Density of imaginary multiplicative chaos via Malliavin calculus

We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\beta := :e^{i\beta \Gamma (x)}:$$\end{document}μβ:=:eiβΓ(x): for a log-correlated Gaussian field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 1$$\end{document}d≥1 dimensions. We prove a basic density result, showing that for any nonzero continuous test function f, the complex-valued random variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\beta (f)$$\end{document}μβ(f) has a smooth density w.r.t. the Lebesgue measure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}$$\end{document}C. As a corollary, we deduce that the negative moments of imaginary chaos on the unit circle do not correspond to the analytic continuation of the Fyodorov-Bouchaud formula, even when well-defined. Somewhat surprisingly, basic density results are not easy to prove for imaginary chaos and one of the main contributions of the article is introducing Malliavin calculus to the study of (complex) multiplicative chaos. To apply Malliavin calculus to imaginary chaos, we develop a new decomposition theorem for non-degenerate log-correlated fields via a small detour to operator theory, and obtain small ball probabilities for Sobolev norms of imaginary chaos.


Introduction
In this paper we study imaginary Gaussian multiplicative chaos, formally written as μ β :=: e iβ (x) :, where is a log-correlated Gaussian field on a bounded domain U ⊂ R d and β a real parameter. The study of imaginary chaos can be traced back to at least [8,12], in case of cascade fields to [5], and to [16,18] in a wider setting of log-correlated fields.
Imaginary multiplicative chaos distributions : e iβ (x) : can be rigorously defined as distributions in a Sobolev space of sufficiently negative index [16]. In the case where is the 2D continuum Gaussian free field (GFF), they are related to the sine-Gordon model [16,19] and the scaling limit of the spin-field of the critical XOR-Ising model B Juhan Aru juhan.aru@cantab.net 1 Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland is given by the real part of : e i2 −1/2 (x) : [16]. Imaginary chaos has also played a role in the study of level sets of the GFF [29], giving a connection to SLE-curves. In [10] it was shown using Wiener chaos methods that certain fields constructed using the Brownian Loop Soup converge to imaginary chaos. Recently, reconstruction theorems have been proved for both the continuum [4] and the discrete version [14] of the imaginary chaos, showing that, somewhat surprisingly, when d ≥ 2 it is possible to recover the underlying field from the information contained in the imaginary chaos in the whole subcritical phase β ∈ (0, √ d). In a wider context, real multiplicative chaos : e γ (x) :, with γ ∈ R has been the subject of a lot of recent progress (see e.g. reviews [24,26]). Complex and in particular imaginary multiplicative chaos appear then naturally, for example, as analytic extensions in γ . Complex variants of multiplicative chaos also come up when studying the statistics of zeros of the Riemann zeta function on the critical line [28].
The main result of this paper is the existence and smoothness of density for random variables of the type μ β ( f ). The main contribution, however, is probably the technique used to prove the main result. Indeed, whereas in the case of imaginary multiplicative cascades [6] and real multiplicative chaos [27] rather direct Fourier methods give the existence of a density, this approach is problematic in the case of imaginary chaos. The main obstacle is the presence of cancellations that are difficult to control without an exact recursive independence structure or monotonicity. We circumvent these problems by turning to Malliavin calculus. Interestingly, in order to apply methods of Malliavin calculus we have to first obtain new decomposition theorems for log-correlated fields, and prove quite technical concentration estimates for tails of imaginary chaos.

The main result: existence of density
Let us now denote by μ = μ β the imaginary chaos with parameter β ∈ (0, √ d) in d dimensions. In the appendix of [20] and in [16] the tails of this random variable were studied and it was shown that P[|μ( f )| > t] behaves roughly like exp(−t 2d/β 2 ) -this basically follows from the fact that using Onsager inequalities, one can obtain a very good control on the moments of imaginary chaos.
In the present article we are interested in the local properties of the law of μ β ( f ) and our main result is that this random variable has a smooth density. The following slightly informal statement is made precise in Theorem 3.6.

Theorem Let be a non-degenerate log-correlated field in an open domain U and let f be a nonzero continuous function with compact support in U . Then the law of μ β ( f ) is absolutely continuous with respect to the Lebesgue measure on C and the density is a Schwartz function.
Moreover, for any η > 0 the density is uniformly bounded from above for β ∈ (η, √ d) and converges to zero pointwise as β → √ d. Finally, the same holds in the case where μ β is the imaginary chaos corresponding to the fieldˆ with covariance E[ˆ (x)ˆ (y)] = − log |x − y| on the unit circle, with f being any nonzero continuous function defined on the circle.

Remark
The reason why the circle field is brought out separately is because it does not satisfy our definition of non-degenerate log-correlated fields, see Sect. 2, and requires a bit of extra work. With similar work other cases of degenerate log-correlated fields could be handled. However, a unified approach to handle a more general class of log-correlated fields is still lacking.
The requirement of compact support for f can also be dropped in many situations. For example, the theorem is also true in the case where is the zero-boundary GFF on a bounded simply connected domain in R 2 and f ≡ 1.
This theorem has already proved to be useful in further study of imaginary chaos 1 , but we also expect this basic result and the method to be useful more generally in the study of complex chaos [18], and in studying the integrability results related to multiplicative chaos [17,25] and the Sine-Gordon model. Not only should one be able to use this technique to prove density results in these more general cases, but as a corollary one can deduce the existence of certain negative moments, which have played important role in the above-mentioned results. In a follow-up work, we will prove by independent methods that the density for imaginary chaos is in fact everywhere positive.

An application to the Fyodorov-Bouchaud formula
Let us mention here one direct application of our results, linking our studies to recent integrability results on the Gaussian multiplicative chaos stemming from Liouville conformal field theory [17,25]. Namely, in [25] the author proved that for real γ ∈ (0, √ 2) the total mass of : e γ (x) :, where is the log-correlated Gaussian field on S 1 with covariance C(x, y) = − log |x − y|, has an explicit density w.r.t. the Lebesgue measure; this was conjectured in [13] and proved by different methods in [11]. Moreover, in Theorem 1.1 of [25] the author proves an explicit expression for the p−th moment of Y γ := 1 2π S 1 : e γ (x) : dx with −∞ < p < 2/γ 2 : where with a slight abuse of notation is here the usual -function. 2 Notice that for any p, the expression is analytic in γ (outside of isolated singularities) and in particular analytic in a neighbourhood around the imaginary axis. So naively one might think that at least as long as the moments are defined for : e iβ (x) :, they would correspond to the expression given by (1.1) with γ = iβ. And indeed, it is not hard to see that for p ∈ N this is the case. Our results however imply that this cannot be true in general, even in the case where the p−th moment is well-defined for the imaginary chaos. In other words, the analytic extension of the moment formulas is in general different from naively changing γ in the Wick exponential. (1). One can then interpret the above corollary as saying that for γ = iβ, the law of Y iβ cannot be given by

Other results: a decomposition of log-correlated fields and Sobolev norms of imaginary chaos
As mentioned, our main tool in the proof of Theorem 3.6 is Malliavin calculus which is an infinite-dimensional differential calculus on the Wiener space introduced by Malliavin in the seventies [21]. Whereas Malliavin calculus has been used to prove density results in various other settings [22], we believe that it is a novel tool in the context of multiplicative chaos and could possibly have further interesting applications-e.g. in proving density results for more general models. In order to apply Malliavin calculus, we need to derive some results that could be of independent interest. First, we derive a new decomposition theorem for non-degenerate log-correlated fields. The following statement is more carefully formulated in Theorem 4.5 and the proof has an operator-theoretic flavour.
Theorem Let be a non-degenerate log-correlated Gaussian field on an open domain U ⊆ R d with covariance kernel given by − log |x − y| + g(x, y) and g subject to some regularity conditions. Then, for every V U we may write (possibly in a larger probability space) where Y is an almost -scale invariant field and Z is a Hölder-regular field independent of Y , both defined on the whole of R d .
Second, we develop a way to study the small ball probabilities of f μ β H −d/2 (R d ) . The precise version of the following statement is given by Proposition 6.7.
This result is closely related to small ball probabilities of the Malliavin determinant of μ β ( f ). To prove it we establish concentration results on the tail of imaginary chaos.

Structure of the article
We have set up the article to highlight how the general theory of Malliavin calculus is applied to prove such a density result and what are the concrete estimates of imaginary chaos needed to apply it. After collecting some preliminaries in Sect. 2, we use Sect. 3 to walk the reader through the relevant notions and results of Malliavin calculus in the context of imaginary multiplicative chaos, thereby building up the backbone of the proof of the main theorem. In that section we state carefully the main result, and prove it up to technical estimates. The remaining proofs are then collected in Sect. 5 and in Sect. 6; the former contains some general lemmas of Malliavin calculus, and the latter deals with concentration results for imaginary chaos, including the proof of the Proposition 6.7 above. In Sect. 4 we prove the decomposition theorem stated above.

Log-correlated Gaussian fields and imaginary chaos
In this section we establish the formal setup for the log-correlated field and of the imaginary chaos associated to , often denoted by : exp(iβ ) : with β ∈ R.

Log-correlated Gaussian fields
Let U ⊂ R d be a bounded and simply connected domain and suppose we are given a kernel of the form where g is bounded from above and satisfies g(x, y) = g(y, x). Furthermore, we assume that g ∈ H d+ε loc (U × U ) ∩ L 2 (U × U ) for some ε > 0. 3 We may also extend C(x, y) as 0 outside of U ×U . Then C defines a Hilbert-Schmidt operator on L 2 (R d ), and hence C is self-adjoint and compact.
Assuming C is positive definite, by spectral theorem there exists a sequence of strictly positive eigenvalues λ 1 ≥ λ 2 ≥ · · · > 0 and corresponding orthogonal eigenfunctions ( f k ) k≥1 spanning the subspace L := (Ker C) ⊥ in L 2 (R d ). We may now construct the log-correlated field with covariance kernel C(x, y) via its Karhunen-Loève expansion This definition makes sense since C 1/2 is an injection on L. We will define the KL-basis (e k ) k≥1 for H by setting e k := √ λ k f k , and we will also write , h H := ∞ k=1 A k h, e k H for h ∈ H . The left hand side in the latter definition is purely formal since / ∈ H almost surely. Let us finally define what we mean by a non-degenerate log-correlated field in all of this paper. Definition 2.1 (Non-degenerate log-correlated field) Consider a kernel C (x, y) = C(x, y) from (2.1) and the associated log-correlated field , given by (2.2). We call the kernel C and the field non-degenerate when C is an injective operator on L 2 (U ), i.e. Ker C = {0}.
Note that for covariance operators injectivity is equivalent to being strictly positive in the sense that C f , f > 0 for all f ∈ L 2 (U ), f = 0. 4 The standard log-correlated field on the circle.
The only degenerate field we will work with in this paper is the standard logcorrelated field on the circle. I.e. it is the field on the unit circle which has the covariance C (x, y) = log 1 |x−y| , where one now thinks of x and y as being complex numbers of modulus 1. Equivalently, we may consider the field on [0, 1] with the covariance in which case we may write where A k and B k are i.i.d. standard normal random variables. This circle field is degenerate because it is conditioned to satisfy 1 0 (e 2πiθ ) dθ = 0 and the operator C maps constant functions to zero. It is however not hard to see that after restricting the domain of the field (e 2πi· ) to I 0 := [−1/4, 1/4] it becomes non-degenerate.

Imaginary chaos
Let us now fix β ∈ (0, √ d). For any f ∈ L ∞ (U ) we may define the imaginary chaos μ tested against f via the regularization and renormalisation procedure where ε is a convolution approximation of against some smooth mollifier ϕ ε . An easy computation shows that the convergence takes place in L 2 ( ). Importantly, the limiting random variable does not depend on the choice of mollifier. Again, one has to be careful however when defining μ( f ) for uncountably many f simultaneously. Indeed, μ turns out to have a.s. infinite total variation, but it does define a random H s (R d )-valued distribution when s < −β 2 /2 [16]. One may also (via a change of the base measure in the proofs of [16]) fix f ∈ L ∞ (R d ) and consider g → μ( f g) as an element of H s (R d ). Although μ is not defined pointwise, we will below freely use the notation U f (x)μ(x) dx to refer to μ( f ).

Malliavin calculus: basic definitions
In this subsection we will collect some very basic notions of Malliavin calculus: the Malliavin derivative and Malliavin smoothness. We will mainly follow [22] in our definitions, making some straightforward adaptations for complex-valued random variables both here and in the following sections.
Let C ∞ p (R n ; R) be the class of real-valued smooth functions defined on R n such that f and all its partial derivatives grow at most polynomially.

Definition 2.2
We say that F is a smooth (real) random variable if it is of the form for some h 1 , . . . , h n ∈ H and f ∈ C ∞ p (R n ; R), n ≥ 1. For such a variable F we define its Malliavin derivative D F by Thus we see that D F is an H -valued random variable and in fact, in the case where F is a smooth random variable, D F corresponds to the usual derivative map: for any h ∈ H , we have that One may also define D m F as a H ⊗m -valued random variable by setting In our case H is a space of functions defined on U and hence H ⊗m can be seen as a space of functions defined on U m . At times it will be convenient to write down the arguments of the function explicitly using subscripts, e.g. for all t 1 , . . . , t m ∈ U we set We extend the above definition in a natural way to complex smooth random variables by setting when F and G are real smooth random variables. Thus in general D will map complex random variables to the complexification of H , which we denote by H C . We will assume that the inner product ·, · H C is conjugate linear in the second variable. From here onwards we will use F for complex-valued Malliavin smooth random variables, unless otherwise stated.
To define D for a larger class of random variables one uses approximation by the smooth functions above. More precisely, we define for any non-negative integer k and real p ≥ 1 the class of random variables D k, p as the completion of (complex) smooth random variables with respect to the norm The spaces D k, p are decreasing with p and k, and we denote D ∞ := p,k≥1 D k, p . Similarly we set D k,∞ := p≥1 D k, p .
Finally, viewing D as an unbounded operator on L 2 ( ; C) with values in L 2 ( ; H C ), we may define its adjoint δ which is also called the divergence operator. More specifically we have

Density of imaginary chaos via Malliavin calculus
Let f be a continuous function of compact support in U . Our goal is to apply Malliavin calculus to show that the random variable M := μ( f ) has a smooth density with respect to the Lebesgue measure on C.
We start by walking through the basic results of Malliavin calculus that we want to apply and we then reduce the proof of Theorem 3.6 to concrete estimates on imaginary chaos. Some useful lemmas of Malliavin calculus are proven in Sect. 5 and the estimates on imaginary chaos are verified in Sect. 6, with input from Sect. 4.
Formally one can write the Malliavin derivative DM of M = μ( f ) as The content of the following proposition is to make the above computations rigorous by truncating the series ∞ n=1 , e n H e n (x) to be able to work with Malliavin smooth random variables, as in Definition 2.2.
The reason we are interested in showing that M belongs to D ∞ is the following classical result of Malliavin calculus, stating sufficient conditions for the existence of a smooth density. For convenience we state it here directly for complex valued random variables. Proof Following [22], the Malliavin matrix of a random vector F = (F 1 , . . . , F n ) ∈ R n is given by γ F := ( D F j , D F k H ) n j,k . We will use Proposition 2.1.5 from [22], which states that if F i ∈ D ∞ and E| det γ F | − p < ∞ for all p ≥ 1, then F has a density w.r.t. the Lebesgue measure on R n which is a Schwartz function.

Proposition 3.2 Let F ∈ D ∞ be a complex valued random variable and let
As Re F, Im F ∈ D ∞ by assumption, it is enough to check that det γ F is equal to the given formula in the case F = (Re F, Im F). This is easy to check by writing and expanding the squares on the right hand side. We leave the details to the reader.
Thus to show that F has a smooth and bounded density it will be enough to show that the negative moments of D F 4 H C − | D F, D F H C | 2 are all finite. In fact this quantity is not straightforward to control directly and to make calculations possible, we first apply the following projection bounds, whose proofs we postpone to Sect. 5: To further show that the density is uniformly bounded in β outside any interval surrounding the origin, we need to have some quantitative control on the densities. We will use the following simple adaption of Lemma 7.3.2 in [23] to the complex case to do this: Bounding δ(A) is again technically not straightforward, but the following general bound could possibly be of independent interest. It is again proved in Sect. 5.

Proposition 3.5 Let F be a complex Malliavin random variable in
Using the above results on Malliavin calculus, we can now reduce Theorem 3.6 to concrete propositions on imaginary chaos. Proving the estimates needed for these propositions is basically the content of Sect. 6.
We start with a precise statement of the main theorem: There are basically two technical chaos estimates needed to deduce the theorem. First, super-polynomial bounds on small ball probabilities of the Malliavin determinant are used both to prove that the density exists and is a Schwartz function, and to show uniformity: Proposition 3.7 Let , f , M = μ( f ) be as in the theorem above. Then we have the following bounds for the Malliavin determinant det γ M . For any ν > 0, there exist constants C, c, a, ε 0 > 0 (which do not depend on β) such that for all ε ∈ (0, ε 0 ) and for all β ∈ (ν, √ d), Here the bound on and We can now prove Theorem 3.6 modulo these propositions.

Proof of Theorem 3.6
To apply Proposition 3.2 to prove that M = μ( f ) has a density w.r.t. Lebesgue measure, and that moreover this density is a Schwartz function, we need to verify two conditions: • That M ∈ D ∞ -this is the content of Proposition 3.1; • And that E| det(γ M )| − p < ∞ for all p ≥ 1 -this follows directly from the bound (3.4) in Proposition 3.7.
Finally, it remains to argue that the density is uniformly bounded from above for β ∈ (η, √ d) for some fixed η > 0, and converges to zero pointwise on R d as β → √ d. This follows from Lemma 3.4, once we show that E|δ(A)| 4 is uniformly bounded in β ∈ (η, √ d) and tends to zero as β → √ d. By Proposition 3.5 By using the inequality (x + y) 4 x 4 + y 4 and then Cauchy-Schwarz we have that We thus conclude from (3.5) in Propositions 3.7 and 3.8.
The proofs of the above-mentioned chaos estimates appear in Sect. 6. More precisely, • In Sect. 6.2 we prove that M is in D ∞ , i.e. Proposition 3.1. This boils down to bounding moments of DM and is a rather standard calculation. Similar computations with small improvements on existing estimates allow to prove Proposition 3.8 in Sect. 6.3. • In Sect. 6.4, we prove Proposition 3.7, which requires a novel approach. It is also in this subsection where we make use of the almost global decomposition theorem for non-degenerate log-correlated fields, proved in Sect. 4.
The missing general results of Malliavin calculus are proved in Sect. 5.

Almost global decompositions of non-degenerate log-correlated fields
It is often useful to try to decompose the log-correlated Gaussian field on the open set U ⊂ R d as a sum of two independent fields Y and Z , where Y is in some sense canonical and easy to calculate with, and Z is regular. In [15] it was shown that such decompositions exist around every point x 0 ∈ U when g ∈ H s loc (U × U ) for some s > d and Y is taken to be a so-called almost -scale invariant field.
Our goal in this section is to establish a more general variant of this decomposition theorem which removes the need to restrict to small balls and works in any subdomain V U (we write A B to indicate that A ⊂ B) by simply assuming that is nondegenerate on V , meaning that C defines an injective integral operator on L 2 (V ), as explained in Sect. 2.
In the context of the present article, the usefulness of this result is strongly interlinked with the following standard comparison result for Cameron-Martin spaces. In the case of Reproducing Kernel Hilbert spaces, this can be found for example in [3].

the Cameron-Martin spaces of and Y respectively. Then H Y ⊂ H and moreover for every h ∈ H Y , we have that h H Y ≥ h H .
Basically, via this Lemma our decomposition allows to meaningfully transfer calculations on the initial field to easier ones on the almost -scale invariant fields Y , where Fourier methods become available.
We will start by recalling the basic definitions related to -scale invariant and almost -scale invariant log-correlated fields. We then state the theorem and discuss heuristics, and finally prove the theorem in two last subsections. In this section all function spaces are the standard function spaces for real-valued functions, i.e. we don't need to consider their complexified counterparts.

Overview of -scale and almost -scale invariant log-correlated fields
To define -scale invariant and almost -scale invariant fields, we first need to pick a seed covariance k. For simplicity we will in what follows make the following assumptions on k: The seed covariance k : R d → R satisfies the following properties: The fact that k is supported in B(0, 1) yields the useful property that distant regions of the associated Gaussian field will be independent.
Let us also remark that an easy way to construct a seed covariance k satisfying the above assumptions is to take a smooth, non-negative and rotationally symmetric function ϕ supported in B(0, 1/2) with ϕ L 2 = 1 and then letting k = ϕ * ϕ be the convolution of ϕ with itself.

Definition 4.3
Let k : R d → R be as above. The -scale invariant covariance kernel C X associated to k is given by Similarly, the related almost -scale invariant covariance kernel C Y = C Y (α) associated to k and a parameter α > 0 is given by We often use approximations Y δ of Y , which can be defined via the stochastic integrals We also define the tail fieldŶ δ := Y − Y δ , which decorrelates at distances bigger than δ. The following lemma then gives basic estimates on the covariance of this tail field. See Appendix A for the proof.

Statement of the theorem and the high level argument
The main theorem of this section can be stated as follows.
where Y is an almost -scale invariant field with seed covariance k and parameter α and Z is a Hölder-regular field independent of Y , both defined on the whole of R d .
Notice that the 2D zero boundary Gaussian free field is a non-degenerate logcorrelated field in the open disk. However, there is no hope to decompose it using an almost -scale invariant field on the whole of D, so in that sense the above theorem is as global as you could hope. 5

Remark 4.6
In [15, Theorem B] it was shown that even for a degenerate log-correlated field , one can for any x ∈ U find a ball B(x, r (x)), restricted to which is nondegenerate and can be decomposed as an independent sum of an almost star-scale invariant field and a Hölder-regular field. In this sense one can see Theorem 4.5 as a generalization in the special case of non-degenerate fields.
Before going to the proof of Theorem 4.5, let us try to illustrate the high level argument in terms of the following toy problem on the unit circle T = {z ∈ C : |z| = 1}: Let be a non-degenerate log-correlated field on T with covariance of the form log 1 |x−y| + g(|x − y|), where now also the g term only depends on the distance between the two points. This means that we can write the covariance using the Fourier series with |dx| denoting the arc-length measure. As is assumed to be non-degenerate, we know that 1 n + g n > 0 for all n ≥ 1. The almost -scale field would correspond to a field with covariance of the form and thus the difference between the tail and the two covariances would be It is now easy to see that if g n = O(n −s ) for some s > 1 + α, the coefficients in the above difference are positive for all large enough n. By further reducing α, we can guarantee that 1 n 1+α + g n > 0 for all n ≥ 1, so that the difference C − C Y is again a positive definite kernel.
The main issue in implementing this strategy for general log-correlated covariances on domains in R d is the fact that in general we do not have a canonical basis such that C and C X would be simultaneously diagonalizable. To still be able to make useful calculations, we thus want to find some universal, non-basis dependent setting, where both can be studied. This is comfortably offered for example by the Fourier transform on spaces L 2 (R d ) and H s (R d ). Thus as a first step we will find a suitable extension of to a log-correlated field on the whole of R d with covariance of the form C X + R where C X is the covariance of a -scale invariant field and R is the kernel of an integral operator which maps . The second step is then to actually make the calculations work, and to do this in the general set-up we make use of some operator-theoretic methods.

Extension of log-correlated fields to the whole space
Let us begin by solving the aforementioned extension problem. In what follows we will denote by the same symbols both the integral operators and their kernels, and C X (resp. C Y (α) ) will always refer to the covariance operator of a -scale (resp. almost -scale) invariant field with a fixed seed covariance k (resp. and parameter α).
First of all, we note the existence of the following partition of unity consisting of squares of smooth functions.

Lemma 4.7 Let U ⊂ R d be an open domain and V U an open subdomain. Then there exists an open set W with V W U and non-negative functions a, b ∈
Secondly we need the following estimates on the covariance operator C X .

Lemma 4.8 For any s ∈ R the operator C X is a bounded invertible operator
Moreover the Fourier transform of the associated kernel Proof We have C X f = K * f , so it is enough to study the Fourier transform of K . We computê Sincek(0) > 0 and alsok(ξ ) = O(|ξ | −α ) for some α > d + 1, we see that the above quantity is bounded from above and below by a constant multiple of (1 + |ξ and one again sees that this is bounded from above and below by a constant multiple of (1 + |ξ Next we note that since k is compactly supported,k is smooth and also |∇k( from which the second claim follows. As a corollary of the following lemma from [15] we can rephrase (2.1) using a -scale invariant covariance instead of pure logarithm.
Let us next prove the extension itself. We emphasise that the kernel R in the proposition below is not necessarily definite positive. Then there exists a bounded integral operator R : L 2 (R d ) → L 2 (R d ) such that C X + R is strictly positive and the corresponding kernels satisfy for all x, y ∈ V . The kernel R is Hölder-continuous with some exponent γ > 0 and moreover, there exists δ > 0 such that R defines a bounded operator be as in Lemma 4.7 and consider the (distribution-valued) Gaussian field Z = a + bX defined on R d . Here and X are independent and have covariance operators C and C X respectively. By using Lemma 4.9 we can write Thus we may write the kernel of the covariance operator of Z as We conclude that G is a bounded operator H r (R d ) → H r +s (R d ).
Let us then consider the operator T with kernel corresponding to the first term in the definition of R.
Note that since a 2 + b 2 = 1 we have y).
for any α ∈ R since a and b − 1 are compactly supported and smooth. Thus it is enough to show that A : We will show the claim for A -the same proof works for B as well since we only use the fact that a is smooth and has compact support and we can again reduce to this situation by replacing b with b − 1.
The boundedness of A : 3) A small computation shows that we can write By using the smoothness of a, we have for . By Cauchy-Schwarz we can therefore bound the first term by This combined with using Lemma 4.8 to bound the second term we get Thus recalling that we want to prove (4.3) we have Now, as r < 0, the first term is bounded by a constant times f 2 H r (R d ) . For the second term we let p(ξ ) := |ξ ||â(ξ )| and note that since Integrating over ζ gives just p L 1 (R d ) and then by using the inequality (1 + |ξ | 2 ) r (1 + |ζ − ξ | 2 ) −r (1 + |ζ | 2 ) r we may also integrate over ξ and ζ separately to see that the above is bounded by a constant times Thus putting things together we obtain (4.3 is the space of δ-Hölder functions vanishing at infinity, thatg is γ -Hölder for some γ > 0. By (4.2) this implies that we only need to show that (a(x)a(y) + b(x)b(y) − 1)C X (x, y) is Hölder-continuous. As this term is compactly supported, we can add a smooth cutoff function ρ such that As a is smooth, the map (x, y) → 1 0 ∇a(x + u(y − x)) du is in particular a Hölder continuous map R 2d → R d . Thus it is enough to show that (x, y) → (y − x) log 1 |x−y| is Hölder-continuous but this follows easily by checking that each component function (y j − x j ) log 1 |x−y| is Hölder continuous in each coordinate. The Hölder constants are also easily seen to be bounded for x, y ∈ supp ρ.
Finally let us note that C Z is strictly positive since if f ∈ L 2 (R d ) is nonzero, then at least one of f | V or f | supp b is nonzero. In the first case a(x)a(y)C (x, y) f (x) f (y) > 0 by the assumption that C was assumed to be injective in V , while in the second case

Deducing the decomposition theorem
Having obtained the desired extension, we are ready to prove the decomposition theorem. The second part of the proof consists in showing that we may subtract C Y (α) from C X + R for some small enough α > 0 and still obtain a positive operator.
To do this, we need to use the following classical stability property of strictly positive operators of the form 1 + K with K compact and self-adjoint that follows directly from the spectral theorem.

Lemma 4.11 Let H be a Hilbert space and T a self-adjoint compact operator on H
and suppose that 1 + T is strictly positive. Then there exists ε > 0 such that 1 + A + T is strictly positive for any self-adjoint A with A H→H ≤ ε.
As a consequence of the above lemma and the smoothing properties of the map R obtained in Lemma 4.10 we first create a necessary lee-room. Notice that C X + R = C 1/2 The following statement is thus effectively saying that in fact

Lemma 4.12
There is some ε > 0 such that Proof We start by observing that the operatorR = C where J is the identity map. Now, due to the fact that R(x, y) has compact support (see Eq. (4.2) and recall that C X (x, y) = 0 for |x − y| > 1) this mapping takes successively  [30]) and as the other maps are bounded, the whole composition is compact.
As R is also self-adjoint on L 2 (R d ), there is an orthonormal basis of L 2 (R d ) consisting of eigenfunctions ofR. To show that 1 +R is strictly positive it is enough to show thatR has no eigenfunctions with eigenvalues ≤ −1. Assume that f is an eigenfunction ofR with nonzero eigenvalue λ. Then by Lemma 4.10 we know that R maps H s (R d ) → H s+2δ (R d ) for any s ∈ [0, d/2] and thus after applyingR to f roughly 1/δ times we see that actually f ∈ H d/2 (R d ). Thus there exists some g ∈ L 2 (R d ) such that f = C 1/2 X g, and we have that by the assumption on C X + R, implying that λ > −1. Thus 1 +R is strictly positive and the claim follows from Lemma 4.11.
The final important technical ingredient is that for any α 0 > 0, converges pointwise to 0 when we let the parameter α of the almost -scale invariant field Y (α) to 0.

Lemma 4.13
For all α > 0 set U α := C X − C Y (α) and let U 0 = C X . Then U for all s ∈ R, and for any α 0 > 0, we have Moreover, for any fixed α 0 > 0 and f ∈ L 2 (R d ) we have Before proving the lemma, let us see how it implies the theorem: Proof of Theorem 4.5: We begin by writing It thus suffices to show that for some sufficiently small α > 0 we have for all nonzero g ∈ L 2 (R d ). Indeed, this implies that C X −C Y (α) + R is a positive integral operator on L 2 (R d ), whose kernel by Lemma 4.10 and [15, Proposition 4.1 (iii)] is Hölder-continuous, and thus the corresponding Gaussian process has an almost surely Hölder-continuous version (see e.g. [2,Theorem 1.3.5]). In addition by Lemma 4.10 and Lemma 4.13 we see that R and To show that 1+R α is positive on L 2 (R d ) on the other hand we may write can be made as small as we wish by choosing α small. AsR α −R is self-adjoint we have By linearity and self-adjointness of C We finally prove the lemma: Proof of Lemma 4.13 Note that U α is a Fourier multiplier operator with the symbol As by assumptionk is non-negative and decays faster than any polynomial, we have that where the hidden constant does not depend on α. In particular for every α < α 0 , we Let us now fix α 0 and consider for α < α 0 the self-adjoint operator T α = U For any fixed ξ the integrand tends to 0 as α → 0. Thus, asû α (ξ ) (1 + |ξ | 2 ) − d+α 0 2 for all α < α 0 , we can apply the dominated convergence theorem to deduce that

General bounds on det M and ı(A)
In this section we prove two (to our knowledge) non-standard lemmas for Malliavin calculus, that we believe could possibly be of independent interest for proving the existence of density and its positivity also in more general settings. Firstly, we prove a certain projection bound for the determinant of complex Malliavin variables. Second, we obtain an estimate on the complex covering fields that is again a much easier starting point for further calculations.

Proof of Proposition 3.3 Let us first expand
By (3.1), we deduce that As we have the following projection inequality the result follows, once we show that for any h ∈ H C , By Cauchy-Schwarz inequality and the triangle inequality we have By now repeating the bound with h in place of h we obtain (5.2) which finishes the proof.

Bounding ı(A) via derivatives in independent Gaussian directions -Proposition 3.5
For a succinct write-up, it is helpful to use directional derivatives in independent random directions, although the proposition could also be proved by first proving a version for smooth random variables and then taking limits. Now, recall that for smooth random variables F, and h ∈ H C we could write We consider directional derivatives in independent random directions, with the law of . More precisely, let X ∼ be an independent Gaussian field defined on a new probability space ( X , F X , P X ) whose expectation we denote by E X . For a Malliavin variable F ∈ D 2,∞ , as D F ∈ H C and X is independent of , one can define and directly conclude from this definition that: We are now ready to prove Proposition 3.5.

Proof of Proposition 3.5 Write
The first term is −1 D F 2 H C |δ(D F)| in absolute value, so it is enough to consider the other two terms. By the product rule for Malliavin derivatives, we may write To bound the first term, we first notice that by Cauchy-Schwarz

D D F, D F H C , D F H C ≤ D D F, D F H C D F H C .
For the first term, it is now helpful to use the averaging in Lemma 5.1 for a quick bound. We write By Cauchy-Schwarz this can be bounded by Similarly, one can bound which we can rewrite as

By Cauchy-Schwarz this expression is bounded by
where we have used the fact (derived in Eq. (5.1)) that Thus the proposition follows from the following claim: Proof of claim Maybe the nicest way to prove this claim is to use derivatives in random directions as above. First, observe that using averaging we can write a neat analogue of Eq. (5.5) : Thus we have By triangle inequality and Cauchy-Schwarz we obtain from which the claim follows.

Estimates for Malliavin variables in the case of imaginary chaos
The aim of this section is to prove the probabilistic bounds needed to apply the tools of Malliavin calculus to M = μ( f ). We start by going through some old and new Onsager inequalities and related integral bounds. In Sect. 6.2, we prove by a rather standard argument that M is in D ∞ , i.e. Proposition 3.1. In Sect. 6.3 we derive bounds on |δ(DM)| and D 2 M H C ⊗H C and deduce Proposition 3.8 by a quite similar argument. Finally, in Sect. 6.4 we prove bounds on the Malliavin determinant of M and this is the main technical input of the paper. Here things get quite interesting -we rely both on the decomposition theorem, Theorem 4.5, and projection bounds for Mallivan determinants from Sect. 5, but also need to find ways to get a good grip on the concentration of M = μ( f ), and on Sobolev norms of the imaginary chaos μ itself.

Onsager inequalities and related bounds
In this section, we collect a few Onsager inequalities and related bounds. To this end, we define for any Gaussian field and x = (x 1 , . . . , x N ), y = (y 1 , . . . , y M ) the quantity Also, we let δ = * ϕ δ be a mollification of where ϕ δ = δ −d ϕ(·/δ) and ϕ is a smooth non-negative function with compact support that satisfies R d ϕ = 1.
The following is a restatement of a standard Onsager inequality from [16]. 6 Moreover, the same holds for the field itself.
We will also need stronger Onsager inequalities for (almost) -scale invariant fields, whose rather standard proof is pushed to the appendix A. Both of these Onsager inequalities are used in conjunction with the following bounds: • for all β > 0, Proof We only sketch the proof, as all the main ideas can be found in proof of [ ). We will next show (6.7). By mimicking the beginning of the proof of [16, Lemma 3.10], we can bound the left hand side of (6.7) by where C > 0 and the second sum runs over all nearest neighbour configurations F such that the induced graph with vertices {1, . . . , N } and edges (i, F(i)) has k components. Of course, the domain on which we integrate is actually much smaller than B(0, 1), but integrating over this larger domain will be enough for our purposes. After integration, we obtain that the left hand side of (6.7) is at most

Now, by Jensen's inequality
concludes the proof of (6.7). We finally turn to the proof of (6.5) and (6.6). By again mimicking the beginning of the proof of [16, Lemma 3.10], we can bound the left hand side of (6.5) by where M k is the number of nearest neighbour functions {1, . . . , N } → {1, . . . , N } with k components and C is some large enough constant. This concludes the proof of (6.5); the proof of (6.6) is similar.

M belongs to D ∞ -proof of Proposition 3.1
The purpose of this section is to prove Proposition 3.1. Before doing so, we collect two auxiliary lemmas from Malliavin calculus.

Lemma 6.4 ([22, Lemma 1.2.3])
Let (F n , n ≥ 1) be a sequence of (complex) random variables in D 1,2 that converges to F in L 2 ( ) and such that sup n E D F n Then F belongs to D 1,2 and the sequence of derivatives (D F n , n ≥ 1) converges to D F in the weak topology of L 2 ( ; H C ).
Second, we need a rather direct consequence of [22, Lemma 1.5.3]: Lemma 6.5 Let p > 1, k ≥ 1 and let (F n , n ≥ 1) be a sequence of (complex) random variables converging to F in L p ( ). Suppose that sup n F n k, p < ∞. Then F belongs to D k, p and F k, p ≤ C k, p lim sup n F n k, p for some C k, p > 0.

Proof of Lemma 6.5 See Appendix A.
We now have the ingredients needed to prove Proposition 3.1. The proof of this result is rather standard, but needs a bit of care as the most convenient way of obtaining Malliavin smooth random variables is truncating the Karhunen-Loève expansion of . Doing so we face the issue that there is no Onsager inequality available for this approximation of the field that we are aware of. We will bypass this difficulty by considering a further convolution of this truncated version of against a smooth mollifier ϕ and then use the Onsager inequality (6.1) for convolution approximations.

Proof of Proposition 3.1
Here, we sketch the proof and give full details in the Appendix B. We start by showing that M belongs to D ∞ . Let n ≥ 1, δ > 0, j ≥ 0 and p ≥ 1. In the following, we will denote M n,δ is a smooth random variable (in the sense of Definition 2.2) and D j M n,δ is equal to Combining Onsager inequalities, (6.4) and Lemma 6.5, one can show by taking the limit n → ∞ that for all k ≥ 1, M δ ∈ D k,2 p and that sup δ>0 M δ k,2 p < ∞.
Details of this are in the appendix. Now, because (M δ , δ > 0) converges in L 2 p towards M, Lemma 6.5 then implies that for all k ≥ 1, M ∈ D k,2 p . This concludes the proof that M ∈ D ∞ .
The proof of the formula for DM now follows via a series of approximation arguments. From the first part by taking n → ∞, one can rather quickly deduce that Next, one argues that (DM δ , δ > 0) converges in L 2 ( ; H ) towards and concludes that it necessarly corresponds to DM by Lemma 6.4. Here one again uses Onsager inequalities and dominated convergence. The full details are found in the appendix.

Bounds on |ı(DM)| and D 2 M H C ⊗H C -proof of Proposition 3.8
The goal of this section is to control the tails of |δ(DM)| and D 2 M H C ⊗H C . We first note that these two random variables can be written explicitly in terms of imaginary chaos.
The proof of (6.9) is very similar to the proof of the formula of DM and we omit the details. The origin of (6.8) can be explained by the following formal computation, that can be turned into a rigorous proof in a very similar manner as what we did in the proof of Proposition 3.1 when we obtained the explicit expression of DM -one needs to use smooth approximations both for the field , and smooth Malliavin variables.
Proof of Proposition 3. 8 We will only write the details for the variable δ(DM) since bounding the moments of D 2 M H C ⊗H C is very similar to bounding the moments of imaginary chaos itself (with the use of (6.6) instead of (6.4)).
Let N ≥ 1 and let K U be the support of f . By Lemma 6.6 we have By a limiting argument, one can justify the formal identity: Let (z 1 , . . . , z 2N ) : = (x 1 , . . . , x N , y 1 , . . . , y N ). By induction one sees that after differentiating w.r.t. the first k of the variables β 1 , . . . , β N , γ 1 , . . . , γ N and expanding one is left with a finite number of terms of the form Note that |C(z a j , z b j )| ≤ C log 4R |z a j −z b j | for some C > 0 and R large enough so that K ⊂ B(0, R). Thus applying Lemma 6.1 to each summand, we can bound the whole sum by By scaling this is less than which by Lemma 6.3 is less than C N (d − β 2 ) 3N .

Small ball probabilities for the Malliavin determinant of M -proof of Proposition 3.7
This section contains the main probabilistic input to Theorem 3.6 -the proof of Proposition 3.7. Roughly, the content of this proposition is to establish super-polynomial We will start by presenting a toy model explaining the strategy; then we explain the proof setup and prove the proposition modulo some technical chaos lemmas. The section finishes by proving the technical estimates.

A toy model: small ball probabilities for : exp(iˇGFF) : H −1 (R 2 )
To explain the strategy of our proof, we consider a toy problem asking about the small ball probabilities for norms of imaginary chaos. For concreteness, let us do it here with the 2D Gaussian free field; see Proposition 6.7 at the end of this section for a more general statement.
Writing out the norm squared, we have that where G is the Dirichlet Green's function on K . Now, the expectation E μ 2 H −1 (K ) is easy to calculate and it is bounded. As all moments exist, one could imagine proving bounds near zero by using concentration results on μ. However, these concentration results do not see the special role of zero and would not suffice for good enough bounds for asymptotics near 0.
The idea is then to use only the decorrelated high-frequency part of to stay away from zero. To make this more precise, denote by δ the part of the GFF containing only frequencies less than δ −1 and letˆ δ = − δ denote the tail of the GFF. Consider now the projection bound A small calculation shows that f δ H −1 (K ) = : e iβ δ (y) : H 1 (K ) . It is further believable that we should have : e iβ δ (y) : H 1 (K ) δ −β 2 /2 δ H 1 (K ) , and that this expression admits Gaussian concentration. As in the concrete case E δ H 1 (K ) δ −1 , we can conclude that the denominator is of order δ −1−β 2 /2 with super-polynomial concentration on fluctuations.
In the numerator, the term of the form K : e iβˆ δ (x) : e β 2 E[ δ (x) 2 ] dx remains. Such a tail chaos is very highly concentrated around its mean which is of order δ −β 2 , with fluctuations of unit order having a super-polynomial cost in δ. Thus the whole ratio will concentrate around with super-polynomial cost for fluctuations on the same scale. Thus setting ε = δ 1−β 2 /2 we obtain super-polynomial decay for P μ H −1 (K ) < ε . Whereas this is good enough for any fixed β, observe that as β → √ 2 the exponent 1 − β 2 /2 goes to 0. Moreover, we have E μ 2 δ H 1 (K ) does not depend on β, we see that we are in fact losing in terms of β 2 − 2 as well.
Illustratively, we are losing in high frequencies because we are replacing After taking expectation, in terms of near-diagonal contributions, as G(x, y) ∼ − log |x−y| near the diagonal, this basically translates to replacing − |x| −β 2 /2 log |x| with |x| −β 2 /2 and results in the loss of a factor of 2 −β 2 as β 2 → 2. Thus we have to tweak our test function f δ further to at the same time guarantee sufficient concentration and not to lose too much on tails.
We will see later on that this strategy gives us more generally the following result.
The same strategy for the determinant requires some extra input, yet the key ideas are present already in this toy model: the projection bound corresponds to the analogue of Malliavin determinants given by Lemma 3.3, the concentration of the numerator to Lemma 6.8 and that of the denominator to Lemma 6.9. The only new technical ingredient will enter as Lemma 6.10.

Proof setup and proof of Proposition 3.7 modulo technical lemmas
Let f be a bounded continuous function whose support is a compact subset of U and set M = μ( f ). Our goal in this section is to obtain lower bounds on P[det γ M ≥ λ], where det γ M is the Malliavin determinant (3.1).
As in the toy problem, it is not so clear how to obtain sharp bounds directly and the idea is to use the projection bound from Lemma 3.3, which says that for any h ∈ H C . A key step is the specific choice of h(x), which needs to at the same time give a precise enough bound and allow for chaos computations. Moreover, we have to ensure that it also belongs to the Cameron-Martin space. Here, one of the technical difficulties is that in general we do not have a good understanding of the Cameron-Martin space of . To deal with that, we will use the decomposition theorem, Theorem 4.5 to be able to work with almost -scale invariant fields. More precisely, let us fix an open set V with V a compact subset of U such that supp f ⊂ V . Then by Theorem 4.5 one can write | V = Y + Z =: X where Y is an almost -scale invariant field with smooth and compactly supported seed covariance k and parameter α, and Z is an independent Hölder-continuous field. Recall further the approximations Y ε of Y of such a field from Sect. 4.1 and the notation for its tail where the right hand side only depends on μ, and thus on , restricted to V . Thus, to obtain bounds on det γ M , we can instead of working with the (complexified) Cameron-Martin space H C = H ,C , just as well work with the Cameron-Martin space of Y + Z , which is defined on the whole plane. Apologising for the abuse of notation, we still denote it by H C . This small trick allows us to use the independence structure of the field Y , and also puts Fourier techniques in our hand.

Definition of h.
Whereas the decomposition theorem and the change of Cameron-Martin space make the computations potentially doable, they become practically doable only with a very careful choice of the test function h. Namely, we set is defined using a smooth indicator g δ of δ-separated squares and the parameter δ will be chosen in a suitable way according to λ. More precisely, let Q δ be the collection of cubes of the form where k 1 , . . . , k d ∈ Z. Note in particular that the cubes are δ-separated and hence the restrictions ofŶ δ to two distinct cubes in Q δ are independent. We then set where ϕ is a smooth mollifier supported in the unit ball and ϕ δ (x) = δ −d ϕ(x/δ). We note that h is indeed almost surely an element of H C , since the Malliavin derivative of (iβ) − is almost surely smooth so multiplying by it shows that for all small enough δ > 0 and all β ∈ (ν, √ d).

Lemma 6.9
For all η > 0 small enough, we can choose C > 0 such that where W is a Y δ -measurable positive random variable. Moreover, we can pick c 1 , c 2 > 0 such that for all δ ∈ (0, 1) and t ≥ c 1 δ −2−η we have for all small enough δ > 0 and all β ∈ (ν, √ d).
We now explain how we deduce Proposition 3.7 from these lemmas, and then in the next subsections turn to their proofs.

Proof of Proposition 3.7 By Lemma 3.3, we have that
so it suffices to bound P Here h δ is as above and we will choose δ depending on ε.
Using Lemma 6.9, we first bound for some η > 0 Hence, taking c to be the constant from Lemma 6.8 we can bound The second term can be bounded using Lemma 6.9 loosely by exp(−c 1 δ −c 1 ) for some c 1 > 0. For the first term, Lemma 6.8 gives that and Lemma 6.10 gives constants c 3 > 0 and we thus obtain the proposition. The case of the standard log-correlated field on circle needs extra attention, and is treated in Sect. 6.4.6.
One can see that a simplified version of the above proof can also be used to prove Proposition 6.7.

Proof of Proposition 6.7
Recall that on the support of f , we can write |V = Y + Z = X , where Y is almost −scale invariant and Z is Holder regular, both defined on the whole space. Note that by Lemma 4.8 and Theorem 4.5 the operators C Y and C Z are bounded from H −d/2 (R d ) to H d/2 (R d ) and hence so is C X . Thus for any so that in particular Using this inequality one can proceed as in the proof of Proposition 3.7 except one does not need to take care of the term D M, h δ .
The rest of this subsection is dedicated to the proofs of Lemmas 6.8, 6.9 and 6.10, and sketching the extension to the case of the circle.

Proof of Lemma 6.8
Proof of Lemma 6.8 Let us fix some ν > 0 small. Note that DM, h δ H C is equal to We can write In particular, by the Paley-Zygmund inequality for any such square Q it holds that P[J Q ≥ λ(d − β 2 ) −2 |Z , E Q ] ≥ p, where λ = C/2 and p > 0 is some constant. In the following, we denote byQ δ the collection of those squares in which f is larger than f ∞ /2 (again, we may consider − f instead of f if needed). Now, recall that Z is a Hölder continuous Gaussian field, and thus by local chaining inequalities (e.g. Proposition 5.35 in [31]), we have that for some universal constant .
As P(E c ) ≤ C exp(−Cδ −2 ) and E ⊆ Q E Q , it remains to only take care of the second term working under the assumption that the event E Q holds for all Q. For any t > 0 to be chosen later, we have where Bin(n, p) denotes the Binomial distribution. In the second line we used the conditional independence of J Q given Z and the conditional probability obtained above; on the last line we used the Hoeffding's inequality Noting that c 1 δ −d ≤ |Q δ | ≤ c 2 δ −d for some c 1 , c 2 > 0, we see that by choosing t = pβλδ d /(2c 2 ) we get

Proof of Lemma 6.9
Proof of Lemma 6. 9 We start with some immediate bounds that allow the usage of inequalities on Sobolev spaces H s C (R d ). First, by Lemma 4.8 we have for some C > 0. On the other hand, by Lemma 4.1, we have that be a non-negative function which equals 1 in the support of g δ (recall that g δ is defined in (6.11)). Set Using the above norm bounds in conjunction with the classical inequality for any ε > 0 (see e.g. Theorem 5.1 in [7]), we can bound h δ H C by some constant times We can bound g δ H d/2+ε To do this, we will use Gaussian concentration inequalities. Namely, by Theorem 4.5.7 in [9], if X is isonormal on a Hilbert space H , and any T : We will make use of this concentration in the case T = · H d/2+ε (R d ) to bound W := T (F). We first apply Theorem A in [1], which gives that :: e iβŶ δ (y) : R δ (x, y)dxdy =: iβ We can then first bound If we expand the 2N -th moment of such a sum, we obtain terms of the form Before taking expectation in each such term we separate the field We can then write each term as where the integration is over x j , y j ∈ Q j and x j , y j ∈ Q j . We bound the expectation by δ N 2 , where q and q denote the vectors of midpoints for the ordered squares Q j and Q j . This can be seen by noting that since the seed covariance k is Lipschitz, we have Thus we obtain the upper bound where now : e iβŶ δ (x) :: e iβŶ δ (y) : R δ (x, y)dxdy.
By Hölder's inequality we can bound By scaling the right hand side equals δ 4Nd By relabeling the points as z 1 , . . . , z 4N and using Lemma 6.2 we then have the upper bound which by Lemma 6.3 is bounded by for some constant C > 0. Hence we can bound E| DM, h δ | 2N by where for convenience we have turned q, q back to x, x by paying the same price. The latter integral is the 2N -th moment of the 2β chaos of field Y δ 1/2 , which by Lemma 6.2 and (6.7) is bounded by Note that for any fixed b, C, ν > 0 we have 2b −1 C log 1 δ < δ −ν and δ small enough. One thus sees that yields the desired upper bound by choosing e.g. N = δ −β 2 /(24d) .

Special case: the standard log-correlated field on the circle
In this section we will briefly explain how to extend the proof of Proposition 3.7 to the case where we are interested in the total mass of the imaginary chaos defined using the field on the unit circle which has the covariance log 1 |x−y| , where one now thinks of x and y as being complex numbers of modulus 1. See Sect. 2 for the precise definitions.
Recall, that the extra complication in this case is that the field is degenerate in the sense that it is conditioned to satisfy 1 0 (e 2πiθ ) dθ = 0. In terms of the proof of Proposition 3.7 this creates some annoyance, as the function h δ we used in the projection bounds does not anymore belong to the Cameron-Martin space H C of , and we will instead need to look at the functionh δ = h δ − h δ (y) dy.
This will let us still use the decomposition X = Y + Z where |I 0 = X |I 0 and streamline most of the proof. In the case of Lemmas 6.8 and 6.10, i.e. in terms DM,h δ H C and DM,h δ H C , this subtraction of the mean introduces the extra term iβ M 1 0 h δ (y) dy. In the case of Lemma 6.9, we have an extra term of the form | for all δ small enough.
Proof We will bound the N -th moment of |M h δ (y)|, use the Chebyshev inequality and optimize over N . Note that by the Cauchy-Schwarz inequality we have 2N 1/2 and by [16,Theorem 1.3] we know that (recall that we are currently in a onedimensional setting) for some C > 0. We mention that, in the article [16], the dependence of the above constant in terms of β was not stated but follows from their approach (see (6.4)). To bound E[| 1 0 h δ (y) dy| 2N ], we note that by Jensen's inequality we have where the right hand side equals We bound |ψ(x)| 2 e −β 2 E[Y δ (x) 2 ] by Cδ β 2 and since R δ (x, y) = 0 whenever x, y do not belong to the same square, we can bound the above expression by By developing the expectation into a multiple integral, using an Onsager inequality associated to the smooth field Z (see (6.3)) and then rewriting the multiple integrals as an expectation, we see that we can get rid of the field Z in the above expectation by only paying a multiplicative price C N .
Thus it remains to bound To bound this expectation, we expand the product and obtain a multiple integral over The expectation of the product of : e iβŶ δ (δ y) : and : e −iβŶ δ (δz) : leads to E(Ŷ δ (δ·); y; z) that we bound using the Onsager inequality (6.2). Since for any fixed y and z, we can first integrate the variables x i and control the remaining integral over y i and Altogether we obtain that which gives us the tail estimates Optimising over N now concludes.
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A Appendix: Some standard proofs
Proof of Lemma 4.4 It is calculationally somewhat easier to work with the rescaled field Y ( ) (x) =Ŷ (δx), which can be expressed using white noise as: e du/2k (e u (t − x)) 1 − δ α e −αu dW (t, u).
The first inequality then follows directly: k(e u (x − y)) du ≤ log 1 |x − y| by the fact that k is supported in B(0, 1) and k(t) ≤ 1 for all t.
For the second inequality we compute from which the claim follows. Finally, the independence comes from the fact that k is supported in B(0, 1) Proof of Lemma 6.2 Let us begin with the field Y ε . Set q j = 1 for 1 ≤ j ≤ N and q j = −1 for N + 1 ≤ j ≤ 2N and note that for all s, t ≤ ε ∧|x − y| and E[Y δ (x) 2 ] ≤ log 1 δ for all δ ∈ (0, 1).
As the fieldŶ ε (εx) has the same distribution as the field Y (ε) (x) from the proof of Lemma 4.4, we have Finally, if R is a regular field then Proof of Lemma 6. 5 We prove this lemma in the context of real-valued random variables. The extension to complex-valued random variables follows immediately.
In page 58 of [22], an operator L on the set of variables with finite second moment is introduced and used to define the norm |F| k, p := E ((I − L) k/2 F) p 1/ p . The norms | · | k, p and · k, p are equivalent (see [22] page 77). Hence sup n E ((I − L) k/2 F n ) p < ∞. By weak compactness of balls in L p ( ), we can extract a subsequence (n(i), i ≥ 1) such that ((I − L) k/2 F n(i) , i ≥ 1) converges weakly towards some element G. Since the L p -norm is weakly lower-semicontinuous, we moreover have Since (e k 1 ⊗ · · · ⊗ e k j , k 1 , . . . , k j = 1 . . . n) is an orthonormal family of H ⊗ j , we deduce that f (x l ) f (y l ) (C * (ϕ δ ⊗ ϕ δ )(x l , y l )) j e β 2 E( δ ;x;y) ≤ C j, p f where K is the support of f . Importantly, the above constant C j, p does not depend on δ. Notice that Hence, if we let ε > 0 be such that β 2 /2+ε < d/2, there exists C j, p > 0 independent of δ such that (e k * ϕ δ )(x)e k .
One can then show that (DM n,δ , n ≥ 1) converges in L 2 ( ; H ) towards On the other hand, the first part of the proof showed that sup n E DM n,δ and the e k , k ≥ 1, form an orthonormal family of H , we have Each single term in the above sum goes to zero as δ → 0. Moreover, using Onsager inequality for convolution approximations (6.1), one can obtain a domination in a similar manner as what we did in the first part of the proof. By the dominated convergence theorem, it implies that (B.4) goes to zero as δ → 0. Secondly, ((e k * ϕ δ )(x) − e k (x))((e k * ϕ δ )(y) − e k (y)) dxdy (B.5) where K is as before the support of f . The above integrand is dominated by the integrable function C |x − y| −β 2 log(c/|x − y|). Dominated convergence theorem thus implies that (B.5) goes to zero as δ → 0. Putting things together, we have shown the aforementioned convergence: (DM δ , δ > 0) converges in L 2 ( ; H ) towards With (B.3), we notice that sup δ E DM δ 2 H C < ∞ and Lemma 6.4 also shows that (DM δ , δ > 0) converges to DM in the weak topology of L 2 ( ; H ). This yields