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Conformal Invariance of Double Random Currents and the XOR-Ising Model I: Identification of the Limit

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Conformal Invariance of Double Random Currents and the XOR-Ising Model I: Identification of the Limit Conformal invariance of double random currents and the XOR-Ising model I: identification of the limit Hugo Duminil-Copin∗y Marcin Lisz Wei Qianx July 29, 2021 Abstract This is the first of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising models on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents with free and wired boundary condi- tions, and in the XOR-Ising models with free and plus/plus boundary conditions. There- fore we establish Wilson's conjecture on the XOR-Ising model [81]. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the Gaussian Free Field. In this paper, we identify uniquely the possible subsequential limits of the loop ensembles. Combined with [28], this completes the proof of conformal invariance. 1 Introduction 1.1 Motivation and overview The rigorous understanding of Conformal Field Theory (CFT) and Conformally Invariant random objects via the developments of the Schramm-Loewner Evolution (SLE) and its relations to the Gaussian Free Field (GFF) has progressed greatly in the last twenty-five years. It is fair to say that once a discrete lattice model is proved to be conformally invariant in the scaling limit, most of what mathematical physicists are interested in can be exactly computed using the powerful tools in the continuum. A large class of discrete lattice models are conjectured to have interfaces that converge in the scaling limit to SLEκ type curves for κ (0; 8]. Unfortunately, such convergence results 2 are only proved for a handful of models, including the loop-erased random walk [68] and the uniform spanning tree [49] (corresponding to κ = 2 and 8), the Ising model [19] and its FK representation [77] (corresponding to κ = 3 and 16=3), Bernoulli site percolation on the triangular lattice [76] (corresponding to κ = 6). Known proofs involve a combination of ∗Institut des Hautes Etudes´ Scientifiques yUniversit´ede Gen`eve zUniversit¨atWien xCNRS and Laboratoire de Math´ematiquesd'Orsay, Universit´eParis-Saclay 1 exact integrability1 enabling the computation of certain discrete observables, and of discrete complex analysis to imply the convergence in the scaling limit to holomorphic/harmonic functions satisfying certain boundary value problems that are naturally conformally covariant. To upgrade the result from conformal covariance of these \witness" observables to the convergence of interfaces in the system, one needs an additional ingredient. In some cases, when properties of the discrete models are sufficiently nice (typically tightness of the family of interfaces, mixing type properties, etc), a clever martingale argument introduced by Oded Schramm enables to prove convergence of interfaces to SLEs and CLEs. This last step involves the spatial Markov properties of the discrete model in a crucial fashion. We refer to the proofs of conformal invariance of interfaces in Bernoulli site percolation, the Ising model, the FK Ising model, or the harmonic explorer for examples. Unfortunately, the discrete properties of the model are sometimes not sufficiently nice to implement this martingale argument and there are still many remaining examples for which the scaling limit of the interfaces cannot be easily deduced from the conformal invariance of certain observables { most notably for the case of the double dimer model, for which an important breakthrough was performed by Kenyon in [46], followed by a series of impressive papers [9, 23]. In [81], Wilson discussed conjectures concerning the critical XOR-Ising model obtained by taking the product of two independent Ising models (see below for a formal definition). While he did not provide any proof of conformal invariance in his paper, Wilson performed advanced numerics and made a number of predictions for the behaviour of the interfaces in the model. We point out that unlike the aforementioned models, the XOR-Ising model does not satisfy any obvious spatial Markov property at the discrete level, thus making the classical martingale argument unlikely to offer a convenient road to prove conformal invariance. This makes it even more remarkable that its scaling limit does satisfy the spatial Markov property (see the end of Section 1.2 for a more detailed discussion). In this paper, we prove convergence of the critical XOR-Ising interfaces on the square lattice in the scaling limit to SLE4 type curves. In the process of proving conformal invariance of XOR interfaces, we also obtain similar results for the double random current model. The double random current model also lacks spatial Markov property in the discrete (at the outer boundaries of the clusters: see end of Section 1.3 for a more detailed discussion). Convergence to SLE4 type curves were previously proved for the harmonic explorer [72], contour lines of the discrete GFF [69], and cluster boundaries of a random walk loop-soup with the critical intensity [12,54]. Nevertheless, all these models are discrete approximations of objects defined in the continuum which are already known to be SLE4 type curves. In this respect, the XOR-Ising and the double random current models are the first discrete lattice models not having any a priori connection to SLE4 whose interfaces are proved to converge to SLE4 type curves. As mentioned above, our proof does not follow the martingale strategy. Instead, it relies on a coupling between the XOR Ising model and a naturally associated height function and can be decomposed into three main steps (see the next sections for more details): (i) Proving the joint tightness of the family of critical XOR interfaces and the height function, as well as certain properties of the joint coupling. (ii) Proving convergence of the height function to the GFF. 1Only approximately for site percolation on the triangular lattice. 2 (iii) In the continuum, identifying the scaling limit of XOR interfaces using properties of the GFF. Each of the three previous steps involves quite different branches of probability. The first one extensively uses percolation-type arguments for dependent percolation models. The second one concerns a height function studied already by Dub´edat[22], and Boutilier and de Tili`ere[11]. However, unlike in [11, 22], it harvests a link between a percolation model (the double random current) and dimers. Moreover, it uses techniques introduced by Kenyon to prove convergence of the dimer height function, but with a new twist as the proof relies heavily on fermionic observables introduced by Chelkak and Smirnov to prove conformal invariance of the Ising model, as well as a delicate result on the double random current model (see below) helping identifying the boundary conditions. Finally, the last step relies on a deep understanding of the so-called local sets of the GFF introduced by Schramm and Sheffield [70], and follows a rather intricate strategy. It makes use of the two-valued sets introduced by Aru, Sep´ulveda and Werner [8], and further establishes new results on local sets including a characterization of local sets with three values. A primary difficulty of this step is to find the right strategy, in particular to single out a set of properties satisfied by every subsequential limit of the discrete model which are both obtainable using techniques in the discrete and also sufficient for proving the final results using techniques in the continuum. Interestingly, some properties which seem inaccessible using discrete techniques can be cleanly established using arguments from the continuum. Part (i) of the proof is postponed to the second paper [28]. In this paper, we focus on (ii) and (iii). Let us finish this motivational part by mentioning that our proof is made possible by the introduction of a new coupling between the XOR-Ising model and the double random current model. We therefore also obtain conformal invariance results for this model. The random current model has proved to be a very powerful tool to understand the Ising model. Its applications range from correlation inequalities [36], exponential decay in the off-critical regime [2, 29, 33], classification of Gibbs states [65], continuity of the phase transition [5], etc. Even in two dimensions, where a number of other tools are available, new developments have been made possible via the use of this representation [6, 27, 55]. For a more exhaustive account of random currents, we refer the reader to [26]. We note that the coupling with the double random current extends to the family of Ashkin-Teller models [51], and it is likely that our argument may generalize to other models as well. More generally, we believe that invoking the connections between discrete height function and loop-ensemble types models is a very promising tool to obtain conformal invari- ance of other planar models at criticality. In the remainder of this introduction, we will state our main results on the convergence of the models. For simplicity, we first present results on each model individually, and not on the related height function. In Section 2, we will describe our proof road-map and reveal the role of the height function. In particular, we will present a range of further results on the joint convergence of the interfaces and the height function. In the continuum, the height function converges to the GFF, and the interfaces to its level lines. One feature of this work is that we make use of couplings between various models in the discrete. A particularly nice consequence of our approach is that the couplings in the discrete can be carried through to the continuum limit: 3 The limits of interfaces of four models (the XOR-Ising models with free and plus/plus • boundary conditions and the double random currents models with free and wired bound- ary conditions) can all be coupled with the same GFF as its level loops at different heights.
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