Antti Kupiainen
email: antti.kupiainen@helsinki.fi

R Émi Rhodes
email: remi.rhodes@wanadoo.fr

Vincent Vargas
email: vincent.vargas@ens.fr

INTEGRABILITY OF LIOUVILLE THEORY: PROOF OF THE DOZZ FORMULA

Keywords: Liouville Quantum Gravity, quantum field theory, Gaussian multiplicative chaos, Ward identities, BPZ equations, DOZZ formula

 proposed a remarkable explicit expression, the so-called DOZZ formula, for the 3 point structure constants of Liouville Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given earlier by F. David and the authors. This result is a fundamental step in the path to prove integrability of LCFT, i.e. to mathematically justify the methods of Conformal Bootstrap used by physicists. From the purely probabilistic point of view, our proof constitutes the first rigorous integrability result on Gaussian Multiplicative Chaos measures.

 [START_REF] Polyakov | Quantum geometry of bosonic strings[END_REF]where he proposed a path integral theory of random two dimensional Riemannian metrics. Motivated by an attempt to solve LCFT Belavin, Polyakov and Zamolodchikov (BPZ hereafter) formulated in their 1984 paper [START_REF] Belavin | Infinite conformal symmetry in two-dimensional quantum field theory[END_REF] the general structure of Conformal Field Theory (CFT hereafter). In the BPZ approach the basic objects of CFT are correlation functions of random fields and solving CFT consists in deriving explicit expressions for them. BPZ proposed to construct the correlation functions of a CFT recursively from two inputs: the spectrum and the three point structure constants. The former summarizes the representation content of the CFT (under the Virasoro algebra) and the latter determine the three point correlation functions, see Section 1.1. The recursive procedure to find higher point correlation functions is called Conformal Bootsrap. Though BPZ were able to find spectra and structure constants for a large class of CFT's (e.g. the Ising model) LCFT was not one of them 1 . The spectrum of LCFT was soon conjectured in [12,8,[START_REF] Gervais | [END_REF] but the structure constants remained a puzzle.

A decade later, Dorn and Otto [START_REF] Dorn | Two and three point functions in Liouville theory[END_REF] and independently Zamolodchikov and Zamolodchikov [START_REF] Zamolodchikov | Structure constants and conformal bootstrap in Liouville field theory[END_REF] (DOZZ hereafter) proposed a remarkable formula for the structure constants of LCFT the so-called DOZZ formula. Even by the physicists' standards the derivation was lacking rigor. To quote [START_REF] Zamolodchikov | Structure constants and conformal bootstrap in Liouville field theory[END_REF]: "It should be stressed that the arguments of this section have nothing to do with a derivation. These are rather some motivations and we consider the expression proposed as a guess which we try to support in the subsequent sections." Ever since these papers the derivation of the DOZZ formula from the original (heuristic) functional integral definition of LCFT given by Polyakov has remained a controversial open problem, even on the physical level of rigor.

Recently the present authors together with F. David gave a rigorous probabilistic construction of Polyakov's LCFT functional integral [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]. This was done using the probabilistic theory of Gaussian Multiplicative Chaos (GMC). Subsequently in [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF] we proved identities for these correlation functions postulated in the work of BPZ (conformal Ward identities and BPZ equations). This provided a probabilistic setup to address the conformal bootstrap and in particular the DOZZ formula. In this paper we address the second problem: we prove that the probabilistic expression given in [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] for the structure constants is indeed given by the DOZZ formula. This result should be considered as an integrability result for LCFT and in particular for GMC. As such it constitutes the first rigorous proof of integrability in GMC theory. Integrability of GMC theory was also conjectured in statistical physics in the study of disordered systems. In this context, a remarkable formula for GMC on the circle has been proposed by Fyodorov and Bouchaud [START_REF] Fyodorov | Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential[END_REF], based on arguments similar to [START_REF] Dorn | Two and three point functions in Liouville theory[END_REF][START_REF] Zamolodchikov | Structure constants and conformal bootstrap in Liouville field theory[END_REF] (soon after, numerous integrability results for GMC on different 1d geometries appeared in the physics literature: see [START_REF] Fyodorov | Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of 1/f Noises generated by Gaussian Free Fields[END_REF] for instance in the case of the unit interval). It turns out that the Fyodorov-Bouchaud formula is a particular case of the conjectured one point bulk structure constant for LCFT on the unit disk with boundary (these formula can be found in Nakayama's review [START_REF] Nakayama | Liouville field theory: a decade after the revolution[END_REF]). We believe that our approach can be adapted to this situation and hence will provide a rigorous derivation of the Fyodorov-Bouchaud formula (work in progress [START_REF] Rémy | The Fyodorov-Bouchaud formula[END_REF]).

It should be noted that the LCFT structure constants and the DOZZ formula have a wider relevance than the scaling limits of planar maps. It has been argued in Ribault's review [START_REF] Ribault | Conformal Field theory on the plane[END_REF] that LCFT seems to be a universal CFT: e.g. the minimal model structure constants (e.g. Ising model, tri-critical Ising model, 3 states Potts model, etc...) originally found by BPZ may be recovered from the DOZZ formula by analytic continuation. In another spectacular development the LCFT structure constants show up in a seemingly completely different setup of four dimensional gauge theories via the so-called AGT correspondence [START_REF] Alday | Liouville Correlation Functions from Four Dimensional Gauge Theories[END_REF] (see the work by Maulik-Okounkov [START_REF] Maulik | Quantum Groups and Quantum Cohomology[END_REF] for the mathematical implications of these ideas).

In the remaining part of this introduction we briefly review LCFT in the path integral and in the conformal bootstrap approach and state the DOZZ formula.

1.1. LCFT in the path integral. In the Feynman path integral formulation, LCFT on the Riemann sphere Ĉ is the study of conformal metrics on Ĉ of the form e γφ(z) |dz| 2 where z is the standard complex coordinate and d 2 z the Lebesgue measure. φ(z) is a random function (a distribution in fact) and one defines an "expectation" (1.1) F := F (φ)e -SL(φ) Dφ where S L is the Liouville Action functional

(1.2) S L (φ) = 1 π C (|∂ z φ(z)| 2 + πµe γφ(z) )d 2 z
see Section 2.1 for the precise formulation 2 . LCFT has two parameters γ ∈ (0, 2) and µ > 0. The positive parameter µ in front of the exponential interaction term e γφ is essential for the existence of the theory (the case µ = 0 corresponds to Gaussian Free Field theory, a completely different theory) but plays no specific role 3 . On the other hand, the parameter γ encodes the conformal structure of the theory; more specifically, one can show that the central charge 4 of the theory is c L = 1 + 6Q 2 with (1.3)

Q = 2 γ + γ 2 .
The basic objects of interest in LCFT are in physics terminology vertex operators

(1.4) V α (z) = e αφ(z)
where α is a complex number and their correlation functions

N k=1 V α k (z k ) .
Their definition involves a regularization and renormalization procedure and they were constructed rigorously in [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] for N 3 and for real α i satisfying certain conditions. The construction of the correlations in [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] is probabilistic and based on interpreting e -1 π C |∂zφ(z)| 2 d 2 z Dφ in terms of a suitable Gaussian Free Field (GFF) probability measure: see subsection 2.1 below for precise definitions and an explicit formula for the correlations in terms of Gaussian Multiplicative Chaos. 2 We use brackets and not E for the linear functional (1.1) since it turns out the measure e -S L (φ) Dφ is not normalizable into a probability measure. 3 LCFT has a scaling relation with respect to µ so that the precise value of this parameter does not matter. 4 In this article, the central charge of LCFT will not appear in a transparent way hence we refer to the work [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] or [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF] for an account on the central charge.

In particular it was proved in [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] that these correlation functions are conformal tensors. More precisely, if z 1 , • • • , z N are N distinct points in C then for a Möbius map ψ(z) = az+b cz+d (with a, b, c, d ∈ C and ad-bc = 1)

(1.5)

N k=1 V α k (ψ(z k )) = N k=1 |ψ ′ (z k )| -2∆α k N k=1 V α k (z k )
where ∆ α = α 2 (Q -α 2 ) is called the conformal weight. This global conformal symmetry fixes the three point correlation functions up to a constant:

(1.6)

3 k=1 V α k (z k ) = |z 1 -z 2 | 2∆12 |z 2 -z 3 | 2∆23 |z 1 -z 3 | 2∆13 C γ (α 1 , α 2 , α 3 )
with ∆ 12 = ∆ α3 -∆ α1 -∆ α2 , etc. The constants C γ (α 1 , α 2 , α 3 ) are called the three point structure constants and they have an explicit expression in terms of the Gaussian Multiplicative Chaos, see Section 2.3. They are also the building blocks of LCFT in the conformal bootstrap approach as we now review.

1.2. LCFT in the conformal bootstrap. The bootstrap approach to Conformal Field Theory goes back to the 70's. It is based on the operator product expansion (OPE) introduced by K. Wilson in quantum field theory. In a conformal field theory, the OPE is expected to take a particularly simple form as was observed in the 70's [START_REF] Ferrara | Tensor representations of conformal algebra and conformally covariant operator product expansion[END_REF][START_REF] Polyakov | Nonhamiltonian approach to conformal quantum field theory[END_REF][START_REF] Mack | Duality in quantum field theory[END_REF]. The simplest CFT's (like the Ising model) contain a finite number of primary fields Φ i i.e. random fields whose correlation functions transform as (1.5). The OPE is the statement that in a correlation function Φ αi (z i ) one may substitute

(1.7) Φ i (z i )Φ j (z j ) = k C k ij (z i , z j )Φ k (z j )
where C k ij (z i , z j ) is an (infinite) sum of linear differential operators which are completely determined up to the three point structure constants C ijk (these are defined in the same way as in (1.6) for LCFT). Furthermore it was argued that the resulting expansion should be convergent. A recursive application of the OPE would then allow in principle to express the N-point function in terms of the structure constants i.e. to "solve" the CFT.

The input in the bootstrap is thus the set of its primary fields, called the spectrum of the theory, and their structure constants. In unitary CFT's such as the ones describing scaling limits of reflection positive statistical mechanics models the spectrum is in principle determined by the spectral analysis of the representation of the generator of dilations acting in the physical Hilbert space of the CFT. This space can be constructed using the Osterwalder-Schrader reconstruction theorem [START_REF] Osterwalder | Axioms for Euclidean Green?s functions I[END_REF][START_REF] Osterwalder | Axioms for Euclidean Green?s functions II[END_REF] and in case of LCFT this is rather straightforward given the results of [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]: see [START_REF] Kupiainen | Constructive Liouville Conformal Field Theory[END_REF] for lecture notes on this.

There is also plenty of evidence what the spectrum of LCFT should be [12,8,[START_REF] Gervais | [END_REF], see in particular Teschner's review [START_REF] Teschner | Liouville Theory Revisited[END_REF] for a thorough discussion. It should consist of the vertex operators V Q+iP with P ∈ R + i.e. there is a continuum of primary fields (unlike say in the Ising model where there are three). Assuming this, one ends up with the following rather explicit formula for the 4 point correlation functions for α i in the spectrum [START_REF] Ribault | Conformal Field theory on the plane[END_REF]:

V α1 (z)V α2 (0)V α3 (1)V α4 (∞) = Q+iR+ C γ (α 1 , α 2 , α)C γ (2Q -α, α 3 , α 4 )|F α,{αi} (z)| 2 dα
where F α,{αi} (z) are explicit meromorphic functions (the so-called universal conformal blocks) which depend only on the parameters α i , α and the central charge of LCFT c L = 1 + 6Q 2 . The integral over α is here the standard Lebesgue integral over P (where α = Q + iP ) and corresponds to the sum in (1.7).

Note that the spectrum of LCFT consists of vertex operators with complex α whereas the probabilistic approach naturally deals with real α. Also, the main application of LCFT to Liouville Quantum Gravity involves real values for α. In the theory of Liouville Quantum Gravity, the scaling limits of e.g. Ising correlations on a random planar map are given in terms of Liouville correlations with real α's and regular planar Ising CFT correlations via the celebrated KPZ relation [START_REF] Knizhnik | Fractal structure of 2D-quantum gravity[END_REF]: for an explicit mathematical conjecture, see [START_REF] David | Renormalizability of Liouville Quantum Gravity at the Seiberg bound[END_REF][START_REF] Kupiainen | Constructive Liouville Conformal Field Theory[END_REF]. Thus the probabilistic and bootstrap approaches are in an interesting way complementary. The basis for the bootstrap approach, the DOZZ formula for C γ (α 1 , α 2 , α 3 ), has a unique meromorphic extension to α 1 , α 2 , α 3 ∈ C 3 . In our probabilistic approach we prove that the probabilistic expressions for C γ (α 1 , α 2 , α 3 ) are analytic functions in the α i around their real values and extend to meromorphic functions in C 3 that coincide with the DOZZ expression.

One should note that the bootstrap approach can be turned to a tool to classify CFT's. By making an ansatz for the spectrum one can compute the four point function Φ α1 (z 1 )Φ α2 (z 2 )Φ α3 (z 3 )Φ α4 (z 4 ) using the OPE by pairing the fields in two different ways. This leads to quadratic relations for the structure constants that one can attempt to solve. This was succesfully carried out by BPZ in [START_REF] Belavin | Infinite conformal symmetry in two-dimensional quantum field theory[END_REF] for the minimal models and has lead to spectacular progress even in three dimensions, e.g. in case of the 3d Ising model [START_REF] El-Showk | Solving the 3D Ising model with the conformal bootstrap[END_REF][START_REF] El-Showk | Solving the 3D Ising model with the conformal bootstrap II. c-minimization and precise critical exponents[END_REF]. In the case of LCFT one can check that the DOZZ expression indeed solves these quadratic equations: see the review [START_REF] Ribault | Conformal Field theory on the plane[END_REF]. Furthermore, given the DOZZ formula the bootstrap expressions are formally defined for all γ ∈ C \ iR 5 . Hence the values of γ for which the theory is defined (within the framework of theoretical physics) is quite general compared to the path integral formulation which makes sense for γ ∈ (0, 2) 6 . We consider the mathematical justification of the bootstrap approach to be a major challenge to probabilists and plan to pursue this in the case of LCFT in the future (see also [START_REF] Smirnov | Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model[END_REF][START_REF] Chelkak | Universality in the 2D Ising model and conformal invariance of fermionic observables[END_REF][START_REF] Hongler | The energy density in the planar Ising model[END_REF][START_REF] Boutillier | The critical Z-invariant Ising model via dimers: the periodic case[END_REF][START_REF] Boutillier | The critical Z-invariant Ising model via dimers: locality property[END_REF][START_REF] Chelkak | Conformal invariance of spin correlations in the planar Ising model[END_REF][START_REF] Dubédat | Exact bosonization of the Ising model[END_REF][START_REF] Chelkak | Discrete stress-energy tensor in the loop O(n) model[END_REF] for recent spectacular progress in relating the critical 2d Ising model to the predictions of the bootstrap approach).

1.3. The DOZZ formula. As mentioned above, an explicit expression for the LCFT structure constants was proposed in [START_REF] Dorn | Two and three point functions in Liouville theory[END_REF][START_REF] Zamolodchikov | Structure constants and conformal bootstrap in Liouville field theory[END_REF] . Subsequently it was observed by Teschner [START_REF] Teschner | On the Liouville three point function[END_REF] that this formula may be derived by applying the bootstrap framework to special four point functions (see section 6). He argued that this leads to the following remarkable periodicity relations for the structure constants:

C γ (α 1 + γ 2 , α 2 , α 3 ) = - 1 πµ A( γ 2 )C γ (α 1 - γ 2 , α 2 , α 3 ) (1.8) C γ (α 1 + 2 γ , α 2 , α 3 ) = - 1 π µ A( 2 γ )C γ (α 1 - 2 γ , α 2 , α 3 ) (1.9) with µ = (µπl( γ 2 4 )) 4 γ 2 πl( 4 γ 2 )
and

A(χ) = l(-χ 2 )l(χα 1 )l(χα 1 -χ 2 )l( χ 2 (ᾱ -2α 1 -χ)) l( χ 2 (ᾱ -χ -2Q))l( χ 2 (ᾱ -2α 3 -χ))l( χ 2 (ᾱ -2α 2 -χ))
. (1.10) where ᾱ = α 1 + α 2 + α 3 and

(1.11) l(x) = Γ(x)/Γ(1 -x).
The equations (1.8), (1.9) have a meromorphic solution which is the DOZZ formula. It is expressed in terms of a special function Υ γ 2 (z) defined for 0 < ℜ(z) < Q by the formula 7

(1.12) ln Υ γ 2 (z) = ∞ 0 Q 2 -z 2 e -t - (sinh(( Q 2 -z) t 2 )) 2 sinh( tγ 4 ) sinh( t γ ) dt t .
The function Υ γ 2 can be analytically continued to C because it satisfies remarkable functional relations: see formula (9.4) in the appendix. It has no poles in C and the zeros of Υ γ 2 are simple (if γ 2 ∈ Q) and given by the discrete set (-

γ 2 N -2 γ N) ∪ (Q + γ 2 N + 2 γ N).
With these notations, the DOZZ formula (or proposal)

C DOZZ γ (α 1 , α 2 , α 3 ) is the following expression (1.13) C DOZZ γ (α 1 , α 2 , α 3 ) = (π µ l( γ 2 4 ) ( γ 2 ) 2-γ 2 /2 ) 2Q-i α i γ Υ ′ γ 2 (0)Υ γ 2 (α 1 )Υ γ 2 (α 2 )Υ γ 2 (α 3 ) Υ γ 2 ( ᾱ-2Q 2 )Υ γ 2 ( ᾱ 2 -α 1 )Υ γ 2 ( ᾱ 2 -α 2 )Υ γ 2 ( ᾱ 2 -α 3 )
. 5 In fact, LCFT was defined for γ ∈ iR in the physics literature quite recently by Ribault-Santachiara [START_REF] Ribault | Liouville theory with a central charge less than one[END_REF] but the theory is very different than LCFT for γ ∈ C \ iR hence we will not discuss this case here. Let us just mention that Iklhef, Jacobsen and Saleur [START_REF] Ikhlef | Three-point functions in c 1 Liouville theory and conformal loop ensembles[END_REF] have shown how 3 point correlation functions of Conformal Loop Ensembles (CLE) relate to the three point structure constants of LCFT with γ ∈ iR discovered by Kostov-Petkova [START_REF] Kostov | Bulk correlation functions in 2D quantum gravity[END_REF] and Zamolodchikov [START_REF] Zamolodchikov | Three-point function in the minimal Liouville gravity[END_REF]. 6 In fact, one can make sense of LCFT in the path integral for γ = 2 but we will not discuss this case here. 7 The function has a simple construction in terms of standard double gamma functions: see the reviews [START_REF] Nakayama | Liouville field theory: a decade after the revolution[END_REF][START_REF] Ribault | Conformal Field theory on the plane[END_REF][START_REF] Teschner | Liouville Theory Revisited[END_REF] for instance.

The main result of the present paper is to show the first important equality between LCFT in the path integral formulation and the conformal bootstrap approach, namely to prove that for γ ∈ (0, 2) and appropriate α 1 , α 2 , α 3 the structure constants C γ (α 1 , α 2 , α 3 ) in (1.6) are equal to C DOZZ γ (α 1 , α 2 , α 3 ) defined by (1.13).

Our proof is based on deriving the equations (1.8), (1.9) for the probabilistically defined C γ . An essential role in this derivation is an identification in probabilistic terms of the reflection coefficient of LCFT. It has been known for a long time that in LCFT the following reflection relation should hold in some sense:

(1.14) V α = R(α)V 2Q-α .
Indeed the DOZZ formula is compatible with the following form of (1.14):

(1.15)

C DOZZ γ (α 1 , α 2 , α 3 ) = R DOZZ (α 1 )C DOZZ γ (2Q -α 1 , α 2 , α 3 ). with (1.16) R DOZZ (α) = -(π µ l( γ 2 4
))

2(Q-α) γ Γ(-γ(Q-α) 2 ) Γ( γ(Q-α) 2 ) Γ(-2(Q-α) γ ) Γ( 2(Q-α) γ
) .

The mystery of this relation lies in the fact that the probabilistically defined C γ (α 1 , α 2 , α 3 ) vanish if any of the α i Q whereas they are nonzero for α i < Q, see Section 2.2.

In our proof R(α) emerges from the analysis of the tail behavior of a Gaussian Multiplicative Chaos observable. We prove that it is also given by the following limit

(1.17) 4R(α) = lim ǫ→0 ǫ C γ (ǫ, α, α)
i.e. R(α) has an interpretation in terms of a renormalized two-point function. We will show that for those values of α such that R(α) makes sense from the path integral perspective, i.e. α ∈

( γ 2 , Q), R(α) = R DOZZ (α).
It turns out that some material related to the coefficient R(α) already appears in the beautiful work by Duplantier-Miller-Sheffield [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF] where they introduce what they call quantum spheres (and other related objects). Quantum spheres are equivalence classes of random measures on the sphere with two marked points 0 and ∞. Within this framework, the reflection coefficient R(α) can naturally be interpreted as the partition function of the theory 8 . Finally, let us stress that the DOZZ formula 1.13 is invariant under the substitution of parameters

γ 2 ↔ 2 γ , µ ↔ µ = (µπℓ( γ 2 4 )) 4 γ 2 πℓ( 4 γ 2 )
. This duality symmetry is at the core of the DOZZ controversy. Indeed this symmetry is not manifest in the Liouville action functional (1.2) though duality was axiomatically assumed by Teschner [START_REF] Teschner | Liouville Theory Revisited[END_REF] in his argument, especially to get (1.9). It was subsequently argued that this duality could come from the presence in the action (1.2) of an additional "dual" potential of the form e 2 γ φ with cosmological constant µ in front of it. As observed by Teschner [START_REF] Teschner | Liouville Theory Revisited[END_REF], this dual cosmological constant may take negative (even infinite) values, which makes clearly no sense from the path integral perspective. That is why the derivation of the DOZZ formula from the LCFT path integral has remained shrouded in mystery for so long 9 . 8 We will not elaborate more on this point as no prior knowledge of the work by Duplantier-Miller-Sheffield [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF] is required to understand the sequel (see [START_REF] Aru | Sun X.: Two perspectives of the 2D unit area quantum sphere and their equivalence[END_REF] for an account of the relation between [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] and [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF]). More precisely, the required background to understand R(α) will be introduced in subsection 2.7 below. 9 Indeed, there are numerous reviews and papers within the physics literature on the path integral approach of LCFT and its relation with the bootstrap approach but they offer different perspectives and conclusions (see [START_REF] Harlow | Analytic Continuation of Liouville Theory[END_REF][START_REF] O'raifeartaigh | The Two-exponential Li-ouville Theory and the Uniqueness of the Three-point Function[END_REF][START_REF] Seiberg | Notes on Quantum Liouville Theory and Quantum Gravity[END_REF] for instance)

1.4. Organization of the paper. The organization of the paper is the following: in the next section, we introduce the necessary background and the main result, namely Theorem 2.11. The next sections are devoted to the proof of the main result. Section 3 is devoted to the study of tail estimates of GMC and their connection with the reflection coefficient. In section 4, we show that the correlation functions of vertex operators are analytic functions of their arguments (α k ) k . We prove technical lemmas on the reflection coefficient in section 5. In Section 6, we prove various OPEs statements with degenerate vertex operators, which are used to derive non trivial relations between three point structure constants and the reflection coefficient. These relations will serve prominently in the proof of the DOZZ formula in sections 7 and 8.

Acknowledgements. The authors wish to thank Francois David, Sylvain Ribault and Raoul Santachiara for fruitful discussions on Liouville Field theory and the conformal bootstrap approach.

Probabilistic Formulation of LCFT and Main Results

In this section, we recall the precise definition of the Liouville correlation functions in the path integral formulation as given in [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF], introduce some related probabilistic objects and state the main results.

Conventions and notations.

In what follows, z, x, y and z 1 , . . . , z N all denote complex variables.We use the standard notation for complex derivatives

∂ x = 1 2 (∂ x1 -i∂ x2 ) and ∂ x = 1 2 (∂ x1 + i∂ x2 ) for x = x 1 + ix 2 .
The Lebesgue measure on C (seen as R 2 ) is denoted by d 2 x . We will also denote | • | the norm in C of the standard Euclidean (flat) metric and for all r > 0 we will denote by B(x, r) the Euclidean ball of center x and radius r.

2.1. Gaussian Free Field and Gaussian Multiplicative Chaos. The probabilistic definition of the functional integral (1.1) goes by expressing it as a functional of the Gaussian Free Field (GFF). Since we want LCFT to have Möbius symmetry the proper setup is the Riemann Sphere Ĉ = C ∪ {∞} equipped with a conformal metric g(z)|dz| 2 . The correlation functions of LCFT will then depend on the metric but they have simple transformation properties under the change of g, the so-called Weyl anomaly formula. We refer the reader to [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] for this point and proceed here by just stating a formulation that will be useful for the purposes of this paper.

We define the GFF X(z) as the centered Gaussian random field with covariance (see [START_REF] Dubédat | SLE and the Free Field: partition functions and couplings[END_REF][START_REF] Sheffield | Gaussian free fields for mathematicians[END_REF] for background on the GFF) Remark 2.1. In the terminology of [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF], consider the metric g(z) = |z| -4 + with scalar curvature R g (z) := -4g -1 ∂ z ∂ z ln g(z) = 4ν with ν the uniform probability measure on the equator |z| = 1. Then X is the GFF with zero average on the equator: X(z)R g (d 2 z) = 0.

For LCFT we need to consider the exponential of X. Since X is distribution valued a renormalization procedure is needed. Define the circle average of X by

(2.2) X r (z) := 1 2πi |w|=e -r X(z + w) dw w
and consider the measure

(2.3) M γ,r (d 2 x) := e γXr(x)-γ 2 2 E[Xr (x) 2 ] |x| -4 + d 2 x.
Then, for γ ∈ [0, 2), we have the convergence in probability (2.4)

M γ = lim r→∞ M γ,r
and convergence is in the sense of weak convergence of measures. This limiting measure is non trivial and is a (up to a multiplicative constant) Gaussian multiplicative chaos (GMC) of the field X with respect to the measure |x| -4 + d 2 x (see Berestycki's work [START_REF] Berestycki | An elementary approach to Gaussian multiplicative chaos[END_REF] for an elegant and elementary approach to GMC and references).

Remark 2.2. For later purpose we state a useful property of the circle averages. First, X 0 (0) = 0, the processes r ∈ R + → X r (0) and r ∈ R + → X -r (0) are two indepedent Brownian motions starting from 0. For z center of a unit ball contained in B(0, 1) c the process r ∈ R + → X r (z) -X 0 (z) is also a Brownian motion starting at 0 and for disjoint balls B(z i , 1) ⊂ B(0, 1) c the processes r → X r (z i )-X 0 (z i ) are mutually independent and independent of the sigma algebra σ{X(z); z ∈ [∪ N k=1 B(z i , 1)] c }. . 2.2. Liouville correlation functions. We may now give the probabilistic definition of the functional integral (1.1):

(2.5)

F := R e -2Qc E F (X -2Q ln |z| + + c)e -µe γc Mγ (C) dc
where E is expectation over the GFF. We refer the reader to [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] (or to [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF] for a brief summary) for the explanation of the connection between (1.1) and (2.5). Briefly, the variable c is essential and stems from the fact that in (1.1) we need to integrate over all φ and not only the GFF X which is defined up to constants. The origin of the factor e -2Qc is topological and depends on the fact that we work on the sphere Ĉ. The

random variable M γ (C) is almost surely finite because EM γ (C) = C |z| -4 + d 2 z < ∞. This implies that • is not normalizable: 1 = ∞.
The class of F for which (2.5) is defined includes suitable vertex operator correlation functions once these are properly renormalized. We set

(2.6) V α,ǫ (z) = e αc e αXǫ(z)-α 2 2 E[Xǫ(z) 2 ] |x| -4∆α + where we recall ∆ α = α 2 (Q -α 2 ). Let z i ∈ C, i = 1, .
. . , N with z i = z j for all i = j. It was shown in [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] that the limit 10 (2.7)

N k=1 V α k (z k ) := 2lim ǫ→0 N k=1 V α k ,ǫ (z k )
exists, is finite and nonzero if and only if the following Seiberg bounds originally introduced in [START_REF] Seiberg | Notes on Quantum Liouville Theory and Quantum Gravity[END_REF] hold:

(2.8)

N k=1 α k > 2Q, α i < Q, ∀i.
The first condition guarantees that the limit is finite and the second that it is nonvanishing. Indeed, if there exists i such that α i Q then the limit is zero. Note that these bounds imply that for a nontrivial correlation we need at least three vertex operators; therefore, we have N 3 in the sequel. The correlation function (2.7) satisfies the conformal invariance property (1.5).

The correlation function can be further simplified by performing the c-integral (see [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]):

N k=1 V α k (z k ) = 2µ -s γ -1 Γ(s) lim ǫ→0 E N k=1 e α k Xǫ(z k )-α 2 k 2 EXǫ(z k ) 2 |z k | -4∆α k + M γ (C) -s (2.9) where (2.10) s = N k=1 α k -2Q γ .
Using the Cameron-Martin theorem (see [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]) we may trade the vertex operators to a shift of X to obtain an expression in terms of the multiplicative chaos:

(2.11) 10 The global constant 2 is included to match with the physics literature normalization which is based on the DOZZ formula (1.13) where (2.12)

N k=1 V α k (z k ) = 2µ -s γ -1 Γ(s) i<j 1 |z i -z j | αiαj E C F (x, z)M γ (d 2 x) -s
F (x, z) = N k=1 |x| + |x -z k | γα k
Thus, up to explicit factors the Liouville correlations are reduced to the study of the random variable C F (x, z)M γ (d 2 x). In particular, the Seiberg bounds α k < Q for all k are the condition of integrability of F against the chaos measure M γ (see [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]).

Finally, we remark that expression (2.11) makes sense beyond the Seiberg bounds i.e. for some s < 0. Indeed, it was shown in [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF] that

(2.13) 0 < E C F (x, z)M γ (d 2 x) -s < ∞ provided (2.14) -s < 4 γ 2 ∧ min 1 k N 2 γ (Q -α k ), α k < Q, ∀k
with s given by (2.10). Under condition (2.14), it is then natural to define the so-called unit volume correlations by (2.15)

N k=1 V α k (z k ) uv = N k=1 V α k (z k ) Γ(s) .
i.e. we divide by the Γ function which has poles at N k=1 α k -2Q ∈ -γN. An important ingredient in our proof of the DOZZ formula is Theorem 4.1 which says that these correlation functions have an analytic continuation in the α i 's to a complex neighbourhood of the region allowed by the bounds (2.14).

Remark 2.3. The DOZZ formula for the structure constants is analytic not only in α i but also in γ. A direct proof of analyticity of the probabilistic correlation functions in γ seems difficult. However, it is an easy exercise in Multiplicative Chaos theory to prove their continuity in γ, a fact we will need in our argument. Actually, it is not hard to prove that they are C ∞ in γ but we will omit this as it is not needed in our argument.

2.3.

Structure constants and four point functions. The structure constants C γ in (1.6) can be recovered as the following limit

(2.16) C γ (α 1 , α 2 , α 3 ) = lim z3→∞ |z 3 | 4∆3 V α1 (0)V α2 (1)V α3 (z 3 )
Combining (2.11) with (2.16) we get (2.17)

C γ (α 1 , α 2 , α 3 ) = 2µ -s γ -1 Γ(s)E(ρ(α 1 , α 2 , α 3 ) -s )
where

ρ(α 1 , α 2 , α 3 ) = C |x| γ(α1+α2+α3) + |x| γα1 |x -1| γα2 M γ (d 2 x
). We will also have to work with the unit volume three point structure constants defined by the formula

(2.18) Cγ (α 1 , α 2 , α 3 ) = C γ (α 1 , α 2 , α 3 ) Γ(s) .
The four point function is fixed by the Möbius invariance (1.5) up to a single function depending on the cross ratio of the points. For later purpose we label the points from 0 to 3 and consider the weights α 1 , α 2 , α 3 fixed:

3 k=0 V α k (z k ) = |z 3 -z 0 | -4∆ -γ 2 |z 2 -z 1 | 2(∆3-∆2-∆1-∆0) |z 3 -z 1 | 2(∆2+∆0-∆3-∆1) (2.19) × |z 3 -z 2 | 2(∆1+∆0-∆3-∆2) G α0 (z 0 -z 1 )(z 2 -z 3 ) (z 0 -z 3 )(z 2 -z 1 ) (2.20)
We can recover G α0 as the following limit

(2.21) G α0 (z) = lim z3→∞ |z 3 | 4∆3 V α0 (z)V α1 (0)V α2 (1)V α3 (z 3 ) .
Combining with (2.11) we get

(2.22) G α0 (z) = |z| γα 1 2 |z -1| γα 2 2 T α0 (z) where, setting s = α0+α1+α2+α3-2Q γ , T α0 (z) is given by T α0 (z) = 2µ -s γ -1 Γ(s)E[R α0 (z) -s ] (2.23) and (2.24) R α0 (z) = C |x| 3 k=0 α k + |x -z| γα0 |x| γα1 |x -1| γα2 M γ (d 2 x).
In this paper we will study the structure constants (2.17) by means of four point functions (2.20) with special values of α 0 .

BPZ equations.

There are two special values of α 0 for which the reduced four point function T α0 (z) satisfies a second order differential equation. That such equations are expected in Conformal Field Theory goes back to BPZ [START_REF] Belavin | Infinite conformal symmetry in two-dimensional quantum field theory[END_REF]. In the case of LCFT it was proved in [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF] that, under suitable assumptions on

α 1 , α 2 , α 3 , if α 0 ∈ {-γ 2 , -2 γ } then T α0 is a solution of a PDE version of the Gauss hypergeometric equation (2.25) ∂ 2 z T α0 (z) + (c -z(a + b + 1)) z(1 -z) ∂ z T α0 (z) - ab z(1 -z) T α0 (z) = 0
where a, b, c are given by

a = α0 2 (Q -α 0 -α 1 -α 2 -α 3 ) - 1 2 , b = α0 2 (Q -α 1 -α 2 + α 3 ) + 1 2 , c = 1 + α 0 (Q -α 1 ). (2.26)
This equation has two holomorphic solutions defined on C\]1, ∞[: c,z) is given by the standard hypergeometric series. Using the facts that T α0 (z) is real, single valued and C 2 in C \ {0, 1} we proved in [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF] (Lemma 4.4) that it is determined up to a multiplicative constant as (2.28)

(2.27) F -(z) = 2 F 1 (a, b, c, z), F + (z) = z 1-c 2 F 1 (1 + a -c, 1 + b -c, 2 -c, z) where 2 F 1 (a, b,
T α0 (z) = λ(|F -(z)| 2 + A γ (α 0 , α 1 , α 2 , α 3 )|F + (z)| 2 )
where the coefficient

A γ (α 0 , α 1 , α 2 , α 3 ) is given by (2.29) A γ (α 0 , α 1 , α 2 , α 3 ) = - Γ(c) 2 Γ(1 -a)Γ(1 -b)Γ(a -c + 1)Γ(b -c + 1) Γ(2 -c) 2 Γ(c -a)Γ(c -b)Γ(a)Γ(b) provided c ∈ R \ Z and c -a -b ∈ R \ Z.
Furthermore, the constant λ is found by using the expressions (2.17) and (2.23) (note that s has a different meaning in these two expressions):

(2.30)

λ = T α0 (0) = C γ (α 1 + α 0 , α 2 , α 3 ). Hence for α 0 ∈ {-γ 2 , -2 γ } T α0 is completely determined in terms of C γ (α 1 + α 0 , α 2 , α 3
). In the case α 0 = -γ 2 we were able to determine in [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF] the leading asymptotics of the expression (2.23)

as z → 0 provided 1 γ + γ < α 1 + γ 2 < Q : (2.31) T -γ 2 (z) = C γ (α 1 - γ 2 , α 2 , α 3 ) + B(α 1 )C γ (α 1 + γ 2 , α 2 , α 3 )|z| 2(1-c) + o(|z| 2(1-c) )
where

(2.32) B(α) = -µ π l(-γ 2 4 )l( γα 2 )l(2 + γ 2 4 -γα 2 )
.

Hence, in view of (2.28) and (2.30), relations (2.31), (2.32) lead to

(2.33) B(α 1 )C γ (α 1 + γ 2 , α 2 , α 3 ) = A γ (- γ 2 , α 1 , α 2 , α 3 )C γ (α 1 - γ 2 , α 2 , α 3 )
which yields the relation (1.8) (after some algebra!) in the case

1 γ + γ < α 1 + γ 2 < Q. Hence (2.34) T -γ 2 (z) = C γ (α 1 - γ 2 , α 2 , α 3 )|F -(z)| 2 + B(α 1 )C γ (α 1 + γ 2 , α 2 , α 3 )|F + (z)| 2 .
The restriction 1 γ + γ < α 1 + γ 2 for α 1 was technical in [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF] and will be removed in section 6. The restriction α 1 + γ 2 < Q seems necessary due to the Seiberg bounds as the probabilistic C γ (α 1 + γ 2 , α 2 , α 3 ) vanishes then. Understanding what happens when α 1 + γ 2 > Q is the key to our proof of the DOZZ formula. Before turning to this we draw a useful corollary from the results of this section.

T -γ 2 (z) = C γ (α 1 - γ 2 , α 2 , α 3 )(|F -(z)| 2 + A γ (- γ 2 , α 1 , α 2 , α 3 )|F + (z)| 2 ).
The hypergeometric equation (2.25) has another basis of holomorphic solutions defined on C \ (-∞, 0):

(2.36) G -(z) = 2 F 1 (a, b, c ′ , 1 -z), G + (z) = (1 -z) 1-c ′ 2 F 1 (1 + a -c ′ , 1 + b -c ′ , 2 -c ′ , 1 -z) where c ′ = 1 + a + b -c = 1 -γ 2 (Q -α 2 ) (i.
e. these are obtained by interchanging α 1 and α 2 and replacing z by 1 -z). The two basis are linearly related

F -(z) = Γ(c)Γ(c -a -b) Γ(c -a)Γ(c -b) G -(z) + Γ(c)Γ(a + b -c) Γ(a)Γ(b) (1 -z) c-a-b G + (z) F + (z) = Γ(2 -c)Γ(c -a -b) Γ(1 -a)Γ(1 -b) G -(z) + Γ(2 -c)Γ(a + b -c) Γ(a -c + 1)Γ(b -c + 1) (1 -z) c-a-b G + (z).
and we get

T -γ 2 (z) = C γ (α 1 - γ 2 , α 2 , α 3 )(D|G -(z)| 2 + E|G + (z)| 2 )
with explicit coefficients D, E (see [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF], Appendix). On the other hand by studying the asymptotics as z → 1 we get

(2.37) T -γ 2 (z) = C γ (α 1 , α 2 - γ 2 , α 3 ) + B(α 2 )C γ (α 1 , α 2 + γ 2 , α 3 )|1 -z| 2(1-c ′ ) + o(|z| 2(1-c ′ ) ).
In view of expression (2.34), exploiting the decomposition of T -γ 2 in the basis |G -(z)| 2 , |G + (z)| 2 leads to the following crossing symmetry relation:

Proposition 2.4. Let α 2 + γ 2 < Q and α 1 + α 2 + α 3 -γ 2 > 2Q. Then (2.38) C γ (α 1 - γ 2 , α 2 , α 3 ) = T (α 1 , α 2 , α 3 )C γ (α 1 , α 2 + γ 2 , α 3 )
where T is the given by the following formula

(2.39) T (α 1 , α 2 , α 3 ) = -µπ l(a)l(b) l(c)l(a + b -c) 1 l(-γ 2 4 )l( γα2 2 )l(2 + γ 2 4 - γα2
2 ) Remark 2.5. The relations (1.8) and (2.38) were derived in the physics literature [START_REF] Teschner | On the Liouville three point function[END_REF] by assuming (i) BPZ equations, (ii) the diagonal form of the solution (2.28) (iii) crossing symmetry (an essential input in the bootstrap approach). We want to stress that our proof makes no such assumptions, in fact (i)-(iii) are theorems.

2.6. Reflection relation. One of the key inputs in our proof of the DOZZ formula is the extension of (2.34) to the case α 1 + γ 2 > Q. In order to appreciate what is involved let us first explain what we should expect from the DOZZ solution. One can check from the DOZZ formula the following identity:

(2.40) B(α) = R DOZZ (α) R DOZZ (α + γ 2 )
.

Combining this with (1.15) we get that (2.34) for

α 1 + γ 2 > Q is compatible with (2.41) T -γ 2 (z) = C γ (α 1 - γ 2 , α 2 , α 3 )|F -(z)| 2 + R(α 1 )C γ (2Q -α 1 - γ 2 , α 2 , α 3 )|F + (z)| 2 where (2.42) R(α) = R DOZZ (α).
We will prove (2.41) when α 1 + γ 2 > Q (and under suitable assumptions on α 1 , α 2 , α 3 ) in Theorem 6.2 with a probabilistic expression for R(α 1 ). Once this is done we need to prove (2.42) and use these facts to derive the DOZZ formula. 2.7. Reflection coefficient. The identity (2.41) follows from a careful analysis of the small z behaviour of

E[R -γ 2 (z) -s ].
This in turn will be determined by the behavior of the integral (2.24) around the singularity at the origin. To motivate the definitions let us consider the random variable

I(α) := B(0,1) |x| -γα M γ (d 2 x). If α > 2 γ
we have E I(α) = ∞ and I(α) has a heavy tail. The reflection coefficient enters in the tail behaviour of I(α). To study this we recall basic material introduced in [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF] and in particular we consider the polar decomposition of the chaos measure. Let X s := X s (0) be the circle average (2.2). We have

X(e -s e iθ ) = X s + Y (s, θ)
where X s is Brownian Motion starting form origin at s = 0 and Y (z) is an independent field with covariance

(2.43) E[Y (s, θ)Y (t, θ ′ )] = ln e -s ∨ e -t |e -s e iθ -e -t e iθ ′ | .
We introduce the chaos measure with respect to Y

(2.44) N γ (dsdθ) = e γY (s,θ)-γ 2 E[Y (s,θ) 2 ] 2 dsdθ.
Then we get (2.45)

I(α) law = ∞ 0 2π 0 e γ(Bs-(Q-α)s) Z s ds. with (2.46) Z s = 2π 0 e γY (s,θ)-γ 2 E[Y (s,θ) 2 ] 2 dθ.
This is a slight abuse of notation since the process Z s is not a function (for γ √ 2) but rather a generalized function. With this convention, notice that Z s ds is stationary i.e. for all t the quality Z t+s = Z s holds in distribution.

It satisfies for all bounded intervals I (2.47)

E I Z s ds p < ∞, -∞ < p < 4 γ 2 .
The following decomposition lemma due to Williams (see [START_REF] Williams | Path Decomposition and Continuity of Local Time for One-Dimensional Diffusions, I[END_REF]) will be useful in the study of I(α):

Lemma 2.6. Let (B s -νs) s 0 be a Brownian motion with negative drift, i.e. ν > 0 and let M = sup s (B sνs). Then conditionally on M the law of the path (B s -νs) s 0 is given by the joining of two independent paths:

• A Brownian motion ((B 1 s + νs)) s τM with positive drift ν > 0 run until its hitting time τ M of M . • (M + B 2
t -νt) t 0 where B 2 t -νt is a Brownian motion with negative drift conditioned to stay negative. Moreover, one has the following time reversal property for all C > 0 (where τ C denotes the hitting time of C)

(B 1 τC -s + ν(τ C -s) -C) s τC law = ( B s -νs) s L-C
where ( B s -νs) s 0 is a Brownian motion with drift -ν conditioned to stay negative and L -C is the last time ( B s -νs) hits -C.

Remark 2.7. As a consequence of the above lemma, one can also deduce that the process

( B L-C +s - ν(L -C + s + C)) s 0 is equal in distribution to ( B s -νs) s 0 .
This lemma motivates defining the process B α s

B α s = B α -s if s < 0 Bα s if s > 0
where B α s , Bα s are two independent Brownian motions with negative drift α -Q and conditioned to stay negative. We may apply Lemma 2.6 to (2.45). Let M = sup s 0 (B s -(Q -α)s) and L -M be the last time

B α s hits -M . Then ∞ 0 e γ(Bs-(Q-α)s) Z s ds law = e γM ∞ -L-M e γB α s Z s+L-M ds law = e γM ∞ -L-M e γB α s Z s ds (2.48)
where we used stationarity of the process Z s (and independence of Z s and B s ). We will prove in section 3 that the tail behaviour of I(α) coincides with that of

J(α) = e γM ∞ -∞ e γB α s Z s ds.
The distribution of M is well known (see section 3.5.C in the textbook [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] for instance):

(2.49)

P(e γM > x) = 1 x 2(Q-α) γ which implies (2.50) P(J(α) > x) ∼ x→∞ E ∞ -∞ e γB α s Z s ds 2(Q-α) γ x -2(Q-α) γ .
This is the tail behaviour that we prove for I(α) and its generalizations in section 3. Define the unit volume reflection coefficient R(α) for α ∈ ( γ 2 , Q) by the following formula

(2.51) R(α) = E[ ∞ -∞ e γB α s Z s ds 2 γ (Q-α) ].
R(α) is indeed well defined as can be seen from the following lemma

Lemma 2.8. Let α ∈ ( γ 2 , Q). Then (2.52) E[ ∞ -∞ e γB α s Z s ds p ] < ∞ for all -∞ < p < 4 γ 2 .
The full reflection coefficient is now defined for all α ∈ ( γ

2 , Q) \ ∪ n 1 { 2 γ -n-1 2 γ} by (2.53) R(α) = µ 2(Q-α) γ Γ(- 2(Q -α) γ ) 2(Q -α) γ R(α)
The function R(α) has a divergence at the points 2 γ -n-1 2 γ with n 1 because of the Γ function entering the definition. Its connection to the structure constants is the following:

Lemma 2.9. For all α ∈ ( γ 2 , Q) \ ∪ n 1 { 2 γ -n-1 2 γ}, the following limit holds lim ǫ→0 ǫ C γ (ǫ, α, α) = 4R(α)
Hence the reflection coefficient should be seen as a 2 point correlation function.

2.8. Main results. The two main results of this paper are exact formulas for the two point correlation function (reflection coefficient) and the three point structure constants of the theory. These formulas were suggested in the context of the conformal bootstrap.

We will first prove the following formula:

Theorem 2.10. For all α ∈ ( γ 2 , Q) one has (2.54) R(α) = R DOZZ (α)
Finally, the main result of this paper is the following identity:

Theorem 2.11. Let α 1 , α 2 , α 3 satisfy the bounds (2.14) with N = 3. The following equality holds

C γ (α 1 , α 2 , α 3 ) = C DOZZ γ (α 1 , α 2 , α 3 ).
From the purely probabilistic point of view, Theorem 2.11 can be interpreted as a far reaching integrability result on GMC on the Riemann sphere; indeed recall that C γ (α 1 , α 2 , α 3 ) has an expression in terms of a fractional moment of some form of GMC: see formula (2.17). There are numerous integrability results on GMC in the physics literature (see the introduction); to the best of our knowledge, theorem 2.11 is the first rigorous non trivial integrability result on GMC and we believe the techniques of this paper will enable to prove many other integrability results for GMC.

Tail estimates for Multiplicative Chaos

In this section, we prove the tail estimates needed in this paper and that involves the reflection coefficient.

W := B(z,1) F (x) |x -z| γα M γ (d 2 x)
for F bounded and C 1 in a neighborhood of z. We assume γ 2 < α < Q and define auxiliary quantities

β = ( 2 γ (Q -α) + 2 γ 2 ) ∧ 4 γ 2 and η by (1 -η)β = 2 γ (Q -α) + η.
Hence η is strictly positive. With these definitions we have Lemma 3.1. For all η < η we have

|P(W > x) -|z| 4α(α-Q) F (z) 2 γ (Q-α) R(α) x 2 γ (Q-α) | C x 2 γ (Q-α)+η
Proof. We will write the integral in polar coordinates of B(z, 1). Define

N = 1 2π 2π 0 X(z + e iθ )dθ.
Then

B s := 1 2π 2π 0 (X(z + e -s e iθ ) -X(z + e iθ ))dθ
is a Brownian motion with B(0) = 0 and we may decompose the field X as

X(z + x) = N + B -ln |x| + Y z (x)
where Y z is a lateral noise centered around z given by

Y z (x) = X(z + x) - 1 2π 2π 0 X(z + |x|e iθ )dθ.
Notice that Y z has same distribution as the lateral noise Y (centered around 0), that Y z and B are independent and N is independent of B. We have

(3.1) |E[Y z (x)N ]| = | ln |z + x| -ln |z|| C|x| and the variance of N is E[N 2 ] = 2 ln |z|.
Hence, we get the following decomposition into independent components (3.2)

X(z + x) = N + B -ln |x| + (Y z (x) - E[Y z (x)N ] E[N 2 ] N ) + E[Y z (x)N ] E[N 2 ] N.
We introduce a variable N distributed as N but independent of N, B, Y z . We can rewrite (3.2) as the following equality in distribution:

(3.3) X(z + x) = N + B -ln |x| + (Y z (x) - E[Y z (x)N ] E[N 2 ] N ) + E[Y z (x)N ] E[N 2 ]
N Consider the random function u(x) given by

u(x) = e E[Yz (x)N ] E[N 2 ] N -E[Yz (x)N ] E[N 2 ] N -γ 2 E[Yz(x)N ] F (z + x) |z + x| 4 .
By (3.1) and since F is C 1 around z we get

|u(x) - F (z) |z| 4 | (C + e C|N |+C| N| )|x|.
We may thus write

W = W 1 + W 2 (3.4) W 1 law = e γ N-γ 2 2 E[ N 2 ] F (z) |z| 4 ∞ 0 e γ(Bs-(Q-α)s) Z s ds
with Z s and N independent and

|W 2 | Ce C( N +N ) B(0,1)
e γB -ln |x| +γYz(x)-

γ 2 E[(Yz (x)+B -ln |x| ) 2 ] 2 |x| γα-1 d 2 x.
One can notice that

B(0,1)
e γB -ln |x| +γYz(x)-

γ 2 E[(Yz (x)+B -ln |x| ) 2 ] 2 |x| γα-1 d 2 x law = ∞ 0 e γ(Bs-(Q-α+ 1 γ )s) Z s ds.
Recall the Williams decomposition Lemma 2.6. Let m = sup s 0 (B s -

(Q -α + 1 γ )s) and let L -m be the largest s s.t. B α -s = -m. Then ∞ 0 e γ(Bs-(Q-α+ 1 γ )s) Z s ds law = e γm ∞ -L-m e γB α+ 1 γ s Z s+L-m ds law = e γm ∞ -L-m e γB α+ 1 γ s Z s (ds) e γm ∞ -∞ e γB α+ 1 γ s Z s ds (3.5)
where we used stationarity of the process Z s .

For all p < ( 2

γ (Q -α) + 2 γ 2 ) ∧ 4 γ 2 = β, we have (3.6) P(|W 2 | x) Cx -p
Indeed, for all p 1 , q 1 > 1 with 1 p1 + 1 q1 = 1 we have by using Hölder and (3.5) that

(3.7) P(|W 2 | x) 1 x p E[|W 2 | p ] C x p E[e Cpp1( N+N ) ] 1/p1 E[(e γm ∞ -∞ e γB α+ 1 γ s Z s ds) pq1 ] 1/q1 C x p
provided q 1 is sufficiently close to 1 and where we used Lemma 2.8 which requires p < 4 γ 2 . We first prove an upper bound for P(W > x). From (3.7) we get for η ∈ (0, 1):

P(W > x) = P(W 1 + W 2 > x) P(W 1 > x -x 1-η ) + Cx -p(1-η)
Proceeding as in (3.5) we get

P(W 1 > x -x 1-η ) P(e γ N -γ 2 2 E[ N 2 ] F (z) |z| 4 e γM ∞ -∞ e γB α s Z s ds > x -x 1-η )
where M = sup s 0 (B s -(Q -α)s). Then (2.49) implies

P(W > x) e (2(Q-α) 2 -γ(Q-α))E[ N 2 ] F (z) |z| 4 2 γ (Q-α) R(α) (x -x 1-η ) 2 γ (Q-α) + Cx -p(1-η) |z| -4α(Q-α) F (z) 2 γ (Q-α) R(α) x 2 γ (Q-α) + Cx -2 γ (Q-α)-η + Cx -p(1-η)
for p < β. Recall that we defined η > 0 by (1 -η)β = 2 γ (Q -α) + η. We conclude

P(W > x) |z| -4α(Q-α) F (z) 2 γ (Q-α) R(α) x 2 γ (Q-α) + Cx -2 γ (Q-α)-η (3.8)
for all η < η. Now, we consider the lower bound. We have

P(W > x) P(W 1 > x + x 1-η ) -P(W 2 < -x 1-η ) P(W 1 > x + x 1-η ) -Cx -2 γ (Q-α)-η (3.9)
for all η < η. By the Williams decomposition we get as in (3.5)

W 1 (Law) = e γ N-γ 2 2 E[ N 2 ] F (z) |z| 4 e γM ∞ -L-M e γB α s Z s ds := W (L -M ).
where

M = sup s 0 (B s -(Q -α)s) and M and B α are independent of Z s . Let η ′ be such that (1 -η ′ ) 4 γ 2 = 2 γ (Q -α) + η ′ . One has η ′ η. Consider the event E defined by e γ N-γ 2 2 E[ N 2 ] F (z) |z| 4 ∞ -L-M e γB α s Z s ds < x 1-η ′ .
We have trivially

P(W 1 > x + x 1-η ) P({W 1 > x + x 1-η } ∩ E).
Under

{W 1 > x + x 1-η } ∩ E we have e γM |x| η ′ . Indeed, if e γM < |x| η ′ then under E we get W 1 < x which is impossible. Thus M -η ′ γ ln |x| whereby L -M L -η ′ γ ln |x| and hence W (L -η ′ γ ln |x| ) W (L -M
). We conclude

P(W 1 > x + x 1-η ) P({W (L -η ′ γ ln |x| ) > x + x 1-η } ∩ E) P(W (L -η ′ γ ln |x| ) > x + x 1-η ) -Cx -(1-η ′ ) 4 γ 2 +ǫ |z| -4α(Q-α) F (z) 2 γ (Q-α) E[( ∞ -L - η ′ γ ln x e γB α s Z s ds) 2 γ (Q-α) ](x + x 1-η ) -2 γ (Q-α) - C x 2 γ (Q-α)+η ′ -ǫ (3.10)
for all ǫ > 0 where in the second step we used Lemma 2.8.

We claim now that

(3.11) E[( ∞ -∞ e γB α s Z s ds) 2 γ (Q-α) ] -E[( ∞ -L - η ′ γ ln x e γB α s Z s ds) 2 γ (Q-α) ] Cx -η ′ .
Combined with (3.10) and (3.9) this yields

P(W > x) |z| -4α(Q-α) F (z) 2 γ (Q-α) R(α) (x -x 1-η ) 2 γ (Q-α) + Cx -p(1-η) |z| -4α(Q-α) F (z) 2 γ (Q-α) R(α) x 2 γ (Q-α) -Cx -2 γ (Q-α)-η (3.12)
for all η < η. (3.12) and (3.8) then finish the proof.

It remains to prove (3.11). By Remark 2.7, the process Bα s defined for s 0 by the relation Bα

s = B α s-L - η ′ γ ln x + η ′ γ ln
x is independent from everything and distributed like (B α s ) s 0 . We can then write

∞ -∞ e γB α s Z s ds = A + x -η ′ B
where

A = ∞ -L - η ′ γ ln x e γB α s Z s ds and B = 0 -∞ e γ Bα s Z s-L - η ′ γ ln x
ds.

We now distinguish two cases:

2 γ (Q -α) 1 and 2 γ (Q -α) > 1. 2 γ (Q -α) 1. We use (1 + u) 2 γ (Q-α) -1 2 γ (Q -α)u for u 0 to bound E[(A + x -η ′ B) 2 γ (Q-α) -A 2 γ (Q-α) ] 2 γ (Q -α)x -η ′ E[BA 2 γ (Q-α)-1 ]
By Hölder's inequality with p ∈ (1, 4 γ 2 ), we get

E[BA 2 γ (Q-α)-1 ] E[B p ] 1/p E[A q( 2 γ (Q-α)-1) ] 1 q < ∞ since B is equal in distribution to 0 -∞ e γ Bα s Z s ds and A
∞ 0 e γB α s Z s ds which has negative moments of all order by Lemma 2.8.

2 γ (Q -α 2 ) > 1. Let p := 2 γ (Q -α).
By triangle inequality we have

E[(A + x -η ′ B) p -A p ] (E[A p ]) 1/p + x -η ′ (E[B p ]) 1/p p -E[A p ] (E[A p ]) 1/p + Cx -η ′ p -E[A p ] Cx -η ′ E[A p ] 1-1/p Cx -η ′
where again we used that A and B have moment of order p.

Remark 3.2. A simple variation of the proof yields the result (2.50).

3.2.

Tail estimate around two insertions. Let for i = 2, 311

W i := B(zi,1) F i (x) |x -z i | γαi M γ (d 2 x).
We will suppose that |z 2 | 2, |z 3 | 2 and |z 2 -z 3 | 3 so that the balls B i = B(z i , 1) are well separated. We denote by η2 and η3 the exponents occurring in the tail estimates of Lemma 3.1 applied to W 2 and W 3 . Set

η 2 = η2 ∧ 1 γ (Q -α 3 ) ∧ 1 2 , η 3 = η3 ∧ 1 γ (Q -α 2 ) ∧ 1 2 . Then we have Lemma 3.3. For all β < β := ( 2 γ (Q -α 2 ) + η 2 ) ∧ ( 2 γ (Q -α 3 ) + η 3 ) |P(W 2 + W 3 > x) - 3 i=2 |z i | 4αi(αi-Q) F i (z i ) 2 γ (Q-αi) R(α i ) x 2 γ (Q-αi) | Cx -β Remark 3.4. The above theorem is useful when β > 2 γ (Q -α 2 ) ∨ 2 γ (Q -α 3 )
. This is the case when α 2 and α 3 are sufficiently close to each other.

Remark 3.5. The proof of Theorem 3.3 is based on the fact that the two variables W 2 and W 3 are "nearly" independent. Along the same lines as the proof of Theorem 3.3, one can in fact show that for all p 2 , p 3 > 0 there exists some constant C > 0 such that

E[W p2 2 W p3 3 ] CE[W p2 2 ]E[W p3 3 ].
Proof. The strategy here is to apply the previous lemma with one insertion. We start with the upper bound.

We have

P(W 2 + W 3 > x) P(W 2 + W 3 > x, W 2 > x 2 ) + P(W 2 + W 3 > x, W 3 > x 2 ) (3.13)
The variables W 2 and W 3 are nearly independent as we now argue. We consider the circle of radius 3 2 centered at z 2 . By the Markov property of the GFF, we have the following decomposition inside B(z 2 , 3 2 ) X(x) = X + P(X)(x)

where P(X)(x) is the Poisson kernel of the ball B(z 2 , 3 2 ) applied to X and X is a GFF with Dirichlet boundary conditions on B(z 2 , 3 2 ) independent of the outside of B(z 2 , 3 2 ). On the smaller ball B(z 2 , 1), the process P(X)(x) is a smooth Gaussian process hence for all p > 0

E[e p sup |x-z 2 | 1 P(X)(x) ] < ∞.
We set H = sup |x-z2| 1 P(X)(x). Of course, we have

W 2 e γH W 2
where W 2 is computed with the chaos measure of X. W 2 , W 3 have moments less than orders 2 γ (Q -α i ) respectively so that for all u, v > 0 and all ǫ ′ > 0 that

P(W 2 > u, W 3 > v) P(e γH W 2 > u, W 3 > v) 1 u 2 γ (Q-α2)-ǫ ′ E[ W 2 γ (Q-α2)-ǫ ′ 2 ]E[e (2(Q-α2)-γǫ ′ )H 1 W3>v ] 1 u 2 γ (Q-α2)-ǫ ′ E[ W 2 γ (Q-α2)-ǫ ′ 2 ]E[e p(2(Q-α2)-γǫ ′ )H ] 1/p P(W 3 > v) 1/q C u 2 γ (Q-α2)-ǫ ′ v 1 q ( 2 γ (Q-α 3 )-ǫ ′ )
for all p, q > 1 such that 1 p + 1 q = 1. By taking q close to 1 we conclude

(3.14) P(W 2 > u, W 3 > v) C u 2 γ (Q-α2)-ǫ v 2 γ (Q-α3)-ǫ
for all ǫ > 0. Therefore, exploiting (3.14) we have for all ǫ > 0

P(W 2 + W 3 > x, W 2 > x 2 ) P(W 2 + W 3 > x, W 2 > x 2 , W 3 √ x) + P(W 2 > x 2 , W 3 > √ x) P(W 2 > x - √ x) + C x 2 γ (Q-α2) x 1 γ (Q-α3)-ǫ
We get a similar bound by interchanging 2 and 3. Inserting to (3.13) we obtain

P(W 2 + W 3 > x) P(W 2 > x - √ x) + P(W 3 > x - √ x) + C x 2 γ (Q-α2) x 1 γ (Q-α3)-ǫ + C x 2 γ (Q-α3) x 1 γ (Q-α2)-ǫ
and then we use Lemma 3.1 on one insertion. Now, we proceed with the lower bound. We have, exploiting (3.14), that for all ǫ > 0

P(W 2 + W 3 > x) P({W 2 > x} ∪ {W 3 > x}) P(W 2 > x) + P(W 3 > x) -P(W 2 > x, W 3 > x) P(W 2 > x) + P(W 3 > x) - C x 2 γ (Q-α2)+ 2 γ (Q-α3)-ǫ
and then we use again Lemma 3.1.

Analytic Continuation of Liouville Correlation Functions

In this section we study the analytic continuation of the unit volume correlations (2.15). These are defined for real weights α = (α 1 , . . . , α N ) satisfying the extended Seiberg bounds (4.1)

U N := {α ∈ R N : 1 γ (2Q - N k=1 α k ) < 4 γ 2 ∧ min 1 k N 2 γ (Q -α k ), ∀k : α k < Q}.
We will prove Proof. By Möbius invariance we may assume |z i | > 2 and |z i -z j | > 2. We use (2.9) to write the unit volume correlation functions as the limit

N k=1 V α k (z k ) uv = 2µ -s γ -1 N k=1 |z k | -4∆α k + lim r→∞ F r (α) (4.2)
where

F r (α) = E N k=1 e α k Xr (z k )-α 2 k 2 EXr (z k ) 2 M γ (C r ) -s (4.3)
where C r := C \ ∪ N k=1 B(z k , e -r ). F r is defined for all α ∈ C N and is complex differentiable in α i , hence defining an entire function in the α i . We show that there is an open V ⊂ C N containing U N s.t. F r converges uniformly on compacts of V . Note that this is nontrivial since for α k = a k + ib k we have

|e α k Xr (z k )-α 2 k 2 E[Xr (z k ) 2 ] | = e a k Xr (z k )-a 2 k 2 E[Xr (z k ) 2 ] e b 2 k 2 EXr (z k ) 2 and e b 2 k 2 EXr (z k ) 2 ∝ e b 2 k
2 r blows up as r → ∞. By Remark 2.2, we know that t ∈ R + → B k r+t := X r+t (z k ) -X r (z k ) are mutually independent Brownian motions and they are independent of σ{X(x); x ∈ C r }. Hence

F r+1 (α) -F r (α) = E N k=1 e α k Xr+1(z k )-α 2 k 2 E[Xr+1(z k ) 2 ] (M γ (C r+1 ) -s -M γ (C r ) -s ) .
Now we apply the Cameron-Martin theorem as in (2.11) to the real parts of the vertex insertions to get

|F r+1 (α) -F r (α)| Ce (r+1) N k=1 b 2 k 2 |E( Cr+1 f r (x)M γ (d 2 x)) -s -E( Cr f r (x)M γ (d 2 x) -s )| (4.4) where f r (x) = e N k=1 γa k Gr+1(x,z k ) and we have defined G r+1 (z, z ′ ) := E[X(z)X r+1 (z ′ )]. We get from (2.1) f (x) := sup r f r (x) C k |x| + |z k | + |x -z k | γα k .
We need to estimate the difference of expectations in (4.4). Let

Y r := Cr+1\Cr f r (x)M γ (d 2 x).
and set also Z r :

= Cr f r (x)M γ (d 2 x). Then |E( Cr+1 f r (x)M γ (d 2 x)) -s -E( Cr f r (x)M γ (d 2 x)) -s )| = |E((Z r + Y r ) -s -Z -s r )| E[1 Yr<ǫ |(Z r + Y r ) -s -Z -s r |] + E[1 Yr ǫ |(Z r + Y r ) -s -Z -s r |]
where ǫ > 0 will be fixed later. The first expectation on the RHS is bounded by

E1 Yr<ǫ |(Z r + Y r ) -s -Z -s r | Cǫ sup t∈[0,1] E(Z r + tY r ) -ℜs-1 Cǫ
uniformly in r. The last bound follows by noting that for -ℜs -1 > 0 the expectation is bounded uniformly in r by CE( f (x)M γ (d 2 x)) -ℜs-1 which is finite due to (2.14) whereas for -ℜs -1 < 0 we may bound it for example by CE( C1\C2 M γ (d 2 x)) -ℜs-1 which is finite as well.

For the second expectation we use the Hölder inequality

E1 Yr ǫ |Z -s r+1 -Z -s r | CP(Y r ǫ) 1/p ((EZ -qℜs r+1 ) 1/q + (EZ -qℜs r ) 1/q ). Taking q > 1 s.t. -qℜ(s) < min j 2 γ (Q -α j ) ∧ 4 γ 2
we may bound the two expectations uniformly in r as in the previous paragraph and then using Markov inequality we get

E1 Yr ǫ |Z -s r+1 -Z -s r | Cǫ -m/p (EY m r ) 1/p .
It remains to bound EY m r for suitable m > 0. We note that C r+1 \ C r = ∪ i A i r where A i r is the annulus centred at z i with radi e -r-1 , e -r . Then we obtain for m < 4

γ 2 EY m r CE( k A k r f (x)M γ (d 2 x)) m C max k e -r(γ(Q-a k )m-γ 2 m 2
2

) := Ce -rθ (4.5)
where in the second step we used the estimate (9.2). Now, let us fix a 0 ∈ U N . Then we can find m > 0 and δ > 0 s.t. θ > 0 for all a with min k |a k -a 0 k | δ. Hence, for α ∈ C M with α k = a k + ib k and ǫ > 0

|F r+1 (α) -F r (α)| Ce (r+1) N k=1 b k 2 (ǫ + ǫ -m/p e -r p θ ).
Taking ǫ = e -δr with δ = θ p+m we then have

|F r+1 (α) -F r (α)| Ce -(δ-N k=1 b k 2 )r .
Hence, F r (α) converges uniformly in a ball around a 0 in C N .

5. Proof of lemmas 2.8 and 2.9 on the reflection coefficient Let now p = -q < 0. Set τ -1 = inf{s 0, B α s = -1}. The process B α s+τ-1 + 1 is a Brownian motion with drift -ν starting from 0 and conditioned to stay below 1. Therefore, we have if B s is a standard Brownian motion starting from 0 and β := sup s 0 (B s -νs)

E[ ∞ 0 e γB α s Z s ds -q ] E[ τ-1+1 τ-1 e γB α s Z s ds -q ] = CE[1 β 1 1 0 e γ(Bs-νs) Z s ds -q ]
since Z s is stationary and B α s is independent from Z. Finally, we conclude by

E[1 β 1 1 0 e γ(Bs-νs) Z s ds -q ] E[1 β 1 e -q inf s∈[0,1] (Bs-νs) ] E( 1 0 Z s ds) -q < ∞
where (2.47) was used.

5.2. Proof of lemma 2.9. We use formula (1.6) to write

C γ (ǫ, α, α) in terms of V ǫ (0)V α (z 2 )V α (z 3 ) where we take |z 2 |, |z 3 | > 2 with |z 2 -z 3 | > 2. Then 2∆ 12 = -2∆ ǫ = -ǫ(Q - ǫ 2 ) → ǫ→0 0. and 2∆ 23 = 2∆ ǫ -4∆ α → ǫ→0 α(α -2Q)
Therefore, we have

lim ǫ→0 ǫC γ (ǫ, α, α) = lim ǫ→0 ǫ|z 2 -z 3 | α(2Q-α) V ǫ (0)V α (z 2 )V α (z 3 ) = |z 2 -z 3 | 2α(Q-α) 2µ 2 γ (Q-α) γ -1 Γ(- 2 γ (Q -α)) lim ǫ→0 ǫE[( C F ǫ (x)M γ (d 2 x)) 2 γ (Q-α)-ǫ γ ]
where

F ǫ (x) = |x| γ(ǫ+2α) + |x| γǫ |x -z 2 | γα |x -z 3 | γα . Let W i,ǫ = B(zi,1) F ǫ (x)M γ (d 2 x) for i = 2, 3 and A ǫ = (B(z1,1)∪B(z2,1)) c F ǫ (x)M γ (d 2 x) so that C F ǫ (x)M γ (d 2 x) = A ǫ + W 2,ǫ + W 3,ǫ .

Now, we have the following inequality if

2 γ (Q -α) 1 E[(W 2,ǫ + W 3,ǫ ) 2 γ (Q-α)-ǫ γ ] E[(A ǫ + W 2,ǫ + W 3,ǫ ) 2 γ (Q-α)-ǫ γ ] E[A 2 γ (Q-α)-ǫ γ ǫ ] + E[(W 2,ǫ + W 3,ǫ ) 2 γ (Q-α)-ǫ γ ]
By the double tail estimate Lemma 3.3 we have

P(W 2,ǫ + W 3,ǫ ) > x) = 2|z 2 -z 3 | -2α(Q-α) R(α)x -2 γ (Q-α) (1 + O(x -η )) for η > 0, uniformly in ǫ. Since ǫE[A 2 γ (Q-α)-ǫ γ ǫ
] converges to 0, this yields

lim ǫ→0 ǫE[( C F ǫ (x)M γ (d 2 x)) 2 γ (Q-α)-ǫ γ ] = 2γ|z 2 -z 3 | -2α(Q-α) R(α)
and then lim

ǫ→0 ǫC γ (ǫ, α, α) = 4µ 2 γ (Q-α) 2(Q-α) γ Γ(- 2(Q-α) γ ) R(α) = 4R(α)
If 2 γ (Q -α) > 1 we have by triangle inequality and ǫ small enough so that p

= 2 γ (Q -α) -ǫ γ > 1 [E(W 2,ǫ + W 3,ǫ ) p ] 1/p [E(A ǫ + W 2,ǫ + W 3,ǫ ) p ] 1/p [E(A ǫ ) p ] 1/p + [E(W 2,ǫ + W 3,ǫ ) p ] 1/p
and we can conclude similarly as the previous case.

The BPZ equations and algebraic relations

This section is devoted to the study of the small z asymptotics of the four point functions T -γ 2 and T -2 γ leading to the proof of (2.34) and (2.41). The proof of the latter is the technical core of the paper and the key input in the probabilistic identification of the reflection coefficient.

+ γ < α 1 + γ 2 < Q or in other words γ 2 + 1 γ < α 1 < 2 γ .
This interval is non empty if and only if γ 2 < 2. In this section we will remove this constraint. The reason for the restriction γ 2 + 1 γ < α 1 was the following. In order to prove (2.34), one must perform the asymptotic expansion of T -γ 2 (z) around z → 0 (2.31) as explained in section 2.4. In the case 1 γ + γ 2 < α 1 , the exponent 2(1 -c) which is equal to γ(Q -α 1 ) is strictly less than 1 hence there are no polynomial terms in z and z in the expansion (2.31) to that order (such terms are present in the small z expansion of |F -(z)| 2 ). In the case α 1 < 1 γ + γ 2 , the asymptotic expansion of T -γ 2 (z) around 0 is more involved. Nonetheless, we prove here: Theorem 6.1. We assume the Seiberg bounds for (-γ 2 , α 1 , α 2 , α 3 ), i.e.

3 k=1 α k > 2Q + γ 2 and α k < Q for all k. If γ 2 < α 1 < 2 γ then (6.1) T -γ 2 (z) = C γ (α 1 - γ 2 , α 2 , α 3 )|F -(z)| 2 -µ π l(-γ 2 4 )l( γα1 2 )l(2 + γ 2 4 -γα1 2 ) C γ (α 1 + γ 2 , α 2 , α 3 )|F + (z)| 2
and the relation (1.8) holds.

Proof. Let first γ 2 < 2. (6.1) was proven in [36, Theorem 2.3] in the case γ 2 + 1 γ < α 1 < 2 γ . This result extends to the interval γ 2 < α 1 < 2 γ by analyticity. Indeed, for fixed γ ∈ (0, √ 2), the interval γ 2 + 1 γ < α 1 < 2 γ is non empty. Furthermore, by Theorem 4.1 both sides of eq. (6.1) are analytic in α 1 (with other parameters fixed) in a neighborhood of the interval γ 2 < α 1 < 2 γ seen as a subset of C. Uniqueness of analytic continuation thus establishes (6.1) for γ 2 < 2. γ 2 = 2 is obtained by continuity in γ (see Remark 2.3 on this).

Let now γ 2 > 2 and γ 2 < α 1 < 2 γ : The proof of (6.1) follows from the study of the function T -γ 2 (z) as z tends to 0. Thus by (2.23) we need to study the function (2.24) with α 0 = -γ 2 .To streamline notation let us set (6.2)

K(z, x) = |x -z| γ 2 2 |x| γ( 3 k=1 α k -γ 2 ) + |x| γα1 |x -1| γα2
and for any Borel set B ⊂ C

(6.3) K B (z) := B K(z, x) M γ (d 2 x).
Then R -γ 2 (z) = K C (z). We will also write K(z) for K C (z). We set p := 1 γ (

3 k=1 α k -γ 2 -2Q).
Taylor expansion yields the relation

E[K(z) -p ] = E[K(0) -p ] + z∂ z E[K(z) -p ] |z=0 + z∂ z E[K(z) -p ] |z=0 + R(z) where R(z) := 1 2 1 0 z 2 ∂ 2 z E[K(tz) -p ] + 2z z∂ z ∂ z E[K(tz) -p ] + z2 ∂ 2 z E[K(tz) -p ] dt.
Notice that the term ∂ z E[K(z) -p ] |z=0 is well defined. Indeed we have

∂ z E[K(z) -p ] |z=0 = -p γ 2 4 E C 1 x K(0, x)M γ (d 2 x)K(0) -p-1 .
Split the the x-integral over C in two parts, over B 1/2 and over B c 1/2 where in this section we use the notation B r = B(0, r).

Then |E B c 1/2 1 |x| K(0, x)M γ (d 2 x)K(0) -p-1 | 2E K(0) -p < ∞
as GMC measures possess negative moments of all orders. For the integral over B 1 2 we use the Cameron-Martin theorem to get

E B 1/2 1 |x| K(0, x)M γ (d 2 x)K(0) -p-1 CE( B 1/2 |x| -1-γα1+ γ 2 2 M γ (d 2 x)K(0) -p-1 ) = C B 1/2 |x| -1-γα1+ γ 2 2 E C K(0, u)e γ 2 G(x,u) M γ (d 2 u) -p-1 d 2 x. (6.4)
To bound the last expectation we note that the integrand is strictly positive on x ∈ B 1 2 and for example u ∈ B(3, 1). This ball is far away from the singularities, hence on u ∈ B(3, 1) the kernel K(0, u)e γ 2 G(x,u) is bounded from below away from 0 . Thus

E C K(0, u)e γ 2 G(x,u) M γ (d 2 u) -p-1 CE(M γ (B(3, 1)) -p-1 < ∞
as the measure M γ possesses moments of negative order. The final integral in (6.4) converges as the constraint

α 1 < 2 γ guarantees that 1 + γα 1 -γ 2 2 < 3 -γ 2 2 < 2 since γ 2 > 2.
The same argument shows that z∂ z E[(K 0 (z)) -p ] |z=0 is well defined.

It remains to investigate the remainder R(z). All the three terms are treated in the same way and we focus on the first. It may be written as a sum of two terms:

R 1 (z) := -p γ 2 4 (1 + γ 2 4 )z 2 1 0 (1 -t)E C 1 (x -tz) 2 K(tz, x)M γ (d 2 x)(K(tz)) -p-1 dt R 2 (z) :=p(p + 1) γ 4 16 z 2 1 0 (1 -t)E C 1 (x -tz) K(tz, x)M γ (d 2 x) 2 (K(tz)) -p-2 dt.
The term R 1 is the leading one contributing and explicit factor c|z| γ(Q-α1) . It is analysed in the same way as a similar term in the proof of [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF]Theorem 2.3] so we will be brief. First as above we may restrict the integral to B 1/2 up to an O(z 2 ) contribution from the integral over B c 1/2 . By Cameron-Martin and the change of variables x → ytz this term equals -p

γ 2 4 (1 + γ 2 4 )|z| 2+ γ 2 2 -γα1 1 0 (1 -t)t γ 2 2 -γα1 B 1 2t|z| |y -1| γ 2 2 (y -1) 2 |y| γα1 |yzt -1| γα2 × E C K(tz, u)e γ 2 G(tyz,u) )M γ (d 2 u) -p-1 dtd 2 y.
Dominated convergence theorem then ensures that this term is equivalent as z → 0 to -p

γ 2 4 (1 + γ 2 4 )|z| γ(Q-α1) 1 0 (1 -t)t γ 2 2 -γα1 dt C |y -1| γ 2 2 (y -1) 2 |y| γα1 d 2 y E C K(0, u)e γ 2 G(0,u) M γ (d 2 u) -p-1
.

The y integral can then be computed thanks to (9.7) and one can show that this term, combined with the corresponding term coming from the ∂ 2 z derivative, gives rise to the structure constant C γ (α 1 + γ 2 , α 2 , α 3 ).

The rest of this proof is dedicated to showing that R 2 is a o(|z| γ(Q-α1) ). We will bound the expectation occuring in R 2 so let us denote

I(B, tz) := E B 1 |x -tz| K(tz, x)M γ (d 2 x) 2 ( C K(tz, u)M γ (d 2 u)) -p-2 . (6.5)
We shall prove I(C, tz) C|tz| γ(Q-α1)-2+η (6.6) with η > 0 which proves our claim since resulting t integral converges at 0 as γ(Q -α 1 ) -2 = γ 2 2 -2 > -1. We can now put t = 1 and we will bound I(C, z) for z small. For z small enough 1 |x-z| is bounded in B c , z)) it suffices to bound I(B 1 2 , z). Next we bound I(A, z) where A is the annulus centered at origin with radii L|z| and 1 2 and L > 1 will be chosen later. First, we use Jensen's inequality in the normalized measure 1 A (x)K(z, x)M γ (d 2 x) to get

I(A, z) E A 1 |x -z| 2 K(z, x)M γ (d 2 x) K A (z) -p-1 .
Next we observe that on A the covariance kernel of X coincides with that of the exact scale invariant kernel constructed in [START_REF] Rhodes | Multidimensional Multifractal Random Measures[END_REF] (in the Gaussian case) up to a global additive constant. In particular, up to an additive independent Gaussian random variable, the restriction of X to B 1 2 can be constructed as an increasing function of some white noise in this ball and therefore we can make use of the FKG inequality (see [27, section 2.2] for the case of countable product) to get

E A 1 |x -z| 2 K(z, x)M γ (d 2 x) K A (z) -p-1 E A 1 |x -z| 2 K(z, x)M γ (d 2 x) E[K A (z) -p-1 ] C A |x -z| γ 2 2 -2 |x| γα1 d 2 x C|z| γ 2 2 -γα1 |y|>L |y -1| γ 2 2 -2 |y| γα1 d 2 y C|z| -2 |z| γ(Q-α1) L -γ(α1-γ 2 )
where the last integral was convergent due to α 1 > γ 2 . This fits to (6.6) provided we take L = |z| -δ with δ > 0.

We are left with estimating I(B L|z| , z). Let us first consider the part not too close to the singularity at z: set S := B L|z| \ B(z, |z| 1+ǫ ) for some ǫ > 0, to be fixed later. We have

E S 1 |x -z| K(z, x)M γ (d 2 x) 2 K(z) -p-2 |z| -2-2ǫ E[K S (z) 2 K(z) -p-2 ].
Then we get, for r ∈ (0, 2) using the fact that K S (z) K(z)

E[K S (z) 2 K(z) -p-2 ] E[K S (z) r (K(z)) -p-r ] C(EK S (z) qr ) 1/q
where in the second step we used Hölder inequality and bounded the negative GMC moment again by a constant. Finally, since |x -z| 2|Lz| on S we get

[E(K S (z) qr )] 1/q C|Lz| γ 2 2 r [E[( B L|z| |x| -γα1 M γ (d 2 x)) qr ]] 1/q C|Lz| γ(Q-α1+ γ 2 )r-1 2 γ 2 qr 2
where the last estimate was an easy consequence of the annulus bound (9.2) (see also [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]Lemma A.1]).

Here we need to assume that rq < 4 γ 2 ∧ 2 γ (Q -α 1 ). Notice that since we assume γ 2 < α 1 then 4 γ 2 > 2 γ (Q -α 1 ) so that given q we need to have 0 < r < 2 γq (Q -α 1 ). The optimal choice for r is

r ⋆ = γ 2 +Q-α1 γq (this is less than 2 γq (Q -α 1 ) for α 1 < 2 γ ), in which case E[K S (z) rq ] 1/q C|Lz| 1 2q ( γ 2 +Q-α1) 2 .
Gathering everything we conclude

|E S 1 |x -z| K(z, x)M γ (d 2 x) 2 K(z) -p-2 CL 1 2q ( γ 2 +Q-α1) 2 |z| -2-2ǫ+ 1 2q ( γ 2 +Q-α1) 2 .
We can now fix δ, q, ǫ. First notice that 1 2

( γ 2 + Q -α 1 ) 2 -γ(Q -α 1 ) = 1 2 (Q -α 1 -γ 2 ) 2 > 0.
Hence choosing q sufficiently close to 1 and then ǫ < ǫ(q) and finally δ < δ(ǫ), I(S, z) can be bounded by (6.6).

We are thus left with proving (6.6) for I(B, z) where B := B(z, |z| 1+ǫ ). An application of the Cameron-Martin theorem gives

I(B, z) = B 2 K(z, x)K(z, x ′ )e γ 2 G(x,x ′ ) (x -z)(x ′ -z) E C K(z, u)e γ 2 G(x,u)+γ 2 G(x ′ ,u) M γ (d 2 u) -p-2 d 2 xd 2 x ′ C B 2 |x -z| γ 2 2 -1 |x ′ -z| γ 2 2 -1 |x| γα1 |x ′ | γα1 |x -x ′ | γ 2 E B 1 2 |u -z| γ 2 2 |u| γα1 |u -x| γ 2 |u -x ′ | γ 2 M γ (d 2 u) -p-2 d 2 xd 2 x ′ (6.7)
where for upper bound we restricted the u integral to B 1 2 . By a change of variables x = zy, x ′ = zy ′ this becomes

I(B, z) C|z| 2-2γα1 B(1,z ǫ ) 2 |y -1| γ 2 2 -1 |y ′ -1| γ 2 2 -1 |y -y ′ | γ 2 A(y, y ′ , z) d 2 yd 2 y ′ (6.8) with A(y, y ′ , z) = E B 1 2 |u -z| γ 2 2 |u| γα1 |u -yz| γ 2 |u -y ′ z| γ 2 M γ (d 2 u) -p-2
.

Note that the only potential divergence in the y, y ′ integral is at y = y ′ since γ 2 > 2. Hence we need to show A(y, y ′ , z) vanishes at diagonal. The behaviour of A(y, y ′ , z) as y → y ′ is controlled by the fusion rules (see [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF]). In the case at hand we have four insertions, located at 0, zy, zy ′ , z that are all close to each other as z → 0. Fusion estimates have been proven in [START_REF] Kupiainen | Local conformal structure of Liouville Quantum Gravity[END_REF] in the case of three insertions. A simple adaptation of that proof to the case of 4 insertions is stated in the Appendix, Lemma 9.1. The estimate for A(y, y ′ , z) depends on the relative positions of the four insertions. In our case we have |zy -z|∨|zy ′ -z|∨|zy -zy ′ | ≪ |z|∧|zy|∧|zy ′ |. This means that the insertions zy, zy ′ , z will merge together way before merging with 0. We will partition the integration region in (6.8) acording to the relative positions of these three points or equivalently the relative positions of y, y ′ , 1. By symmetry in y, y ′ we have then three integration regions in (6.8) to consider

• Let A 1 := {|y -1| |y ′ -1| |y -y ′ |}. Then on B ∩ A 1 we have by Lemma 9.1 A(y, y ′ , z) C|1 -y ′ | 1 2 ( 3γ 2 -Q) 2 |z| 1 2 ( 3γ 2 +α1-Q) 2 . Since 2 -2γα 1 + 1 2 ( 3γ 2 + α 1 -Q) 2 = -2 + γ(Q -α 1 ) + 1 2 (α -2 γ ) 2 we get I(B ∩ A 1 , z) C|z| -2+γ(Q-α1)+ 1 2 (α-2 γ ) 2 B(1,|z| ǫ ) 2 1 |y -y ′ | 2-1 2 ( 3γ 2 -Q) 2 d 2 yd 2 y ′ . (6.9)
The integral is convergent if

( 3γ 2 -Q) 2 > 0 which is the case if γ 2 = 2. • Let A 2 := {|y -1| |y -y ′ | |y ′ -1|}. Then on B 1 ∩ A 2 we have by Lemma 9.1 A(y, y ′ , z) C|y -y ′ | 1 2 ( 3γ 2 -Q) 2 |z| 1 2 ( 3γ 2 +α1-Q) 2 .
Hence we end up with the bound (6.9) for I(B ∩ A 2 ) as well (since

γ 2 2 -1 > 0) . • Let A 3 := {|y -y ′ | |y -1| |y ′ -1|}. Then on B 1 ∩ A 3 we have by Lemma 9.1 A(y, y ′ , z) C|y -y ′ | 1 2 (2γ-Q) 2 |y -1| 1-γ 2 2 |z| 1 2 ( 3γ 2 +α1-Q) 2 . Hence I(B ∩ A 3 ) C|z| -2+γ(Q-α1)+ 1 2 (α-2 γ ) 2 B(1,|z| ǫ ) 2 1 |y -y ′ | γ 2 -1 2 (2γ-Q) 2 d 2 yd 2 y ′ . The integral converges since γ 2 -1 2 (2γ -Q) 2 = 4 -1 2 Q 2 < 2.
6.2. Fusion with reflection. In this section we uncover the probabilistic origin of the reflection relation (1.14), (1.15). We prove the following extension of Theorem 6.1 to the case

α 1 + γ 2 > Q: Theorem 6.2. Let i α i -γ 2 > Q and α i < Q for all i. There exists η > 0 s.t. if Q -α 1 < η then (6.10) T -γ 2 (z) = C γ (α 1 - γ 2 , α 2 , α 3 )|F -(z)| 2 + R(α 1 )C γ (2Q -α 1 - γ 2 , α 2 , α 3 )|F + (z)| 2 .
Proof. We use the notations introduced in the proof of (6.1). We will prove

E K(z) -p -E K(0) -p = µ p γΓ(p) -1 R(α 1 )|C γ (2Q -α 1 - γ 2 , α 2 , α 3 )|z| γ(Q-α1) + o(|z| γ(Q-α1) ).
Note that since now γ(Q -α 1 ) < 1 we need Taylor series only to 0th order. The leading asymptotics will result from the integral defining K in a small ball at origin. Let us denote B := B |z| 1-ξ with ξ ∈ (0, 1) to be fixed later. We define (6.11)

T 1 := E K B c (z) -p -E K(0) -p and T 2 := E K(z) -p -E K B c (z) -p so that (6.12) E K(z) -p -E K(0) -p = T 1 + T 2 .
We first show that

T 1 = o(|z| γ(Q-α1)
). Interpolating we get

|T 1 | p 1 0 E |K B c (z) -K(0)| tK B c (z) + (1 -t)K(0) -p-1 dt CE |K B c (z) -K(0)| K B c (0) -p-1 (6.13)
where we used

K B c (z) CK(0) K B c (0) since |x -z| γ 2 2 C|x| γ 2 2 on B c . Since K(0) = K B (0) + K B c (0) we obtain |T 1 | C(A 1 + A 2 )
where

A 1 = E[K B (0)K B c (0) -p-1 ] and A 2 = E[|K B c (z) -K B c (0)|K B c (0) -p-1 ].
Using Cameron-Martin theorem, we get for A 1 that

A 1 C |x| |z| 1-ξ |x| γ 2 2 -γα1 E |u|>|z| 1-ξ K(0, u)|u -x| -γ 2 M γ (du) -p-1 d 2 x.
Since |u -x| 2|u| we may bound the expectation by

E |u|>|z| 1-ξ |u| -γα1-γ 2 2 M γ (d 2 u) -p-1 C|z| (1-ξ) (α 1 + γ 2 -Q) 2 2 (6.14)
where we used the GMC estimate (9.3) in the Appendix. We conclude that

A 1 C|z| (1-ξ) (α 1 + γ 2 -Q) 2 2 +γ(Q-α1) . (6.15) Hence A 1 = o(|z| γ(Q-α1) ) if e.g. ξ < 1
2 and η is small enough. Next we bound A 2 . Let A be the annulus

A := {x ∈ C; |z| 1-ξ |x| 1/2}. We can split the numerator in A 2 into |K B c 1/2 (z) -K B c 1/2 (0)| and |K A (z) -K A (0)| by means of the triangular inequality. On B c 1/2 we can use ||x -z| γ 2 2 -|x| γ 2 2 | C|x| γ 2 2 |z| to get E[|K B c 1/2 (z) -K B c 1/2 (0)|K B c (0) -p-1 ] C|z|E[K B c 1/2 (0)K B c (0) -p-1 ] C|z|E[K B c (0) -p ] C|z|. Finally, using ||x -z| γ 2 2 -|x| γ 2 2 | C|x| γ 2
2 -1 |z| on A and then applying Cameron-Martin, we get

E[|K A (z) -K A (0)|K B c (0) -p-1 ] C|z| A |x| γ 2 2 -1-γα1 E |u|>|z| 1-ξ K(0, u)|u -x| -γ 2 M γ (d 2 u) -p-1 d 2 x.
Since |z| 1-ξ |x| we can bound

|u|>|z| 1-ξ K(0, u)|u -x| -γ 2 M γ (d 2 u) C |u|>|x| |u| -γα1-γ 2 2 M γ (d 2 u)
and then the GMC estimate (9.3) in the Appendix gives

E[|K A (z) -K A (0)|K B c (0) -p-1 ] C|z| A |x| γ 2 2 -1-γα1+ 1 2 (α1+ γ 2 -Q) 2 d 2 x (6.16) C|z| ξ+(1-ξ) γ(Q-α1)+ 1 2 (α1-2 γ ) 2 = o(|z| γ(Q-α1
) ) (6.17) for ξ < 1 2 and η small enough (since

α 1 -2 γ > γ 2 -η). Hence T 1 = o(|z| γ(Q-α1) ).
Now we focus on T 2 . First we show that it suffices to restrict K to the complement of the annulus A h := {x ∈ C; e -h |z| |x| |z| 1-ξ } where h > 0 is fixed: it will serve as a buffer zone to decorrelate the regions {x ∈ C; |x| e -h |z|} and {x ∈ C; |x| > |z| 1-ξ }. Interpolating as in (7.11) we deduce

|E[K(z) -p -K A c h (z) -p ]| E[K A h (z)K B c (0) -p-1
]. Using the Cameron-Martin theorem we get

|E[K(z) -p -K A c h (z) -p ]| C A h |x -z| γ 2 2 |x| -γα1 E B c K(0, u)|u -x| -γ 2 M γ (du) -p-1 d 2 x. (6.18)
The expectation was estimated in (6.14) so that we get

|E[K(z) -p -K A c h (z) -p ]| C|z| (1-ξ) γ(Q-α1)+ 1 2 (Q-α1-γ 2 ) 2 . (6.19)
For ξ < 1 2 and η small this yields

|E[K(z) -p -K A c h (z) -p ]| = o(|z| γ(Q-α1)
). Therefore, we just need to evaluate the quantity

E[K A c h (z) -p ]-E[K B c (z) -p ]
where we recall the definitions

B c = {|x| |z| 1-ξ } and A c h = B c ∪ B e -h |z| . Hence K A c h (z) = K B c (z) + K B e -h |z| (z).
We use the polar decomposition of the chaos measure introduced in Section 2.7. Let |z| = e -t . Then

K B c (z) = 2π 0 (1-ξ)t -∞ e γ(Bs-(Q-α1)s) |e -s+iθ -z| γ 2 2 |1 -e -s+iθ | γα2 (e -s(γ(α1+αa+α3-γ 2 ) ∨ 1)N γ (dsdθ) := K 1 K B e -h |z| (z) = 2π 0 ∞ t+h e γ(Bs-(Q-α1)s) |e -s+iθ -z| γ 2 2 |1 -e -s+iθ | γα2 N γ (dsdθ) := K 2 .
The lateral noises in K 1 and K 2 are weakly correlated. Indeed, from (2.43) we get (6.20)

-e -ξt E[Y (s, θ)Y (s ′ , θ ′ )] 2e -ξt .
for all s < (1 -ξ)t, s ′ > t + h and θ, θ ′ ∈ [0, 2π]. Define then the process

P (s, θ) := Y (s, θ)1 {s<(1-ξ)t} + Y (s, θ)1 {s>t+h} .
Let Y be independent of everything with the same law as Y and define the process

P (s, θ) := Y (s, θ)1 {s<(1-ξ)t} + Y (s, θ)1 {s>t+h} .
Then we get

E[ P (s, θ) P (s ′ , θ ′ )] -e -ξt E[P (s, θ)P (s ′ , θ ′ )] E[ P (s, θ) P (s ′ , θ ′ )] + 2e -ξt .
Let N be a unit normal variable independent of everything. Then the above means that the covariance of P + e -ξt N dominates that of P and the covariance of P + √ 2e -1 2 ξt N dominates that of P . Since K| P +aN = e γaN -1 2 γ 2 a 2 K| P we get by Kahane's convexity inequality (see [START_REF] Rhodes | Gaussian multiplicative chaos and applications: a review[END_REF]Theorem 2.1]) with the convex function

x ∈ R + → x -p e -C|z| 2ξ E[(K 1 + K 2 ) -p ] E[(K 1 + K 2 ) -p ] e C|z| ξ E[(K 1 + K 2 ) -p ]
where K 2 is computed with Y . Let

ǫ := e γB t+h -γ(Q-α1)(t+h)-γ 2 2 t . (6.21)
Then by the Markov property of Brownian motion

K 2 =ǫ 2π 0 ∞ 0 e γ( Bs-(Q-α1)s) |e -s-h+iθ -z |z| | γ 2 2 |1 -|z|e -s-h+iθ | γα2 N γ (d(h + t + s), dθ) (6.22)
where B is a Brownian motion independent of everything. Moreover, by stationarity of Y and its independence of everything we may replace N γ (d(h + t + s), dθ) by N γ (ds, dθ). As a consequence

E[(K 1 + ǫc -K 3 ) -p ] E[(K 1 + K 2 ) -p ] E[(K 1 + ǫc + K 3 ) -p ] (6.23)
where

K 3 = ∞ 0 e γ( Bs-(Q-α1)s) Z s ds and (6.24) c ± := (1 ± e -h ) γ 2 2 (1 ∓ |z|e -h ) γα2 .
By the Williams path decomposition Lemma 2.6 and (2.48) we deduce (6.25)

K 3 law = e γM ∞ -L-M e γB α 1 s Z s ds
where we recall M = sup s ( B s -(Q -α 1 )s) and L -M is the last time B α1 s hits -M . We discuss the lower and upper bounds in (6.23) in turn.

Lower bound. Let us use the notation

J B = B e γB α 1
s Z s ds and J for J R . We have

E[(K 1 + ǫc -K 3 ) -p ] E[(K 1 + ǫc -e γM J) -p ]
Now we can use the standard fact that M has exponential law with parameter 2(Q -α 1 ) to get

E[(K 1 + ǫc -K 3 ) -p ] -E[(K 1 ) -p ] 2(Q-α1) γ ∞ 1 E K 1 + ǫc -vJ -p -(K 1 ) -p v -1-2 γ (Q-α1) dv 2(Q-α1) γ c 2 γ (Q-α1) - E (ǫJ) 2 γ (Q-α1) (K 1 ) -p-2 γ (Q-α1) ∞ 0 (1 + w) -p -1 w -1-2 γ (Q-α1) dw = 2(Q-α1) γ c 2 γ (Q-α1) - Γ(-2 γ (Q -α 1 ))Γ(p + 2 γ (Q -α 1 )) Γ(p) E[J 2 γ (Q-α1) ]E ǫ 2 γ (Q-α1) (K 1 ) -p-2 γ (Q-α1)
where in the second step we made a change of variables w = ǫc-J K 1 v and for lower bound took the integration over w 0. In the last step we used Lemma 9.3 in the appendix to compute the integral and independence of J from everything. We end up with

E[(K 1 + ǫc -K 3 ) -p ] -E[(K 1 ) -p ] W c 2 γ (Q-α1) - E ǫ 2 γ (Q-α1) (K 1 ) -p-2 γ (Q-α1)
where we have have set

W := µ -2 γ (Q-α1) R(α 1 ) Γ(p + 2 γ (Q -α 1 )) Γ (p
) and R(α 1 ) is the reflection coefficient defined in (2.53). The remaining expectation can be computed thanks to the Cameron-Martin theorem applied to the term ǫ

2 γ (Q-α1) . Since t + h > (1 -ξ)t we get E ǫ 2 γ (Q-α1) (K 1 ) -p-2 γ (Q-α1) = |z| γ(Q-α1) E K B c (z) -p-2 γ (Q-α1)
where we defined for D ⊂ C (6.26)

K D (z) := D |x -z| γ 2 2 |x| γ(2Q-α1) |x -1| γα2 |x| γ(2Q-α1-γ 2 +α2+α3) + M γ (d 2 x)
and in the case D = C we will write K(z) for K C (z). Next, we claim

E K B c (z) -p-2 γ (Q-α1) -E K B c (0) -p-2 γ (Q-α1) = o(|z| γ(Q-α1) ). (6.27)
Indeed, the LHS is just T 1 in (6.11) computed with a larger p and |x| γα1 replaced by |x| γ(2Q-α1) . It is readily checked from (6.15) and (6.16) that (6.27) holds.

Summarizing all of our considerations related to the lower bound, we have shown that

E[(K 1 ) -p ] -E[(K 2 ) -p ] e -C|z| ξ |z| γ(Q-α1) c 2 γ (Q-α1) - W E K(0) -p-2 γ (Q-α1) + o(|z| γ(Q-α1) ) -(1 -e -C|z| ξ )E[K B c (z) -p ].
The second term on the RHS is

O(|z| ξ ) = o(|z| γ(Q-α1
) ) provided we take ξ > γ(Q-α 1 ) (this is the condiition that fixes ξ) so that recalling (6.24), we deduce

lim inf z→0 |z| -γ(Q-α1) E[K B e -h |z| (z) -p -K B c (z) -p ] (1 + e -h ) γ(Q-α1) W E K(0) -p-2 γ (Q-α1) .
Since h is arbitrary, it can be chosen arbitrarily large so as to get

lim inf z→0 |z| -γ(Q-α1) E[K B e -h |z| (z) -p -K B c (z) -p ] W E K(0) -p-2 γ (Q-α1) = µ p γΓ(p) -1 R(α 1 )|C γ (2Q -α 1 - γ 2 , α 2 , α 3 )
which is the desired lower bound.

Upper bound: For the upper bound we go back to the formula (6.25) where we need to face the integration region lower value L -M . For A > 0 fixed, we consider first the quantity

L(z) := E K 1 + ǫc + e γM ∞ -L-M e γB α 1 s Z s (ds) -p -(K 1 ) -p 1 {M A}
and we want to show that L(z) = o(|z| γ(Q-α1) ). Indeed, interpolating and using Cameron-Martin for the ǫ we get

|L(z)| c + E ǫe γM ∞ -L-M e γB α 1 s Z s ds K B c (z) -p-1 1 {M A} Ce γA E R e γB α 1 s Z s ds E ǫK B c ∩B(0, 1 2 ) (z) -p-1 C|z| γ(Q-α1) e γA E R e γB ν s Z s ds E |z| 1-ξ |z| 1 2 |x| -γα1-γ 2 2 M γ (d 2 x) -p-1 Ce γA |z| γ(Q-α1) |z| 1-ξ 2 (Q-α1-γ 2 ) 2
where we used the GMC estimate (9.3) and Lemma 2.8

It remains to investigate the quantity

U (z) :=E K 1 + ǫc + e γM ∞ -L-M e γB α 1 s Z s (ds) -p -(K 1 ) -p 1 {M A} E K 1 + ǫc + e γM J(A) -p -(K 1 ) -p 1 {M>A}
where we set J(A) = ∞ -L-A e γB α 1 s Z s ds. Using again the law of M , which is exponential with parameter 2(Q -α 1 ), and making the change change of variables ǫc+JA K 1 e γv = y we get

U (z) 2(Q-α1) γ E ∞ A K 1 + ǫc + e γv J A -p -(K 1 ) -p e -2(Q-α1)v dv = 2(Q-α1) γ c 2 γ (Q-α1) + E J 2 γ (Q-α1) A ǫ 2 γ (Q-α1) ∞ e γA ǫc + J A K 1 (1 + y) -p -1 (K 1 ) -p-2 γ (Q-α1) y -2 γ (Q-α1)-1 dy .
Now we can use Cameron-Martin as in the case of the lower bound to get that the above expectation can be rewritten as (recall (6.26))

E J 2 γ (Q-α1) A ǫ 2 γ (Q-α1) ∞ e γA ǫc + J A K 1 (1 + y) -p -1 (K 1 ) -p-2 γ (Q-α1) y -2 γ (Q-α1)-1 dy =|z| γ(Q-α1) E J 2 γ (Q-α1) A ∞ e γA ǫ(|z|e -h ) -2γ(Q-α 1 ) c + J A K B c (z) (1 + y) -p -1 K B c (z) -p-2 γ (Q-α1) y -2 γ (Q-α1)-1 dy .
Recalling (6.21) we have ǫ(|z|e -h ) -2γ(Q-α1) = e γB t+h +γ(Q-α1)(t+h)-γ 2 2 t and thus ǫ(|z|e -h ) -2γ(Q-α1) → 0 almost surely as z → 0 provided α 1 + γ 2 > Q which is the case. Dominated convergence theorem then ensures that the latter expectation converges to

E[J 2 γ (Q-α1) A ]E[ K B c (z) -p-2 γ (Q-α1) ] ∞ 0 (1 + y) -p -1 y -2 γ (Q-α1)-1 dy.
We can then conclude as for the lower bound by letting h, A → ∞.

6.3. The 4 point function with -2 γ insertion. In this section, we prove an analogue of Theorem 6.2 for the other degenerate insertion with weight -2 γ :

Theorem 6.3. We assume the Seiberg bounds for (-2 γ , α 1 , α 2 , α 3 ), i.e.

3 k=1 α k > 2Q + 2 γ and α k < Q for all k. There exists η > 0 such that for all α 1 , α 2 , α 3 ∈ (Q -η, Q) (6.28) T -2 γ (z) = C γ (α 1 - 2 γ , α 2 , α 3 )|F -(z)| 2 + R(α 1 )C γ (2Q -α 1 - 2 γ , α 2 , α 3 )|F + (z)| 2 .
Proof. The proof follows the proof of Theorem 6.2 almost word by word and we keep the same notation with the following obvious modifications. The function K in (6.29) is replaced by

(6.29) K(z, x) = |x -z| 2 |x| γ( 3 k=1 α k -2 γ ) + |x| γα1 |x -1| γα2
i.e. most important, the factor |x -z|

γ 2 2
is replaced by |x -z| 2 . Furthermore the exponent p is now given by p = (α 1 + α 2 + α 3 -2 γ -2Q)/γ and it is positive. We will fix η > 0 and ξ > 0 so that the following conditions hold for all

α 1 ∈ (Q -η, Q) 4 γ (Q -α 1 ) <(1 -ξ)(4 -γα 1 -2γη) (6.30) 4 γ (Q -α 1 ) <ξ. (6.31)
Note that for ξ = η = 0 (6.30) holds since 4 -γQ = 2 -γ 2 2 > 0 and therefore by continuity for small enough η and small enough ξ >4 γ η they hold as well. As in the proof of Theorem 6.2 we start with the splitting (6.12) to T 1 and T 2 given by (6.11) and we first show that

T 1 = o(|z| 4 γ (Q-α1) ). We obtain again |T 1 | C(A 1 + A 2 )
with the same definitions for A i . The Cameron-Martin bound for A 1 becomes

A 1 C |x| |z| 1-ξ |x| 2-γα1 E |u|>|z| 1-ξ K(0, u)|u -x| -γ 2 M γ (du) -p-1 d 2 x
and as the expectation is bounded by a constant we conclude that

A 1 C|z| (1-ξ)(4-γα1) = o(|z| 4 γ (Q-α1
) ) (6.32) by (6.30).

Next, for A 2 the bound (6.16) is replaced by

E[|K A (z) -K A (0)|K B c (0) -p-1 ] C|z| A |x| 1-γα1 d 2 x C|z| 1+(1-ξ)(3-γα1) = o(|z| 4 γ (Q-α1) ) (6.33)
again by (6.30). Hence T 1 = o(|z| Now we proceed with T 2 , again with the obvious changes (e.g. γ 2 2 in the definitions for K 1 , K 2 and c ± replaced by 2). Hence replacing (6.26) by

K D (z) := D |x -z| 2 |x| γ(2Q-α1) |x -1| γα2 |x| γ(2Q-α1-2 γ +α2+α3) + M γ (d 2 x)
we obtain instead of (6.27) the bound

E K B c (z) -p-2 γ (Q-α1) -E K B c (0) -p-2 γ (Q-α1) = o(|z| 4 γ (Q-α1) ). (6.34)
Indeed, the LHS is T 1 computed with a larger p and |x| γα1 replaced by |x| γ(2Q-α1) . Hence from (6.32) and (6.33) we get the bound

E K B c (z) -p-2 γ (Q-α1) -E K B c (0) -p-2 γ (Q-α1) C|z| (1-ξ)(4-γ(2Q-α1)) Since 4 -γ(2Q -α 1 ) = 4 -γα 1 -2γ(Q -α 1 )
4 -γα 1 -2γη and so (6.34) holds. The rest of the arguments for the lower and the upper bounds for T 2 follow then word by word.

6.4. Crossing relations. Proposition 2.4 now follows from Theorem 6.1 as explained in Section 2.5. Let us state it in the form we will apply it and also for the unit volume structure constants:

Proposition 6.4. Let ǫ ∈ ( γ 2 , 2 γ ) and α, α ′ < Q s.t. α + α ′ + ǫ -γ 2 > 2Q. Then (6.35) C γ (α ′ - γ 2 , ǫ, α) = T (α ′ , ǫ, α)C γ (α ′ , ǫ + γ 2 , α)
where T is the given by the following formula

(6.36) T (α ′ , ǫ, α) = -µπ l(a)l(b) l(c)l(a + b -c) 1 l(-γ 2 4 )l( γǫ 2 )l(2 + γ 2 4 -γǫ 2 ) where (6.37) a = γ 2 ( α ′ 2 - Q 2 ) + γ 2 ( ǫ 2 + α 2 - γ 2 ) - 1 2 b = γ 2 ( α ′ 2 - Q 2 ) + γ 2 ( ǫ 2 - α 2 ) + 1 2 and (6.38) c = 1 + γ 2 (α ′ -Q).
where a, b, c are given by

a = γ 4 (α ′ + α + ǫ -Q - γ 2 ) - 1 2 b = γ 4 (α ′ -α + ǫ -Q) + 1 2 c = 1 - γ 2 (Q -α ′ ). (6.39)
The above relation can be rewritten under the following form for the unit volume correlations (see (2.18) for the definition)

(6.40) Cγ (α ′ - γ 2 , ǫ, α) = T (α ′ , ǫ, α) Cγ (α ′ , ǫ + γ 2 , α)
where T is given by

(6.41) T (α ′ , ǫ, α) = Γ( 1 γ (α + α ′ + ǫ + γ 2 -2Q) Γ( 1 γ (α + α ′ + ǫ -γ 2 -2Q) T (α ′ , ǫ, α).
Along the same lines as Proposition 6.4, by exploiting Theorem 6.3 with the -2 γ insertion, one can show the following crossing symmetry relations: Proposition 6.5. Let when α, ǫ and α ′ close but strictly less than Q with α + α ′ + ǫ > 2Q + 2 γ . Then

(6.42) C γ (α - 2 γ , ǫ, α ′ ) = T (α, ǫ, α ′ )R(ǫ)C γ (α, 2Q -ǫ - 2 γ , α ′ )
where T is the given by the following formula

(6.43) T (α, ǫ, α ′ ) = l(a)l(b) l(c)l(a + b -c) where (6.44) a = 2 γ ( α 2 - Q 2 ) + 2 γ ( ǫ 2 + α ′ 2 - 2 γ ) - 1 2 b = 2 γ ( α 2 - Q 2 ) + 2 γ ( ǫ 2 - α ′ 2 ) + 1 2
and

(6.45) c = 1 + 2 γ (α -Q).

Proof of formula (2.54) on the reflection coefficient

We will suppose that γ 2 ∈ Q. This is no restriction since the general case can be deduced from this case by continuity in γ (Remark 2.3). The proof of formula (2.54) for the reflection coefficient is made of several steps and relies on proving that R satisfies the same shift equations (9.5) and (9.6) as R DOZZ . For the benefit of the reader we give here the summary of the structure of the argument (recall (2.53) and (2.52)): Subsection 7.1: We prove that R is analytic in the interval ( γ 2 , Q). More specifically, we show that for all compact intervals I ⊂ ( γ 2 , Q) there exists β > 0 such that R can be extended to a holomorphic function in I × (-β, β). Subsection 7.2: We prove that R satisfies the following shift equation for α ∈ (γ, Q)

(7.1) R(α - γ 2 ) = -µπ R(α) l(-γ 2 4 )l( γα 2 -γ 2 4 )l(2 + γ 2 2 -γα 2 )
In fact, we will show this relation in a small interval of the form (Q -η, Q) with η > 0. By analyticity, this leads to the same relation for α ∈ (γ, Q). One can choose an interval I ⊂ ( γ 2 , Q) of length greater than γ 2 such that R can be extended to a holomorphic function in I × (-β, β). The functional relation (7.1) enables to extend R to R × (-β, β); the extension that we also denote R is meromorphic with simple poles on the real line located at { 2 γ -γ 2 N} ∪ { γ 2 -2 γ N}. Subsection 7.4: We prove that R satisfies the following shift equation (inversion relation)

(7.2) R(α)R(2Q -α) = 1
One key ingredient is the analytic continuation result of subsection 7.3, the so-called gluing lemma, i.e. lemma 7.3. In fact, we will prove this relation for α in a small interval of the form (Q -η, Q) with η > 0. By analyticity, the condition will extend everywhere.

Subsection 7.5: We prove that R (as a meromorphic function on R × (-β, β)) satisfies the following shift equation

(7.3) R(α) = -c γ R(α + 2 γ ) l(-4 γ 2 )l( 2α γ )l(2 + 4 γ 2 -2α γ )
where c γ = γ 2 4 µπR(γ) = 0 is an unknown constant depending on γ (and µ). Recall that from the DOZZ solution we expect that (7.4) c γ = (µπl( γ 2 4 ))

4 γ 2 l( 4 γ 2 )
.

This indeed is the case since, quite miraculously, we show that the three equations (7.1), (7.2), (7.3) fully determine R and in particular lead to the above relation for c γ . Now R DOZZ satisfies (7.1) and (7.3) with (7.4); these shift equations fully determine R DOZZ and therefore we conclude R is equal to R DOZZ .

7.1. Proof of analycity of R in the interval ( γ 2 , Q). We start with the consequence (6.40) of the BPZ equations (with α = α ′ )

(7.5) Cγ (α - γ 2 , ǫ, α) = T (α, ǫ, α) Cγ (α, ǫ + γ 2 , α)
which holds for α close but strictly less than Q and ǫ ∈ ( γ 2 , 2 γ ) with 2α + ǫ -γ 2 > 2Q. In the same way as in the proof of lemma 2.9, we get for α ∈ ( γ 2 , Q)

lim ǫ↓ γ 2 (ǫ - γ 2 ) Cγ (α - γ 2 , ǫ, α) = µ 2 γ (Q-α) 4(Q -α) γ R(α). By Theorem 4.1, for ǫ > γ 2 , Cγ (α -γ 2 , ǫ, α) is analytic in α ∈ ( γ 2 , Q) and for ǫ γ 2 , Cγ (α, ǫ + γ 2 , α) is analytic in α ∈ ( γ 4 + ǫ 2 , Q).
Hence the relation (7.5) holds for all ǫ > γ 2 and α ∈ (

γ 4 + ǫ 2 , Q). Since lim ǫ↓ γ 2 (ǫ - γ 2 ) T (α, ǫ, α) = µπ 2 γ l( γ 2 α -γ 2 4 -1) l(1 + γ 2 (α -Q))l(-γ 2 4 )l( γ 2 4 ) Γ( 2 γ (α -Q) + 1) Γ( 2 γ (α -Q)) we conclude that for all α ∈ ( γ 2 , Q) (7.6) R(α) = µ 1-2 γ (Q-α) π γ l( γ 2 α -γ 2 4 -1) l(1 + γ 2 (α -Q))l(-γ 2 4 )l( γ 2 4 ) Γ( 2(α-Q) γ + 1) Γ( 2(α-Q) γ ) 2 γ (Q -α) Cγ (α, γ, α)
which proves our claim since Cγ (α, γ, α) is analytic in α ∈ ( γ 2 , Q). 7.2. Proof of the γ 2 shift equation. We use again (6.40)

(7.7) Cγ (α ′ - γ 2 , ǫ, α) = T (α ′ , ǫ, α) Cγ (α ′ , ǫ + γ 2 , α) which holds for ǫ ∈ ( γ 2 , 2 γ ) and α + α ′ + ǫ -γ 2 > 2Q with α, α ′ < Q. Let now α ′ = Q -η, α = 2 γ + η
where we fix η such that

(7.8) η ∈ (0, 2 γ -γ 2 ) ∧ γ 8
and consider both sides of (7.7) as a function of ǫ. From Theorem 4.1 we infer that both sides are analytic in ǫ in the region ǫ > 2η and hence the identity (7.7) extends to this region. As in the proof of Lemma 2.9 we have lim

ǫ↓2η (ǫ -2η) Cγ (α ′ - γ 2 , ǫ, α) = µ 2 γ (Q-α) 4(Q -α) γ R(α).
This indicates Cγ (α ′ -γ 2 , ǫ, α) has a pole at ǫ = 2η. We will now extract this pole.

Fix points z 2 , z 3 ∈ C such that |z 2 | 2, |z 3 | 2 and |z 2 -z 3 | 3. We have Cγ (α ′ - γ 2 , ǫ, α) = G ǫ,α,α ′ (z 2 , z 3 )E Z C (ǫ) 1-ǫ γ where G ǫ,α,α ′ (z 2 , z 3 ) = 2µ 1-ǫ γ γ -1 i<j 1 |z i -z j | αiαj +2∆ij with z 1 = 0, (α 1 , α 2 , α 3 ) = (ǫ, α, α ′ -γ 2 ) = (ǫ, 2 γ + η, 2 γ -η) and for A ⊂ C (7.9) Z A (ǫ) := A |x| γ(ǫ+α2+α3) + |x| γǫ |x -z 2 | γα2 |x -z 3 | γα3 M γ (d 2 x). Define next F (ǫ) := E Z C (ǫ) 1-ǫ γ -(Z B1(z2) (ǫ) + Z B1(z3) (ǫ)) 1-ǫ γ .
Importantly, notice that Z B1(z2) (ǫ) and Z B1(z3) (ǫ) do not depend on ǫ since for x ∈ B 1 (z 2 ) or x ∈ B 1 (z 3 ) we have We want to show that F (ǫ) is analytic over a neighborhood of the pole at ǫ = 2η (observe that (7.8) entails 2η < 1 γ ).

Lemma 7.1. F (ǫ) is analytic in ǫ ∈ (-2η -δ, 1 γ ) for some δ > 0. Proof. Let us fix δ > 0 such that (7.10) 2η + δ < 4 γ -γ and 4η + δ < γ which is possible because of (7.8). As in the proof of Theorem 4.1 we construct F as the uniform limit as t → ∞ of analytic functions F t in a neighborhood of (-2η -δ, 1 γ ). Let us denote C t = {z : |z| e -t } and define (recall that B r stands for the ball B r (0))

F t (ǫ) = E e ǫXt(0)-tǫ 2 2 Z t (0) 1-ǫ γ -(Z B1(z2) + Z B1(z3) ) 1-ǫ γ .
Let us first show that for each t, ǫ → F t (ǫ) is an analytic function of ǫ in an open neighborhood of (-2η -δ, 1 γ ). Let R 1 := Z B1(z2) + Z B1(z3) and R 2 := Z t (0) -R 1 . By (2.13) and (2.14) R 1 admits moments of order q for q < 2 γ (Q -η -2 γ ) and R 2 has moments of order q for q < 4 γ 2 . We interpolate

(7.11) E[e ǫXt(0) (R 2 + R 1 ) 1-ǫ γ -R 1-ǫ γ 1 ] = 1 0 E[e ǫXt(0) R 2 (sR 2 + R 1 ) -ǫ γ ] ds Let ǫ = ǫ 1 + iǫ 2 .
If ǫ 1 > 0 then since Ee pǫXt(0) < ∞ for all p < ∞ and since chaos has negative moments by Hölder we can bound the integrand by

CE[R q 2 ] 1 q for any q > 1. If ǫ 1 < 0 we bound |E[R 2 (sR 2 + R 1 ) -ǫ γ ]| C(E[R 1- ǫ 1 γ 2 ] + E[R 2 R - ǫ 1 γ 1 ]
). Hence we need 1 -ǫ1 γ < 4 γ 2 and by (slight variant of) Remark 3.5 we need

-ǫ1 γ < 2 γ (Q -η -2 γ ) = 1 -2η
γ . These conditions hold due to (7.10). F t is easily seen to be complex differentiable in ǫ.

Let us show that the family F t is Cauchy for the topology of uniform convergence over compact subsets. For this we will bound F t+1 -F t . First observe that because Z B1(z2) (ǫ) and Z B1(z3) (ǫ) are independent of X t (0) these terms cancel out in F t+1 -F t . Furthermore, Girsanov theorem gives

Ee ǫXt(0)-tǫ 2 2 Z t (0) 1-ǫ γ = Ee iǫ2Xt(0)+ tǫ 2 2 2 Z t (ǫ 1 ) 1-ǫ 1 γ
Hence as in the proof of Theorem 4.1 we get

|F t+1 -F t | e (t+1)ǫ 2 2 2 E|Z t+1 (ǫ 1 ) 1-ǫ 1 γ -Z t (ǫ 1 ) 1-ǫ 1 γ |.
From now on, since ǫ 1 is fixed we suppress it in the notation and denote Z t (ǫ 1 ) by Z t . We proceed as in the proof of Theorem 4.1. Let Y t := Z t+1 -Z t . We fix θ > 0 and write

E|Z 1- ǫ 1 γ t+1 -Z 1- ǫ 1 γ t | E1 Yt e -θt |Z 1- ǫ 1 γ t+1 -Z 1- ǫ 1 γ t | + E1 Yt e -θt |Z 1- ǫ 1 γ t+1 -Z 1- ǫ 1 γ t |
Interpolating the first term is bounded by

E1 Yt<e -θt |(Z t + Y t ) 1-ǫ 1 γ -Z 1- ǫ 1 γ t | Ce -θt sup s∈[0,1] E(Z t + sY t ) -ǫ 1 γ Ce -θt E(Z C (ǫ 1 ) -ǫ 1 γ )
The last expectation is finite since

-ǫ1 γ < 2 γ (Q -η -2 γ ) = 1 -2η
γ holds by (7.10). For the second term we use in turn Hölder's inequality (with conjugate exponents p, q), the mean value theorem and the Markov inequality (for some m ∈ (0, 1) to ensure finiteness of the expectation last expectation below) to get

E1 Yt e -θt |Z 1- ǫ 1 γ t+1 -Z 1- ǫ 1 γ t | [P Y t e -θt ] 1/p [E|Z 1- ǫ 1 γ t+1 -Z 1- ǫ 1 γ t | q ] 1 q [P Y t e -θt ] 1/p sup s∈[0,1] [EY q t (Z t + sY t ) -q ǫ 1 γ ] 1 q C t (q)e -θm p t E[Y m t ] 1/p C t (q)e 1 p (γ(Q-ǫ1-θ)m- m 2 γ 2 2 )
where we used Remark 3.5 and defined (7.12)

C t (q) = sup s∈[0,1] [EY q t (Z t + sY t ) -q ǫ 1 γ ] 1 q .
Summarizing, we have shown

(7.13) |F t+1 -F t | Ce tǫ 2 2
2 (e -θt + C t (q)e 1 p (γ(Q-ǫ1-θ)m-

m 2 γ 2 2 )
).

Now we have to optimize with respect to the free parameters p, q, θ, m. Let us first fix q (hence p). We first fix q to make (7.12) finite. Let first ǫ 1 > 0. By existence of negative moments of chaos we get for all r > q C t (q) C(r)

[EY r t ] 1 r . Hence sup t C t (q) < ∞ if q < 2 γ (Q -ǫ 1 ) ∧ 4 γ 2 = 4 γ 2 . If ǫ 1 < 0 we bound Y t Z B1 (ǫ 1 ) and Z t+1 Z B1 (ǫ 1 ) + Z B c 1 (ǫ 1 ) to get C t (q) [EY q t Z -q ǫ 1 γ t+1 ] 1 q C[EZ B1 (ǫ 1 ) q(1-ǫ 1 γ ) + EZ B1 (ǫ 1 ) q Z B c 1 (ǫ 1 ) -q ǫ 1 γ ] 1 q .
The first expectation is finite if q 1 + 2η+δ γ < 4 γ 2 and by Remark 3.5 the second one is finite if

q 2η+δ γ < 2 γ (Q -η -2 γ ) = 1 -2η γ .
Due to (7.10) we can find q > 1 such that this condition holds and hence sup t C t (q) < ∞.

Next, we choose θ > 0 such that Q -2 γ -θ > 0 and then m ∈ (0, 1) small enough such that

κ := p -1 (γ(Q - 2 γ -θ)m -γ 2 2 m 2 ) > 0.
As we have ǫ 1 < 2 γ we get from (7.13)

(7.14) |F t+1 -F t | Ce tǫ 2 2
2 (e -θt + e -κt ). Hence for ǫ 2 2 min(θ, κ) the sequence F t converges uniformly in compacts of a neighborhood of (-2η -δ, 1 γ ). Finally observe that F (ǫ) = lim t→∞ F t (ǫ) for ǫ ∈ R with ǫ ∈ (2η, Q). Lemma 7.2. We have

G ǫ,α,α ′ (z 2 , z 3 )E((Z B1(z2) (ǫ) + Z B1(z3) (ǫ)) 1-ǫ γ ) = γ -2η ǫ -2η R( 2 γ + η) + F 2 (ǫ, z 2 , z 3 )
where F 2 is analytic in ǫ on (-2η, 1 γ ) and lim ǫ→-2η

(ǫ + 2η)F 2 (ǫ) = 2µ 1+ 2η γ (1 + 2η γ ) R( 2 γ -η).
Proof. Let us first make the following general observation. Consider a random variable Y 0 admitting a "maximal" moment of order β > 0, namely E[Y α ] < +∞ for α < β and E[Y β ] = +∞. Then the mapping s ∈ C → E[Y s ] is holomorphic on the set {s ∈ C; 0 < ℜ(s) < β}. Furthermore it diverges at s = β. The point is to find an analytic continuation and, for this, we need to know more about the shape of the tail of the random variable Y .

The simplest situation is when we know that |P(Y > t)-c 1 t -β | c 2 t -β-δ for some constants c 1 , c 2 , β, δ > 0 and for all t t 0 . In this case, we start from the following standard decomposition for s < β

E[Y s ] = ∞ 0 P(Y > t)t s-1 dt that we rewrite as E[Y s ] = ∞ 0 P(Y > t) -c 1 t -β 1 {t 1} t s-1 dt - c 1 s -β .
In the above right-hand side, the parametrized integral is holomorphic on the set {s ∈ C; 0 < ℜ(s) < β + δ} as can be shown by the theorem of holomorphicity of such integrals and our tail estimate. This provides the desired analytic continuation for E[Y s ] as well as identification of the pole at s = β.

The argument can be extended to situations when we are able to give a more precise asymptotic expansion of the tail. In our lemma, this corresponds to the case when we have a tail estimate of the type

|P(Y > t) -c 1 t -β1 -c 2 t -β2 | c 3 t -β2-δ
for some constants c 1 , c 2 , c 3 , δ > 0 and 0 < β 1 < β 2 . In this case, the analytic continuation (still denoted by E[Y s ]) is given by the expression

E[Y s ] = ∞ 0 P(Y > t) -(c 1 t -β1 + c 2 t -β2 )1 {t 1} t s-1 dt - c 1 s -β 1 - c 2 s -β 1 .
One again the integral is holomorphic for s < β 2 + δ and we have two poles. The second pole at s = β 2 can be recovered by taking the limit

lim s→β2 (s -β 2 ) E[Y s ] + c 1 s -β 1 ) = c 2 .
The statement of our lemma is nothing but the above argument applied to the random variable Y := Z B1(z2) (ǫ) + Z B1(z3) (ǫ). The shape of the tail of this random variable is given by Lemma 3.3 (when α 2 , α 3 are both sufficiently close to Q as indicated in Remark (3.4)).

Combining we get that the function

f (ǫ) := Cγ ( 2 γ -η, ǫ, 2 γ + η) - γ -2η ǫ -2η R( 2 γ + η) is analytic on ǫ ∈ (-2η, 1 γ ). By (7.7) on ǫ ∈ (2η, 1 γ ) f = g where g(ǫ) := T (Q -η, ǫ, 2 γ + η) Cγ (Q -η, ǫ + γ 2 , 2 γ + η) - γ -2η ǫ -2η R( 2 γ + η)
Thus by analytic continuation g is analytic in ǫ on (-2η, 1 γ ). Consider now the limit ǫ → -2η. We get from Lemma 7.2 (7.15) lim

ǫ→-2η (ǫ + 2η)f (ǫ) = 2µ 1+ 2η γ (1 + 2η γ ) R( 2 γ -η) = R( 2 γ -η) Γ(-1 -2η
γ ) and then (7.16) lim ǫ→-2η

(ǫ + 2η)g(ǫ) = 2T (Q -η, -2η, 2 γ + η)R(Q -η) Γ(-1 -2η γ )
.

Now, we have for

ǫ = -2η = -γ 2 + α ′ -α that a = γ 2 α ′ - γ 2 2 -1 b = γ 2 (α ′ -α) - γ 2 4 = γǫ 2 and c = 1 + γ 2 (α ′ -Q) a + b -c = γ 2 (α ′ -α) - γ 2 2 -1 = 1 -(2 + γ 2 4 - γǫ 2 )
Therefore, we have l(b) = l( γǫ 2 ) and l(a

+ b -c)l(2 + γ 2 4 -γǫ 2 ) = 1 and T (Q -η, -2η, 2 γ + η) = -µπ l( γ 2 α ′ -γ 2 2 -1) l(1 + γ 2 (α ′ -Q))l(-γ 2 4 )
In conclusion, we get the following relation from the fact that (7.15) and (7.16) are equal:

(7.17) R(α ′ - γ 2 ) = -µπ l( γ 2 α ′ -γ 2 2 -1) l(1 + γ 2 (α ′ -Q))l(-γ 2 4 ) R(α ′ )
We have proven (7.17) for α ′ close to Q but since by previous subsection R is analytic on ( γ 2 , Q) it extends to α ′ ∈ (γ, Q). Combined with expression (7.6) for R, we can use the relation (7.17) to analytically continue R to a neighborhood of R of the form R × (-β, β) with β > 0. In the sequel, we will consider the meromorphic extension of R to R × (-β, β). Since R DOZZ also satisfies (7.17) and 0 < R(α) R DOZZ (α) < ∞ for α ∈ ( γ 2 , Q), one 7.4. Proof of the inversion relation. We use the relation from Proposition 6.5:

(7.21) C γ (α - 2 γ , ǫ, α ′ ) = T (α, ǫ, α ′ )R(ǫ)C γ (α, 2Q -ǫ - 2 γ , α ′ )
which was proven for α, ǫ and α ′ close but strictly less than Q with α + α ′ + ǫ > 2Q + 2 γ .

The case γ < √ 2. By analytic continuation of the unit volume three point structure constants Cγ (α -2 γ , ǫ, α ′ ), Cγ (α, 2Q -ǫ -2 γ , α ′ ) (given by (2.18)) and R(ǫ), the relation is valid for

α = Q -η, ǫ = 2 γ + η ′ , α ′ = 2 γ
for η ′ in the interval (0, η). One can notice that (7.21) is also valid within this range of parameters with no poles since

α - 2 γ + ǫ + α ′ = γ 2 + 4 γ + η ′ -η > 4 γ + 2η ′ and α + 2Q -ǫ - 2 γ + α ′ = Q -η + γ -η ′ + 2 γ = 2Q + γ 2 -η -η ′
Now, we wish to analytically continue α to the value Q + η by using the gluing lemma 7.3. We can indeed use the gluing lemma when the 4 point correlation function

V -γ 2 (z)V 2 γ +η ′′ (0)V γ-η ′ (1)V 2 γ (
∞) stays well defined for all η ′ ∈ (0, η) and all η ′′ ∈ (-η, η) (indeed, one can show that the existence of the above 4 point correlation function implies that one can apply the gluing lemma). More specifically, we must check that

2Q + γ 2 - 2 γ -η ′′ -γ + η ′ - 2 γ < γ ∧ (γ -2η ′′ ) ∧ ( 4 γ -γ + 2η ′ )
This is possible as soon as γ 2 < γ ∧ ( 4 γ -γ), i.e. γ < 8 3 . Therefore, the relation (7.21) becomes for α = Q + η, ǫ = 2 γ + η ′ , α ′ = 2 γ for η sufficiently small and η ′ ∈ (0, η)

(7.22) C γ (α - 2 γ , ǫ, α ′ ) = T (α, ǫ, α ′ )R(ǫ)R(α)C γ (2Q -α, 2Q -ǫ - 2 γ , α ′ ) (7.23) C γ (Q + η - 2 γ , ǫ, 2 γ ) = T (α, ǫ, α ′ )R(ǫ)R(α)C γ (Q -η, 2Q -ǫ - 2 γ , 2 γ )
Now, we consider the limit of (7.23) as ǫ increases to Q -η. From (2.17) we infer

lim ǫ↑Q-η (Q -η -ǫ)C γ (α - 2 γ , ǫ, 2 γ ) = -2, lim ǫ↑Q-η (Q -η -ǫ)C γ (Q -η, 2Q -ǫ - 2 γ , 2 γ ) = 2
Indeed, one can notice that the two limits above correspond to insertions such that s goes to 0 in expression (2.17). Also we get lim ǫ↑Q-η

(Q -η -ǫ)l(a) = -γ, lim ǫ↑Q-η (Q -η -ǫ)l(b) = 1 γ and lim ǫ↑Q-η l(c)l(a + b -c) = l(1 + 2η γ )l(- 2η γ ) = 1 This yields R(Q -η)R(Q + η) = 1 for η small hence R(2Q -α)R(α) = 1 on the strip R × (-β, β)
where R admits a meromorphic extension.

The case γ > √ 2. By analytic continuation, the relation is valid for

α = Q -η, ǫ = γ + η ′ , α ′ = 2 γ
for η ′ around 0. Indeed, in this case, we have

α + 2Q -ǫ - 2 γ + 2 γ = Q -η -η ′ + 4 γ > 2Q and (in C γ (α -2 γ , ǫ, α ′ ) the biggest insertion is at ǫ = γ + η ′ since γ > √ 2) α - 2 γ + ǫ + α ′ = γ 2 -η + γ + η ′ + 2 γ > 2γ
In fact, both relations above are valid for η and η ′ in a small interval. We want to analytically continue α = Q -η to α = Q + η by using the gluing lemma. In order to do so, we must check that the 4 point correlation function

V -γ 2 (z)V 2 γ +η ′′ (0)V 2 γ -η ′ (1)V 2 γ (
∞) is well defined during the procedure, i.e. for η ′′ ∈ [-η, η] and η ′ small. We have

- γ 2 + 2 γ + η ′′ + 2 γ -η ′ + 2 γ > 4 γ + η
for η small. Hence, we can analytically continue to α = Q + η which leads to the following relation

C γ (α - 2 γ , ǫ, α ′ ) = T (α, ǫ, α ′ )R(ǫ)R(α)C γ (2Q -α, 2Q -ǫ - 2 γ , α ′ )
or equivalently for all η ′ small

C γ ( γ 2 + η, γ + η ′ , 2 γ ) = T (α = Q + η, ǫ = γ + η ′ , α ′ = 2 γ )R(γ + η ′ )R(Q + η)C γ (Q -η, 2 γ -η ′ , 2 γ )
Now, we can conclude as in the previous case by letting ǫ = γ + η ′ increase to Q -η.

7.5. Proof of the 2 γ shift equation. We get that for all ǫ, α, α ′ close to but strictly less than Q that

(7.24) R(ǫ)C γ (2Q -ǫ - 2 γ , α, α ′ ) = l(c -1)l(c -a -b + 1) l(c -a)l(c -b) R(α)C γ (ǫ, 2Q -α - 2 γ , α ′ ) where (7.25) a = 2 γ ( ǫ 2 - Q 2 ) + 2 γ ( α + α ′ 2 - 2 γ ) - 1 2 b = 2 γ ( ǫ 2 - Q 2 ) + 2 γ ( α -α ′ 2 ) + 1 2 and (7.26) c = 1 + 2 γ (ǫ -Q) c -a -b + 1 = 2 - 2 γ (α - 2 γ )
This leads to the following values

(7.27) c -a = 3 2 + 2 γ ( ǫ 2 - Q 2 ) - 2 γ ( α + α ′ 2 - 2 γ ) c -b = 1 2 + 2 γ ( ǫ 2 - Q 2 ) - 2 γ ( α -α ′ 2 ) and (7.28) c -1 = 2 γ (ǫ -Q) c -a -b + 1 = 2 - 2 γ (α - 2 γ )
We fix η very small. Now, we can analytically continue equation (7.24) to the following values

(7.29) ǫ = γ 2 + η ′ , α = γ 2 + η ′′ , α ′ = 2 γ + η with η ′ ∈ [η, 2η] and η ′′ ∈ [6η, 7η]
. This is indeed possible if each side is well defined without taking out poles. Indeed, we have with these values that

2Q -ǫ - 2 γ + α + α ′ = 2Q -η ′ + η ′′ + η > 2Q so C γ (2Q -ǫ -2 γ , α, α ′
) satisfies the standard Seiberg bounds. Also, we have

ǫ + 2Q -α - 2 γ + α ′ = 2Q + η ′ -η ′′ + η therefore the quantity C γ (ǫ, 2Q -α -2 γ , α ′ ) exists provided that the largest insertion does not blow up. The largest insertion is 2Q-α-2 γ = Q-η ′′ which has a moment of order 2 γ η ′′ . The moment of C γ (ǫ, 2Q-α-2 γ , α ′ ) is given by 1 γ (2Q -(2Q + η ′ -η ′′ + η)) = 1 γ (η ′′ -η ′ -η) < 2 γ η ′′
Now, we want to let η ′ go to -η. To do so, we must use the gluing lemma which correponds to the fact that the following 4 point correlation function is defined during the procedure (2Q -ǫ -

2 γ = Q -η ′ ) V -γ 2 (z)V 2 γ -η ′ (0)V γ 2 +η ′′ (1)V 2 γ +η (∞) This well defined if for all η ∈ (-2η, η) we have (biggest insertion is 2 γ + η) 2Q + γ 2 - 2 γ -η - γ 2 -η ′′ - 2 γ -η < γ -2η which is equivalent to η + η ′′ + η > 2η which holds since η ′′ ∈ [6η, 7η]. Therefore, we get for ǫ = γ 2 -η that for all η ′′ ∈ [6η, 7η] (7.30) R(ǫ)R(2Q -ǫ - 2 γ )C γ (ǫ + 2 γ , α, α ′ ) = l(c -1)l(c -a -b + 1) l(c -a)l(c -b) R(α)C γ (ǫ, 2Q -α - 2 γ , α ′ )
Thanks to the inversion relation, we have and therefore we expect that c γ = (πµl( γ 2 4 ))

4 γ 2 l( 4 γ 2 )
. However, at this stage of the proof, we can not determine the constant c γ . It will be determined indirectly in subsection 7.6.

It remains prove the limit (7.32), i.e. we study (recall expression (2.17))

E ρ(Q -η ′′ , γ 2 -η, 2 γ + η)
η ′′ γ as η ′′ goes to 0 with (α 1 , α 2 , α 3 ) = Q -η ′′ , γ 2 -η, 2 γ + η, i.e. we put the Q -η ′′ insertion at 0. We decompose

ρ(Q -η ′′ , γ 2 -η, 2 γ + η) = R η ′′ + S η ′′
where R η ′′ is the reminder

R η ′′ = |x| 1 2 |x| γ(α1+α2+α3) + |x| γα1 |x -1| γα2 M γ (d 2 x)
and S η ′′ is the rest. We have the following lemma: From these inequalities, we can conclude that lim

η ′′ →0 E[R η ′′ p γ η ′′ ] = 2 2-p .
Now, we have

E[R η ′′ γ η ′′ ] E[ρ(Q -η ′′ , γ 2 -η, 2 γ + η) η ′′ γ ] E[R η ′′ γ η ′′ (1 + S η ′′ R η ′′ ) η ′′ γ ]
The left hand side in the above inequality converges to 2. The right hand side satisfies for all p, q > 1 such that

1 p + 1 q = 1 E[R η ′′ γ η ′′ (1 + S η ′′ R η ′′ ) η ′′ γ ] E[R pη ′′ γ η ′′ ] 1/p E[(1 + S η ′′ R η ′′ ) qη ′′ γ ] 1/q
Now for all q > 1, E[(1

+ S η ′′ R η ′′ ) qη ′′ γ ] 1/q → η ′′ →0
1 hence we the result (7.32) follows from the Lemma since p can be taken arbitrary close to 1. for some constant C γ . If γ 2 and 2 γ are independent over the rationals i.e. if γ 2 / ∈ Q then we conclude that C γ = 1 and ψ(α) = ψ is constant in α. From (2.51) we see that R(Q) = 1 and from (2.53) since Γ(-x)x → -1 as x → 0 we get R(Q) = -1. On the other hand, from (1.16) follows R DOZZ (Q) = -1 hence the constant ψ = 1.Hence R(α) = R DOZZ (α) for all α. The case γ 2 ∈ Q follows by continuity. This concludes the proof.

Proof of the DOZZ formula

We suppose that γ 2 / ∈ Q; the general case follows by continuity. Let us fix α 2 , α 3 in (Q -η, Q) for η sufficiently small and consider the function F : α 1 → C γ (α 1 , α 2 , α 3 ). Let us collect what we have proven about F . By Theorem 4.1 F is analytic on (2η, Q) and by Theorem 6.1 it satisfies the the γ 2 shift equation (1.8), for γ 2 +2η < α 1 < 2 γ . Therefore F extends to a meromorphic function on a strip of the form R×(-β, β) with β > 0 satisfying (1.8). We call this extension F too. Now, using the exact expression for R (or relation (7.3) with c γ = µπl( γ 2 4 )) By the gluing Lemma, the extension F is given in a neighborhood of α = Q by F (α) = R(α)F (2Q -α).

Hence, one can infer from the above expression the shift equation (1.9) for α 1 ∈ R× (-β, β) (same argument as the one used to derive (1.8)). Hence F satisfies both (1.8) and (1.9). Now, we consider the function ψ α2,α3 : α 1 → Cγ (α1,α2,α3) C DOZZ γ (α1,α2,α3) in the strip R × (-β, β). This function is holomorphic since C γ and C DOZZ γ are meromorphic with the same simple poles and zeros (which can be read off the γ 2 shift equation (1.8)). Furthermore, ψ α2,α3 is γ and 4 γ periodic since C γ and C DOZZ γ both satisfy (1.8) and (1.9). Therefore ψ α2,α3 (α 1 ) = c α2,α3 for some constant c α2,α3 depending on α 2 , α 3 .

Since C γ and C DOZZ γ are symmetric in their arguments we obtain ψ α2,α3 (α 1 ) = ψ α1,α3 (α 2 ) = ψ α1,α2 (α 3 ) for α 1 , α 2 , α 3 ∈ (Q-η, Q). Hence c α2,α3 is constant in α 2 , α 3 . Therefore C γ (α 1 , α 2 , α 3 ) = a γ C DOZZ γ (α 1 , α 2 , α 3 ) for α 1 , α 2 , α 3 satisfying (2.14) with N = 3 for some constant a γ (by analycity). Finally a γ = 1 since both C γ and C DOZZ γ satisfy Lemma 2.9. 9. Appendix 9.1. Chaos estimates. We list in this Appendix estimates for chaos integrals that are used frequently in the paper. Let A(z, ǫ) be the annulus with radi ǫ, 2ǫ and centre z. Then, for m ∈ [0, dx 2 + µ coth(µx) d dx which is the generator of (B t + µt) conditioned to be positive. We denote this process B µ t . We have the following comparison principle: Lemma 9.2. There exists a probability space such that for all µ < µ ′ , we have B µ t B µ ′ t almost surely. Proof. For all x > 0, we consider the drift ϕ x (µ) = µ coth(µx). A straightforward computation yields ϕ ′

x (µ) = e 4µx -4µxe 2µx -1 (e 2µx -1) 2 .

Therefore ϕ ′ x (µ) 0 since e u -ue u 2 -1 0 for all u 0. We will need another comparison principle. We want to show that B µ t starting from 0 stochastically dominates (B t + µt) starting from 0 and conditioned to be above -A with A > 0. This can also be read off the drift. Indeed, for µ, x fixed, we consider ψ µ,x (A) = µ coth(µ(x + A)). We have ∀x -A, ψ ′ µ,x (A) = µ 2 (1 -coth(µ(x + A)) 2 ) 0

By taking µ to 0 in lemma 9.2 , the above comparison principle can in fact be extended to B 0 t where B 0 t denotes the standard 3d Bessel process. 9.3. Functional relations on Υ γ 2 and R DOZZ . The function Υ γ 2 defined by (1.12) can be analytically continued to C and it satisfies the following remarkable functional relations for z ∈ C (9.4)

Υ γ 2 (z + γ 2 ) = Γ( γ 2 z) Γ(1 -γ 2 z) ( γ 2 ) 1-γz Υ γ 2 (z), Υ γ 2 (z + 2 γ ) = Γ( 2 γ z) Γ(1 -2 γ z) ( γ 2 ) 4 γ z-1 Υ γ 2 (z).
The function Υ γ 2 has no poles in C and the zeros of Υ γ 2 are simple (if γ 2 ∈ Q) and given by the discrete set (-γ 2 N -2 γ N) ∪ (Q + γ 2 N + 2 γ N): for more on the function Υ γ 2 and its properties, see the reviews [START_REF] Nakayama | Liouville field theory: a decade after the revolution[END_REF][START_REF] Ribault | Conformal Field theory on the plane[END_REF][START_REF] Teschner | Liouville Theory Revisited[END_REF] for instance.

(2. 1 )

 1 E[X(x)X(y)] = ln 1 |x -y| + ln |x| + + ln |y| + := G(x, y) where we use the notation |z| + = |z| if |z| 1 and |z| + = 1 if |z| 1.

2. 5 .

 5 Crossing relation. Let us suppose α 1 < Q and α 2 + γ 2 < Q. We have from the previous subsection(2.35)

3. 1 .

 1 Tail estimate around one insertion. Let |z| > 2 and consider the random variable

Theorem 4 . 1 .

 41 Fix N 3 and disjoint points z 1 , . . . , z N ∈ C N . The unit volume correlation function (2.15) extends to an analytic function of α defined in a complex neighborhood of U N in C N .

5. 1 .Ze

 1 Proof of lemma 2.8. By symmetry, it is enough to show thatE[ α s Z s ds p 1/pWe set ν = Q -α. The process B α s is stochastically dominated by a Brownian motion with drift -ν starting from origin and conditioned to stay below 1 (see Appendix); hence we have if B s is a standard Brownian motion starting from 0 s ds) p ]E[e γp sup s∈[n,n+1] (Bs-Bn) ]E[1 Bn-νn 1 e γp(Bn-νn) ] CE[1 Bn-νn 1 e γp(Bn-νn) ] = Cn -γpy e -(y+νn) 2 2n dy where we used(2.47). Considering separately y < -νn 2 and y ∈ [-νn 2 , 1] the last integral is seen to be exponentially small in n and the claim follows.

1 2 and we have I(B c 1 2 ,

 122 z) < C. Since I(C, z) 2(I(B 1 2 , z) + I(B c 1 2

  |x| γǫ + |x| γǫ = 1. Hence we denote them by Z B1(z2) .

  α ′ ) = l(c -1)l(c -a -b + 1) l(c -a)l(c -b) R(α)C γ (ǫ, 2Q -α -2 γ , α ′ )Now, by analytic continuation, we can let η ′′ converge to 0, i.e. α converge to γ 2 . We get the following limit when η ′′ goes to 0l(c -1)l(c -a -b + 1) l(c -a)l(c -b) R(α) ∼ η ′′ →0

Lemma 7 . 5 . 1 1 1 1e

 7511 For all p ∈ (1, 2), we have It is obvious that we can replace R η ′′ with|x| |x| γα1 M γ (d 2 x)We set M = sup s 0 (B s -(Q -η ′′ )s) where B s is the circle average of X of radius e -s . We set L -M the last time B α1 -s hits -M . Recall that P(M > v) = e -2η ′′ v and therefore E[(e γM ) η ′′ p γ ] = 2 2-p . We have in distribution|x| |x| γα1 M γ (d 2 x) law = e γM ∞ -L-M e γB α 1s Z s ds.Therefore, we have the following inequality with α = 2 γ for instance and R s a two sided Bessel process by stochastic domination (see the section 9.2 in the appendix on diffusions) -γRs Z s ds.

7. 6 .

 6 Proof that R = R DOZZ . Let ψ(α) = R(α) R DOZZ (α) . ψ is meromorphic in the strip R × (-β, β).Since R and R DOZZ obey the same γ 2 shift equation, the function ψ is γ 2 periodic. ψ is strictly positive in ( γ 2 , Q) so by periodicity ψ is strictly positive on R. By the 2 γ shift equation, one has for all α ∈ R ψ(α) = C γ ψ(α + 2 γ )

4 γ 2 l( 4 γ 2 ) - 1 ) 2 , α 3 )

 42123 Theorem 6.3 can be written asT -2 γ (z) = C γ (α 1 -2 γ , α 2 , α 3 )|F -(z)| 2 -|F + (z)| 2

4 γ 2 ) 2 and 2 2 2 ( 9 . 3 ) 9 . 1 .|y 1 -y 3 | |y 1 -y 4 | 1 2 (α1+α2+α3-Q) 2 |y 1 -y 4 | 1 2 2 ]C |y 1 -y 2 | |y 1 -y 3 | 1 2 1 2- α 2 3 2 |y 1 -y 4 | 1 2 2 d 2 dx 2 2 d 2 dx 2 - 2 d 2

 22229391131121121211131122222222 (9.1)E(M γ (A(z, ǫ) m ) Cǫ γQm-γ 2 m |x -z| -α M γ (d 2 x) m Cǫ γ(Q-α)m-γ 2 mFor negative moments we have for α > Q:E |x-z|>ǫ |x -z| -α M γ (d 2 u)Lemma Assume (α i ) i=1,...,4 are real numbers satisfying α i < Q and p:= γ -1 ( i α i -2Q) > 0. Consider y 1 , y 2 , y 3 , y 4 ∈ C are such that |y 1 -y 2 | |y 1 -y 3 | |y 2 -y 3 | min i∈{1,2,3} |y 4 -y i |. 1) If α 1 + α 2 < Q, α 1 + α 2 + α 3 0 and α 4 0 then E B(y1,10) 4 i=1 |u -y i | -γαi M γ (d 2 u) (α1+α2+α3+α4-Q) 2 . 2) If α 1 + α 2 > Q, α 3 0 and α 3 + α 4 0 then E B(y1,10) 4 i=1 |u -y i | -γαi M γ (d 2 u) -p-(α1+α2-Q) 2 |y 1 -y 3 | |y 1 -y 4 | (α1+α2+α3-Q) 2 (α1+α2+α3+α4-Q) 2 .9.2. A reminder on diffusions.A drifted Brownian motion (B t + µt) with µ > 0 is a diffusion with generator G µ = 1 + µ d dx . When seen until hitting b > 0, the dual process Y b of B t + µt is a diffusion with generator 1 µ coth(µ(b -x)) d dx . Therefore, b -Y b has generator 1

  with c γ = γ2 8 µπR(γ) and where we have used l(x)l(-x) = -1 x 2 and the first shift equation ′′ . Now, we can conclude by looking at the limit as η ′′ goes to 0 ofC γ ( γ 2 -η, Q -η ′′ , 2 γ + η). We prove below that This yields that C γ ( γ 2 -η, Q -η ′′ , 2 γ + η) ∼

			η ′′ →0	-4 η ′′ and therefore we get the shift equation
	(7.33)	R(ǫ) R(ǫ + 2 γ )	= -	c γ γ )l(2 + 4 γ 2 )l( 2ǫ l(-4 γ 2 -2ǫ γ )
	with c γ = γ 2 4 µπR(γ) = 0.
	Remark 7.4. A straightforward computation yields that
		γ 2 4	µπR DOZZ (γ) =	(πµl( γ 2 4 )) l( 4 γ 2 )	4 γ 2
		1 γ )l(2 + 4 γ 2 )l( 2ǫ l(-4 γ 2 -2ǫ γ ) 1 l(-4 γ 2 )l( 2ǫ γ )l(2 + 4 γ ) γ 2 -2ǫ 1 l(-4 ∼ η ′′ →0 = γ 2 )l( 2ǫ γ )l(2 + 4 γ 2 -2ǫ γ ) c γ	(-(-	η ′′ γ η ′′ γ	)R(α) )( 2 γη ′′	µπR(γ) l(-γ 2 4 )l( γ 2 4 )	)
	R(α + γ 2 ) 4 )l( γα l(-γ 2 2 )l(2 + γ 2 4 -γα 2 ) E ρ(Q -η ′′ , R(α) = -µπ 2 2 , α, α ′ ) ∼ It is clear that C γ (ǫ + γ η ′′ →0 η (7.32) lim η ′′ →0 γ 2 -η, 2 γ + η) η ′′ γ = 2.

The indices 2, 3 occur in the applications of this estimate in the main text.

γ (Q-α1) ).

Supported by the Academy of Finland, 2 Research supported in part by ANR grant Liouville (ANR-15-CE40-0013)

can see that R and R DOZZ have their poles and zeros located at the same place. For instance, the poles of R are located at { 2 γ -γ 2 N} ∪ { γ 2 -2 γ N}.

7.3. The gluing lemma. We introduce the following condition:

3. We suppose that α 2 , α 3 satisfies condition (7.18). Then the function

is the restriction on the real line of a holomorphic function defined on a neighborhood of Q. This holomorphic function is explicitely given for α = α 1 + γ 2 by

)) In particular, the function S is real analytic in the neighborhood of Q.

Proof. We will show that α 1 → S( γ 2 +α 1 ) is the restriction on the real line of a holomorphic function defined on the neighborhood of 2 γ . Clearly, one can find ǫ > 0 and ǫ < 2 γ -γ 2 such that for all

As a consequence of Theorem 6.1, we get the crossing symmetry relation (1.8); more specifically, one can find η > 0 such that for all α 2 , α 3 ∈ (Q -η, Q) and all α 1 in an interval I ⊂ ( γ 2 , 2 γ ) of size η we have 2Q + γ 2 -α 1 -α 2 -α 3 < 0 and

.

where ¯ α = α 1 + α 2 + α 3 . By analytic continuation from α 1 , α 2 , α 3 to α 1 , α 2 , α 3 (see Theorem 4.1), we get for all

.

where ᾱ = α 1 + α 2 + α 3 . Now, recall that we are working with R extended to a strip R × (-β, β) thanks to relation (7.17). As a consequence of Theorem 6.2 we get the following crossing symmetry relation (derived similarly to (1.8)): one can find η > 0 such that for all

.

where ¯ α = α 1 + α 2 + α 3 . By analytic continuation from α 1 , α 2 , α 3 to α 1 , α 2 , α 3 (see Theorem 4.1 and use the fact that R is analytic), we get for all

.

where ᾱ = α 1 + α 2 + α 3 . This yields the result.

With definition (1.16) and a little algebra, one can show that R DOZZ (α) satisfies the following shift equation for all α ∈ C (9.5)

2 ) as well as the dual shift equation for all α ∈ C (9.6)

Recall that the function Υ γ 2 satisfies the shift equations (9.4). According to the DOZZ formula (1.13), since Υ γ 2 (0) = 0, we get for α > γ 2 and using the above relations

))

))

9.5. An integral formula. We have:

Lemma 9.3. For all p > 0 and a ∈ (1, 2) the following identity holds

Proof. We set ā = -a + 1 and b = p + a -1. We have

Now, we have

Now, we use the following formula (see [30]):

This yields the desired relation since Γ(z + 1) = zΓ(z). 9.6. Some identities. We have the following identity for all z

.

By taking the ∂ zz derivative, we get

.

Hence for z = 1 this yields (9.7)

(1 -u) .