Supercritical sharpness for the random cluster representation of real-valued spin models

Abstract: We study the supercritical regime $β&gt;β_c$ of a large family of real-valued spin models on $\mathbb Z<sup>d$, including the Blume-Capel and $P(\varphi)$ models. We consider their random cluster representations and prove that they are well-behaved, in the sense that local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This implies, among other things, a surface-order exponential bound for the (lower) large deviations of the empirical magnetisation. These results were previously known only in the cases of the Ising and $\varphi<sup>4$ models.

• The paper proves uniform local uniqueness throughout the supercritical phase for admissible real-valued spin models in every dimension d≥2, using Burton–Keane arguments, half-space measure uniqueness, and surface-tension estimates.
• The method replaces model-specific Lee–Yang and random-current techniques with a general Lebowitz-inequality approach, yielding disconnection probabilities that decay exponentially in surface area and renormalized percolation close to one.
• The results support surface-order magnetization large deviations, plus-state stability under weak positive boundary conditions, and surface-order spectral-gap bounds outside a countable exceptional set of parameters.

Setting and scope

The paper establishes supercritical sharpness for the random cluster (Fortuin–Kasteleyn) representation of a broad family of real-valued spin systems on $\mathbb Z^d$, $d\ge 2$. The class is indexed by admissible single-site measures $\rho$ on $\mathbb R$: even, super-Gaussian ($\int e^{au^2}\,d\rho<\infty$ for all $a>0$, which is necessary and sufficient for the partition function to be finite), with non-trivial support. This includes the Ising model, the Blume–Capel model, the $\varphi^4$ model, general Griffiths–Simon distributions, and lattice $P(\varphi)$ models with even polynomial potentials of degree at least four. The Gaussian free field is explicitly excluded. The critical point is defined via spontaneous magnetisation $m^*(\beta)=\langle\tau_0\rangle_\beta^+$; it is known that $\beta_c\in(0,\infty)$ for all these models in every dimension $d\ge 2$0.

Prior to this work, supercritical sharpness of the random cluster representation was known only for the Ising model (Bodineau) and the $d\ge 2$1 model, where the proofs rely on either the Lee–Yang theorem or the random (tangled) current representation — tools unavailable beyond those models. The present paper replaces them by a soft percolation argument in the spirit of Burton–Keane, requiring essentially only the Lebowitz inequality.

Main result

The central theorem states that for every admissible $d\ge 2$2, every $d\ge 2$3, and every $d\ge 2$4,

$d\ge 2$5

where $d\ge 2$6 is the local uniqueness event: some cluster crosses the annulus $d\ge 2$7, and any two crossing paths of the inner annulus are connected within it, uniformly over boundary conditions. Via standard coarse-graining (the Liggett–Schonmann–Stacey domination argument), this implies that the renormalised process stochastically dominates Bernoulli site percolation with parameter arbitrarily close to one, so qualitative behaviour throughout the supercritical phase matches that of a strongly perturbative regime.

Two structural results feed into applications. First, the set $d\ge 2$8 of non-uniqueness of infinite-volume random cluster measures is at most countable — proved by a Lebowitz–Martin-Löf style energy-concentration argument comparing plus and free two-point functions across parameters. Second, under uniqueness (i.e., for $d\ge 2$9), the authors obtain:

• A surface-order large deviation bound for the lower tail of empirical magnetisation: for $\rho$0, $\rho$1 is sandwiched between $\rho$2 and $\rho$3. The upper tail is always volume-order. This bound also yields surface-order spectral gap estimates for Glauber dynamics.
• A characterisation of the plus state: moderately positive boundary conditions $\rho$4 with $\rho$5 converge weakly to $\rho$6. The lower threshold is sharp, since $\rho$7 converges instead to the free measure.

These applications are stated without full proofs, following verbatim the corresponding arguments from the earlier $\rho$8 paper.

Uniqueness of half-space measures

The main methodological innovation is an almost-everywhere uniqueness result for half-space random cluster measures: for almost every $\rho$9, $\mathbb R$0, where $\mathbb R$1 is a constant magnetic field on the boundary plane. For Ising and $\mathbb R$2 the analogous statement holds for every $\mathbb R$3; here only almost-everywhere uniqueness is obtained, which suffices because the subsequent differentiation argument integrates over $\mathbb R$4.

The proof strategy adapts Lebowitz–Martin-Löf to the half-space, where translation symmetry is broken: changing $\mathbb R$5 on the orbit of an edge under the half-space automorphism group now costs only surface order, but so does changing boundary conditions, and the two effects must be disentangled. The key idea is to work under the Edwards–Sokal coupling with boundary conditions induced on thin hyperrectangles $\mathbb R$6 by the full-space wired and free measures, using that $\mathbb R$7 when $\mathbb R$8. Two approximation lemmas show that, up to $\mathbb R$9 errors: (i) induced bulk absolute values agree under the monotone coupling, and (ii) induced bulk partitions agree in terms of cluster counts. Point (ii) is non-obvious since the induced partition is a global object; it is established by a Burton–Keane-type argument showing that, far from the boundary, the number of clusters connecting a small plate to the top of a box is at most two with high probability, via a trifurcation counting argument combined with a spanning-forest surgery adapted to the half-space geometry.

A technical remark notes that both uniqueness proofs adapt directly to the standard random cluster model with cluster weight $\int e^{au^2}\,d\rho<\infty$0, where they simplify considerably.

Surface tension and positivity

For unbounded spins there is no canonical maximal finite-volume measure, so the authors construct finite-volume plus measures using a random positive boundary condition: a measure $\int e^{au^2}\,d\rho<\infty$1 supported on $\int e^{au^2}\,d\rho<\infty$2 with all Gaussian moments, chosen once and for all on compact parameter ranges. These measures converge to $\int e^{au^2}\,d\rho<\infty$3 and remain $\int e^{au^2}\,d\rho<\infty$4-regular up to the boundary, and satisfy FKG, monotonicity, and volume-monotonicity properties.

The surface tension is defined through partition functions on vertical strips $\int e^{au^2}\,d\rho<\infty$5 with $\int e^{au^2}\,d\rho<\infty$6 versus $\int e^{au^2}\,d\rho<\infty$7 boundary conditions, and equals (via an Edwards–Sokal computation) minus the log-probability of disconnection under the wired strip measure. Strict positivity follows from a differentiation argument: the derivative of the surface tension is bounded below by a sum of magnetisation squares, using the Lebowitz inequality

$\int e^{au^2}\,d\rho<\infty$8

proved here in full generality for inhomogeneous single-site measures that are even or supported on $\int e^{au^2}\,d\rho<\infty$9, by rotating a duplicated system and reducing to Griffiths' first inequality for an associated ferromagnetic Ising system on $a>0$0. The result is the quantitative bound $a>0$1, which translates into exponential decay of the wired disconnection probability: crossings of the infinite strip occur with probability at least $a>0$2.

From wired to free boundary conditions

The final step transfers the crossing estimate from wired to free measures. With a large magnetic field $a>0$3 on the top and bottom faces, free-measure crossings hold with probability $a>0$4 (via regularity, FKG, and edge-closing surgeries whose cost is controlled by choosing the strip aspect ratio small). Removing the boundary field uses a Russo-type logarithmic derivative in $a>0$5: the bulk contribution vanishes precisely because the half-space wired and free measures coincide for almost every $a>0$6 — this is where Theorem on half-space uniqueness enters — while the corner contribution is made arbitrarily small by restricting to points far from lateral boundaries. Integrating in $a>0$7 gives free-boundary crossings with probability $a>0$8, whence the disconnection bound $a>0$9 for $\varphi^4$0, completing the proof of sharpness. Local uniqueness then follows from RSW theory in $\varphi^4$1 and from slab percolation plus sprinkling and onion-peeling in $\varphi^4$2.

Limitations and open questions

Several restrictions are explicit. The super-Gaussian assumption is necessary for well-definedness but excludes the Gaussian free field. The applications (surface-order LDP, plus-state characterisation, dynamics) require $\varphi^4$3; uniqueness is only known to fail on a countable set, and the authors conjecture that $\varphi^4$4, with $\varphi^4$5 if and only if the transition is continuous — known for the Blume–Capel model on $\varphi^4$6, open in general. Half-space uniqueness holds only for almost every $\varphi^4$7 rather than everywhere, leaving open whether the Lee–Yang-free methods can be strengthened. Quantitative exponential mixing (Conjecture on mixing), which together with the main result would imply exponential decay of truncated correlations in the whole supercritical phase, remains open outside the Ising model; its resolution is identified as the remaining obstacle to completing the low-temperature sharpness programme for this class. Continuity of the phase transition for the Ising and $\varphi^4$8 models on transitive graphs beyond $\varphi^4$9 also remains unresolved.

Conclusion

The paper extends supercritical sharpness of random cluster representations from the Ising and $P(\varphi)$0 models to all real-valued spin systems with $P(\varphi)$1 symmetry satisfying a super-Gaussian moment condition, replacing model-specific analytic tools (Lee–Yang, random currents) by a probabilistic Burton–Keane-type uniqueness argument for half-space measures, combined with a Lebowitz-inequality-based positivity proof for the surface tension. The resulting local uniqueness theorem, uniform in boundary conditions, provides the renormalisation backbone for surface-order large deviations of the magnetisation, plus-state stability, and spectral gap bounds throughout the supercritical phase, modulo the countable exceptional set $P(\varphi)$2.