Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos

We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. As smaller loops are included in the count, that sum diverges logarithmically with the diameter of the loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener-It\^{o} chaos expansion of linear functionals of vertex-like fields.


Contents
In this article we are primarily concerned with the Brownian loop soup (BLS) [35,36,62] and the Gaussian multiplicative chaos (GMC) [5,27,28,30,49,51], two objects that have attracted intense attention recently, because of their intrinsic interest and their numerous applications and connections to other models and areas of mathematics, including Euclidean field theory [17,26], the Schramm-Loewner Evolution (SLE) [2,53], the statistical analysis of eigenvalues of large random matrices and of zeros of the Riemann zeta function [6,31,43,50,59], growth models [8], disordered systems [15], the study of lattice models of statistical mechanics, of scaling limits of random planar maps and of the geometry of two-dimensional quantum gravity [16,20,29,48]. More precisely, we study a two-parameter family of random generalized fields, denoted by V * β,λ , obtained from the BLS in a way that is inspired by the construction of vertex operators in conformal field theory (see, e.g., [17]), as we explain below. The conformal properties of such fields were first analyzed in [14], which in turn was partly motivated by [22]; here, we show that, when β → 0 and λ → ∞ in an appropriate way, V * β,λ converges to a version of the imaginary GMC [27,30]. The BLS, introduced in [36], is closely related to SLE and to the scaling limit of interfaces of statistical mechanical models at the critical point, such as the critical Ising model. It is a Poisson process of planar loops with intensity measure given by a multiple λµ loop of the Brownian loop measure µ loop . The latter, studied by Werner in [62], is a measure on simple loops in the plane, and on Riemann surfaces, which is conformally invariant and also invariant under restriction (i.e., the measure on a Riemann surface S that is contained in another Riemann surface S is just the measure on S restricted to those loops that stay in S ). Werner showed [62] that, up to a multiplicative constant, there exists a unique such measure. The Brownian loop measure is related [62] to the outer boundaries of critical percolation clusters in the scaling limit and to the outer boundaries of Brownian loops, from which it derives its name.
In [14], Camia, Gandolfi and Kleban initiated the study of certain conformally covariant Euclidean fields obtained from the BLS. Roughly speaking, the fields studied in [14] are defined as exponentials, of the form e iβN , of operators N that compute properties of the Brownian loop soup, where β is a positive constant and λ > 0 is the intensity of the Brownian loop soup. The operator N (z) counts either (i) the total number of windings of all loops around a point z, in which case e iβN is called a winding field, or (ii) the algebraic sum of distinct loops that disconnect z from infinity, where each loop contributes with a random sign, in which case e iβN is called a layering field. This second type of operator was first introduced in [22], as a model for an eternally inflating universe, using a scale invariant Poisson Boolean model instead of the BLS. In this paper, we call the model introduced in [22] the "disk model." The variants that use the BLS were introduced in [14] as conformally covariant versions of the original disk model. Whether or not such models will eventually find applications to cosmology, they show interesting properties of their own, particularly those based on the BLS, and their study reveals analogies with the analysis of more classical Euclidean field theories, in the tradition of constructive field theory, and in particular of conformal field theories. These analogies provide one of the motivations for our work. Indeed, the families of fields studied here and in [14] provide examples of nontrivial Euclidean field theories that are amenable to a mathematically rigorous analysis. These field theories do not seem to have been studied before and present novel features, such as a periodic spectrum of conformal dimensions.
Another motivation for our work is a calculation in [22] suggesting that, in infinite volume, the disk model should have a non-interacting (free-field) limit as the intensity λ of the Poisson Boolean model tends to infinity. In this paper, we consider the disk model in the unit disk and the layering models based on the BLS and massive BLS in any planar domain conformally equivalent to the unit disk. For all such models, we rigorously show the existence of a Gaussian limit by proving convergence of the corresponding layering fields to versions of the imaginary GMC. However, somewhat surprisingly, the limits we obtain are not free, meaning that, although the limiting fields can be expressed as exponentials of Gaussian fields (i.e., GMCs), the covariance of those Gaussian fields is not given by the Green's function of the Laplacian.
A third motivation is to provide a new example of the deep and very rich connection between conformal field theories and conformally invariant stochastic models such as the BLS. In doing so, we also prove some results about the Brownian loop measure µ loop which may be of independent interest (see Section 4).
We believe that an important contribution of the present paper is the introduction of the Wiener-Itô chaos expansion as a tool to study asymptotic limits of random fields. The chaos expansion turns out to be a very useful tool in this context, effectively reducing the asymptotic analysis of the fields V * λ,β to that of their one-point functions, and allowing us to obtain our main results as a consequence of a multivariate central limit theorem.

Main results
The fields e iβN studied here and in [14,22] are inspired by the vertex operators defined as exponentials of free fields times an imaginary coefficient (see, e.g. [17,26]), and have similar properties of conformal covariance. However, one difficulty arises immediately in defining the fields e iβN , since N (z) itself is not well-defined. Indeed, because the BLS is conformally (and therefore scale-)invariant, with probability one, infinitely many loops disconnect from infinity any given point of the plane, making N (z) a. s. infinite; the same occurs for the other models considered in this paper. It is easy to see that the number of loops of diameter between δ > 0 and R < ∞ that disconnect from infinity any given point of the plane diverges logarithmically as δ → 0 or R → ∞. This explains why it is natural to consider fields of the form e iβN , and why these need to be defined using cutoffs in order to make the number of loops around a point z finite. In general, two types of cutoff are necessary: an "ultraviolet" cutoff that removes small loops, and an "infrared" cutoff that removes large loops.
A particularly interesting way to introduce an infrared cutoff is to use the restriction property of the Brownian loop soup (and other Poissonian models) and consider only the loops that are contained in a finite domain D. An ultraviolet cutoff δ > 0 can be introduced by keeping only loops with diameter at least δ. The resulting random fields V δ λ,β (z) = e iβN δ λ (z) are then well defined. As δ → 0, the n-point correlation functions of these fields, δ −2∆ loop λ,β E(V δ λ,β (z 1 ) . . . V δ λ,β (z n )), normalized by an appropriate power 2∆ loop λ,β (the factor of 2 is a convention, in keeping with the physics literature on conformal field theory) of the cutoff δ depending on the model, are shown [14] to converge to real-valued functions that depend on the domain D in a conformally covariant way.
The existence of the limit as δ → 0 for the normalized fields δ −2∆ loop λ,β V δ λ,β themselves, was obtained first in [12] and later in [39] for fields of the winding type, provided that the intensity of the BLS is not too large. The limit exists in an appropriate Sobolev space of negative index, meaning that the limiting winding fields are random generalized functions, measurable with respect to the BLS (see Theorem 5.1 of [12]). It is also shown in [12] that the limiting fields are not Gaussian fields (see Proposition 5.2 of [12]).
In this paper, following [12], we prove the existence of the limiting layering fields, which we denote V * λ,β,D = V * λ,β , obtained from δ −2∆ * λ,β V * ,δ λ,β as δ → 0. In the notation, we often suppress D, which is considered fixed, while * = loop, disk, m will distinguish between the three different models considered in this paper, based respectively on the BLS of Lawler and Werner [36], on the disk model mentioned above, and on the massive Brownian loop soup, in which a killing factor is introduced in the BLS (see [13] for a detailed introduction). Each model has different characteristics: for * = loop the model is scale and conformally invariant, for * = disk the model is scale but not conformally invariant, for * = m the model is not even scale invariant since the killing rate introduces a scale. Despite these differences, all models can be analyzed using our methods. Each layering field V * λ,β is based on a Poisson process of loops (i.e., closed planar curves) with intensity measure λµ * , where µ * is either the Brownian loop measure µ loop , the disk measure dµ disk = dzdr r 3 , where dz denotes the two-dimensional Lebesgue measure and r is the radius of a random disk, or the massive Brownian loop measure where γ denotes a planar loop and the "mass function" m : C → R + introduces a killing of Brownian loops according to their length. As shown in Section 4.2 below, the divergence of the massive Brownian loop measure µ m is the same as that of µ loop . This follows from the fact that the divergence of both µ m and µ loop is due to small loops, whose measure is not affected much by a bounded mass function m. As a consequence, the measureμ defined by is finite over bounded domains. For µ loop , µ disk and µ m , we show that, as λ → ∞ and β → 0 in such a way that λβ 2 tends to some parameter ξ 2 > 0, V * λ,β converges to a "tilted" imaginary GMC [27,30], as explained below. Assuming that X : D → C is a generalized Gaussian process with covariance kernel where η > 0 and g is a bounded continuous function over D × D, in this article, we say that a random generalized function M ξ,D is an imaginary Gaussian multiplicative chaos in D ⊂ C with parameter ξ > 0, written if there exists a sequence {X δ } δ of Gaussian fields converging to X such that the functions are well defined and converge, as δ → 0, to M ξ,D , where convergence can be understood in probability in some Sobolev space H −α (D) of negative index.
Theorem 1.1 (Existence of the limit as a GMC). Take λ > 0 and β ∈ [0, 2π), let * = loop, m and let D C be any domain conformally equivalent to the unit disk D.
For each z ∈ D, let d z = dist(z, ∂D). Then, as λ → ∞ and β → 0 in such a way that λβ 2 → ξ 2 < 5, V * λ,β has a limit W * ξ,D in the sense of convergence of finite dimensional distributions. Moreover, and M * ξ,D is an imaginary Gaussian multiplicative chaos in D with parameter ξ and covariance kernel K * D (z, w) = µ * (γ : γ is in D and disconnects z, w from ∂D) for every z, w ∈ D. The same conclusions hold for * = disk with D = D, ξ 2 < 1/π and Proof. The theorem is an immediate consequence of Theorems 4.9 and 6.2 below.
1. The action of W * ξ,D on ϕ ∈ C ∞ 0 (D) will be denoted by and similarly for other (generalized) fields. With this notation, Equation (1.6) means that, for any ϕ ∈ C ∞ 0 (D), 2. We note that (1.6) is deterministic, so all the randomness of the field W * ξ,D is captured by the imaginary GMC M * ξ,D .
3. Because of the definition of imaginary GMC, the theorem implicitly implies that µ * (γ : γ is in D and disconnects z, w from ∂D) can be written as η log + 1 |z−w| + g(z, w) for some bounded continuous function g. This is proved in Lemma 4.4 below.
4. For * = disk, Theorem 1.1 is stated for the unit disk D for simplicity, but its conclusions can be easily extended to general bounded simply connected domains with a C 1 boundary. The same considerations apply to Theorems 4.5 and 4.9 below, as one can easily deduce from Theorems A.3 and A.4 in the appendix.
For * = loop, and taking D to be the upper half plane H, an explicit formula for µ loop (γ : γ is in H and disconnects z, w from ∂H) was predicted by Gamsa and Cardy [23] using conformal field theory methods. The correct formula, which coincides with the Gamsa-Cardy prediction up to an additive term, was obtained rigorously in [25]. Given the conformal invariance of µ loop , from the formula for D = H, one can in principle obtain a formula for K loop D for any domain conformally equivalent to H. In the case of the unit disk D, one has the following formula (see [25]): where 3 F 2 is the hypergeometric function andσ =σ(z, w) = |z−w| 2 |1−zw| 2 . The limiting fields W * ξ,D are conformally covariant in the sense of the result below. for each z = f (w) ∈ D. Then, with the notation of the previous theorem, we have that, for every ϕ ∈ C ∞ 0 (D), and where d = stands for equality in distribution. Moreover, if f is a Möbius transformation of the unit disk into itself, then Proof. The theorem follows immediately from Theorem A.6 and Remark A.5 in the appendix.
The result above can be expressed in the following way: and In keeping with the physics literature on conformal field theory, the powers to which |f (w)| is raised can be written as 2 − 2∆ loop ξ and 2 − 2∆ disk ξ , where ∆ loop ξ = ξ 2 /20 and ∆ disk ξ = πξ 2 /4 can be interpreted as the conformal/scaling dimension of the fields W * ξ,D .

Relations to the existing literature
Brownian loop soup. The use of random paths in the analysis of Euclidean field theories and statistical mechanical models was initiated by Symanzik, who introduced a representation of the φ 4 Euclidean field as a "gas" of weakly interacting random paths in his seminal work on Euclidean quantum field theory [57]. The approach was later developed by various authors, most notably Brydges, Fröhlich, Spencer and Sokal [10,11], and Aizenman [1] (see [21] for a comprehensive account). The probabilistic analysis of Brownian and random walk paths and associated local times was carried out by Dynkin [18,19]. More recently, "gases" or "soups" (i.e., Poissonian ensembles) of Brownian and random walk loops have been extensively studied in connection with SLE and the Gaussian free field (see, e.g., [13, 36-38, 58, 60, 61]). The Brownian loop soup [36] is, roughly speaking, a Poisson point process with intensity measure λµ, where λ is a positive constant and the Brownian loop measure µ [62] is uniquely determined (up to a multiplicative constant) by its conformal restriction property, a combination of conformal invariance and the property that the measure in a subdomain is the original measure restricted to loops that stay in that subdomain). A realization of the loop soup consists of a countable collection of loops. Given a bounded domain D, there is an infinite number of loops that stay in D; however, the number of loops in D of diameter at least ε > 0 is finite. A consequence of conformal invariance is scale invariance: if A is a realization of the Brownian loop soup and each loop is scaled in space by 1/N and in time by 1/N 2 , the resulting configuration also has the distribution of the Brownian loop soup.
The Brownian loop soup exhibits a connectivity phase transition in the parameter λ > 0 that multiplies the intensity measure. When λ ≤ 1/2, the loop soup in D is composed of disjoint clusters of loops [54,60,61] (where a cluster is a maximal collection of loops that intersect each other). When λ > 1/2, there is a unique cluster [54,60,61] and the set of points not surrounded by a loop is totally disconnected (see [9]). Furthermore, when λ ≤ 1/2, the outer boundaries of the loop soup clusters are distributed like Conformal Loop Ensembles (CLE κ ) [52,54,60] with 8/3 < κ ≤ 4. The latter are conjectured to describe the scaling limit of cluster boundaries in various critical models of statistical mechanics, such as the critical Potts models for q ∈ (1,4]. More precisely, if 8/3 < κ ≤ 4, then 0 < (3κ−8)(6−κ) 4κ ≤ 1/2 and the collection of all outer boundaries of the clusters of the Brownian loop soup with intensity parameter λ = (3κ−8)(6−κ) 4κ is distributed like CLE κ [54]. For example, the continuum scaling limit of the collection of all macroscopic boundaries of critical Ising spin clusters is conjectured to correspond to CLE 3 and to a Brownian loop soup with λ = 1/4.
Gaussian Multiplicative Chaos. The theory of Gaussian multiplicative chaos was initiated by Kahane [28] in an attempt to define rigorously a random measure of the form e αφ(x) σ(dx) where α > 0 is a real parameter, φ is a log-correlated, centered Gaussian field on a domain D and σ is an independent measure on D. The theory has been developed and expanded extensively in recent years by various authors (see, e.g. [5,30,48,49,51]) and has applications that range from mathematical finance to the study of turbulent flows (Kahane's work was motivated by a desire to make rigorous a model of energy dissipation in turbulent flows developed by Mandelbrot [40]). A case that has recently attracted particular attention is that of dimension two when φ is the massless Gaussian free field (GFF) (i.e., the Gaussian field whose covariance is given by the Green's function associated with the Laplace operator), called the Euclidean (bosonic) massless free field in quantum field theory. In this case, the random measure is called Liouville Quantum Gravity and was first discussed by Polyakov [46] and others [3,4]. The Complex Gaussian Multiplicative Chaos (CGMC) [30] corresponds to taking a complex parameter α. In this case, e αφ σ(dx) is no longer a measure but can still be defined as a random field. The special case in which α is purely imaginary has been studied in [27], where it is shown to be related to the scaling limit of the spin field of the XOR-Ising model and to the sine-Gordon field. Moreover, as pointed out in [27], at least at a formal level, the imaginary GCM appears to play a central role in the study of so-called imaginary geometry (see [41]).
Notation. In what follows, and for the rest of the paper, every random object is defined on a common probability space (Ω, G , P). The expectation with respect to P is denoted by E(·) or by the brackets · , according to notational convenience. Whenever required by the context, we will write · D instead of · , in order to indicate that some underlying random field is indexed by a given domain D. In several parts of the paper, we will deal with three specific random models based on the Brownian loop soup, the disk model and the massive Brownian loop soup on a given domain D; to emphasize the dependence on the law of the underlying model, we will sometimes denote expectation with the symbols · loop D , · disk D and · m D (m stands for 'massive'), respectively, or with the symbol · * D where * stands for either loop, disk or m. In the notation for the random fields we discuss in this paper, we will also sometimes include the domain D, but only when we want to emphasize the dependence of the fields on the domain; most of the time the domain D can be considered fixed and will be suppressed from the notation. By an abuse of notation, we will sometimes treat objects that are not defined pointwise as if they were. For example, we will write V * λ,β (z) for lim δ→0 δ −2∆ loop λ,β V δ λ,β (z) even though V * λ,β is not defined pointwise. The lowercase letter ω always denotes generic elements of the set Ω. We will use | · | to denote Euclidean distance.
Organization of the paper. In the next section we give some preliminaries on Poisson layering fields and the Brownian loop soup. In Section 3, following [12], we prove a convergence result for a general class of fields, which contains the fields studied in this paper. In Section 4, we apply the result of the previous section to the Poisson and Gaussian layering fields that are the main focus of this paper. We also characterize the Gaussian layering fields as "tilted" imaginary GMC. In Section 5, we show that the fields constructed in the previous section admit a Wiener-Itô chaos expansion. In Section 6, using the Wiener-Itô chaos expansion presented in the previous section, we prove that the Poisson layering fields converge to the Gaussian layering fields in the appropriate limit of the parameters β → 0 and λ → ∞. The Appendix contains various technical results, including some background on Sobolev spaces (Appendix A.1) and several of the proofs of the results of Section 4.2 (see Appendix A.3). Appendix A.4 contains a useful representation for the one-point functions of Poisson and Gaussian layering fields, while in Appendix A.5, we prove the existence and conformal covariance of Poisson and Gaussian layering fields for general domains.

Preliminaries on Poisson layering fields and the Brownian loop soup
In order to motivate the reader -and to fix ideas -we will now provide a succinct presentation of a class of Poisson layering fields associated with the Brownian loop soup. Understanding the limiting behavior of such fields -that are special elements of the general class introduced in Section 3 -is one of the main motivations of the present paper. The layering model underlying the definition of such fields first appeared in [14], where properties of its n-point correlation functions were studied. As explained in the introduction, such layering model is defined by first imposing a cutoff and then removing it via a limiting procedure, but only the existence of the limit for the n-point functions is proved in [14]. The existence of a layering field, i.e., a random generalized function whose n-point functions are those derived in [14], will follow from Theorem 3.3 below. We observe that the Poisson layering fields discussed in this section have to be distinguished from the Gaussian layering fields introduced in Section 4.5.

Loops
In this section (and for the rest of the paper) we identify C and and t f •γ = s(t γ ). As a special case, consider the mapping Φ a,b (w) = aw + b, a = 0: then, Φ a,b • γ is the loop γ scaled by |a|, rotated around the origin by arg a and shifted by b, with time parametrization s(t) = |a| 2 t, and length t Φ a,b •γ = |a| 2 t γ . An unrooted loop is an equivalence class of loops under the equivalence relation γ ∼ θ r γ for every r ∈ R, where θ r γ(s) = γ(s + r mod t γ ). We denote by M the class of unrooted loops in R 2 and, for a given domain D, we write M D to indicate the class of unrooted loops in D.
As explained in [36, Section 4.1], the class M can be made into a metric space, and the corresponding Borel σ-field is denoted by M. For a given domain D, the trace σ-field of M associated with M D is written M D (note that M D is a σ-field of subsets of M D ). For our purposes, we can work with any σ-field M (resp. M D ) that contains all sets of the form where δ, R ≥ 0 and z ∈ C, and where U ⊆ V are two Borel subsets of R 2 (resp. a Borel subset of D) and z ∈ U . Any measure µ on (M, M) or (M D , M D ) is called a measure on unrooted loops.

The Brownian loop measure and the Brownian loop soup
We use the symbol µ br to denote the complex Brownian bridge measure, i.e., the probability measure on loops rooted at 0 and of time length 1, induced by the process where W t is a standard complex Brownian motion initialized at 0. For z ∈ C and t > 0, we write µ br z,t to indicate the complex Brownian bridge measure on loops rooted at z of time length t, i.e., the measure The Brownian loop measure is a σ-finite measure on loops given by where dz denotes the Lebesgue measure on C which, for our purposes and as already mentioned, can be identified with R 2 . This measure is clearly translation invariant and it is easy to check that it is scale invariant, meaning that µ loop = µ loop • Φ a,b for any a > 0 and b ∈ C. Since µ loop inherits rotational invariance from the complex Brownian motion, we actually have that µ loop = µ loop • Φ a,b for any a, b ∈ C, a = 0. To recover the full conformal invariance of Brownian motion one has to regard the push-forward measure of µ loop on the space (M, M) of unrooted loops defined above (as it is customary, and by a slight abuse of notation, we continue to use the symbol µ loop for such a push-forward measure). If D is a domain, then by µ loop D we denote the measure µ loop restricted to (M D , M D ). Let D, D be two simply connected domains, and let f : D → D be a conformal equivalence. The full conformal invariance of µ loop is expressed by the fact that µ loop D • f = µ loop D . A proof of this fact can be found in [35,36]. The Brownian loop soup L D = L D,λ with intensity parameter λ > 0 is a Poissonian collection of unrooted loops with intensity measure λµ loop D . We write L = L C . The Brownian loop soup inherits all invariance properties of the Brownian loop measure. In particular, for D, D and f as above, one has that L D has the same distribution as the image of L D under f .

Poisson layering fields
Following [14], we now define the layering model arising from the Brownian loop soup. Letγ be the hull of the loop γ, i.e., the complement of the unique unbounded connected component of the complement of γ. Regarding γ as a subset of C, we say that γ covers z if z ∈γ. We restrict our attention to bounded domains D and declare each loop in L D to be of type 1 or type 2 with equal probability, independently of all other loops. For any z ∈ D, the scale invariance of the BLS implies that the number of loops of each type covering z ∈ D is almost surely infinite. Because of this, we introduce an ultraviolet cutoff δ > 0 on the size of the loops and let Similarly, we let µ δ D denote the measure µ D restricted to loops of diameter larger than δ. We note that µ δ D is a finite measure when D is bounded and δ > 0. This implies that, almost surely, each point z is covered by only finitely many loops from L δ D , so that the number N δ j (z) of loops of type j = 1, 2 covering z ∈ D is almost surely finite. We can now define Note that the loops which do not cover z do not contribute to N δ D (z), and therefore N δ D (z) is finite almost surely. The layering field with cutoff δ > 0 and parameters λ, β > 0 is defined by The correlation functions of such random fields were explicitly studied in [14, Theorem 4.1] in the limit δ → 0. There, it was proved that the one-point function It is therefore natural to study the "renormalized" field δ −2∆ V δ λ,β as δ → 0. Observe however that, as δ → 0 and in view of the equality |δ −2∆ V δ λ,β (z)| = δ −2∆ , the field δ −2∆ V δ λ,β does not converge as a function on D. As a consequence, in order to deduce well-defined convergence results, one has to regard δ −2∆ V δ λ,β as an element of a topological space larger than any classical function space. This is usually achieved by regarding δ −2∆ V δ λ,β as a random generalized function, i.e., a random continuous functional on some appropriately chosen space of test functions where the action of δ −2∆ V δ λ,β on a test function ϕ is given by Such a strategy was successfully implemented in [12,Section 5] for winding fields. The aim of the next section is to establish a class of general convergence results, extending the conclusions [12] to a large class of layering models.

A general class of fields
In this chapter we set up a general framework for dealing simultaneously with random fields which are exponentials of Poisson or Gaussian random measures having a σ-finite control with infinite mass.

Random measures
Let (A, A ) be a measurable space, and let ν be a σ-finite positive measure on (A, A ). We will assume that ν(A) = +∞, and also that ν has no atoms (at the cost of some technicalities, one can remove such an assumption with essentially no bearings on our results).
For every λ > 0, we write to indicate a Poisson measure on (A, A ) with intensity λν(·). Remember that the distribution of N λ is completely characterized by the following two properties: (a) for every collection B 1 , ..., B n ∈ A of pairwise disjoint sets, one has that the random variables N λ (B 1 ), ..., N λ (B n ) are stochastically independent, and (b) for every B ∈ A the random variable N λ (B) is distributed according to a Poisson law with parameter λν(B); here, we adopt the standard convention that a Poisson random variable with parameter 0 (resp. ∞) equals 0 (resp. ∞) with probability one. It is easily checked that, for P-almost every ω, the mapping B → N λ (B)(ω) defines an integer-valued measure on (A, A ) such that, for every a ∈ A, the value of N λ ({a})(ω) is either 0 or 1 (this last property comes from the fact that ν has no atoms). It is also convenient to introduce the quantity The mapping B → N λ (B) is called a compensated Poisson measure.
We will also denote by a Gaussian measure with control ν on (A, A ). This means that G is a centered Gaussian family, indexed by the elements B of A such that ν(B) < ∞, with covariance Background material on Poisson and Gaussian measures can be found in [32-34, 44, 45]. We record the following facts (see e.g. [45,Chapter 5]): are well defined as limits in L 2 (P) of finite sums of the type In this case, one has also the crucial isometric relations for every f, h ∈ L 2 (ν).

Kernels
We keep the notation from the previous subsection and, for the rest of Section 3, we let D denote a domain of C, which we identify with R 2 .
We write B w,r to indicate the open disk of radius r > 0 centered at w ∈ R 2 . Two distinguished points in R 2 are the origin, denoted by 0, and the point (0, 1) denoted by 1. For z, w ∈ D, we define where ∂D stands for the boundary of D, and d is the Euclidean distance. We also indicate the diameter of a subset U of D by diam(U ).
In what follows, we consider collections of jointly measurable kernels of the type where δ 0 > 0 is a fixed parameter (possibly infinite, in which case δ ∈ (0, δ 0 ] has to be read as δ > 0). The objects appearing in (3.7) have the property of locally exploding in the δ → 0 limit, that is: for every z ∈ D, x → h δ z (x) ∈ L 1 (ν) ∩ L 2 (ν) for δ > 0, and, as δ → 0, the kernel h δ z converges ν-almost everywhere to some h z such that ν(|h z |) = ν(|h z | 2 ) = ∞, where we adopted the standard notation ν(g) := A g(x)ν(dx).  .7), a small parameter family of ν controlled complex random fields associated with h is defined to be a collection of random objects of the type ζ ∈ R, and X is either a Gaussian measure G with control ν, or a Poisson measure N λ with intensity λν, for some λ > 0. It will be always implicitly assumed that the kernels in h are such that the mapping (ω, z) → U δ (z)(ω) is jointly measurable (such an assumption is needed e.g. to apply Fubini-type arguments).

Remark 3.2.
In the case of the layering fields introduced in Section 2, A is the space of marked loops, where each loop γ appears in two copies, each with a mark = 1 or −1. In this context, letting x = (γ, ) ∈ A and A δ (z) = {γ : z ∈γ, diam(γ) > δ}, ν is 1 2 µ loop times the uniform measure on {−1, 1} and h δ z (x) = 1 A δ (z) (γ). As a consequence, is the difference of two independent Poisson r.v.'s with intensity equal to half of the µ loop -mass of the loops surrounding z that are contained in D and have radius larger than δ. See Section 4 for more details.
The "explosive" nature of the kernels h of interest implies that X(h δ z ) is not well defined for δ = 0 and that some care is needed when removing the cutoff δ. As δ → 0, the existence of the limit of the fields U δ , when normalized by an appropriate power of the cutoff δ, is discussed in the next section.

A general existence result
In this section we identify general conditions for the existence of the limiting fields of a small parameter family of ν controlled complex random fields, as the cutoff fields are renormalized and the cutoff is removed.
The next theorem is a generalization of Theorem 5.1 of [12]. Our formulation of Theorem 3.3 has been devised in order to facilitate the connection with the formalism adopted in [12]. For the convenience of the reader, definitions and basic facts about negative Sobolev spaces are gathered together in Section A.1. Theorem 3.3. Let D be a bounded simply connected domain with a C 1 boundary and let U = {U δ (z) : δ ∈ (0, δ 0 ], z ∈ D} be a small parameter family of ν controlled complex random fields, as in (3.7)-(3.9), such that Assume that there exist ∆ ∈ (0, 1/2) and a real-valued nonnegative function φ with domain D such that, for every z, w ∈ D with z = w, Moreover, suppose that there exist a constant c(D) < ∞ and a real-valued nonnegative function with d z,w as in (3.6). Then, for every α > 3 2 , the cutoff field z → δ −2∆ U δ (z) converges as δ → 0 in second mean in the Sobolev space H −α , in the sense that there exists a H −α -valued random distribution U , measurable with respect to the σ-field generated by X (as appearing in (3.9)) and such that Proof. The proof is obtained by extending the techniques used in the proofs of Theorem 5.1 and Proposition 5.3 of [12]. For any z = w, the limit lim exists pointwise by (3.11). We will now show that convergence in L 1 (D × D) also holds, by virtue of the Bounded Convergence Theorem. For this purpose, note that if δ ≥ d z,w z,w as τ δ z,w ≥ 0; then, for any z, w ∈ D and any δ, δ , from (3.12), (3.13), (3.3), we infer that Hence, in order to prove prove L 1 (D × D) convergence, it suffices to show that . Note that, since D has a C 1 smooth boundary, f has a continuous extension on to D and hence f L ∞ (D) < ∞, see [47,Theorem 3.5].
By the corresponding change of variables, the integral in (3.17) is equal to with the inequality holding as ∆ Then, a straightforward calculation (or Lemma A.1 in the appendix) shows that whenever ∆ < 1 2 , from which (3.17) follows. We now show that δ −2∆ U δ is a Cauchy sequence in the Banach space S := S(α, D) of H −α -valued square-integrable random elements that are measurable with respect to the σ-field generated by X. Notice that, for every z, the random variable δ −2∆ U δ (z) is square-integrable, as it is bounded. Let {f i } denote an eigenbasis of the Laplacian on D with Dirichlet boundary conditions, and let λ 1 ≤ λ 2 ≤ · · · → ∞ be the corresponding sequence of eigenvalues. Then, writing δ −2∆ U δ =:Ũ δ to simplify the notation, Note that the last inequality follows from the uniform bound on the norm f i L ∞ (D) ≤ cλ 1 4 i , see Theorem 1 of [24]. The sum in the last expression is finite whenever α > 3 2 . This is seen by using Weyl's law [63], which says that the number of eigenvalues λ i , that are less than is proportional to with an error of o( ). From the L 1 (D × D) convergence of (δδ ) −2∆ U δ (z)U δ (w) and, by inspection of the form of the integrand in (3.22), one deduces that 1. Theorem 3.3 can be used in order to study the convergence of Poisson winding models, similarly to [12,Section 5]. In the next section, we will use such a result to extend the analysis of [12] to the case of Poisson layering models.
2. As shown in [12], the conclusion of Theorem 3.3 does not hold for the winding model when ∆ ≥ 1/2, so the condition ∆ ∈ (0, 1/2) cannot be removed from the theorem without further assumptions. The aim of the present section is to define a class of layering fields -containing the objects discussed in Section 2 as special cases -to which the content of Theorem 3.3 directly applies. In order not to disrupt the flow of our discussion, proofs are deferred to Section A.3.

Three measures on unrooted loops
The three examples of a measure µ * on the space (M, M) (of unrooted loops) that are more relevant for the present paper are: (I) the Brownian loop measure, in which µ * = µ loop , as defined in [36] and summarized in Section 2 (in this case, µ loop has full conformal invariance); (II) the scale invariant disk distribution, see [9,22], in which µ * = µ disk is defined as follows: consider the set D = {B : B = B y,r , for some y ∈ R 2 and r ∈ R + }; where R m (γ) = tγ 0 m 2 (γ(t))dt (where γ is any element of the equivalence class corresponding to a given unrooted loop and t γ denotes the time length of γ). Observe that there is another representation for the massive Brownian loop measure, obtained by taking one realization of the BLS, then a mean one exponential random variable T γ for each loop, and, finally, removing the loops for which tγ 0 m 2 (γ(t))dt > T γ (see [13,Proposition 2.9]). The massive Brownian loop measure µ m is not scale invariant; on the other hand, the mass of the loops with diameter bounded from below and which wind around a point is finite, even when one does consider the full measure µ m (without restricting it to M D ).
We will typically deal with some particular sets of loops, so we introduce some ad hoc notation, adapted from [14]. Each set of interest is identified by the symbol A, decorated with appropriate indices; such sets are all measurable since our reference σ-field contains sets of the type (2.2). The (possibly decorated) symbol α * is used below to build expressions useful for denoting the measure µ * of a given set; we stress that, even if the measure µ * appearing below is a generic measure on loops, in this paper the star * has to be mainly regarded as a placeholder for either one of the three symbols, loop, disk and m. Here is our notation: -for two real values δ and R with 0 ≤ δ < R, and for z ∈ R 2 , we let (note that, in the previous display and similarly for the forthcoming definitions, the last equality implicitly defines the set A δ,R (z)); -for a real number δ ≥ 0, z ∈ D and a set V ⊆ D, we let (it will be always be clear from the context whether an expression of the type α * δ,Q refers to a real number Q > 0 or to a set Q ⊆ D); -for real numbers 0 ≤ δ < δ, points z, w ∈ D and a set V ⊆ D, we let -for two points z, t ∈ D and δ > 0 -for a set V we write so that, for instance, α * ¬B 0,1 (0|1) is the measure of the set of loops not restricted to the disk of radius 1 centered at the origin, containing the origin but not the point (0, 1).
1. As already observed, in the three cases µ * = µ loop , µ disk , µ m , using the notation (4.4), we have that is: for the Brownian loop measure, the disk distribution and the massive Brownian loop measure, µ * charges with infinite mass the set of loops in D covering any fixed point z ∈ D. In general, the behaviour of the function α * δ,R (z) for small δ's is needed to identify the conformal dimensions of the fields of interest.
2. To further illustrate the use of the notation, let us recall that the following equalities are shown in [14, Lemma A.1]: (4.14)

Some useful one-and two-point computations
In this section we discuss the divergence of the measures we are interested in. In particular, we show that the divergence of the massive Brownian loop measure µ m is the same as that of µ loop . This is a consequence of the fact that the divergence of both µ m and µ loop is due to small loops, whose measure is not much affected by a bounded mass function m. Some of the results in this section can be expressed in terms of the measureμ defined by . In order to make our discussion more streamlined, the proofs of the results of this section are gathered together in Section A.3.
Lemma 4.1. Given a bounded mass function m ≤ m ∈ R, we have that for z ∈ D, 0 < δ < R, and some c = c(z, m, D) > 0, For later purposes, it is useful to introduce the following notation:α 0,R (z) :=μ(γ : where the measureμ has been defined in (4.15).
We also introduce the concept of thinness of a loop soup, taken from [42] (see We say that a loop soup with intensity measure λµ * is thin if µ * is thin. An equivalent formulation, the one we will use in this paper, is that lim R→∞ µ * (γ : γ ∩ D = ∅, diam(γ) ≥ R) = 0. The thinness of the BLS is proved in [42]. The thinness of the disk model and the massive BLSs can be proved easily, as shown below. Proof. For µ disk , using elementary geometric and trigonometric considerations, we have that By translation and rotation invariance, this implies that the disk model is thin. For the massive BLS, we trivially have that which concludes the proof.
Next, we focus on two-point estimates. We will show that in the case of a thin loop soup, if z 1 and z 2 are close to each other, then one can relate the divergence of α * D (z 1 , z 2 ), as z 2 → z 1 , to that of α * δ,R (z 1 ) as δ → 0, as demonstrated by the next theorem.
Theorem 4.2. If * = loop or disk, then for each z 1 ∈ D the following limit exists and is finite, and it defines a continuous function of z 1 ∈ D: . Then, for each z 1 ∈ D, the following limit exists: We also record the following continuity property, holding in the special case of the (massive) Brownian loop and disk measures; it will be exploited in Section 4.5.
Then, the three mappings g loop , g disk , g m : D × D → R are continuous.
We now focus on kernels defined on loop spaces.

Kernels on unrooted loops and Poisson layering models
We will specify the exact form of the kernels h appearing in (3.7) that is needed in order to obtain a collection of general Poisson layering fields. We shall consider exclusively kernels yielding a mechanism for assigning a random sign to each loop; although centered random variables more general than a random symmetric sign could be used, we stick to this case for simplicity and in view of [14,22]. This completes the definition of the truncated fields and brings us back to the abstract setting of Section 3. We consider the following elements: (i) the set A = M D × R; the elements of A are generally denoted by x = (γ, ); where δ a denotes the Dirac mass at a (see also [14]), and µ * D is a measure on unrooted loops contained in D, verifying µ * D (γ ⊂ D : z ∈ γ) = ∞; (iv) a locally exploding collection of kernels h of the type (3.7), where h δ z (x) = 1 A δ,D (z) (γ), with x = (γ, ) ∈ A and z ∈ D, and the definition of the set A δ,D (z) is given in (4.4). We observe in particular that, for every z ∈ D and for ν-almost every x = (γ, ) ∈ A, For notational convenience, we will sometimes write to indicate the law of a symmetric random sign.
where the measure ν and the kernel h are as in items (iii) and (iv) above. 1. For Poisson layering models, the choice of replacing the letter ζ with β in (3.9) is made in order to facilitate the connection with [14].

2.
A more precise terminology than the one used in Definition 4.3 would make reference to the measure µ * D ; this would lead to expressions such as "µ * D -(or µ loop D -, µ disk D -, µ m D -) Poisson layering models and fields," but we prefer the terminology introduced above for the sake of simplicity. In the next section, we will use the same kernels as here, but take X = G, where G is a Gaussian measure with control µ * D ; we will call the resulting fields Gaussian layering fields.

Convergence of renormalized Poisson layering fields in negative Sobolev spaces
We can finally present the main result of the section, which states that the Poisson layering models V δ λ,β with intensity measure λν = λµ * D ⊗ S, where we have used (4.28), and * = loop, m or disk and β ∈ [0, 2π), when renormalized with an appropriate scaling factor, converge in second mean in any Sobolev space H −α with α > 3/2. The proof, which is a direct application of Theorem 3.3, requires a certain amount of control on the functions α * δ,R (z), in order to verify the condition of Theorem 3.3. This is where the restriction of the general framework of Definition 4.3 to the measures µ * with * = loop, m, disk plays a crucial role.
(regarded as a random generalized function) converges in second mean in the Sobolev space H −α , in the sense of (3.14), to a random generalized function V * λ,β , which we call a Poisson layering field.
Proof. If D is a bounded simply connected domain with a C 1 boundary, the result corresponds to Theorem A.3. For more general domains, it follows from Theorem A.6.

Gaussian layering fields
We consider the elements A = M D × R, A = M D ⊗ B, ν = µ * D ⊗ S, and h introduced at item (i)-(iv) of Section 4.3 (recall notation (4.28)), and also consider a Gaussian measure G on (A, A ) with control ν, as defined in (3.2).
Remark 4.7. In the discussion to follow, we will be only interested in the properties of the Gaussian measure B → G(f B ), (4.31) where B ∈ M D is such that µ * D (B) < ∞ and, for x = (γ, ), f B (x) := 1 B (γ). A direct covariance computation shows that the Gaussian measure defined in (4.31) has the same distribution as where G 0 is a Gaussian measure on (M D , M D ) with control µ * D , in such a way that the use of the measure ν on the product space M D × R is somehow redundant. Our choice of using the measure ν for expressing the law of the underlying Gaussian integrators is motivated by computational convenience -see e.g. Section 6.
With this notation, according to the discussion following (3.2) one has that, for each z ∈ D, G(h δ z ) is a centered Gaussian random variable with variance α * δ,D (z), and {G(h δ z )} z∈D is a Gaussian field with covariance kernel K * δ (z, w) = α * δ,D (z, w). We also introduce the generalized Gaussian field w). The field G * can be realized as a random element of the Sobolev space H −α (D), for some α > 0, whose law is given by the centered Gaussian distribution determined by the covariance formula for all test functions f, g in the space H α (D) dual to H −α (D). We can also interpret the Gaussian field {G(h δ z )} z∈D as a random element of H −α (D) by defining G δ (f ) := D G(h δ z )f (z)dz for each f ∈ H α (D). We can then observe that, using Fubini's theorem and the fact that lim δ→0 α * δ,D (z, w) = α * 0,D (z, w) for every z, w ∈ D, (4.34) By an argument analogous to that in the proof of Theorem 3.3, this implies that the field G δ converges to G * , as δ → 0, in second mean in the Sobolev space H −α (D) for every α > 3/2. Next, we introduce the main object of interest in the present subsection.
Definition 4.8 (Gaussian layering model). A Gaussian layering model over D with parameter ξ ∈ R\{0} is a small parameter family W ξ = W ξ,D of ν controlled complex exponential random fields, as in Definition 3.1 with ζ replaced by ξ and with X = G a Gaussian measure with control ν. More explicitly, where the measure ν and the kernel h are as in items (iii) and (iv) above.
The definition above is interesting mainly because, as δ → 0, W δ ξ converges to a (deterministically) "tilted" imaginary Gaussian multiplicative chaos [27,30], as defined in the introduction just before Theorem 1.1.
(regarded as a random distribution) converges in square mean in the negative Sobolev space H −α , in the sense of (3.14), to a random distribution W * ξ,D , which we call a Gaussian layering field. Moreover, dz,D (z), and M * ξ,D is an imaginary Gaussian multiplicative chaos in D with parameter ξ and covariance kernel K * D (z, w) = α * D (z, w).
Proof. If D is a bounded simply connected domain with a C 1 boundary, the result corresponds to Theorem A.4. For more general domains, it follows from Theorem A.6.

Chaos expansions
In this section we show that the action of the fields constructed in Sections 4.4 and 4.5 on smooth test functions admit a Wiener-Itô chaos expansion. To accomplish this task, we first briefly recall the theory of Wiener-Itô chaos expansions of general Poisson and Gaussian measures, and then present the main results of this section. A detailed discussion of Wiener chaos can be found in [32] (Poisson case), [44] (Gaussian case) and [45] (both Gaussian and Poisson).
In the next section, we will show that each term of the Wiener-Itô chaos expansion of the Poisson layering field converges to the corresponding term of the Wiener-Itô chaos expansion of the Gaussian layering field, which implies convergence of the Poisson layering field to the Gaussian layering field in the sense of finite dimensional distributions.

Chaos expansion for the Poisson layering field
Until further notice, we will now consider the general situation of a Poisson random measure N λ , defined on a measurable space (A, A ), with control λν, where λ > 0 and ν is σ-finite and non-atomic -see Section 3.1. We also denote by G a Gaussian measure on (A, A ) with control ν.
One of the pivotal points of our analysis (allowing us to effectively connect Poisson and Gaussian structures) will be the following elementary consequence of the multivariate Central Limit Theorem (CLT): as λ → ∞, the standardized compensated random measure λ −1/2 N λ converges to G in the following sense: for every d ≥ 1 and every where =⇒ indicates convergence in law. The classical Wiener-Itô representation property of N λ (see e.g. [32,33,45]) states that, for every random variable Y ∈ L 2 (σ(N λ )), there exists a unique sequence of (ν qalmost everywhere) symmetric kernels {f q : q = 1, 2, ...}, such that f q ∈ L 2 (A q , A q , ν q ) = L 2 (ν q ) and where the series converges in L 2 (P) and stands for the multiple Wiener-Itô integral of order q of f q with respect to N λ . For every f ∈ L 2 (ν), we write I N λ 1 = A f (x) N λ (dx) := N λ (f ). One should observe that chaotic decompositions are usually stated and proved in the case of real-valued random variables Y ; however, the result extends immediately to the case of a square-integrable random variable with the form Y C = Y R + iY I (with i = √ −1), by separately applying , then the integral I q (f ) is simply defined as I q (f ) = I q (f R ) + iI q (f I ). All general results discussed below apply both to the real and complex case, so that we will only distinguish the two instances when there is some risk of confusion. One crucial relation we are going to exploit in the sequel is the following isometric property of multiple stochastic integrals (see e.g. [45,Proposition 5.5.3]): if f ∈ L 2 (ν q ) and g ∈ L 2 (ν p ) are two symmetric kernels, with p, q ≥ 1, then where δ pq is Kronecker's symbol.
We will now explain how to explicitly compute the decomposition (5.2) for a given random variable Y . For every x ∈ A, the add-one cost operator at x, written D x , is defined as follows: for every σ(N λ )-measurable random variable Y = Y (N λ ) where δ x stands for the Dirac mass at x. As explained in [33,34], the mapping on A × Ω given by (x, ω) → D x Y (N λ (ω)) is always jointly measurable. For every q ≥ 2, one then recursively defines the operators (that also automatically define jointly measurable mappings -see [33]), where we set D = D 1 .
In the discussion to follow, we will extensively apply a formula by Last and Penrose [34, Theorem 1.3] stating that, if Y is a square-integrable functional of N λ , then automatically, for every q ≥ 1, the mapping on A q is well-defined, symmetric and square-integrable with respect to ν q and moreover, for every integer q ≥ 1, one can choose the symmetric kernel f q in (5. 2) as We will also make use of the elementary identities contained in the next statement (the proof is by recursion).

8)
where h ∈ L 1 (ν) is real-valued. Then, for every q ≥ 1, where ∅ in (5.9) indicates the point measure with empty support, that is, the zero measure.
Note that, for the models studied in this paper, A h dν = 0, so triviallyN λ (h) = N λ (h).
We are now ready to present and prove one of the main results of this section.
In order to conclude the proof, we need to prove the upper bound (5.19). To do this, using Equation (4.2) of [14], we write For * = loop, disk, using (4.13) and (4.14), respectively, we then have that as desired.

Chaos expansion for the Gaussian layering field
As before, in what follows G is a Gaussian measure on (A, A ), with σ-finite and nonatomic control ν. Similarly to what we explained in the previous section, for every F ∈ L 2 (σ(G)) (real or complex), there exists a unique sequence of symmetric kernels {w q : q = 1, 2, ...}, such that w q ∈ L 2 (ν q ) and where the series converges in L 2 (P) and is a multiple Wiener-Itô integral of order q of w q with respect to G. For every w ∈ L 2 , we write as before I G 1 (w) = A w(x)G(dx) := G(w). See, e.g., [44,45] for a detailed discussion of these classical objects and results. We will also use the following consequence of [45,Proposition 5.5.3]): if f ∈ L 2 (ν q ) and g ∈ L 2 (ν p ) are two symmetric kernels, with p, q ≥ 1, then where δ pq is Kronecker's symbol.
In order to carry out our analysis, we will make use some Gaussian ersatz of formula (5.7) and Lemma 5.1. The needed result corresponds to the celebrated Stroock's formula for smooth Gaussian functionals (first proved in [55]). Since the general statement of Stroock's formula requires a fair amount of preliminary technical considerations, we content ourselves with one of its corollaries, stated in the lemma below. This yields a Gaussian counterpart to the content of Lemma 5.1.

Lemma 5.2.
For every real-valued h ∈ L 2 (ν) and every real ξ, one has that the chaotic decomposition (5.30) of F = e iξG(h) is given by There exists a direct connection between Equation (5.12) and Equation (5.33) that we shall implicitly exploit in the discussion to follow. To see this, take h ∈ L 1 (ν) ∩ L 3 (ν) (and therefore h ∈ L 2 (ν)) such that ν(h) = 0. Then, it is easily seen that, if λ → ∞ and β → 0 in such a way that λβ 2 → ξ 2 , one has that and, for every fixed x 1 , ..., x q ∈ A, The following statement is the Gaussian analog of Theorem 5.1. The proof is very similar to that of of Theorem 5.1, but there are some differences, so we spell it out for completeness and for the reader's convenience.
ξ ∈ H −α be as in Theorem 4.9. Then, for every ϕ ∈ C ∞ 0 (D), V * ξ (ϕ) admits a Wiener-Itô chaos expansion (5.30) with kernels w q = w ϕ q = w ϕ q,ξ ∈ L 2 (ν q ) given by where we have used the fact that W δ ξ and W * ξ are continuous linear functionals from H α 0 to R, combined with elementary properties of norms. The random variable δ −2∆ * ξ W δ ξ (ϕ) is in L 2 (σ(G)) and therefore admit a chaos expansions. Moreover, because of the L 2 convergence above, the same is true for W * ξ (ϕ). In addition, it follows that the kernels w δ,ϕ q of the chaos expansion of δ −2∆ * ξ W δ ξ (ϕ) converge in L 2 (ν q ) to the kernels w ϕ q of the chaos expansion of W * ξ (ϕ). We will now show that this last statement implies (5.36). Using Lemma 5.2 and a stochastic Fubini theorem to interchange stochastic and deterministic integration, we can write Comparing the last expression with (5.30), we see that the kernels of the chaos expansion of δ −2∆ * ξ W δ ξ (ϕ) can be written as Using the equality (see, e.g., [45, Example 5.3.6 (i)]) pointwise for all z ∈ D. Moreover, using (5.40) and the definition of h δ z (see Section 4.5), we see that Comparing this with (5.22), and using the arguments following that equation, we obtain the uniform bound |W δ,ϕ q;x 1 ,...,xq (z)| ≤ C * |ϕ(z)|d where C * = 1 for * = loop, disk and C * = e 2λ(1−cos β)μ(γ:z∈γ,γ⊂D) for * = m, as in Equation (5.19). Since C * does not depend on z and ϕ is in for every (x 1 , . . . , x q ) ∈ A q . Since w δ,ϕ q converges in L 2 (ν q ) to w ϕ q , as δ → 0, and ν q is σ-finite, the considerations above imply that (5.36) holds.

Convergence of layering fields to complex Gaussian multiplicative chaos
In this section, using the Wiener-Itô chaos expansion presented in the previous section, we prove that the Poisson layering fields constructed in Theorem 4.5 converge to the Gaussian layering fields constructed in Theorem 4.9 in the appropriate limit of the parameters β → 0 and λ → ∞. The kind of converge we will be dealing with is considerably weaker than the one described in (3.14) (and encountered several times in the previous sections). Obtaining convergence in a stronger sense (e.g., as in (3.14)) seems to be a challenging problem, and we leave it open for future research.
Proof. We only focus on * = loop, m since the proof for * = disk is completely analogous. Thanks to Theorem A.6, it suffices to consider fields defined on the unit disk D. Accordingly, for the rest of the proof we restrict our attention to the case D = D. We will subdivide the proof into two parts. For each ϕ ∈ C ∞ 0 (D), let and be the Wiener-Itô chaos expansions of V * λ,β (ϕ) and W * ξ (ϕ), respectively, as in Theorems 5.1 and 5.3. In the second part of the proof, we will show that, for every ϕ ∈ C ∞ 0 (D), the following conditions hold.
Before showing that conditions 1-4 above hold, we now prove how they imply that, for every ϕ ∈ C ∞ 0 (D), The proof of this claim consists in a generalization of the arguments used in [56, p. 337-339]. For every q ≥ 1, we say that an element g q of L 2 (ν q ) is simple if g q is symmetric and has the form where R is a finite integer, c( ) are constants and, for fixed , the A j 's are pairwise disjoint sets with finite ν-measure. Note that, if Z is either one of the random measures G or N λ , then I Z q (g q ) is the homogeneous polynomial of degree q given by We now observe that, in view of (6.6) and (6.8), for every ε > 0 there exist N 0 ≥ 1 and λ 0 , β 0 > 0, such that: (a) and (c) as a consequence of (6.7), 14) The key point is now that, by a standard density argument (see e.g. [45,Lemma 5.5.2]), for N 0 as above, there exist simple kernels g q ∈ L 2 (ν q ), q = 1, ..., N 0 , such that and consequently, for λ ≥ λ 0 and β ≤ β 0 , Now consider a bounded 1-Lipschitz mapping ψ : C → R. We have that, for fixed ε and N 0 and for λ ≥ λ 0 and β ≤ β 0 as above, The previous bound is obtained by applying twice the triangle inequality, and then by exploiting the estimate (deduced by virtue of the Lipschitz nature of ψ and of the isometric property (5.4)) together with (6.13), (6.16) and (6.12), as well as a similar bound (based instead on the isometric property (5.32)) for the quantity Using (6.5) and (5.1), we eventually infer that as λ → ∞. In view of this, since ε is arbitrary, the Portmanteau theorem implies the desired conclusion (6.9). We are now going to verify that Conditions 1-4 are satisfied in our setting. To this end, recall the notation (4.7), applied below in the case D = D, as well as (4.27) We now use the inequality α loop D (z, t) ≤ α loop B z,|z−t| ,B z,2 (z) and (4.13) to obtain (6.20) which implies, together with Lemma A.1, which gives again the desired conclusion. 3. Using (5.13) and (5.36), and the obvious inequality |a + b| 2 ≤ 2|a| 2 + 2|b| 2 , we have that Letting and we have For * = loop, using Fubini's theorem and the independence between the loops γ k and their signs k under ν q , as well as (A.27), the first term in the right hand side of (6.28) can be bounded as follows: under the second integral in the right hand side of (6.29). The second term on the right hand side of (6.29) is the same as the double integral in the right hand side of (6.19), which has already been shown to be finite; the first term is clearly finite and tends to 0 as λ → ∞, β → 0 with λβ 2 → ξ 2 . Hence, we conclude that We now turn to the second summand on the right-hand side of (6.28). For * = loop, using again Fubini's theorem and the independence between the loops γ k and their signs k , as well as (A.27), the second term on the right-hand side of (6.28) can be bounded as follows: For * = m, one can use (A.26) and (A.28) to obtain an upper bound of the same type, with some additional factors that are bounded over D. To conclude, we will now use the fact that Indeed, since ξ 2 < 5 by assumption, one has that, for λ large enough and β small enough, the term A defined in (6.32) verifies A ≤ η q , for some η < 5. Exploiting (6.33) together with the fact that x q < q!e x for every q ≥ 1 and x > 0, we conclude that, for λ large and β small enough, dzdt Such a relation, together with the dominated convergence theorem and Lemma A.1, yields immediately that (6.34) 4. For * = loop, using (5.13) and Fubini's theorem, we have that, for every n ≥ 1, under the integral signs; the resulting upper bound is denoted by R m (λ, β, n). Now, reasoning as above, for λ large and β small enough one has that λβ 2 < η, for some η < 5, and also 2∆ loop λ,β = λ 5 (1 − cos β) < 1/2. These considerations imply that, for every n ≥ 1, where H n (x) := ∞ q=n x q q! , x > 0. Since H n (x) → 0, as n → ∞ and H n (x) < e x , by combining (6.33) with Lemma A.1 and dominated convergence, we deduce that and the proof is therefore concluded.

A.1 Background on Sobolev spaces
We follow [12,Section 5], and refer the reader e.g. to [7, for a classical presentation of the material discussed in the present section. We fix a bounded simply connected domain D ⊂ R 2 , and denote by H 1 0 the classical Sobolev Hilbert space associated with D, which is obtained as the closure of C ∞ 0 (D) (the class of smooth functions on D having compact support) with respect to the norm f 2 There exists an orthonormal basis {u i : i ≥ 1} of L 2 (D) composed of (normalized) eigenfunctions of the Laplace operator on D with Dirichlet boundary condition; one has also that {u i } is an orthogonal basis of H 1 0 , and Green's formula implies that As a consequence of these facts, each f ∈ H 1 0 admits a unique orthogonal decomposition f = i a i u i , and f 2 admits an orthogonal decomposition f = i a i u i in H 1 0 . By analogy, one defines H α 0 to be the closure of C ∞ 0 (D) with respect to the norm f 2 We emphasize the following inclusions, that are used throughout the paper: for α > 0, Remark A.1. In the previous discussion, we have implicitly used the fact that, if f ∈ C ∞ 0 (D), then where a i = f, u i L 2 (D) as before. In order to prove (A.2), one can assume without loss of generality that α ≥ 1 is an integer. For such an α and for every f ∈ C ∞ 0 (D), one has that ∆ α f ∈ C ∞ 0 (D), and consequently ∆ α f = i ∆ α f, u i L 2 (D) u i , where the series converges in L 2 (D). Integration by parts yields moreover that ∆ α f, u i L 2 (D) = (−1) α f, ∆ α u i L 2 (D) = λ α i f, u i L 2 (D) , from which we deduce that as claimed.

A.2 A useful lemma
The following lemma is used several times in the paper.
Lemma A.1. Let D be the unit disk of C, and a, b, c ∈ [0, 1); then Proof. As the integrand is nonnegative, the double integral is well-defined and Tonelli's theorem implies that it is actually equal to an iterated integral, where the order of integration can be interchanged. We subdivide the domain of the integral into several regions. Fix n ∈ N, n > 6, so that 2 sin( π n ) < 1, and consider the points u k = exp( ik2π n ), k = 0, . . . , n − 1. Notice that d(u k , u k+1 ) = 2 sin( π n ), and that for η ≤ 2 sin( π n ) 3 4 we have We fix such an η for the rest of the proof. For each z ∈ D \ B 0,1−η , let k(z) be such that u k(z) minimizes d(z, u k ) over k = 0, . . . , n − 1; we observe that the quantity k(z) is uniquely defined, except for a subset of D with zero Lebesgue measure, and that we can restrict each integral to the complement of such a set without loss of generality. Next, we consider the following (overlapping) regions, covering D 2 : .., n − 1. We now show that the restriction of the integral (A.4) to each of the regions introduced above is finite. (I) Using symmetry and polar coordinates from the origin for both variables (with obvious notation), we have dr z < ∞ as both a, b < 1.
(II) If t ∈ B u k(z) ,2 sin( π n ) then (since |z| ≥ 1 − η) one has that |z − t| ≥ 2 sin( π n ); as a consequence, (III) By invariance of the Lebesgue measure, it is sufficient to consider the case k = 0. We use polar coordinates centered at u k = u 0 : this time, r z = |u 0 − z|, r t = |u 0 − t| and θ z is the angle between u 0 0 and u 0 z; to simplify the discussion we consider that the angle θ z lives in the interval (−π, π], in such a way that θ z ∈ [0, π] whenever z is situated on, or above, the segment u 0 0. First, we observe that |z − t| ≥ |r t − r z |, by virtue of the triangle inequality. Next, we will bound from below 1 − |z|; an analogous bound holds for 1 − |t|. Let u(z) be the point at the intersection of B u 0 ,rz and B 0,1 such that sign(θ u(z) ) = sign(θ z ) (where θ u(z) is the angle in the polar coordinates representation of u(z), as described above); then, θ u(z) = arccos( rz 2 )sign(θ z ). It follows that, for each value of r, the angle θ z satisfies |θ z | ∈ 0, arccos r z 2 for all z such that r z = r. In addition, we have Using the change of variables u = 2 cos θ − r, we have, for each b < 1, Then, for some constant c 1 > 0, as sin( π n ) < 1/2. This concludes the proof.

A.3 Proofs from Section 4.2
The following estimate is needed in the proof of Lemma 4.1.
Proof. By the scale invariance of Brownian motion and the definition of the Brownian loop soup (see Section 2.2), for each 0 < a < b, we have that as desired.
Proof of Lemma 4.1. The first inequality in (4.16) is a consequence of the definition. Then, using the obvious inclusion where we used Lemma A.2 with a = R 2 n+1 and b = T = R 2 n , and, in the inequality before the last one, the fact that α loop δ 1 ,δ 2 (z) = 1 5 log δ 2 δ 1 . As the bound above does not depend on δ, and since α loop δ,R (z) − α m δ,R (z) ≥ 0 is monotone in δ, we have that Finally, as D is bounded, As a consequence, which concludes the proof.
Proof of Theorem 4.2. Following the proof of Lemma A.1 in [14], scale invariance implies that, for z ∈ D and 0 < δ < R, we have so that the first two terms are equal. Next, setting ρ = |z 1 − z 2 |, we have where the last equality follows from a direct application of (A.8). Letting B z 1 (D) denote the largest open disc centered at z 1 and contained in D, with radius r z 1 (D), and using the translation, rotation and scale invariance of µ * , the last term of the last line of (A.9) can be written as To show the existence of the limit, note that lim ρ→0 µ * (γ : 0 ∈γ, γ ⊂ B 0,1 , γ ⊂ B 0,rz 1 (D)/ρ , 1 / ∈ γ) = α ¬B 0,1 (0|1), (A. 10) where α ¬B 0,1 (0|1) < ∞ because of the thinness of µ * . Using again the translation, rotation and scale invariance of µ * , again by the thinness of µ * .
The first and third terms on the right hand side of the last equality tend to zero as z 2 → z 1 because of the thinness of µ * , as in (A.11). The second and fourth terms can be treated in the same way, and we focus only on the second. Using the elementary fact that d z 2 ≤ d z 1 + |z 1 − z 2 | and either (4.13) or (4.14), we have that This concludes the proof of the theorem.
Proof of Lemma 4.4. First of all, we observe that, in view of (4.21) and (4.22), using (4.13), we have that, for * = loop, m, For the disk model, using (4.14), we have lim z 2 →z 1 g disk (z 1 , z 2 ) = lim The continuity of α * D (z 1 , z 2 ) in its two arguments, for z 1 = z 2 , can be inferred from the following upper bound: All terms in the upper bound tend to zero as w 1 → z 1 and w 2 → z 2 . We only show that this is the case for the first term, since all the others can be treated analogously. Using the translation, rotation and scale invariance, of µ loop , we have that where the limit follows from the thinness of µ loop . For µ m , it suffices to note that For µ disk , we can use the same argument as for µ loop , given that µ disk is thin, as show in Lemma 4.2. It remains to check that g * (z, w) → g * (z 1 , z 1 ) as z, w → z 1 . For * = loop, using (4.13), we have that as z, w → z 1 by the continuity of d z and Theorem 4.2. For * = disk one can do exactly the same calculation using (4.14) instead of (4.13). For * = m, using again (4.13), one has that as z, w → z 1 by the continuity of d z and Lemma 4.3. With this, the proof of the lemma is concluded.

A.5 Existence and conformal covariance of layering fields
In this section, we prove the existence and the conformal covariance of the Poisson and Gaussian layering fields. This can be seen as an extension of a result of [14], where existence and conformal covariance are proved for the n-point functions of Poisson layering fields. We starts with the existence of Poisson layering fields in bounded domains with a C 1 boundary.
Theorem A.3. If D is a bounded simply connected domain with a C 1 boundary, then Theorem 4.5 holds.
Proof. By an application of Theorem 3.3, it suffices to check conditions (3.11), (3.12), and (3.13). This can be done using elementary properties of Poisson point processes and arguments analogous to those in the proofs of Theorem 4.1 of [14] and Proposition 5.3 of [12]. (Note, however, that the notation in those papers is different. For instance, what we would denote α loop δ,D (z|w) in this paper would be denoted α δ,D (z) in [14].) Using the definition of U δ λ,β , we have that which verifies condition (3.12) with τ δ z,w = λα * δ,D (z|w)(1 − cos β). Moreover, if δ ≤ δ < min(d z,w , d w,z ), noting that α * δ ,δ (z, w) = 0 when δ < |z − w|, we can write , which verifies (3.13). For * = disk, (4.14) gives analogous results with the quantity The next result concerns the existence of Gaussian layering fields in bounded domains with a C 1 boundary.
Theorem A.4. If D is a bounded simply connected domain with a C 1 boundary, then Theorem 4.9 holds.
Proof. By Theorem 3.3, it suffices to check (3.11), (3.12), and (3.13). Using the fact that Cov(G(h δ z )G(h δ w )) = α * δ,D (z, w), (A. 43) which follows from the definition of the Gaussian process G (see (3.2) and the discussion right after), we have that for all z, w ∈ K and all 1 ≥ δ > 0, as well as for δ = 0, provided that z = w. We observe that F K is bounded for δ > 0 and that, when δ = 0, it can be extended continuously by setting, for each z ∈ K, for all 0 ≤ δ ≤ 1 and all z, w ∈ D provided z = w. We note that, for all δ > 0 and every z ∈ D, lim w→z H(z, w; δ) = −∞. For δ = 0, in view of (4.21) and (4.22), using To conclude the proof for * = loop, we note that, in order to apply Theorem 1.1 of [27], we also need the condition ξ/ √ 5 < √ 2, which is equivalent to ∆ loop ξ < 1/2. Applying Theorem 1.1 of [27], we can now conclude that where the convergence is in probability in the Sobolev space H −α for any α > 1.
For * = m, using (A.49), we can write In the rest of this section, we show the existence of layering fields for general domains D conformally equivalent to the unit disk D and prove their conformal covariance. Let f : D → D be a conformal map. In view of our use of the notation U D (ϕ) = D U D (z)ϕ(z)dz, the conformal covariance proved below can be (formally) expressed in the following way: Remark A.5. For the disk model, an analog of (A.66) and of Theorem A.6 below hold when D = D and f is a Möbius transformation. The proof is the same as that of Theorem A.6.
The following statement is the main result of the section. Proof. In this proof, * = loop or m and we will use the notation d z,D instead of d z to denote dist(z, ∂D) to avoid confusion. We first deal with the Poisson layering fields V loop λ,β,D and V m λ,β,D . Let N B z,δ λ,D (z) denote the random variable corresponding to γ γ where the sum is over loops γ from a Brownian loop soup winding around z, contained in D, but not completely contained in B z,δ , where each γ is 1 or −1 with equal probability. Then, we can write Now consider a test function ϕ ∈ C ∞ 0 (D). The considerations above apply to all z ∈ supp(ϕ) provided that δ < dist(supp(ϕ), ∂D). Hence, we have that where the limits are in distribution and the existence of the limit in (A.81) is guaranteed by Theorem A.3. The proof for the Gaussian layering fields W loop ξ,D and W m ξ,D is similar; we sketch the key argument for completeness. Assume again that z ∈ D is such that d z,D > δ and d f −1 (z),D > |(f −1 ) (z)|δ, let w = f −1 (z), s = |(f −1 ) (z)| = 1/|f (w)|, and recall the notation A δ,V (z) = {γ : z ∈ γ, δ ≤ diam(γ), γ ⊆ V }) (A.83) and A U,V (z) = {γ : z ∈ γ, γ ⊆ U, γ ⊆ V }. (A.84) With this notation, we can write A δ,D (z) and A sδ,D (w) as a disjoint unions, as follows: . Now consider a test function ϕ ∈ C ∞ 0 (D). The considerations above apply to all z ∈ supp(ϕ) provided that δ < dist(supp(ϕ), ∂D). Hence, we have that