-arm event) For every $p \in (0,1)$ and every $R\le L(p)$ ,

(1.8) $$ \begin{align} \pi_1(R)\asymp \pi_1(p;R). \end{align} $$

The stability of arm event probabilities given by equation (1.8) extends to more general arm events. Moreover, an improved version may be formulated; see Remark 7.2 for details.

The strategy for proving Theorem 1.4 is related to Kesten’s original one in that it uses Theorem 1.6 below to study the behaviour of derivatives of crossing events. Nevertheless, several additional difficulties occur, mainly due to the replacement of pivotality by influence in the differential formulas for probabilities of events: recall [Reference GrimmettGri06, Theorem 2.46] the general formula, valid for every $q>0$ ,

(1.9) $$ \begin{align} \tfrac{\mathrm{d}}{\mathrm{d}p}\phi_{p}[\mathcal{C}(\mathcal{D})] =\tfrac1{p(1-p)} \sum_{e \in \mathbb{E}} \mathrm{Cov}_p(\omega_e, \mathcal{C}(\mathcal{D})), \end{align} $$

where $\mathrm {Cov}_p$ denotes the covariance under $\phi _p$ . For $q=1$ , the sum of covariances gets nicely rewritten in terms of pivotal edges: edges that when switched from closed to open, change the occurrence of the event. In particular, it is possible to prove that for crossing events of a rectangle of size R and edges that are far from the boundary of the rectangle, the probability of being pivotal is of the order of the probability $\pi _4(p;R)$ that in the ball of radius R around a given edge e, the two extremities of e belong to different clusters that both reach the boundary of said ball (when $p=p_c$ , we simply write $\pi _4(R)$ ). This property was used crucially in [Reference KestenKes87] and ultimately leads to Kesten’s scaling relation $L(p)^2 \pi _4(L(p)) \asymp (p-p_c)^{-1}$ . The description in terms of pivotal edges is wrong for random-cluster models with $q>1$ as the covariance between an edge and crossing events at scale R is no longer of the order of $\pi _4(p;R)$ ; see [Reference Duminil-Copin, Garban and PeteDGP14].

Driven by this different phenomenology, in this paper we introduce a new interpretation of the covariance valid for every $q>1$ encoding how much an edge is influenced by boundary conditions at a distance R or, equivalently, how fast the model mixes.

Definition 1.5 (Mixing rate)

For $1< q\le 4$ , $1 \le r < R$ , $p\in (0,1)$ and e an edge incident to the origin, write

(1.10) $$ \begin{align} \Delta_p(R) &:= \phi_{\Lambda_R,p}^1[\omega_e] - \phi_{\Lambda_R,p}^0[\omega_e], \end{align} $$

(1.11) $$ \begin{align} \Delta_p(r,R) &:= \phi_{\Lambda_R,p}^1[\mathcal{C}(\Lambda_r)] - \phi_{\Lambda_R,p}^0[\mathcal{C}(\Lambda_r)]. \end{align} $$

The quantity $\Delta _p(R)$ , to which we now refer as the mixing rate, will be crucial in our study, as it will replace the amplitude $\pi _4(p;R)$ of standard pivotal events in the study of Bernoulli percolation. As such, it is very important to derive some of its properties.

Theorem 1.6 (Properties of the mixing rate)

Fix $1<q\le 4$ .

(i) (Mixing rate/covariance connection) For every $\eta>0$ , every $p\in (0,1)$ and every $\eta $ -regular quad $(\mathcal {D},a,b,c,d)$ at scale $R\le L(p)$ (below, the constants in $\asymp $ depend on $\eta $ ),

(1.12) $$ \begin{align} \kern-10pt\mathrm{Cov}_{p}[\omega_e;\mathcal{C}(\mathcal{D})]&\asymp\Delta_p(R),\qquad\qquad\qquad\qquad\forall e\in\Lambda_{2R}\text{ at a distance } \eta R \text{ of } \partial \mathcal{D}, \end{align} $$

(1.13) $$ \begin{align} \mathrm{Cov}_{p}[\omega_e;\mathcal{C}(\mathcal{D})]&\asymp\Delta_p(|e|)^2/\Delta_p(R)\qquad\qquad\qquad\qquad\qquad\quad\ \ \forall e\in \Lambda_{L(p)}\smallsetminus\Lambda_{2R}, \end{align} $$

(1.14) $$ \begin{align} \mathrm{Cov}_{p}[\omega_e;\mathcal{C}(\mathcal{D})]&\stackrel{<}{\scriptstyle{\frown}}\exp[-c(\eta)|e|/L(p)]\Delta_p(L(p))^2/\Delta_p(R)\qquad\qquad\quad \forall e\notin \Lambda_{L(p)}. \end{align} $$

(ii) (Quasi-multiplicativity) For every $p\in (0,1)$ and $1\le r\le \rho \le R\le L(p)$ ,

(1.15) $$ \begin{align} \Delta_p(r,\rho)\Delta_p(\rho,R)\asymp\Delta_p(r,R). \end{align} $$

(iii) (Stability below the characteristic length) For every $p\in (0,1)$ and $1\le R\le L(p)$ ,

(1.16) $$ \begin{align} \Delta_p(R)\asymp\Delta_{p_c}(R). \end{align} $$

(iv) (Comparison to pivotality) There exists $\varepsilon>0$ such that for every $1\le r\le R\le L(p)$ ,

(1.17) $$ \begin{align} \Delta_{p}(r,R) \stackrel{>}{\scriptstyle{\frown}} (R/r)^{\varepsilon}\pi_4(R)/\pi_4(r). \end{align} $$

(v) (Mixing interpretation) For every $1\le 2r\le R\le L(p)$ ,

(1.18) $$ \begin{align} \Delta_p(r,R)\asymp\max\Big\{\Big|\frac{\phi_p[A\cap B]}{\phi_p[A]\phi_p[B]}-1\Big|\ :\ A\in\mathcal F(\Lambda_r)\text{ and }B\in\mathcal F(\mathbb Z^2\smallsetminus\Lambda_R)\Big\}, \end{align} $$

where $\mathcal F(S)$ is the $\sigma $ -algebra generated by the edges with both endpoints in S.

The proof of this theorem is the main innovation of the paper. It is based on new increasing couplings between random-cluster models. While coupling Bernoulli percolation at different parameters is fairly straightforward, coupling different random-cluster models can be quite elaborate. In this paper, we develop several increasing couplings between random-cluster models (typically one at $p_c$ and one at p, or one at $p_c$ and another at $p_c$ , but conditioned on an event) that satisfy various properties.

In the previous theorem, Properties (i)–(v) have crucial interpretations. Property (i) will have the following important application: it states that the covariance between an edge that is deep inside an $\eta $ -regular quad and the crossing event of said quad is of the order of $\Delta _p(R)$ . Some more refined results (in particular near the boundary) can be obtained; see Lemma 5.3 and the remarks below. Property (ii) is an analogue of the quasi-multiplicativity of probabilities of arm events and will be popping up everywhere in the applications of $\Delta _p(R)$ , in particular when trying to estimate the covariance between a crossing event and an edge close to the boundary of the quad. Property (iii) states the stability below the characteristic length for the mixing rate, analogously to that proved by Kesten for the four-arm event probability. Property (iv) shows that replacing $\pi _4(p;R)$ by $\Delta _p(R)$ is really necessary, as the trivial bound stating that the covariance is larger than or equal to pivotality is polynomially far from being sharp for any $q>1$ . Finally, (v) justifies the reference to mixing in the name of $\Delta _p$ , as it links this quantity to the error term in the ratio-weak mixing.

We finish our comments on this theorem with a crucial observation. When trying to compute asymptotics for the covariance between an edge and a crossing event, (i) and (ii) imply that it suffices to understand for every $\varepsilon>0$ the limit of $\Delta _{p_c}(\varepsilon R,R)$ as R tends to infinity. Indeed, these limits allow us to estimate the covariance up to arbitrarily small polynomial terms and therefore estimate the critical exponent. This is very useful as the covariance itself is not easily expressed in terms of properties of large interfaces of the critical system, while $\Delta _{p_c}(\varepsilon R,R)$ (which is equal to $1-2\phi _{\Lambda _R,p_c}^0[\mathcal {C}(\Lambda _{\varepsilon R})]+o(1)$ by duality) is a quantity that can be derived from the scaling limit of the critical model, for instance using the conjectural convergence to the Conformal Loop Ensemble – see also Question 3 below.

It is tempting to deduce from the previous theorem that when $q>1$ , the derivative of crossing probabilities of $\eta $ -regular quads at scale $R\le L(p)$ is of order $R^2\Delta _p(R)$ (exactly like it is of order $R^2\pi _4(R)$ for Bernoulli percolation). This statement is not always correct and illustrates the subtle but deep difference with Bernoulli percolation. Indeed, there is a competition on the right-hand side of equation (1.9) between two possible scenarios:

• • The collective contribution of edges in $\mathcal {D}$ is the main part of the right-hand side. In this case, we expect the derivative at $p_c$ to exist and to be of order $R^2\Delta _{p_c}(R)$ . Moreover, it may be proved in this case that the derivative is stable within the critical window.

• • The collective contribution of edges far from $\mathcal {D}$ is the main part of the right-hand side. In this case, the derivative at $p_c$ is infinite. For $p\ne p_c$ , the contribution comes mostly from edges at distance $L(p)$ , and the derivative is of order $L(p)^2\Delta _p(L(p))^2/\Delta _p(R)$ .

Taking into account estimates from Lemma 5.3 to handle covariances with edges near the boundary, an accurate estimate of the derivative, valid in all scenarios, is therefore given by the following statement.

Corollary 1.7. Fix $1<q\le 4$ and $\eta>0$ . For every $p\in (0,1)$ and every $\eta $ -regular quad $(\mathcal {D},a,b,c,d)$ at scale $R\le L(p)$ ,

(1.19) $$ \begin{align} \tfrac{\mathrm{d}}{\mathrm{d}p}\phi_{p}[\mathcal{C}(\mathcal{D})]\asymp R^2\Delta_p(R)+\sum_{\ell=R}^{L(p)}\ell \Delta_p(\ell)\Delta_p(R,\ell), \end{align} $$

where the constants in $\asymp $ depend on $\eta $ .

Looking at this sum formula for the derivative, one sees that whether the derivative is governed by the collective contribution of edges in or close to $\mathcal {D}$ or by that of edges far from $\mathcal {D}$ is related to the way $\ell \Delta _p(\ell )$ decays or not as $\ell $ tends to infinity. This can also be related to whether the specific heat blows up or not at $p_c$ , as will be seen in the next section. Note that this up-to-constant formula unravelled a third possible scenario where each scale contributes the same amount. This scenario happens for the random-cluster model with $q=2$ , in which the derivative blows up logarithmically in $L(p)$ – the logarithmic blowup of the specific heat may be deduced from the explicit form of the free energy [Reference OnsagerOns44].

To derive this corollary, one will need an important result that is reminiscent of the classical claim that the four-arm exponent is strictly smaller than 2 for Bernoulli percolation.

Proposition 1.8 (Lower bound on $\Delta _{p}(r,R)$ )

For every $1<q\le 4$ , there exists $\delta>0$ such that for every $1\le r\le R\le L(p)$ ,

(1.20) $$ \begin{align} \Delta_{p}(r,R)\stackrel{>}{\scriptstyle{\frown}} (r/R)^{2-\delta}. \end{align} $$

While the rest of the paper relies on fairly generic assumptions on the percolation model at hand, the previous proposition harvests a much more specific property of the random-cluster model on $\mathbb Z^2$ , namely the parafermionic observable. For $1\le q<4$ , the result will follow from crossing estimates that were recently obtained in [Reference Duminil-Copin, Manolescu and TassionDMT20] using this observable. These crossing estimates are uniform in boundary conditions and (possibly fractal) domains. The byproduct of the analysis in [Reference Duminil-Copin, Manolescu and TassionDMT20] is that $\pi _4(R)/\pi _4(r)$ is bounded from below by $(r/R)^{2-\delta }$ and therefore, by (iv), so is $\Delta _p(r,R)$ . For $q=4$ , the crossing estimates are not uniform in boundary conditions, and a more specific analysis, also based on the parafermionic observable, must be performed. It is the subject of Section 6.2 in this paper.

1.4 Scaling relations

In continuous phase transitions, natural observables of the model decay algebraically. The behaviour at and near criticality is thus expected to be encoded by various critical exponents $\alpha $ , $\beta $ , $\gamma $ , $\delta $ , $\eta $ , $\nu $ , $\zeta $ , $\iota $ , $\xi _1$ and $\xi _4$ defined as follows (below, $o(1)$ denotes a quantity tending to 0):

$$ \begin{align*} f''(p)&=|p-p_c|^{-\alpha+o(1)}&\text{ as } p\rightarrow p_c,\\ \theta(p)&=(p-p_c)^{\beta+o(1)}&\text{as } p\searrow p_c, \\ \chi(p)&=|p-p_c|^{-\gamma+o(1)}&\text{as } p\rightarrow p_c,\\ \phi^1_{p_c,h}[0\longleftrightarrow\mathfrak g]&=h^{1/\delta+o(1)}&\text{as } h\rightarrow 0,\\ \phi_{p_c}[0\longleftrightarrow x]&=|x|^{-\eta+o(1)}&\text{as } |x|\rightarrow\infty,\\ \pi_1(R)&=R^{-\xi_1+o(1)} &\text{as } R\rightarrow\infty,\\ \xi(p)&=|p-p_c|^{-\nu+o(1)}&\text{as } p\rightarrow p_c,\\ \phi_{p_c}[|\mathsf{C}|\ge n]&=n^{-\zeta+o(1)} &\text{as } n\rightarrow \infty,\\ \Delta_{p_c}(R)&=R^{-\iota+o(1)} &\text{as } R\rightarrow\infty,\\ \pi_4(R)&=R^{-\xi_4+o(1)} &\text{as } R\rightarrow\infty, \end{align*} $$

where all the quantities above were already defined in previous sections, except that $0\longleftrightarrow \mathfrak g$ is the event that 0 is connected to the ghost by an open path, $|\mathsf {C}|$ is the number of vertices in the cluster $\mathsf {C}$ of the origin, and $f(p)$ and $\chi (p)$ are the thermodynamical quantities, respectively called the free-energy and the susceptibility definedFootnote ¹ by

Let us mention that the first equation only applies when $f''(p)$ diverges as p approaches $p_c$ , which is to say that the phase transition is of second order.

These exponents are quantities of central interest in physics and have been the object of many studies. A beautiful prediction (again, see e.g., [Reference Essam and FisherEF63, Reference FisherFis64, Reference WidomWid65]) is that these exponents should depend on each other via scaling relations:

(R1) $$\begin{align} \eta&=2\xi_1, \end{align} $$

(R2) $$\begin{align} \zeta&=\xi_1/(2 - \xi_1), \end{align} $$

(R3) $$\begin{align} \delta &= (2 - \xi_1)/\xi_1, \end{align} $$

(R4) $$\begin{align} \beta&=\nu\xi_1 , \end{align} $$

(R5) $$\begin{align} \gamma&=(2-2\xi_1)\nu, \end{align} $$

(R6) $$\begin{align} \alpha&=2-2\nu. \end{align} $$

An important feature of the relations above is that they are independent of q: the exponents vary from model to model, but not the formulae. Relations (R1–5) were proved for Bernoulli percolation (i.e., cluster-weight $q=1$ ) in a milestone paper by Harry Kesten [Reference KestenKes87] without any of them being computed or, indeed, even shown to exist (see also [Reference NolinNol08, Reference Garban, Pete and SchrammGPS18, Reference Duminil-Copin, Manolescu and TassionDMT20b]). For the random-cluster with $q=2$ , critical exponents were calculated independently [Reference McCoy and WuMW83, Reference Duminil-Copin, Garban and PeteDGP14, Reference Duminil-CopinDum13] and observed to satisfy (R1–6). Let us mention that similar relations should hold in all dimensions that are below the so-called upper-critical dimension (with certain values of $2$ replaced by the dimension d). We refer to a paper by Borgs, Chayes, Kesten and Spencer [Reference Borgs, Chayes, Kesten and SpencerBCKS99] (see also [Reference Borgs, Chayes, Kesten and SpencerBCKS01]) for a discussion of this phenomenon for Bernoulli percolation.

Kesten’s analysis in the case of Bernoulli percolation relied on another scaling relation, sometimes referred to as Kesten’s scaling relation, stating that $\nu (2-\xi _4)=1$ for $q=1$ . It was observed in [Reference Duminil-Copin, Garban and PeteDGP14] that this equality fails for $q=2$ (one can also check this using the table gathering the predicted exponents below). We will show that it should be replaced by the following generalised Kesten’s scaling relation:

(R7) $$\begin{align} \nu(2-\iota)=1. \end{align} $$

Note that the second property of Theorem 1.6 shows that $\xi _4>\iota $ so that $\nu (2-\xi _4)=1$ fails not only at $q=2$ but for every $q>1$ .

Before discussing the main results, let us mention the predicted values for the different exponents. The three first scaling relations enable us to express $\delta $ , $\eta $ and $\zeta $ in terms of $\xi _1$ only. This is particularly interesting since $\xi _1$ is measurable in terms of the scaling limit of interfaces at criticality. The relations given by equations ( R6 ) and ( R7 ) link $\alpha $ , $\nu $ and $\iota $ . This is again very useful since it was noted in the previous section how $\iota $ can be obtained from the understanding of the scaling limit of interfaces. An alternative approach to computing these three exponents would be to first obtain $\alpha $ , which may perhaps be derived using exact integrability of the random-cluster model; see [Reference BaxterBax89, Section 12.8] and Section 1.6 for more details. Finally, equations ( R4 ) and ( R5 ) express $\beta $ and $\gamma $ in terms of $\xi _1$ and $\nu $ so that one can obtain all the exponents from $\xi _1$ and $\iota $ (or $\alpha $ ).

Conformal invariance enables us to predict that the scaling limit of the random-cluster model with cluster-weight $q\in [0,4]$ is related to CLE $(\kappa )$ (see [Reference Schramm, Sheffield and WilsonSSW09] and the discussion in [Reference Garban and WuGW19]), from which $\xi _1$ and $\iota $ can be deduced. This leads to the following table, where all the exponents are expressed in terms of $\kappa $ .

In this paper, we prove (R1–7) for the random-cluster model with general cluster-weights $q\in [1,4]$ , except for equation ( R6 ) when $\alpha $ is negative. We insist on the fact that the random-cluster models belong to different universality classes when q varies from $q=1$ to $q=4$ , so that this paper provides the first generic derivation of these relations for different universality classes. As in [Reference KestenKes87], we do not claim to show that any of these exponents exist, nor do we compute their values; the actual statements of the scaling relations with no reference to the exponents are given in the following three theorems. Nonetheless, if one makes the assumption of algebraic decay with the proper exponent, the next statements imply the scaling relations mentioned above.

Below, we assume that $1<q\le 4$ as the case $q=1$ is already known. We start with the two simplest scaling relations given by equations ( R1 ) and ( R2 ) involving only quantities at $p=p_c$ and $h=0$ . The theorem is an easy consequence of uniform crossing estimates obtained for the random-cluster at criticality; see, for example, [Reference Duminil-Copin, Sidoravicius and TassionDST17]. While the result is not especially complicated, we chose to include it here for completeness. Introduce the following quantity for every $n>0$ ,

(1.21) $$ \begin{align} \varphi(n):=\min\{r \in \mathbb{N} :\, r^2\pi_1(r)\ge n\}. \end{align} $$

Theorem 1.9 (Scaling relations at criticality)

Fix $1 < q\le 4$ . For $x\in \mathbb {Z}^2$ and $n\ge 1$ ,

(1.22) $$ \begin{align} \phi_{p_c}[0\longleftrightarrow x]&\asymp \pi_1(|x|)^2, \end{align} $$

(1.23) $$ \begin{align} \phi_{p_c}[|\mathsf{C}|\ge n]&\asymp \pi_1(\varphi(n)). \end{align} $$

We now turn to the scaling relation given by equation ( R3 ) involving the magnetic field. For $q=2$ , equation ( R3 ) was proved in [Reference Camia, Garban and NewmanCGN14] using the GKS inequality, but this inequality is not available for general random-cluster models. A fact that came as a surprise to us is that equation ( R3 ) can be derived for every $1\le q\le 4$ without referring to any other result of the paper (see Section 8.3).

Theorem 1.10 (Scaling relation with magnetic field)

Fix $1 < q\le 4$ . For $h>0$ ,

(1.24) $$ \begin{align} \phi_{p_c,h}[0\longleftrightarrow \mathfrak g]&\asymp \pi_1(\varphi(\tfrac1h)). \end{align} $$

The scaling relations in equations R4 – R7 are the most difficult as they involve the random-cluster model at p near $p_c$ and rely heavily on the stability in the near-critical regime.

Theorem 1.11 (Scaling relations near-critical regime)

Fix $1< q \le 4$ . For $p>p_c$ (and $p\ne p_c$ for the second),

(1.25) $$ \begin{align} \theta(p)&\asymp\pi_1(\xi(p)), \end{align} $$

(1.26) $$ \begin{align} \chi(p)&\asymp\xi(p)^2\pi_1(\xi(p))^2, \end{align} $$

(1.27) $$ \begin{align} \Delta_{p_c}(\xi(p))&\asymp \xi(p)^{-2}|p-p_c|^{-1}, \end{align} $$

(1.28) $$ \begin{align} f''(p)&\asymp\sum_{\ell\le \xi(p)}\ell\Delta_{p_c}(\ell)^2. \end{align} $$

Note that assuming that $\iota $ exists, we get very different behaviour depending on whether it is smaller or larger than $1$ or, correspondingly, whether $\alpha $ is positive or negative: that is, whether the random-cluster model undergoes a second-order or higher-order phase transition. The former occurs when $\nu <1$ : that is, conjecturally when $q\in (2,4]$ . When $\nu =1$ , which is conjectured to correspond only at $q=2$ , all the exponents are known and $f''(p)$ blows up logarithmically (in particular, it satisfies the scaling relation as well). When $\nu>1$ , $f''(p)$ should remain bounded in the vicinity of $p_c$ , and the phase transition becomes of third order (or higher). The exponent $\alpha $ may still be defined using the third derivative of f, which is supposed to diverge at $p_c$ . We are currently only able to derive an upper bound on $f'''$ ; the lower bound is unavailable even for Bernoulli percolation. We refer to Remark 8.4 for details.

1.5 Three complementary results on the Potts models

This section gathers three satellite results that are of interest on their own and do not necessarily fit into the story line of the previous sections. For $q=2$ , it is already known that the value of $\xi _1$ is equal to $1/8$ , so our paper provides a new proof of the following immediate corollary using the Edwards-Sokal coupling [Reference GrimmettGri06, Section 1.4]. We include it since the Ising model with a magnetic field, contrary to the case $h= 0$ , is not integrable and hence is notoriously difficult to study. As mentioned previously, it was obtained in [Reference Camia, Garban and NewmanCGN14] using alternative arguments.

Theorem 1.12. Let $m(\beta ,h)$ be the spontaneous magnetisation of the Ising model on $\mathbb {Z}^2$ at inverse-temperature $\beta $ and magnetic field h. For every $h\in (0,1)$ ,

(1.29) $$ \begin{align} m(\beta_c,h)\asymp h^{1/15}. \end{align} $$

Let us mention here that for $q=2$ , we can derive from the estimate $\xi (p)\asymp |p-p_c|^{-1}$ (this result was obtained numerous times; see [Reference Beffara and Duminil-CopinBD12b] for a short proof) the following result.

Theorem 1.13. For $q=2$ , we have that for every p,

(1.30) $$ \begin{align} L(p)&\asymp |p-p_c|^{-1} \end{align} $$

and for every $R\le L(p)$ and e and f two edges at a distance R from each other,

(1.31) $$ \begin{align} &\Delta_p(R)\asymp R^{-1},\quad\ \,\end{align} $$

(1.32) $$ \begin{align} \mathrm{Cov}_p(\omega_e,\omega_f)&\asymp \Delta_{p_c}(R)^2\asymp R^{-2}. \end{align} $$

In particular, we deduce that

(1.33) $$ \begin{align} f''(p)&\asymp \log\big(\tfrac1{|p-p_c|}\big). \end{align} $$

Of course, the last estimate can be obtained by differentiating the exact formula for the free energy due to [Reference OnsagerOns44]. To the best of our knowledge, the other estimates are new.

When $1\le q\le 3$ , it was proved in [Reference Duminil-Copin, Manolescu and TassionDMT20] that $\pi _1(R)\pi _4(R)\ge cR^{c-2}$ for every $R\ge 1$ and some constant $c> 0$ . In Remark 6.8, we show that $\pi _1(R)\Delta _{p_c}(R)\ge cR^{c-2}$ also for $q=4$ . From these inequalities, using equations (1.25), (1.27) and (1.20), one may deduce the result below, which should be understood as $\beta < 1$ for $1\le q\le 3$ and $q=4$ . The result for $q = 1$ (that is, for Bernoulli percolation) was already obtained by Kesten and Zhang [Reference Kesten and ZhangKZ87]; we expect $\beta < 1$ to be valid for all $1 \le q \le 4$ .

Theorem 1.14 (Nondifferentiability of the order parameter)

For every $1\le q\le 3$ or $q=4$ , there exists $c>0$ such that for every $p\ge p_c$ ,

(1.34) $$ \begin{align} \theta(p)&\ge c(p-p_c)^{1-c}. \end{align} $$

In particular, the spontaneous magnetisation $m(\beta )$ of the 2-, 3- and 4-state Potts models satisfies $m(\beta )\ge c(\beta -\beta _c)^{1-c}$ for $\beta \ge \beta _c$ .

1.6 Open questions

The present paper opens many doors in the study of the critical regime of random-cluster models (and, more generally, planar-dependent percolation models). We now mention a few open questions that in our opinion deserve attention. We refrain from asking the obvious question of proving conformal invariance of the model and focus on questions directly related to the current work.

Let us start with a question concerning scaling relations: whether one can prove equation ( R6 ) when $q\le 2$ . As mentioned above, in this case $f''(p)$ is expected to remain bounded when p tends to $p_c$ , but one may consider the behaviour of $f'''(p)$ to make sense of $\alpha $ . Remark 8.4 of the present paper shows that the critical exponent $\alpha $ defined like this satisfies $2\nu \le 2-\alpha $ , leaving the following question open (note that this is also open for $q=1$ ).

Another natural question is to derive critical exponents for random-cluster models. The scaling relations enable us to deduce certain exponents from others, and we may therefore choose which exponents to try to derive. From this point of view, the exponent $\alpha $ is particularly tempting since it directly implies $\nu $ and also since exact integrability often provides physicists and mathematicians with closed formulae that may lead to $\alpha $ . We refer to [Reference BaxterBax89] for more details on this and summarise the discussion in the following question.

Another approach consists in deriving the exponents having as a basis the assumption of conformal invariance. As already discussed, the scaling limit of the family of boundaries of clusters should then be described by a Conformal Loop Ensemble. The parameter of the CLE may be identified via crossing probabilities and self-duality as the unique $\kappa \geq 4$ that satisfies $\sqrt q = - 2 \cos (4\pi /\kappa )$ [Reference Miller and WernerMilWer18].

Due to the present paper, $\xi _1$ and $\iota $ are sufficient to derive the other exponents. Moreover, they both should be computable using the scaling limit of the critical model, while the exponents $\beta $ , $\gamma $ , $\delta $ and $\nu $ involve values of $p\ne p_c$ and should not therefore be accessible directly from the critical scaling limit. The question of deriving $\xi _1$ assuming the convergence to CLE of the interfaces has already been discussed in the literature [Reference Lawler, Schramm and WernerLSW02]. In light of the quasi-multiplicativity property of $\Delta _{p_c}(r,R)$ , the following question seems tractable.

Let us finish this section by mentioning that [Reference Duminil-Copin, Garban and PeteDGP14] emphasises a self-organised mechanism in the way new edges occur as p increases in Grimmett’s monotone coupling (see [Reference GrimmettGri06, Reference Duminil-Copin, Garban and PeteDGP14] for details). The authors argued that edges appear in clouds and that the understanding of these clouds would be crucial towards the construction of the near-critical scaling limit; would anybody manage to construct the conformally invariant scaling limit at $p_c$ ? The current work answers a number of questions and conjectures asked in this paper (including Conjectures 4.1 and 4.2 since $\xi (p)$ is explicitly known; see, for example, [Reference Beffara and Duminil-CopinBD12b] and references therein) but does not provide direct insight on the structure of these clouds. We therefore include the following question.

Almost everything is known about the random-cluster model with $q=2$ on the square lattice, since the conformal invariance of the model and its interfaces was proved [Reference SmirnovSmi10, Reference Chelkak, Duminil-Copin, Hongler, Kemppainen and SmirnovCDHKS14]. It is therefore only natural to discuss the question of the construction of the near-critical scaling limit in this context, especially since one expects subtle differences with the corresponding result for Bernoulli percolation (see [Reference Garban, Pete and SchrammGPS18]).

Recently, the rotational invariance of the critical random-cluster model was obtained in [Reference Duminil-Copin, Kozlowski, Krachun, Manolescu and OulamaraDKKMO20]. This rotational invariance is expected to carry over to the near-critical regime. The arguments developed here, combined with those of [Reference Duminil-Copin, Garban and PeteDGP14], should be relevant for the next question.

Organisation of the paper

Section 2 provides the necessary background to our paper. Section 3 studies the dependency of crossing probabilities on boundary conditions (see Theorem 3.6) and introduces the notion of boosting pairs of boundary conditions. Section 4 contains the proof of points (ii), (iv) and (v) of Theorem 1.6. This is the core of our paper and indeed its biggest innovation. Section 5 initiates the connection between the quantity $\Delta _p$ and covariances, in particular proving Theorem 1.6(i). Section 6 shows the lower bound on $\Delta _p$ given by Proposition 1.8. Section 7 contains the proof of the stability below the correlation length: Theorem 1.4 and Theorem 1.6(iii). Finally, Section 8 contains the derivation of the scaling relations.

A word about constants

We will often work with $p\in (0,1)$ and a spatial scale $R \leq L(p)$ . Unless stated otherwise, constants $c, (c_i)_{i\ge 0}, C$ and $(C_i)_{i\ge 0}$ are assumed uniform in $(p;R)$ as above, with the assumption that p is not close to $0$ or $1$ . They are, however, allowed to depend on the threshold $\delta _L$ used in the definition of $L(p)$ ; recall that this threshold is assumed small but fixed. We do not discuss the dependence in q of constants, but the careful reader will notice that they may be rendered uniform in q, potentially outside the vicinity of $1$ and $4$ .

We reiterate that the constants in the notation $\asymp $ , $\stackrel{<}{\scriptstyle{\frown}}$ and $\stackrel{>}{\scriptstyle{\frown}}$ also follow the same principle.