Excursion decomposition of the XOR-Ising model

TL;DR

This work constructs a continuum excursion decomposition for the two-dimensional critical XOR-Ising model by identifying the XOR-Ising field with a trigonometric function of a Gaussian free field and exploiting two-valued level sets and imaginary chaos. It then proves that the discrete XOR-Ising decomposition on the square lattice converges, under the double random current representation and a master coupling, to the continuum decomposition, including joint convergence with the cosine and sine Wick chaos. The authors develop a robust framework handling subcritical and critical two-valued sets, establish thinness results, and prove local finiteness and measurability properties essential for the decomposition. They also discuss a natural generalization to the Ashkin-Teller polarisation, conjecturing continuum excursion decompositions along the AT critical line with a parameter $\alpha(U)$. Overall, the paper provides a rigorous bridge between discrete XOR-Ising geometry and continuum GFF-based excursion decompositions, with potential implications for AT models and related bosonisation frameworks.

Abstract

We study the excursion decomposition of the two-dimensional critical XOR-Ising model with either $+$ or free boundary conditions. In the first part, we construct the decomposition directly in the continuum. This construction relies on the identification of the XOR-Ising field with the cosine or sine of a Gaussian free field (GFF) $φ$ multiplied by $α= 1/\sqrt{2}$, and is obtained by an appropriate exploration of two-valued level sets of the GFF. More generally, the same construction applies to the fields $:\! \cos(αφ) \!:$ and $:\! \sin(αφ) \!:$ for any $α\in (0,1)$. In the second part, we show that the continuum excursion decomposition arises as the scaling limit of the double random current decomposition of the critical XOR-Ising model on the square lattice. To this end, we exploit the rich Markovian structure of the discrete decomposition and strengthen the convergence of the double random current height function to the continuum GFF by establishing joint convergence with its cosine and sine. We conjecture that for $α\in [1/2,\sqrt{3}/2)$ the continuum excursion decompositions arise as the scaling limit of those of the Ashkin-Teller polarisation field along its critical line.

Figures (7)

• Figure 1: Excursions for the XOR-Ising model, with free/free and $+/+$ boundary conditions respectively, shaded in gray. Iterations are only showed inside certain loops, but should be done within all loops.
• Figure 2: Schematic diagram of the excursion decomposition for $:\! \sin(\alpha\phi) \!:$. Within all loops of macroscopic diameter (i.e. bigger than some fixed $\rho>0$), we iterate the same procedure.
• Figure 3: Schematic diagram of the excursion decomposition for $:\! \cos(\alpha\phi) \!:$. Within all loops of macroscopic diameter (i.e. bigger than some fixed $\rho>0$), we iterate the procedure in Figure \ref{['fig:sin-CLE-decomp']}.
• Figure 4: The $\varepsilon$-grid in black, where the interior of each red square marks the region in which $f_\delta$ is supported. The loops of $\mathds A_{-2\lambda, 2\lambda}$ are drawn in blue. The dashed loops are those not in $\mathds A_{-2\lambda, 2\lambda}^\varepsilon$, thus they all have diameter smaller than $\sqrt{2}\varepsilon$. The solid loops can have any diameter, but those intersecting the support of $f_\delta$, marked in bold, must have diameter greater than $\delta$.
• Figure 5: Diagram of the restriction property used in the proof of Lemma \ref{['lem:uniform_Bound_diam']}. The unshaded region corresponds to a realization of $D^*$ when $D=(1-\delta)\mathbb D$ and $D'=\mathbb D$. The red loops correspond to loops of diameter smaller than $\rho$. The collection of loops satisfies the (positive probability) event that all loops of $\mathrm{CLE}_4(\mathbb D)$ with diameter greater than $\rho$ are contained in $(1-\delta)\mathbb D$.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: spin-perc-height, DRC-1
• Theorem 1.4
• Corollary 1.5
• Remark 1.6
• Conjecture 1.7
• Remark 2.1
• Definition 2.2: Local sets
• Definition 2.3: Two-valued sets