Renormalised two-point functions of CLE$_4$ gaskets

Juhan Aru; Titus Lupu

arxiv: 2604.15146 · v2 · pith:3LLGIJ32new · submitted 2026-04-16 · 🧮 math.PR · math-ph· math.MP

Renormalised two-point functions of CLE₄ gaskets

Juhan Aru , Titus Lupu This is my paper

Pith reviewed 2026-05-10 09:53 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords CLE4Gaussian free fieldAshkin-Teller modelBrownian loop souptwo-point functionsrenormalisationIsing correlations

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The pith

Renormalised probabilities that two points belong to the same or outermost CLE₄ gasket equal the two-point functions of the Ashkin-Teller scaling limit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the renormalised probability that two points lie in the same CLE₄ gasket and the probability they lie in the outermost gasket, for nested CLE₄ in a simply-connected domain. These calculations are performed probabilistically using Brownian loop soups together with the geometry of two-valued sets of the continuum Gaussian free field. The resulting quantities are identified with the two-point correlation functions of the conjectured scaling limit of single spins in the Ashkin-Teller model on the critical line. The same method yields more general renormalised probabilities when CLE₄ gaskets alternate with certain two-valued sets of the Gaussian free field. At the decoupling point the formulas reduce to the known Ising-model correlations.

Core claim

We consider nested CLE₄ in a simply-connected domain and compute the renormalised probabilities that two points belong to the same CLE₄ gasket and that two points belong to the outermost CLE₄ gasket. These quantities correspond to the two-point function of the conjectured scaling limit of the AT single spins on the critical line. At the decoupling point, our results recover the Ising model correlations and suggest a CLE₄-based FK representation of the AT spin model.

What carries the argument

The renormalised two-point probability for two points to belong to the same CLE₄ gasket (or the outermost one), obtained from the geometry of nested CLE₄ loops and the two-valued sets of the Gaussian free field via Brownian loop soups.

If this is right

The explicit formulas supply the two-point correlations for the scaling limit of the Ashkin-Teller model.
At the decoupling point the formulas coincide with the Ising-model two-point function.
The construction furnishes a CLE₄-based loop representation for the Ashkin-Teller spin model analogous to the FK representation for Potts models.
The same renormalisation procedure extends immediately to correlations involving alternating gaskets and two-valued sets of the Gaussian free field.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same loop-soup and GFF machinery could be used to extract higher-point functions or other observables in the Ashkin-Teller model.
Lattice simulations of CLE₄ via Brownian loop soups could furnish numerical checks of the derived formulas.
The explicit connection to two-valued sets of the Gaussian free field opens a route to analogous calculations in other models whose interfaces are described by level lines of the GFF.

Load-bearing premise

The computed renormalised probabilities from CLE₄ gaskets and GFF two-valued sets exactly equal the two-point functions of the conjectured scaling limit of Ashkin-Teller single spins.

What would settle it

A direct numerical computation of spin-spin correlations in the Ashkin-Teller model on a fine lattice at the critical point, compared against the explicit formulas obtained from the CLE₄ construction.

read the original abstract

We first consider nested CLE$_4$ in a simply-connected domain and compute some exact renormalised probabilities: the probability that two points belong to the same CLE$_4$ gasket and the probability that two points belong to the outermost CLE$_4$ gasket. The resulting conformally covariant formulas have a non-trivial modular dependence, expressed explicitly in terms of Jacobi theta functions. We are guided by the conformal field theory of the Ashkin-Teller (AT) model, but our proofs are purely probabilistic using Brownian loop soups, gauge-twisted Gaussian free fields (GFFs) and the level line geometry of the 2D continuum GFF. More generally, we also calculate renormalised probabilities that two points belong to CLE$_4$ gaskets sampled in alternation with certain two-valued sets of the Gaussian free field. These quantities correspond to the two-point function of the conjectured scaling limit of the AT single spins on the critical line. At the decoupling point, our results recover the Ising model correlations; they also suggest a CLE$_4$-based continuum FK representation of the AT single-spin fields along a whole segment of the critical line.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Aru and Lupu compute renormalized CLE4 gasket probabilities probabilistically via loop soups and GFF, recovering Ising at decoupling, but tie the AT spin interpretation to an unproven scaling conjecture.

read the letter

Aru and Lupu compute the renormalized probabilities that two points lie in the same CLE4 gasket or in the outermost gasket for nested CLE4, using Brownian loop soups and GFF geometry. They also treat the alternating case with GFF two-valued sets. The calculations are presented as a purely probabilistic alternative to CFT integrability methods, and they recover the known Ising correlations at the decoupling point. This supplies explicit formulas for these specific quantities, which is new in the CLE literature. The probabilistic route is a clear strength, as it avoids heavy reliance on integrability and gives a direct geometric handle on the events. The recovery of Ising serves as a useful consistency check. The soft spot is the central interpretation: the abstract states that these quantities correspond to the two-point functions of the conjectured scaling limit of Ashkin-Teller single spins. The paper does not establish the scaling limit or verify the full functional form for generic AT parameters; the link rests on the conjecture plus the special-case match. Readers can take the CLE4 computations at face value, but the AT connection remains conjectural. This paper is for specialists working on CLE, loop soups, and conformal probability. A reader focused on probabilistic derivations of conformal invariants will find the explicit formulas and the alternative method useful. It deserves serious peer review because the calculations are concrete, the approach is distinct, and the results could support further work even if the broader claim needs additional justification.

Referee Report

1 major / 2 minor

Summary. The manuscript computes renormalised probabilities that two points belong to the same CLE₄ gasket or to the outermost CLE₄ gasket in a simply-connected domain. It employs Brownian loop soups and the geometry of the 2D Gaussian free field to obtain these quantities, and extends the calculation to cases where CLE₄ gaskets are sampled in alternation with certain two-valued sets of the GFF. These probabilities are identified with the two-point functions of the conjectured scaling limit of Ashkin-Teller single spins on the critical line; at the decoupling point the results recover Ising correlations and suggest a CLE₄-based FK representation of the AT spin model.

Significance. If the probabilistic derivations hold, the work supplies explicit expressions for quantities that, under the AT scaling-limit conjecture, yield the two-point functions of AT spins. The purely probabilistic route via loop soups and GFF geometry constitutes an independent calculation that contrasts with CFT integrability methods. Recovery of the known Ising correlations at the decoupling point provides concrete support for the approach and indicates a possible new representation for the AT model.

major comments (1)

[Introduction] Introduction and abstract: The central interpretive claim that the computed renormalised probabilities 'correspond to' the two-point functions of the conjectured scaling limit of AT single spins rests on an unproven conjecture. While the decoupling-point recovery of Ising correlations is verified, the manuscript does not supply a direct comparison of the derived functional forms against known CFT expressions for generic AT parameters; this leaves the strength of the identification dependent on the conjecture without further independent checks.

minor comments (2)

[§3] The renormalisation procedure (cut-off and limiting process) should be stated with an explicit equation in the main text rather than being referenced only in passing, to make the existence of the limits fully transparent.
Notation for the various renormalised probabilities (same-gasket, outermost-gasket, alternated) would benefit from a consolidated table or displayed equation early in the paper to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work, and recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: [Introduction] Introduction and abstract: The central interpretive claim that the computed renormalised probabilities 'correspond to' the two-point functions of the conjectured scaling limit of AT single spins rests on an unproven conjecture. While the decoupling-point recovery of Ising correlations is verified, the manuscript does not supply a direct comparison of the derived functional forms against known CFT expressions for generic AT parameters; this leaves the strength of the identification dependent on the conjecture without further independent checks.

Authors: We agree that the identification of the computed probabilities with the two-point functions of AT single spins is conjectural, as stated explicitly in the abstract and introduction: the quantities 'correspond to the two-point function of the conjectured scaling limit'. The manuscript's primary contribution is the independent probabilistic derivation via loop soups and GFF geometry, which contrasts with CFT integrability methods and recovers the known Ising correlations at the decoupling point as a consistency check. A direct comparison of the explicit functional forms against CFT expressions for generic AT parameters lies outside the scope of this work, since it would require separate derivation of those CFT formulae rather than the probabilistic route pursued here. We will add a clarifying sentence in the introduction to underscore the conjectural nature of the scaling-limit identification while highlighting the value of the probabilistic computation and the decoupling-point verification. revision: partial

Circularity Check

0 steps flagged

Probabilistic derivation of renormalised CLE₄ probabilities is independent of conjectural AT interpretation

full rationale

The paper computes explicit renormalised probabilities for two points belonging to the same or outermost CLE₄ gasket (and alternated with GFF two-valued sets) via Brownian loop soups and GFF geometry. This is explicitly contrasted with CFT methods and presented as a self-contained probabilistic calculation. The correspondence to AT single-spin two-point functions is stated as matching the conjectured scaling limit, with verification only at the decoupling point recovering Ising correlations; no step reduces the derived expressions to fitted parameters, self-citations, or inputs by construction. The central results stand on the probabilistic framework alone.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculations rest on standard properties of CLE4, Brownian loop soups, and the 2D GFF; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

domain assumption Existence and conformal invariance properties of nested CLE4 in simply-connected domains
Invoked as the starting object whose gasket probabilities are to be computed.
domain assumption Geometric correspondence between CLE4 gaskets and two-valued sets of the Gaussian free field
Used to extend the calculation to alternating gaskets.

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