Near-critical Ornstein--Zernike theory for the planar random-cluster model

TL;DR

This work develops a near-critical Ornstein--Zernike theory for the two-dimensional FK/random-cluster model with $1\le q<4$, producing a uniform asymptotic for the two-point function in the subcritical regime that blends subcritical decay with near-critical prefactors. The core strategy is a dynamical, correlation-length-scale exploration of the cluster in a fixed direction, reformulated as a killed Markov renewal process whose mass-gap and renewal structure yield OZ-type formulas, strict convexity of the inverse correlation length, and invariance principles. Key contributions include a uniform OZ formula $\phi_p[0\leftrightarrow \lfloor n\vec v\rfloor] \asymp \pi_1(\xi_p(\vec v))^2 \sqrt{\xi_p(\vec v) n} e^{-n\xi_p(\vec v)}$, along with corollaries on the pure-exponential survival, endpoint concentration, and strict Wulff-shape convexity, all established at the correlation-length scale. The approach departs from skeleton/diamond decompositions, employing a direct, cone-mominated exploration that yields a robust, flexible framework for near-critical OZ analysis in two dimensions.

Abstract

We develop an Ornstein--Zernike theory for the two-dimensional random-cluster model with $1 \leq q <4$ that also applies in its near-critical regime. In particular, we prove an asymptotic formula for the two-point function which holds uniformly for~$p < p_c$ and blends the subcritical and near-critical behaviours of the model. The analysis is carried out by studying the renewal properties of a subcritical percolation cluster, \emph{at the scale of the correlation length}. More precisely, we explore sequentially the cluster in a given direction, by slices of thickness comparable to the correlation length. We show that this exploration satisfies the properties of a {\em killed Markov renewal process} -- a class of processes that may be analysed independently and have Brownian behaviour. In addition to the two-point function estimate, we derive other consequences of the Ornstein--Zernike theory such as an invariance principle for the rescaled cluster and the strict convexity of the inverse correlation length -- all at the scale of the correlation length, uniformly in~$p<p_c$. Finally, our approach differs from that of earlier papers of Campanino, Ioffe, Velenik and others, with the cluster being dynamically explored rather than constructed from its diamond decomposition.

Figures (9)

• Figure 1: Left: in the proof of \ref{['eq:u_k_rec']} we consider two translates $H_+$ and $H_-$ of the event in $u_k$, one in the upper half-plane, one in the lower one. Here the red path ensures that $H_+$ occurs. The blue paths (also part of $\omega^* \cup A^*$) produce the events $B_L$ and $B_R$. When at least one of $H_+$ and $H_-$ occur, and both $B_L$ and $B_R$, then the two horizontal translates of $\Lambda_{N/2}$ connect to each-other. Right: To produce a path separating $\Lambda_{2^k L}$ from $\partial \Lambda_{2^{k+1} L}$, it suffices for two vertical translates of the events in $u_{k-1}$ to occur, and for the blue paths to occur in $\omega^* \cup A^*$. As for $B_L$ and $B_R$, the latter occur with exponentially high probability in $2^k$.
• Figure 2: Left: the dotted line represents the points at equal distance from $\mathcal{D}$ and $\partial \mathcal{H}_\geq 0$; the solid black lines represent wired boundary conditions. The blue segment is the top half of a batch $S_k$. For it to be connected to $\mathcal{D}$, the annulus surrounding it should be crossed, which occurs with an exponentially small probability in $k$. For small values of $k$, use \ref{['eq:RSWnc']} to complete the dual path separating $\partial\mathcal{H}_{\geq 0}$ form $\mathcal{D}$. Right: For the second part of the argument, consider a domain $\Omega$ containing $\mathcal{D}$ and prove that the probability of $B$ with arbitrary boundary conditions on $\Omega$ is not much larger than that with free boundary conditions --- we use here the first estimate of \ref{['eq:cone_to_cone']}. One bound of \ref{['eq:mixing_bc']} follows directly. The opposite bound is proved by considering $\Omega$ to be the complement of the cluster of $\partial \mathcal{H}_{\geq 0}$ and using the second bound of \ref{['eq:cone_to_cone']} to show that $\mathcal{D} \subset \Omega$ with positive probability.
• Figure 3: Left: The occurence of $S_a$ conditionally on $\Lambda_{ L } \leftrightarrow \mathcal{H}_{\geq n}$ induces an "unforced" connection between $\mathcal{Y}_{\rm in}$ and $\mathcal{Y}_{\rm out}^c$, which has an exponential cost in $k$. Middle: To bound $S_b$, consider the right-most translate of $\mathcal{Y}_{\rm in}$ that contains a connection between $\Lambda_L$ and $\mathcal{H}_{\geq n}$. Its boundary intersects a "pivotal" box $\Lambda_L(x)$, that is a box that is connected to $\Lambda_L$ and $\mathcal{H}_{\geq n}$ inside $\mathcal{Y}(x)$ by distinct clusters. Right: When working with an arbitrary potential past $A$, the connection between $\Lambda_L$ and $\mathcal{H}_{\geq n}$ may use the left half-plane. Nevertheless, there exists a vertical translate of $\Lambda_L$ that is connected to $\mathcal{H}_{\geq n}$ inside $\mathcal{H}_{\geq 0}$. We will consider the closest such translate to $0$.
• Figure 4: Left: An example of a successful exploration (purple), with the arcs $\chi_0$ and $\chi_1$ being the top and bottom sides of the interface and the tubes depicted in grey. The solid blue lines form $\mathcal{A}$, while $\mathcal{B}$ is the solid black segment above $y_-$. An exploration is needed when either $\mathcal{A}$ or the dual arc between $x_+$ and $y_-$ are inaccessible. Right: For a successful exploration, $\chi_1$ may be connected to $\mathcal{H}_{\geq n}$ by a primal cluster that does not intersect $\mathcal{B}$ with probability at least $s_{n-t}$. After exploring the cluster $\mathsf C_{\chi}$ of $\chi_1$ (red) $\mathcal{B}$ may be separated from $\mathcal{H}_{\geq n}$ with uniformly positive probability due to our assumption on $\mathsf C_{\leq t}$.
• Figure 5: The different scenarios that may occur when exploring $\Gamma$, and potentially $\tilde{\Gamma}$. The exploration up to $\tau$ and the resulting connection is drawn in red, the subsequent construction is in purple. The red and purple arrow refer connections to the paths from $\Lambda_{R}(x + (5R,\pm R))$ to $\mathcal{H}_{\geq n}$ and $\infty$, which occur with probability comparable to $s_{n-t}$. From left to right: scenarios 1(a), 2(a), 2(b) and 3. Note that in scenario 2(b), the exploration path $\Gamma$ never exposes an accessible primal arc. That is why we do not explore $\Gamma$ between $\tau$ and $\tau'$ (grey) but rather explore $\tilde{\Gamma}$ up to $\tau"$ (purple).

Theorems & Definitions (48)

• Theorem 1.1: Ornstein--Zernike asymptotics at the scale of the correlation length
• Remark 1.2
• Lemma 2.1
• Proposition 2.2: RSW in the critical window
• Theorem 2.3
• proof : Proof of Lemma \ref{['lem:L(p)expdecay']}
• Definition 3.1
• Theorem 3.2
• Remark 3.3
• Remark 3.4