Riad Khalil

Catherine Faron Bouzidi

Bruno Zucker

Nhan Fies

Le Thanh

Khalil Riad Bouzidi
email: khalil-riad.bouzidi@cstb.fr

Catherine Faron-Zucker
email: catherine.faron-zucker@unice.fr

Bruno Fies
email: bruno.fies@cstb.fr

Nhan Le Than
email: nhan.le-thanh@unice.fr

An ontological approach for modeling technical standards for compliance checking

Keywords: Ontology, Semantic Web, Knowledge Management, Building Industry, E-regulations, E-Government

This paper gives an overview of a formal semantic-based approach of modeling some regulations in the photovoltaic field to help the delivering of technical assessments at the French scientific center on Building Industry (CSTB). Starting from regulatory texts, we first explicit SBVR rules and then formalize them into ontology-based rules in the SPARQL language. These are exploited in the modeling of the compliance checking process required for the delivering of technical assessments.

Introduction

The French initiative "Grenelle de l'Environment", launched in France in 2007 is an open multi-party debate aiming to define the key points of public policy on ecological and sustainable development issues. It will generate additional information of technical regulatory whose analysis is viewed by professionals as an increasing burden. Many rapid changes of regulations and technical documents will intervene to keep the commitment made mainly with regard to optimizing the energy aspects of building.

In this general context, the research communities of Knowledge Engineering and Semantic Web have a key role to play to provide models and techniques to simplify access to technical regulatory information, facilitate its appropriation, support professionals in its implementation, and facilitate the writing of new regulations while taking into account constraints expressed in the existing regulatory corpus.

Our work aims to propose dedicated semantic services to support the process of modeling the compliance checking or regulations. We focus on the process of acquiring expressions in the Semantics of Business Vocabulary and Rules (SBVR) language from textual technical standards and the transformation of these SBVR expressions into interpretable ontology-based rules.

In the next section we present our approach to model technical standards in SBVR. We detail in section 3 the transformation of SBVR rules into ontology-based rules represented in the SPARQL language. In section 4, we present our approach to interpret compliance checks results. We conclude in section 5.

Modeling Business Rules

SBVR stands for "Semantics of Business Vocabulary and Business Rules". It is an OMG standard whose ultimate objective is to provide a meta-model that allows establishing data exchange interfaces for tools that create, organize, analyze and use vocabularies and business rules [START_REF]The Object Management Group OMG. Semantics of Business Vocabulary and BusinessRules (SBVR)[END_REF], [START_REF] Chapin | Semantics of Business Vocabulary & Business Rules (SBVR)[END_REF]. SBVR rules are based upon the Predicate Logic: they capture their semantics and not the way they must be executed. SBVR is not an executable formalism. It is particularly addressed to business experts. It uses a controlled natural language that all business experts understand. It does not have a specific rule format.

The development of an SBVR base is done in two steps: the development of a business vocabulary and the writing of business rules based on the terms and concepts defined in the vocabulary. SBVR controlled vocabularies consist in hierarchies of concepts specific to some domain, their relationships, definitions and synonyms.

We built an ontology for technical documents. OntoDT is extracted manually from the weft provided by CSTB for these documents. This weft contains all information necessary for a process, a product or a material to be studied by a specialized group at CSTB who will be responsible for delivering a Technical Assessments. This ontology of technical document primarily consisted in a hierarchy of primitive concepts in OWL Lite. We merged OntoDT with the terms of the REEF thesaurus developed by CSTB for the Building Industry. By reusing the REEF we built a controlled vocabulary standard enabling to link technical assessments (ATec) to the REEF. As a result; our ontology of technical document contains 50 classes and 26 properties. 35% of these classes are created from REEF terms; the remaining 65% are concepts more specific than those of the REEF thesaurus which contains general concepts of the building industry. In its current state, it lacks specific terms relative to a particular field (Photovoltaic). However, it remains in constant evolution.

Transforming standards into SBVR rules

Technical standards can be understood in different ways and this is why the manual intervention of a domain expert is essential. We argue that NLP approaches of knowledge extraction from regulatory texts can significantly alleviate the task of domain experts but cannot replace them. In our work, we do not consider linguistic analysis of texts and focus on the representation of expert knowledge. CSTB experts helped us to identify and classify the constraints expressed in the photovoltaic standards and then the rules which represent them.

The extraction of rules from standards or statutory text is a tedious job, it often requires to structure the information. The descriptions used have been detailed enough to show how the content of standards can be converted into SBVR vocabulary and business rules. However, a clarification of the text was needed before the transformation into SBVR. The steps below are necessary in order to produce an understandable SBVR text:

1 Disambiguation: Let us start from the following sentence as regulatory text:

Classe A: Accès général, tension dangereuse, applications de puissance dangereuses : Les modules assignés pour utilisation dans cette classe d'application peuvent être utilisés dans des systèmes fonctionnant à plus de 120 V en courant continu. Les modules requis pour la sécurité dans la présente partie de l'EN 61730 et l'EN 61730-2 dans cette classe d'application sont considérés comme satisfaisant aux exigences de la classe II de sécurité.

This text means that a photovoltaic module is of class A if its system runs over more than120 volt and satisfies the requirements of security class II. The first step is to rewrite this text into SBVR rules by using the ontology of technical document so that the text will be read as follows:

If a module has a system which runs over more than 120 volt, and this module is satisfying security class 2, then it is of class A

The concepts identified in this fragment are Module, System, Class, which belong to the ontology of technical document.

1 Reformulation.

Let us consider the following regulatory text:

Les dimensions du châssis principal doivent être : Largeur intérieur : (847 ± 5) mm Hauteur intérieur :(1910 ± 5) mm

This standard extract expresses conditions that are difficult to read by non expert readers. It needs a reformulation to be understood:"The maximum width of a main frame must be lower or equal to 853mm and the minimum width higher or equal to 842mm. The maximum height of a main frame must be lower or equal to 1915mm and the minimum height greater than or equal to 1905mm".

If a frame has a minimum width higher or equal to 842mm and a minimum height higher or equal to 1905mm and a maximum width less than or equal to 853mm and a maximum height less than or equal to 1915mm, then it is a main frame 1 Formulations to avoid.

Let us consider the following regulatory text:

« Un module doit avoir une isolation assignée pour un minimum de 90 °C, avec un calibre et des caractéristiques de tension acceptables pour l'application... » This case contains information not only ambiguous but constraints which cannot be interpreted: "with a size and voltage characteristics acceptable for the application" This reformulation is considered non interpretable which avoids its reformulation in SBVR.

Transformation of SBVR rules into ontology-based SPARQL rules

SBVR describes business concepts and requirements without addressing their implementation. To validate our model, we developed a rule base for checking the compliance of technical documents to regulatory standards in the photovoltaic field. Our rules are formalized in the SPARQL language and annotated in the RDF language. This enables us to both automatically reason on the representation of regulatory standards and to model the process of compliance checking.

With the help of CSTB experts we have defined orders of processing business rules based on priorities existing among standards of the same domain, among the constraints of a standard and the additional knowledge that specifies the rules of the same constraint. We aim to formalize the experts reasoning like this:

• In the same domain: Standards that describe a device (comprising multiple components) have priority on those that describe unique components. Example: Standard NF1 61730 which describes "the requirement of PV module" has priority over NF12600 which describes "the classification of glass"

• In the same standard: Constraints that describe the compliance of a product have priority on those relative to its components. Example: in standard NF61730, application class constraints of PV module have priority on electrical cable constraints used in this module.

• In the same constraint: The representation of a constraint can involve multiple business rules. In that case, there are priority orders that can be extracted from standards when identified by experts. We represent it in RDF in a so-called annotation of standards requirement (ASR) and we rely on these ASR to schedule rules in the conformance checking process. These annotations are based on OntoDT; the requirement scheduling is mainly based on priorities among components. For instance, in Standard NF 61730 described in Fig. 1, priority requirements are: solar energy, solar cell, etc.

<Dt:Standard rdf:ID="NF61730"> <Dt:hasForRequirement rdf:parseType="Collection"> <rdf:Description rdf:about="#SolarEnergie"/> <rdf:Description rdf:about="#SolarCell"/> ... </Dt:hasForRequirement > </Dt:Standard>

Fig. 1. Extract of an ASR

We use SPARQL query patterns like that in Fig. 2 in order to identify requirement orders by querying ASR annotations before browsing the explicit dependency rules.

Interpretation of compliance checking results

The implementation of our compliance checking model is based upon the matching of standards representations with those of a technical document, i.e. the matching of SPARQL queries with RDF annotations if there are conditions for applying the standard. For this purpose, we use the CORESE semantic engine [START_REF] Corby | Querying the SemanticWeb with Corese Search Engine[END_REF].

One major problem when automating the compliance checking process is to justify the decisions taken by the systemcompliance or noncompliance of the product. Technical standards that validate the products are modeled as rules of compliance checking. The failure of these rules means the noncompliance of some component.

We propose an approach based on reaction rules that set off as soon as noncompliance is identified. This kind of rules helps to identify failing compliance checking rules and the component or property responsible of noncompliance.

We use reaction rules to explain the "why" of the negative decision. These rules are triggered by Event occurrences and require satisfaction of Conditions to perform Actions. They are therefore called ECA rules. According to several works, [START_REF] Bry | Realizing Business Processes with ECA Rules: Benefits, Challenges, Limits. In 4th International Workshop, PPSWR 2006[END_REF], [START_REF] Knolmayer | Modeling Processes and Workflows by Business Rules[END_REF], ECA rules are best adapted to describe the logic of a process by a set of rules. They are extensions of production rules (rules where the event part is missing; noted Condition-Action "CA").We use ECA rules to represent in a declarative manner the logic of a process and explain negative decision. Our model of reaction rule is as follows:

1 Event: represents the context of implementation of an action. It is defined by the noncompliance of a component caused by the failure of a compliance checking rule.

1 Condition: is checked during the identification of noncompliant products. CORESE has the notions of Event and Event Listener that enable to catch some predefined events. Events are related to SPARQL query processing. In our case, we will implement an Event Listener to capture the cause of failure, i.e. of non compliance.

1 Action: A message display to an external user to explain the result of applying the condition. In our case it would represent the "why" of making a negative decision.

The originality of our approach is to use reaction rules to justify the decision taken in case of noncompliant of a photovoltaic product. The triggering event is the failure of one or more compliance checking rules of a specific product "R noncompliance ". In the condition we check the failure rules events in CORESE engine to identify the real cause of noncompliance "Check Events ". The action is a response message displayed that explains the noncompliance of the product "Response".

We will use a knowledge base containing a list of explanations produced by experts. Each explanation or response is unique to a single state of noncompliance. The answer will be extracted by using a SPARQL query that takes as input the noncompliant component and displays as output the appropriate response. Let us consider the example of a noncompliant application class A of a photovoltaic module. First, the process is triggered when a noncompliance of the photovoltaic module to Application Class A is identified. The Application Class rules contain two parameters: the Voltage and the Security Class. The noncompliance of one of these two parameters leads to the failure of the rules. The verification of which parameter are noncompliant is carried out. We identify the failure request by catching the Event which demonstrates it.

Conclusion

The main contribution of our research is twofold. First, we propose a method to formalize paper-based regulation texts using SBVR standards and ontology based rules. Second, we propose a model based on ECA rules to explain negative decisions taken by the system.

We aim to use simple and unambiguous standard language conventions to specify regulation. SBVR modeling represents a quality assurance of the technical standards text and decrease later modification needs. It uses an implicit model (SBVR vocabulary) and expresses definitions, concepts and restrictions with a clear associated semantics.

This controlled vocabulary is an ontology called OntoDT formalized in OWL-Lite and coupled with the REEF thesaurus for the building industry. SBVR defines correspondence between SBVR rules and implementation rules which can help to formalize the SBVR concepts in terms of existing rule languages. In our case we use SPARQL as SBVR implementation rule language. Our model integrates expert knowledge in the process of compliance checking.

This knowledge is expressed into annotations representing standards in rules format. Annotations are also used to establish a compliance report that interprets the matching between compliance rules and technical document representations, especially in case of noncompliance. To explain the noncompliance we use reaction rules based on the ECA model. This work is still at a premature phase, experimentation and full evaluation must be completed.

http://www.afnor.org


Bertrand Duplantier
email: bertrand.duplantier@cea.fr.

R Émi Rhodes

Scott Sheffield
email: sheffield@math.mit.edu.

Vincent Vargas
email: vargas@ceremade.dauphine.fr.

Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula

Gaussian Multiplicative Chaos is a way to produce a measure on R d (or subdomain of R d ) of the form e γX(x) dx, where X is a log-correlated Gaussian field and γ ∈ [0, √ 2d) is a fixed constant. A renormalization procedure is needed to make this precise, since X oscillates between -∞ and ∞ and is not a function in the usual sense. This procedure yields the zero measure when γ = √ 2d. Two methods have been proposed to produce a non-trivial measure when γ = √ 2d. The first involves taking a derivative at γ = √ 2d (and was studied in an earlier paper by the current authors), while the second involves a modified renormalization scheme. We show here that the two constructions are equivalent and use this fact to deduce several quantitative properties of the random measure. In particular, we complete the study of the moments of the derivative martingale, which allows us to establish the KPZ formula at criticality.

Introduction 1.1 Overview

In the eighties, Kahane [START_REF] Kahane | Sur le chaos multiplicatif[END_REF] developed a continuous parameter theory of multifractal random measures, called Gaussian multiplicative chaos. His efforts were followed by several authors [START_REF] Allez | Lognormal ⋆-scale invariant random measures, Probability Theory and Related Fields[END_REF][START_REF] Bacry | Log-infinitely divisible multifractal processes[END_REF][START_REF] Barral | Multifractal products of cylindrical pulses[END_REF][START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF][START_REF] Fan | Sur le chaos de Lévy d'indice 0 < α < 1[END_REF][START_REF] Rhodes | ⋆-scale invariant random measures[END_REF][START_REF] Rhodes | Multidimensional multifractal random measures[END_REF][START_REF] Robert | Gaussian Multiplicative Chaos revisited[END_REF] coming up with various generalizations at different scales. This family of random fields has found many applications in various fields of science, especially in turbulence and in mathematical finance.

Roughly speaking, a Gaussian multiplicative chaos on R d or on a bounded domain of R d (with respect to the Lebesgue measure) is a random measure that can formally be written as:

M(dx) = e γX(x)-γ 2 2 E[X(x) 2 ] dx (1)
where X is a centered Gaussian distribution and γ a nonnegative parameter. The situation of interest is when the field X is log-correlated, that is when

K(x, y) def = E[X(x)X(y)] = ln 1 |x -y| + g(x, y) (2)
for some bounded continuous function g. In this case, X is a Gaussian random generalized function (a.k.a. distribution) on R d that cannot defined as an actual function pointwise. Kahane showed that one can nonetheless give a rigorous definition to [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF]. Briefly, the idea is to cut off the singularity of the kernel (2) occurring at x = y (sometimes referred to as ultraviolet cutoff). The cut-off strategy we will use here is based on a white noise decomposition of the Gaussian distribution X. In short, we will assume that the covariance kernel of X can be written as:

K(x, y) = +∞ 1 k(u(x -y)) u du
for some continuous covariance kernel k. Though not covering the whole family of kernels of type [START_REF] Allez | Lognormal ⋆-scale invariant random measures, Probability Theory and Related Fields[END_REF], this family of fields is quite natural since it possesses nice scaling relations (see [START_REF] Allez | Lognormal ⋆-scale invariant random measures, Probability Theory and Related Fields[END_REF]). A white noise decomposition of X involves formally writing X as

X(x) = ∞ 1 R d g(y + x) √ s W (ds, dy)
where W is a space-time white noise and g is a convolution square root of k. Then a cut-off of X at level t can be written as

X t (x) = e t 1 R d g(y + x) √ s W (ds, dy).
This is a Gaussian field with continuous covariance kernel that approximates X in the sense that we recover the distribution X when letting t go to ∞. Having applied this cutoff, it is now possible to define the approximate measure

M t (dx) = e γXt(x)-γ 2 2 E[Xt(x) 2 ] dx. (3)
The family (M t ) t is a positive martingale, so it converges almost surely. The measure M is then understood as the almost sure limit of this martingale. The limiting measure M is non trivial if and only if γ 2 < 2d (see [START_REF] Kahane | Sur le chaos multiplicatif[END_REF]). For γ 2 2d, the measure M as defined by (1) thus vanishes, giving rise to the issue of constructing non trivial objects for γ 2 2d in any other possible way.

In this paper, we pursue the effort initiated in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] to understand the critical case, that is when γ 2 = 2d. It is shown in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] that the natural object at criticality is the derivative multiplicative chaos, which can be formally written as

M ′ (dx) = ( √ 2dE[X(x) 2 ] -X(x))e √ 2dX(x)-dE[X(x) 2 ] dx. (4)
It is a positive atomless random measure. It can be rigorously defined via cut-off approximations in the same spirit as in [START_REF] Alvarez-Gaumé | A proposal for strings at D > 1[END_REF]. More precisely, the approximations are obtained by differentiating (3) with respect to γ at the value γ 2 = 2d, hence the term "derivative". Nevertheless, it is expected that derivative Gaussian multiplicative chaos at criticality, that is M ′ , can be recovered via a properly renormalized version of [START_REF] Alvarez-Gaumé | A proposal for strings at D > 1[END_REF]. In this paper, inspired by analog results in the case of branching Brownian motion [START_REF] Lalley | A conditional limit theorem for frontier of a branching Brownian motion[END_REF] or branching random walk [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF], we prove that

√ t M t (dx) → 2 π M ′ (dx), in probability as t → ∞. (5)
This renormalization procedure is sometimes called Seneta-Heyde scaling [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF]. Property [START_REF] Bacry | Log-infinitely divisible multifractal processes[END_REF] establishes the important fact that the derivative martingale also appears as the limit of the natural renormalization of the vanishing martingale (3) for γ 2 = 2d. Beyond this unifying perspective, this renormalization approach to criticality is convenient to complete the description of the main properties of the derivative martingale initiated in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF]. In particular, the Seneta-Heyde renormalization turns out to be crucial to applying Kahane's convexity inequalities at criticality. From this, we complete the study of the moments of the derivative martingale and compute its power-law spectrum. These properties are a prerequisite for establishing the celebrated Knizhnik-Polyakov-Zamolodchikov (KPZ) formula at criticality arising in Liouville quantum gravity. Let us stress that our proof of the KPZ formula shows that it is valid for any log-correlated Gaussian field for which one can prove a critical Senete-Heyde renormalization theorem of the type [START_REF] Bacry | Log-infinitely divisible multifractal processes[END_REF].

Recently, the authors in [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF] constructed a probabilistic and geometrical framework for two dimensional Liouville quantum gravity and the KPZ equation [START_REF] Knizhnik | Fractal structure of 2Dquantum gravity[END_REF], based on the two-dimensional Gaussian free field (GFF) (see [START_REF] David | Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge[END_REF][START_REF] Francesco | 2D gravity and random matrices[END_REF][START_REF] Distler | Conformal Field Theory and 2-D Quantum Gravity or Who's Afraid of Joseph Liouville?[END_REF][START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF][START_REF] Ginsparg | Lectures on 2D gravity and 2D string theory[END_REF][START_REF] Knizhnik | Fractal structure of 2Dquantum gravity[END_REF][START_REF] Nakayama | Liouville Field Theory -A decade after the revolution[END_REF] and references therein for physics considerations). It consists in taking X equal to the GFF in [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF]. In this context, the KPZ formula has been proved rigorously [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF][START_REF] Rhodes | KPZ formula for log-infinitely divisible multifractal random measures[END_REF] (see also [START_REF] Benjamini | KPZ in one dimensional random geometry of multiplicative cascades[END_REF] in the context of Mandelbrot's multiplicative cascades). This was done in the standard case of Liouville quantum gravity, namely strictly below the critical value of the GFF coupling constant γ in the Liouville conformal factor, i.e, for γ 2 < 4 (recall that the phase transition occurs at γ 2 = 2d in dimension d, producing a phase transition for γ 2 = 4 in dimension 2).

Extensions of our results: the Liouville measure at criticality in dimension 2

Up to minor modifications, the techniques we developped in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] and the present paper enable to prove analogous results in the case where X is the GFF on a bounded domain D ⊂ R 2 . Thus we can give a rigourous definition to the standard Liouville measure at criticality (see discussion below). More precisely, our techniques apply straightforwardly within the following approach. The Green function is given by the formula:

G D (x, y) = π ∞ 0 p D (t, x, y)dt.
where p D is the (sub-Markovian) semi-group of a Brownian motion B killed upon touching the boundary of D, namely

p D (t, x, y) = P x (B t ∈ dy, T D > t)
with

T D = inf{t 0, B t ∈ D}. Note that the π term ensures that G D (x, y) ∼ |x-y|→0
-ln |x -y|. The most direct way to construct the GFF is then to consider a white noise W distributed on D × R + and define:

X(x) = √ π D×R + p D ( s 2 , x, y)W (dy, ds). One can check that E[X(x)X(x ′ )] = π ∞ 0 p D (s, x, x ′ ) ds = G D (x, x ′
). The corresponding cut-off approximations are given by:

X t (x) = √ π D×[e -2t ,∞[ p D ( s 2 , x, y)W (dy, ds).
One can then easily show that:

M ′ t (dx) = (2E[X t (x) 2 ] -X t (x))e 2Xt(x)-2E[Xt(x) 2 ] dx.
converges to a non trivial positive and atomless random measure M ′ . One can also show that:

√ t M t (dx) → 2 π M ′ (dx), in probability as t → ∞.
and that all the other results of this paper apply to M ′ (KPZ, existence of moments, etc...). Let us nevertheless stress two points. Firstly, this approach is one way of constructing the critical Liouville measure but it does not show that all the other approaches yield the same measure. One must then extend the theories developped in [START_REF] Kahane | Sur le chaos multiplicatif[END_REF][START_REF] Robert | Hydrodynamic Turbulence and Intermittent Random Fields[END_REF][START_REF] Robert | Gaussian Multiplicative Chaos revisited[END_REF] or the one developped in [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF] to show that there is a unique critical measure, i.e. independent of the limiting procedure. Secondly, by analogy with the star scale invariant case, we believe that the (properly shifted and normalized) law of the maximum of the GFF on a discrete lattice with mesh going to 0 converges in law to the sum of ln M ′ (D) and an independent Gumbel variable (up to some constant c; see our conjecture 12 in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF]). Indeed, this should be the case if the discrete lattice is triangular as the discrete GFF on a triangular grid is a projection of the continuous GFF on the functions which are affine in the triangles (see [START_REF] Schramm | A contour line of the continuum Gaussian free field, Probability Theory and related Fields[END_REF]). There is no reason to believe that the form of the grid plays any specific role, at least if the lattice is regular enough. This gives in particular an explicit candidate to the limit law of [START_REF] Bramson | Convergence in law of the maximum of the twodimensional discrete Gaussian free field[END_REF].

Remarks about atomic measures at criticality

In dimension two, the Liouville quantum gravity measure on a domain D is sometimes interpreted as the image of the intrinsic measure of a random surface M under a conformal map that sends M to D. This type of "surface" is highly singular (not a manifold in the usual sense). In certain limiting cases where the surface develops singular "bottlenecks", one expects the image measure on D to become an atomic measure. In a certain sense (that we will not explain here), constructing these atomic measures requires one to replace γ < 2 by a "dual value" γ ′ > 2 satisfying γγ ′ = 4.

It is interesting to consider the analogous atomic measure in the critical case γ = γ ′ = 2 and to think about what its physical significance might be. We believe that the both the γ = 2 measures (treated in this paper) and their γ ′ = 2 "atomic measure" variants (see below) have been studied in the physics literature before. However, when reading the physics literature about γ = γ ′ = 2 Liouville quantum gravity, it is sometimes difficult to sort out which physical constructions correspond to which mathematical objects. The remainder of this subsection will describe some of the history of these constructions and their relationship to the current work. This discussion can be safely skipped by the reader without specific background or interest in this area.

The issue of mathematically constructing singular Liouville measures beyond the phase transition, namely for γ 2 > 4, and deriving the corresponding (non-standard dual) KPZ formula has been investigated in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF], giving the first mathematical understanding of the so-called duality in Liouville quantum gravity (see [START_REF] Alvarez-Gaumé | A proposal for strings at D > 1[END_REF][START_REF] Ambjørn | A solvable 2d gravity model with γ > 0[END_REF][START_REF] Das | New critical behavior in d = 0 large-N matrix models[END_REF][START_REF] Duplantier | Conformal fractal geometry and boundary quantum gravity[END_REF][START_REF] Durhuus | Multi-spin systems on a randomly triangulated surface[END_REF][START_REF] Jain | World-sheet geometry and baby universes in 2-D quantum gravity[END_REF][START_REF] Klebanov | Touching random surfaces and Liouville gravity[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Klebanov | Wormholes, Matrix Models, and Liouville Gravity[END_REF][START_REF] Kostov | Boundary Loop Models and and 2D Quantum Gravity[END_REF] for an account of physics motivations). It thus remains to complete the mathematical Liouville quantum gravity picture at criticality, i.e. for γ 2 = 4. From the physics perspective, Liouville quantum gravity at criticality has been investigated in [START_REF] Brézin | Scaling violation in a field theory of closed strings in one physical dimension[END_REF][START_REF] Ginsparg | 2D gravity + 1D matter[END_REF][START_REF] Gross | One-dimensional string theory on a circle[END_REF][START_REF] Gross | A nonperturbative solution of D = 1 string theory[END_REF][START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects 2000, JHEP Proceedings; A matrix model for the two-dimensional black hole[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Kostov | Loop amplitudes for nonrational string theories[END_REF][START_REF] Kostov | Strings with discrete target space[END_REF][START_REF] Kostov | Multicritical phases of the O(n) model on a random lattice[END_REF][START_REF] Parisi | On the one dimensional discretized string[END_REF][START_REF] Polchinski | Critical behavior of random surfaces in one dimension[END_REF][START_REF] Sugino | Critical behavior in c = 1 matrix model with branching interactions[END_REF]. The reader is also referred to [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] for a brief summary about the physics literature on Liouville quantum gravity at criticality. Let us just stress that the critical case γ = 2 corresponds to the value c = 1 of the so-called central charge c of the conformal field theory coupled to gravity, via the famous KPZ result [START_REF] Knizhnik | Fractal structure of 2Dquantum gravity[END_REF]:

γ = ( √ 25 -c - √ 1 -c)/ √ 6, c 1.
The Liouville measure at criticality presents an unusual dependence on the Liouville field ϕ (equivalent to X here) of the so-called "tachyon field" T (ϕ) ∝ ϕ e 2ϕ [START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects 2000, JHEP Proceedings; A matrix model for the two-dimensional black hole[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Polchinski | Critical behavior of random surfaces in one dimension[END_REF]. Its integral over a "background" Borelian set A generates the quantum area A = A T (ϕ)dx, that we can recognize as the formal heuristic expression for the derivative measure (4). The possibility at criticality of another tachyon field of the atypical form T (ϕ) ∝ e 2ϕ nevertheless appears in [START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects 2000, JHEP Proceedings; A matrix model for the two-dimensional black hole[END_REF][START_REF] Klebanov | Wormholes, Matrix Models, and Liouville Gravity[END_REF]. This form seems to heuristically correspond to a measure of type (1) (which actually vanishes for γ = 2). At first sight, our result (5) here then seems to suggest that, up to the requested renormalization (5) of (1), the atypical tachyon field would actually coincide with the usual ϕ e 2ϕ tachyon field. However, this atypical tachyon field e 2ϕ in Liouville quantum gravity has been associated to another, non-standard, form of the critical c = 1, γ = 2 random surface models. Indeed, the introduction of higher trace terms in the action of the c = 1 matrix model of two-dimensional quantum gravity is known to generate a new critical behavior of the random surface [START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects 2000, JHEP Proceedings; A matrix model for the two-dimensional black hole[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Sugino | Critical behavior in c = 1 matrix model with branching interactions[END_REF], with an enhanced critical proliferation of spherical bubbles connected one to another by microscopic "wormholes".

In order to model this non-standard critical theory, it might be necessary to modify the measures introduced here by explicitly introducing "atoms" on top of them, using the approach of [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF] for adding atoms to γ < 2 random measures M γ in the description of the dual phase of Liouville quantum gravity. The "dual Liouville measure" corresponding to γ < 2 involves choosing a Poisson point process from η -α-1 dηM γ (dx), where α = γ 2 /4 ∈ (0, 1), and letting each point (η, x) in this process indicate an atom of size η at location x. When γ = 2 and α = 1, we can replace M γ with the derivative measure M ′ (4) (i.e., the limit ( 5)), and use the same construction; in this case (since α = 1) the measure a.s. assigns infinite mass to each positive-Lebesgue-measure set A ∈ B(R d ). It is nonetheless still well-defined as a measure, and all of its (infinite) mass resides on a countable collection of atoms, each with finite mass. Alternatively, one may use standard Lévy compensation (intuitively, this amounts to replacing an "infinite measure" with an "infinite measure minus its expectation", interpreted in such a way that the result is finite) to produce a random distribution whose integral against any smooth test function is a.s. a finite (signed) value. One may expect that this construction yields the continuum random measure associated with the non-standard c = 1, γ = 2 Liouville random surface with enhanced bottlenecks, as described in [START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects 2000, JHEP Proceedings; A matrix model for the two-dimensional black hole[END_REF][START_REF] Sugino | Critical behavior in c = 1 matrix model with branching interactions[END_REF], thus giving a mathematical interpretation to the (formal) tachyon field e 2ϕ that differs from the renormalized measure (5).

Setup

Notations

For a Borelian set A ⊂ R d , B(A) stands for the Borelian sigma-algebra on A. All the considered fields are constructed on the same probability space (Ω, F , P). We denote by E the corresponding expectation. Given a Borelian set A ⊂ R d , we denote by A c its complement in R d . The relation f ≍ g means that there exists a positive constant c > 0 such that c -1 f (x) g(x) cf (x) for all x.

⋆-scale invariant kernels

Here we introduce the Gaussian fields that we will use throughout the paper. We consider a family of centered stationary Gaussian processes ((X t (x)) x∈R d ) t 0 where, for each t 0, the process (X t (x)) x∈R d has covariance given by:

K t (x) = E[X t (0)X t (x)] = e t 1 k(ux) u du (6)
for some covariance kernel k satisfying k(0) = 1, of class C 1 and vanishing outside a compact set (actually this latter condition is not necessary but it simplifies the presentation).

We also assume that the process (X t (x) -X s (x)) x∈R d is independent of the processes (X u (x)) x∈R d u s for all s < t. In other words, the mapping t → X t (•) has independent increments. Such a construction of Gaussian processes is carried out in [START_REF] Allez | Lognormal ⋆-scale invariant random measures, Probability Theory and Related Fields[END_REF]. For γ 0, we consider the approximate Gaussian multiplicative chaos M γ t (dx) on R d :

M γ t (dx) = e γXt(x)-γ 2 2 E[Xt(x) 2 ] dx (7)
It is well known [START_REF] Allez | Lognormal ⋆-scale invariant random measures, Probability Theory and Related Fields[END_REF][START_REF] Kahane | Sur le chaos multiplicatif[END_REF] that, almost surely, the family of random measures (M γ t ) t>0 weakly converges as t → ∞ towards a random measures M γ , which is non-trivial for γ 2 < 2d. The purpose of this paper is to investigate the phase transition, that is γ 2 = 2d. Remind that we have [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF][START_REF] Kahane | Sur le chaos multiplicatif[END_REF]: Proposition 1. For γ 2 = 2d (and also for γ 2 > 2d), the standard construction (7) yields a vanishing limiting measure:

lim t→∞ M γ t (dx) = 0 almost surely. (8)
One of the main purposes of this article is to give a non trivial renormalization of the family (M √ 2d t ) t . We stress that a suitable renormalization should yield a non trivial solution to the lognormal star-equation: Definition 2. Log-normal ⋆-scale invariance. A random Radon measure M is said to be lognormal ⋆-scale invariant if for all 0 < ε 1, M obeys the cascading rule

M(A) A∈B(R d ) law = A e ωε(r) M ε (dr)) A∈B(R d ) (9)
where ω ε is a stationary stochastically continuous Gaussian process and M ε is a random measure independent from ω ε satisfying the scaling relation

M ε (A) A∈B(R d ) law = M( A ε ) A∈B(R d ) . (10)
Let us mention that the authors in [START_REF] Allez | Lognormal ⋆-scale invariant random measures, Probability Theory and Related Fields[END_REF] have proved that, for γ 2 < 2d, the measure M γ is lognormal ⋆-scale invariant with

ω ε (r) = γX ln 1 ε (r) - γ 2 2 + d ln 1 ε , (11)
where X ln 1 ε is the Gaussian process introduced in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF]. Furthermore this scaling relation still makes perfect sense when the scaling factor ω ε is given by [START_REF] Bramson | Convergence in law of the maximum of the twodimensional discrete Gaussian free field[END_REF] for the value γ 2 = 2d. Therefore, to define a natural Gaussian multiplicative chaos at the value γ 2 = 2d, one has to look for a solution to this equation when the scaling factor is given by ( 11) with γ 2 = 2d and conversely, each random measure candidate for being a Gaussian multiplicative chaos at the value γ 2 = 2d must satisfy these relations. In [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF], a non trivial solution has been constructed, called the derivative martingale. Since it is conjectured that all the non trivial ergodic solutions to this equation (actually we also need to impose a sufficient decay of the covariance kernel of the process ω ε , see [START_REF] Allez | Lognormal ⋆-scale invariant random measures, Probability Theory and Related Fields[END_REF] for further details) are equal up to a multiplicative factor, it is expected that a non trivial renormalization of the family (M

√ 2d t
) t converges towards the derivative martingale. Proving this is the first purpose of this paper. The second purpose is to prove that the derivative martingale satisfies the KPZ formula.

The organization of this paper is as follows. In Section 3, we introduce the derivative martingale and remind the reader of its properties as stated in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF]. In Section 4, we explain how to renormalize (M

√ 2d t
) t to obtain the derivative martingale. This entails many non trivial moment estimates for the derivative martingale. In Section 5, we prove that these moment estimates allow to obtain the KPZ formula at criticality. Remark 3. As observed in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF], we stress that the main motivation for considering ⋆scale invariant kernels is the connection between the associated random measures and the ⋆-equation. Nevertheless, our proofs can be easily generalized. First, we stress that the assumption about the compact support of k involved in (6) may be relaxed, provided that one imposes some restrictions about the decay of k at infinity. Generally speaking, these restrictions are rather weak but may require some tedious extra computations. For instance, it is not very difficult to see that the case when k is C 1 with an exponential decay for k and ∇k does work. One may also wonder about the case of more general Gaussian multiplicative chaos of log-correlated Gaussian fields "à la Kahane" [START_REF] Kahane | Sur le chaos multiplicatif[END_REF]. It is not difficult to see that these other chaos can be written as in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF], with a kernel k depending also on the scale u. Then the same restrictions about the decay of k(u, •) at infinity apply. Let us just add that one can carry out the renormalization approach associated to exact scale invariant kernels as constructed in [START_REF] Bacry | Log-infinitely divisible multifractal processes[END_REF][START_REF] Rhodes | Multidimensional multifractal random measures[END_REF].

Derivative martingale

A way of constructing a solution to the ⋆-equation at the critical value γ 2 = 2d is to introduce the derivative martingale M ′ t (dx) defined by:

M ′ t (dx) = ( √ 2d t -X t (x))e √ 2dXt(x)-dE[Xt(x) 2 ] dx.
It is plain to see that, for each open bounded set A ⊂ R d , the family (M ′ t (A)) t is a martingale. Nevertheless, it is not nonnegative. It is therefore not obvious that such a family converges towards a (non trivial) positive limiting random variable. The following theorem has been proved in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF]:

Theorem 4.
For each bounded open set A ⊂ R d , the martingale (M ′ t (A)) t 0 converges almost surely towards a positive random variable denoted by M ′ (A), such that M ′ (A) > 0 almost surely. Consequently, almost surely, the (locally signed) random measures (M ′ t (dx)) t 0 converge weakly as t → ∞ towards a positive random measure M ′ (dx). This limiting measure has full support and is atomless. Furthermore, the measure M ′ is a solution to the ⋆-equation ( 9) with γ = √ 2d.

Renormalization

The main purpose of this paper is to establish that the derivative martingale can be seen as the limit of a suitable renormalization of the family (M

√ 2d t ) t . Theorem 5. The family ( √ tM √ 2d t
) t converges in probability as t → ∞ towards a non trivial limit, which turns out to be the same, up to a multiplicative constant, as the limit of the derivative martingale. More precisely, we have for all bounded open set A:

√ tM √ 2d t (A) → 2 π M ′ (A), in probability as t → ∞.
The main advantage of this renormalization approach is to make the derivative martingale appear as a limit of integrals over exponentials of the field: this is useful to use Kahane's convexity inequality (see [START_REF] Benjamini | KPZ in one dimensional random geometry of multiplicative cascades[END_REF]). We can then prove: Corollary 6. The positive random measure M ′ (dx) possesses moments of order q for all q < 1. Furthermore, for all q > 0 and every non-empty bounded open set A, we have

sup t 1 E 1 √ tM √ 2d t (A) q < +∞.
We can then determine the power law spectrum of the random measure M ′ : Corollary 7. The power law spectrum of the random measure M ′ is given for 0 q < 1 by

ξ(q) = 2dq -dq 2 . ( 12
)
More precisely, for each bounded open set A of R d , we have

∀q < 1, E[M ′ (λA) q ] ≍ C q λ ξ(q)
when λ goes to 0, and where the coefficient C q depends on q.

KPZ formula 5.1 The KPZ formula

In this section, we investigate the KPZ formula for the derivative martingale, which corresponds to the natural construction of Gaussian multiplicative chaos at criticality γ 2 = 2d. The KPZ formula is a relation between the Hausdorff dimensions of a given set A as measured by the Lebesgue measure or M ′ . So we first recall how to define these dimensions. Given an atomless Radon measure µ on R d and s ∈ [0, 1], we define

H s,δ µ (A) = inf k µ(B k ) s
where the infimum runs over all the covering (B k ) k of A with open Euclidean balls centered at A with radius r k δ. Clearly, the mapping δ > 0 → H s,δ µ (A) is decreasing. Hence we can define the s-dimensional µ-Hausdorff outer measure:

H s µ (A) = lim δ→0 H s,δ µ (A).
The limit exists but may be infinite. H s µ is a metric outer measure on R d (see [START_REF] Falconer | The geometry of fractal sets[END_REF] for definitions). Thus H s µ is a measure on the σ-field of H s µ -measurable sets, which contains all the Borelian sets.

The µ-Hausdorff dimension of the set A is then defined as the value

dim µ (A) = inf{s 0; H s µ (A) = 0}. ( 13
)
Notice that dim µ (A) ∈ [0, 1]. Since µ is atomless, the Hausdorff dimension is also characterized by: dim µ (A) = sup{s 0;

H s µ (A) = +∞}. ( 14
)
Since M ′ does not possess atoms, this relation is valid for M ′ . This allows to characterize the Hausdorff dimension as the threshold value at which the mapping s → H s µ (A) jumps from +∞ to 0.

In what follows, given a compact set K of R d , we define its Hausdorff dimensions dim Leb (K) and dim M ′ (K) computed as indicated above with µ respectively equal to the Lebesgue measure or M ′ . So, a priori, the value of dim M ′ (K) is random. Nevertheless, a straightforward 0 -1 law argument shows that dim M ′ (K) is actually deterministic. We reinforce this intuition by stating: Theorem 8. KPZ at criticality γ 2 = 2d. Let K be a compact set of R d . Almost surely, we have the relation

dim Leb (K) = ξ(dim M ′ (K)) d = 2d dim M ′ (K) -d dim M ′ (K) 2 .
Remark 9. Let us stress that our proof also allows one to choose K random but independent of the measure M. We could also consider sets K depending on M through the first scales of M, that is depending on (X s (x)) x∈R d ,s T for some T > 0. Strengthening this dependence would give rise to non trivial additional difficulties.

Heuristics and open questions on the KPZ formula

Here, we give a direct and heuristic derivation of Theorem 8 for the following reasons:

• it gives a quick intuitive idea of why Theorem 8 is valid;

• it enlightens the idea behind the proof of Theorem 8 (which involves introducing very particular Frostman measures);

• it leads in a natural way to open questions which can be seen as generalizations (or complements to) Theorem 8.

In fact, we will work in the subcritical case γ 2 < 2d (a similar heuristic can be derived for the case γ 2 = 2d). Recall that lognormal ⋆-scale invariance for M, defined by (1) with γ 2 < 2d, amounts to the following equivalent:

M(B(x, r)) ∼ r d e γX ln 1 r (x)-γ 2 2 ln 1 r
where ∼ denotes that both quantities are equal up to multiplication by a random factor of order 1 which does not depend on r (note that the random factor depends on x).

If we set ξ γ (s) = (d + γ 2 2 )s -γ 2 2 s 2
, it is thus tempting to write the following equivalents for a set K:

H s M (K) = lim δ→0 inf k M(B(x k , r k )) s , K ⊂ ∪ k B(x k , r k ), |r k | δ ∼ lim δ→0 inf k (r d k e γX ln 1 r k (x k )-γ 2 2 ln 1 r k ) s , K ⊂ ∪ k B(x k , r k ), |r k | δ = lim δ→0 inf k e sγX ln 1 r k (x k )-s 2 γ 2 2 ln 1 r k r ξγ(s) k , K ⊂ ∪ k B(x k , r k ), |r k | δ ∼ K e sγX(x)-s 2 γ 2 2 E[X(x) 2 ] H ξγ (s)/d (dx),
where the last term is a Gaussian multiplicative chaos applied to the Radon measure

H ξγ(s)/d (K ∩ dx) (at least if H ξγ (s)/d (K) < ∞).
This heuristics shows that the quantum Hausdorff measure H s M should not be too far from . e sγX(x)-s 2 γ 2 2 E[X(x) 2 ] H ξγ(s)/d (dx) (though up to possible logarithmic corrections). In particular, H s M (K) is of order 1 if and only if H ξγ(s)/d (K) is of order 1. Note that the e sγX(x)-s 2 γ 2 2 E[X(x) 2 ] H ξγ(s)/d (dx) measure appears in the physics litterature on KPZ in the so-called "gravitational dressing" (see, e.g., [START_REF] Ginsparg | Lectures on 2D gravity and 2D string theory[END_REF]), and in the rigorous context of the coupling of Schramm-Loewner Evolution to Liouville quantum gravity [START_REF] Duplantier | Schramm-Loewner Evolution and Liouville Quantum Gravity[END_REF]. Naturally, one could ask to what extent the above heuristics can be made rigorous.

A. Auxiliary results

We first state the classical "Kahane's convexity inequalities" (originally written in [START_REF] Kahane | Sur le chaos multiplicatif[END_REF] , see also [START_REF] Allez | Lognormal ⋆-scale invariant random measures, Probability Theory and Related Fields[END_REF][START_REF] Robert | Gaussian Multiplicative Chaos revisited[END_REF] for a proof in English): Lemma 10. Let F : R + → R be some convex function such that

∀x ∈ R + , |F (x)| M(1 + |x| β ),
for some positive constants M, β, and σ be a Radon measure on the Borelian subsets of R d . Given a bounded Borelian set A, let (X r ) r∈A , (Y r ) r∈A be two continuous centered Gaussian processes with continuous covariance kernels k X and k Y such that

∀u, v ∈ A, k X (u, v) k Y (u, v). Then E F A e Xr-1 2 E[X 2 r ] σ(dr) E F A e Yr-1 2 E[Y 2
r ] σ(dr) .

If we further assume

∀u ∈ A, k X (u, u) = k Y (u, u)
then for any increasing function F : R + → R:

E F sup x∈A Y x E F sup x∈A X x .

B. Proofs of Section 4

We denote by F t the sigma algebra generated by {X s (x); s t, x ∈ R d } and by F the sigma algebra generated by t F t . Given a fixed open bounded set A ⊂ R d and parameters t, β > 0, we introduce the random variables

Z β t (A) = A ( √ 2dt -X t (x) + β)1I {τ β x >t} e √ 2dXt(x)-dt dx R β t (A) = A 1I {τ β x >t} e √ 2dXt(x)-dt dx
where, for each x ∈ A, τ β x is the stopping time adapted to the filtration

G t = σ(X s (x); s t, x ∈ R d ) defined by τ β x = inf{u > 0, X u (x) - √ 2d u > β}.
For x ∈ R d , we also define

f β t (x) = ( √ 2d t -X t (x) + β)1I {τ β x >t} e √ 2dXt(x)-dt .
It is plain to check that for each β > 0 and each bounded open set A, (Z β t (A)) t is a nonnegative martingale with respect to (F t ) t such that E[Z β t (A)] = β|A|. It is proved in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] that it is uniformly integrable and therefore almost surely converges towards a non trivial limit. We first stress that, for each x fixed, the process t → X t (x) is standard Brownian motion. We will repeatedly use this fact throughout the proof without mentioning it again.

B.1 Rooted measure

Since for each x, (f β t (x)) t is a martingale, we can define the probability measure Θ β t on B(A) ⊗ F t by

Θ β t = f β t (x) β|A| dx dP. ( 15
)
We denote by E Θ β t the corresponding expectation. In fact, since the above definition defines a pre-measure on the ring t F t , one can define the rooted measure Θ β on B(A)⊗F by using Caratheodory's extension theorem. We recover Θ β |B(A)⊗Ft = Θ β t . We observe that Θ β t (Z β t (A) > 0) = 1 for any t. Similarly, we construct the probability measure Q β on F by setting:

Q β |Ft = Z β t (A) β|A| dP,
which is nothing but the marginal law of (ω, x) → ω with respect to Θ β t . Since (Z β t (A)) t 0 is a uniformly integrable martingale which converges to a limit Z β (A), we can also define Q β directly on F by:

Q β = Z β (A) β|A| dP.
We state a few elementary results below. The conditional law of x given F t is given by:

Θ β t (dx|F t ) = f t (x) Z β t (A) dx.
If Y is a B(A) ⊗ F t -measurable random variable then it has the following conditional expectation given F t :

E Θ β t [Y |F t ] = A Y (x, ω) f β t (x) Z β t (A) dx.
In particular, for any event E ∈ F t , we have

E Q β [1I E E Θ β t [Y |F t ]] = E Q β [1I E A Y (x, ω) f β t (x) Z β t (A) dx] = E Θ β t [1I E Y ].
Under Θ β t , the law of the random process (β + √ 2ds-X s ) s t is that of a 3-dimensional Bessel process started at β. In what follows, we will use the notation:

Y β s (x) = β + √ 2ds -X s (x). ( 16
)
Of course, Y 0 s (x) simply stands for √ 2ds -X s (x). The proof is inspired from [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF]. Mainly, we follow their argument. Nevertheless, we make two remarks. First, most of the auxiliary estimates obtained in [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF] about the minimum of the underlying random walk are much easier to obtain in our context because of the Gaussian nature of our framework (in particular, the random walk conditioned to stay positive is here a Bessel process). Second, the continuous structure makes correlations much more intricate to get rid of: we have no spinal decomposition at our disposal, no underlying tree structure, etc. We adapt some arguments developed in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] at this level.

B.2 Proofs under the rooted measure

The first step is to prove the convergence under Q β . This subsection is thus entirely devoted to the proof of the following result: Proposition 11. Given β > 0, we have

E Q β R β t (A) Z β t (A) ≃ 2 πt as t → ∞ (17)
and

E Q β R β t (A) Z β t (A) 2 ≃ 2 πt as t → ∞. (18)
As a consequence, under Q β , we have

lim t→∞ √ t R β t (A) Z β t (A) = 2 π in probability. ( 19
)
Proof. It is clear that (17)+ [START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF] imply that the variance of the ratio

√ tR β t (A) Z β t (A) under Q β
goes to 0 a t goes to ∞. Therefore, under Q β , this ratio converges in quadratic mean towards 2 π , and hence in probability, thus proving [START_REF] Duplantier | Conformal fractal geometry and boundary quantum gravity[END_REF]. It is plain to establish the relation [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF]. First observe that:

E Q β R β t (A) Z β t (A) = 1 β|A| E[R β t (A)] = 1 β E[1I {τ β x >t} e √ 2dXt(x)-dt ]
Let us denote by B a standard one-dimensional Brownian motion. By using the Girsanov transform, we have:

E Q β R β t (A) Z β t (A) = 1 β P( sup 0 u t B u β) = 1 β P(|B t | β) ≃ 2 πt as t → ∞.
Relation ( 17) is established. The proof of ( 18) is much more involved and will be carried out in several steps. To begin with, it may be worth sketching the strategy of the proof:

1. Observe that R β t (A) Z β t (A) = E Θ β t 1 Y β t (x) |F t . (20)
2. Deduce that

E Q β R β t (A) Z β t (A) 2 = E Θ β t R β t (A) Z β t (A) × 1 Y β t (x) . (21)
3. The third step consists in subtracting a ball centered at x with radius e -ht to the set A, call it B(x, e -ht ). This is convenient because if the radius is well chosen the ratio R β t (A\B(x,e -h t )) Z β t (A\B(x,e -h t )) and the weight 1 Y β t (x) will be "almost" independent. Roughly speaking, the reason why we can subtract a ball is that the measure does not possess atom. So, at least if the radius of the ball is small enough, it is always possible to subtract a ball centered at x without radically affecting the behaviour of the quantity [START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF]. In the forthcoming rigorous proof, we won't base our argument on the fact that the measure is atomless. Instead, we will use estimates on the process Y β t (x) under the rooted measure, which amounts to the same (as proved in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF]).

4. The last step consists in factorizing (21) because of (almost) independence:

E Q β R β t (A) Z β t (A) 2 ≃ E Θ β t R β t (A \ B x,t ) Z β t (A \ B x,t ) × 1 Y β t (x) ≃ E Θ β t R β t (A \ B x,t ) Z β t (A \ B x,t ) E Θ β t 1 Y β t (x) ≃ E Q β R β t (A) Z β t (A) E Q β R β t (A) Z β t (A)
.

This last quantity is equivalent for t large to 2 πt . Actually, most of the forthcoming computations are made to justify that the factorization can be made rigorously. This point is highly technical and the related computations may appear tedious to the reader. Now we begin with the rigorous proof of [START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF]. Recall that we use the shorthand [START_REF] Distler | Conformal Field Theory and 2-D Quantum Gravity or Who's Afraid of Joseph Liouville?[END_REF]. Let us first claim: Lemma 12. We have

E Q β R β t (A) Z β t (A) 2 = O 1 t as t → ∞.
Proof. By Jensen inequality, we have:

E Q β R β t (A) Z β t (A) 2 = E Q β E Θ β t 1 Y β t (x) |F t 2 E Q β E Θ β t 1 Y β t (x) 2 |F t E Θ β t 1 Y β t (x) 2
Since, under Θ β t , the law of the process (Y β s (x)) s t is that of a 3-dimensional Bessel process starting at β > 0, the lemma follows. Now we will decompose the space in two parts: a part, call it E t , where we have strong estimates on the process (Y β s (x)) s t and an other part that we want to be "small". More precisely, let E t be a B(A) ⊗ F t -measurable event such that

E Θ β t (1I Et ) → 1 as t goes to ∞. Let ξ t = E Θ β t 1I E c t 1 Y β t (x) |F t . Notice that R β t (A) Z β t (A) = E Θ β t 1 Y β t (x) |F t = ξ t + E Θ β t 1I Et 1 Y β t (x) |F t . (22)
We deduce:

E Q β R β t (A) Z β t (A) 2 = E Q β R β t (A) Z β t (A) ξ t + E Θ β t R β t (A) Z β t (A) 1I Et 1 Y β t (x) . ( 23
)
We will treat separately the two terms in the above right-hand side. The Cauchy-Schwarz inequality and Lemma 12 yield

E Q β R β t (A) Z β t (A) ξ t E Q β R β t (A) Z β t (A) 2 1/2 E Q β ξ 2 t 1/2 C √ t E Q β ξ 2 t 1/2 . ( 24
)
If we can prove that

E Q β ξ 2 t = o 1 t then (24) will tell us that E Q β R β t (A) Z β t (A) ξ t = o 1
t . Therefore, Proposition 11, in particular [START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF], is a consequence of the two following lemma: Lemma 13. Let β > 0 and E t be an event such that E Θ β t (1I Et ) → 1 as t goes to ∞. We have

E Q β ξ 2 t = o 1 t .
Lemma 14. Let β > 0. There exists a family of events E t such that E Θ β t (1I Et ) → 1 as t goes to ∞ and

E Θ β t R β t (A) Z β t (A) 1I Et Y β t (x) 2 πt + o 1 t
Proof of Lemma 13. Fix ε > 0. By Jensen's inequality, we have

E Q β ξ 2 t E Θ β t 1I E c t 1 Y β t (x) 2 E Θ β t 1I E c t Y β t (x) 2 1I {ε √ t Y β t (x)} + E Θ β t 1 Y β t (x) 2 1I {ε √ t>Y β t (x} E Θ β t 1I E c t Y β t (x) 2 1I {ε √ t Y β t (x} + 1 β E 1I {τ β x >t} Y β t (x) 1I {ε √ t>Y β t (x)} e √ 2dXt(x)-dt E Θ β t 1I E c t ε 2 t + 1 β E 1I {sup [0,t] Xu(x) β} (β -X t (x)) 1I {Xt(x)>β-ε √ t}
Using the joint law of a Brownian motion together with its maximum, we prove that, for some constant C independent of t, ε, we have:

1 β E 1I {sup [0,t] Xu(x) β} (β -X t (x)) 1I {Xt(x)>β-ε √ t} εC/t. Since E Θ β t 1I E c t → 0 as t → ∞, we deduce: lim sup t→∞ tE Q β ξ 2 t εC.
Since ε can be chosen arbitrarily small, the proof of the lemma follows.

Proof of Lemma 14. We consider a function h such that h t → ∞ as t → ∞. We define

R β,h t (x) = B(x,e -h t )∩A 1I {τ β z >t} e √ 2dXt(z)-dt dz R β,h,c t (x) = R β t (A) -R β,h t (x) Z β,h t (x) = B(x,e -h t )∩A f β t (z) dz Z β,h,c t (x) = Z β t (A) -Z β,h t (x).
Now we precise the choice of the set E t :

E 1 t = {h 1/3 t Y 0 ht (x) h t } ∩ {h 1/6 t inf u∈[ht,t] Y 0 u (x)} E 2 t = {Z β,h t (x) t -2 } E 3 t = {R β,h,c t (x) Z β,h,c t (x)} ∩ {R β,h t (x) Z β,h t (x)} E t = E 1 t ∩ E 2 t ∩ E 3 t
Let us admit for a while the following lemma:

Lemma 15. Let β > 0. There is a constant a > 0 such that for every function h t satisfying lim t→∞ h t ln t a = +∞ and lim

t→∞ h t √ t = 0, ( 25
)
then lim t→∞ Θ β t (E t ) = 1, lim t→∞ inf u∈[h 1/3 t ,ht] Θ β t (E t |Y 0 ht (x) = u) = 1. ( 26
)
So we pursue the proof of Lemma 14 while assuming now that the conditions and conclusions of Lemma 15 are in force. In what follows, C will denote a constant that may change from line to line and that does not depend on relevant quantities. On E t , we have

R β,h t Z β,h t t -2 . Therefore E Θ β t R β,h t Z β t (A) 1 Y β t (x) 1I Et t -2 E Θ β t 1 Z β t (A) 1 Y β t (x) 1I Et . On E t , in particular on E 1 t , we have Y β t (x) β + h 1/6 t . Hence E Θ β t R β,h t Z β t (A) 1 Y β t (x) 1I Et t -2 E Θ β t 1 Z β t (A) 1 (β + h 1/6 t ) 1 t 2 (β + h 1/6 t ) E Q β 1 Z β t (A) 1 β 2 |A|t 2 .
Here we have used

E Q β 1 Z β t (A) = 1
β|A| . Thus we have

E Θ β t R β,h t Z β t (A) 1 Y β t (x) 1I Et = o 1 t . ( 27
)
Let us treat the quantity:

E Θ β t R β,h,c t Z β t (A) 1 Y β t (x) 1I Et = 1 β|A| A E R β,h,c t (x) Z β t (A) 1 Y β t (x) 1I Et f β t (x) dx 1 β|A| A E R β,h,c t (x) Z β,h,c t (x) 1 Y β t (x) 1I {h 1/3 t Y 0 h t (x) ht} f β t (x) dx.
We know that the covariance kernel k appearing in (6) vanishes outside a compact set. Without loss of generality, we may assume that k vanishes outside the ball B(0, 1). Let us introduce the sigma algebra G t generated by the random variables {X s (y); y ∈ R d , s t}.

Conditionally to x, G ht , the random variables

R β,h,c t (x) Z β,h,c t (x) and 1 Y β t (x) f β
t (x) are independent. Therefore we have:

E Θ β t R β,h,c t (x) Z β t (A) 1 Y β t (x) 1I Et E Θ β t R β,h,c t (x) Z β,h,c t (x) 1
Y β t (x) 1I {h 1/3 t Y 0 h t (x) ht} = E Θ β t 1I {h 1/3 t Y 0 h t (x) ht} E Θ β t R β,h,c t (x) Z β,h,c t (x) |x, F ht E Θ β t 1 Y β t (x) |x, F ht
To complete the proof of Lemma 14, we admit for a while the two following lemma, the proof of which are gathered in the next subsubsection.

Lemma 16. We have

E Θ β t 1 Y β t (x) |x, G ht 2 πt .
Lemma 17. We have

E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I {h 1/3 t Y 0 h t (x) ht} 2 πt (1 + ε(t)),
for some function ε such that lim t→∞ ε(t) = 0.

We conclude the proof of Lemma 14. With the help Lemma of 16 and 17, we obtain:

E Θ β t 1I {h 1/3 t Y 0 h t (x) ht} E Θ β t R β,h,c t (x) Z β,h,c t (x) |x, G ht E Θ β t 1 Y β t (x) |x, G ht 2 πt E Θ β t 1I {h 1/3 t Y 0 h t (x) ht} E Θ β t R β,h,c t (x) Z β,h,c t (x) |x, G ht = 2 πt E Θ β t 1I {h 1/3 t Y 0 h t (x) ht} R β,h,c t (x) Z β,h,c t (x) 2 πt (1 + ε(t))
for some function ε such that lim t→∞ ε(t) = 0.

B.2.1 Proofs of auxiliary lemmas

Proof of Lemma 15. Under Θ β t , the process (Y β s (x)) s t is a 3-dimensional Bessel process. It is plain to deduce that lim t→∞

Θ β t (E 1 t ) = 1. ( 28
)
Let us prove that lim t→∞ Θ β t (E 3 t ) = 1. We define the random variables

S t = sup x∈A X t (x) - √ 2dt + 1 4 √ 2d ln(t + 1), S = sup t 0 S t ,
and the event B = S < +∞ .

It is proved in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] that P(B) = 1. It is plain to deduce that Θ β (B) = 1. We have

Z β,h,c t (x) = A\B(x,e -h t ) Y β t (z)1I {τ β z >t} e √ 2dXt(z)-dt dz A\B(x,e -h t ) β + √ 2dt -sup x∈A X t (x) 1I {τ β z >t} e √ 2dXt(z)-dt dz = β + 1 4 √ 2d ln(t + 1) -S t R β,h,c t (x) β + 1 4 √ 2d ln(t + 1) -S R β,h,c t (x). Therefore {β + 1 4 √ 2d ln(t + 1) -S 1} ⊂ {Z β,h,c t (x) R β,h,c t (x)},
from which we deduce:

lim t→∞ Θ β t Z β,h,c t (x) R β,h,c t (x) = 1.
With the same argument, we prove

lim t→∞ Θ β t Z β,h t (x) R β,h t (x) = 1. Hence lim t→∞ Θ β t (E 3 t ) = 1. ( 29
)
For E 2 t , we will use some computations made in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF]. By using the Markov inequality and by conditioning on an event B, we get:

Θ β t (Z β,h t (x) > t -2 ) t 2 E Θ β t [Z β,h t (x)|B] + Θ β t (B c ) ( 30
)
for every measurable set B. We set for R > 0

B R = ∀t 0; √ t R(ln(2 + t)) 2 Y β t (x) R(1 + t ln(1 + t)) .
We choose B = B R in [START_REF] Gross | A nonperturbative solution of D = 1 string theory[END_REF]. It is proved in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] that the quantity Θ β t (B c R ) can be made arbitrarily close to 0 when choosing R large enough. Intuitively, this is just a precise statement corresponding to the fact that a Bessel β t process goes to infinity with speed rate √ t as t goes to ∞. So it just remains to prove that, for R fixed, the quantity

t 2 E Θ β t [Z β,h
t |B R ] goes to 0 as t goes to ∞. To that purpose, it suffices to estimate the quantity

t 2 E Θ β t [Z β,h t (x)|(X s (x)) s t , x, B R ]
and prove that it goes to 0 when t goes to ∞ uniformly with respect to the inputs (X s (x)) s t , x. We first rewrite this quantity as:

E Θ β t [Z β,h t (x)|(X s (x)) s t , x, B R ] =E Θ β t A∩B(x,e -h t ) Y β t (z)1I {τ β z >t} e √ 2dXt(z)-dt dz (X s (x)) s t , x, B R .
This expectation has been computed in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] thanks to explicit formulae for the conditional expectations of the process (X s (z)) s given (X s (x)) s t . Before making these formulae precise, we clarify a few points. The kernel k involved in ( 6) vanishes outside a compact set. So, without loss of generality, we assume that k vanishes outside the ball B(0, 1). We divide the ball B(x, e -ht ) in two areas: the ring C(t) = B(x, e -ht ) \ B(x, e -t ) and the ball B(x, e -t ). The results of the computations are the following. Concerning the ring, we have:

E Θ β t A∩C(t) Y β t (z)1I {τ β z >t} e √ 2dXt(z)-dt dz|(X s (x)) s t , x, B R A∩C(t) e D |x -z| d G ln 1 |x -z| dz
for some function G given by

G(y) = 1 + y ln(1 + y) 2 e - √ 2dD √ y ln 2 (2+y)
and some constant D > 0 depending on irrelevant quantities. Therefore, by making a change of polar coordinates, we get (C stands for a constant, more precisely the area of the d-dimensional sphere):

E Θ β t A∩C(t) Y β t (z)1I {τ β z >t} e √ 2dXt(z)-dt dz|(X s (x)) s t , x, B R C e -h t e -t e D r d G ln 1 r r d-1 dr = Ce D t ht G(y) dy.
Now we just want to make sure that there exists a > 0 such that the condition lim t→∞ ht ln t a = +∞ implies that the integral t 2 e D t ht G(y) dy goes to 0 as t → ∞. Observe that for y large enough, the function G is decreasing. Therefore, a rough bound of the quantity under study is

t 2 e D t ht G(y) dy t 3 e D G(h t ).
Now, for y large enough again, we observe that G is less than e - √ 2dDy 1/3 . Therefore it is plain to check that any a > 9 √ 2dD suits. On the ball B(x, e -t ), we have

E Θ β t A∩B(x,e -t ) Y β t (z)1I {τ β z >t} e √ 2dXt(z)-dt dz|(X s (x)) s t , x, B R e D G(t)
for some irrelevant constant D, which may be different from that involved in the bound on the ring. Once again, the condition lim t→∞ ht ln t a = +∞ for some well chosen a > 0 implies that the term G(t) goes to 0 as t → ∞. Finally we deduce that

Θ β t (Z β,h t (x) t -2 ) → 1, as t → ∞. (31)
By gathering ( 28)+( 29)+( 31), we have proved that

Θ β t (E t ) → 1, as t → ∞.
It just remains to prove that

lim t→∞ inf u∈[h 1/3 t ,ht] Θ β t (E t |Y 0 ht (x) = u) = 1.
We first prove that lim t→∞

Θ β t ((E 3 t ) c |Y 0 ht (x) = u) = 0 uniformly with respect to u ∈ [h 1/3 t , h t ]. Observe that Θ β t ((E 3 t ) c |Y 0 ht (x) = u) Θ β t ((E 3 t ) c |Y 0 ht (x) = u, sup x∈A -Y 0 t (x) -1) + Θ β t (sup x∈A -Y 0 t (x) > -1) = 0 + Θ β t ( inf x∈A Y 0 t (x) < 1).
It results from the observations made at the beginning of the proof of this lemma that the above quantity becomes arbitrarily small as t gets large. Concerning E 2 t , observe that

Θ β t (Z β,h t (x) > t -2 |Y 0 ht (x) = u) Θ β t (Z β,h t (x) > t -2 |Y 0 ht (x) = u, B R ) + Θ β (B c R ).
Now we observe that we have already proved that Θ

β t (Z β,h t (x) > t -2 |Y 0 ht (x) = u, B R ) → 0 as t → ∞. Therefore lim sup t→∞ sup u∈[h 1/3 t ,ht] Θ β t (Z β,h t (x) > t -2 |Y 0 ht (x) = u) Θ β (B c R ).
Since Θ β (B c R ) can be made arbitrarily close to 0 for R large enough, we conclude that:

lim sup t→∞ sup u∈[h 1/3 t ,ht] Θ β t (Z β,h t (x) > t -2 |Y 0 ht (x) = u) = 0.
It just remains to prove that

lim t→∞ Θ β t (E 1 t |Y 0 ht (x) = u) = 1 uniformly with respect to u ∈ [h 1/3 t , h t ]. This is obvious since (Y 0 t ) t is a 3-dimensional Bessel process.
Proof of Lemma 16. By using standard tricks of changes of probability measures, we have:

E Θ β t 1 Y β t (x) |x, G ht = 1 f β ht (x) E f β t (x) Y β t (x) |G ht .
Let us compute the latter conditional expectation:

E f β t (x) Y β t (x) |G ht = 1I {τ β x >ht} e √ 2dX h t (x)-dht E 1I {sup [0,t] Xs(x)-ds β} e √ 2d(Xt(x)-X h t (x))-d(t-ht) |G ht = 1I {τ β x >ht} e √ 2dX h t (x)-dht F (Y β ht (x))
where

F (y) = E 1I {sup [0,t-h t ] Xs(x)-ds y} e √ 2d(X t-h t (x))-d(t-ht) .
By using the Girsanov transform, we get:

F (y) =E 1I {sup [0,t-h t ] Xs(x) y} .
This quantity is plain to compute since the process s → X s (x) is a Brownian motion:

F (y) =P(|X t-ht (x)| y) = 1 √ 2π y √ t-h t -y √ t-h t e -u 2 /2 du 2 π y √ t -h t .
The lemma follows.

Proof of Lemma 17. We have:

E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I Et E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I {h 1/3 t Y 0 h t (x) ht} inf u∈[h 1/3 t ,ht] Θ β t E t |Y 0 ht (x) = u From Lemma 15, we have Θ β t E t |Y 0 ht (x) = u → 1 as t → ∞ uniformly with respect to u ∈ [h 1/3 t , h t ]. Therefore E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I {h 1/3 t Y 0 h t (x) ht} (1 + ε(t))E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I Et (32)
for some function ε such that lim t→∞ ε(t) = 0. On E t we have R β,h,c t (x) Z β,h,c t (x), from which we get:

E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I Et E Θ β t R β,h ,c t (x) Z β,h,c t (x)
1I Et∩{Z β t (A)>1/t} + E Θ β t [1I {Z β t (A) 1/t} ]. (33)
By using the Markov inequality, we have:

E Q β [1I {Z β t (A) 1/t} ] 1 t E Q β [ 1
Z β t (A) ] 1 t . (34)
By gathering [START_REF] Jain | World-sheet geometry and baby universes in 2-D quantum gravity[END_REF] and (34), we deduce

E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I Et E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I Et∩{Z β t (A)>1/t} + 1 t . Since Z β,h t (x)1I Et t -2 , we have on the set E t ∩{Z β t (A) > 1/t} the estimate Z β,h t (x) 1 t Z β t (A). Therefore, on the set E t ∩ {Z β t (A) > 1/t}, we have Z β,h,c t (x) (1 -t -1 )Z β t (A).
We deduce

E Θ β t R β,h,c t (x) Z β,h,c t (x) 1I Et 1 1 -t -1 E Θ β t R β,h,c t (x) Z β t (A) 1I Et∩{Z β t (A)>1/t} + 1 t 1 1 -t -1 E Θ β t R β t (A) Z β t (A) + 1 t . (35)
The result follows by gathering ( 17)+( 32)+ [START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects 2000, JHEP Proceedings; A matrix model for the two-dimensional black hole[END_REF].

B.3 Proof of Theorem 5

In the previous subsection, we have proved the convergence of √ t

R β t (A) Z β t (A) in probability towards 2
π under the measure Q β for any β > 0. This is the content of Proposition 11. Our objective is now to use this convergence under Q β to establish the convergence under the original probability measure P.

Proposition 11 ensures that, for any ε > 0,

Q β √ t R β t (A) Z β t (A) - 2 π > 2 π ε → 0, as t → ∞. Equivalently E Z β t (A)1I √ t R β t (A) Z β t (A) - √ 2 π > √ 2 π ε → 0, as t → ∞.
From [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF], we know that sup

t max x∈A X t (x) - √ 2d t < ∞ almost surely. By setting E R = {sup t max x∈A X t (x) - √ 2d t < R},
we obtain an increasing family such that

P R>0 E R = 1.
From the convergence (Z β t (A) is nonnegative)

E Z β t (A)1I √ t R β t (A) Z β t (A) - √ 2 π > √ 2 π ε 1I E R → 0, as t → ∞,
we deduce the convergence of 36), we deduce that, necessarily on E R :

Z β t (A)1I √ t R β t (A) Z β t (A) - √ 2 π > √ 2 π ε 1I E R → 0 (36) in probability as t → ∞. Fix β > R. On E R we have R β t (A) = M √ 2d t (A). Concerning Z β t (A), we observe that, for β > R, we have ∀t > 0, βM √ 2d t (A) + M ′ t (A) = Z β t (A). Therefore lim t→ Z β t (A) = M ′ (A) > 0 on E R for β > R (recall that M √ 2d t (A) → 0 as t → ∞). From (
1I √ t R β t (A) Z β t (A) - √ 2 π > √ 2 π ε → 0, as t → ∞ that is 1I √ t M √ 2d t (A) M ′ t (A) - √ 2 π > √ 1 π ε → 0, as t → ∞.
The proof of Theorem 5 is over.

B.4 Other proofs of Section 4.

We use the comparison with multiplicative cascades set out in the appendix of [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF]. The idea is to compare moments of discrete lognormal multiplicative cascades to moments of the family (M t ) t thanks to Lemma 10. More precisely and sticking to the notations in [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF], we have

E Z √ nM √ 2d n ln 2 ([0, 1] d ) α E √ n T e Xn(t)-1 2 E[Xn(t) 2 ] σ(dt) α , (37)
where

• Z is a Gaussian random variable with fixed mean and variance (thus independent of n), and independent of the family (M t ) t ,

• the parameter α belongs to ]0, 1[, making the mapping x → x α concave,

• T e Xn(t)-1 2 E[Xn(t) 2 ] σ(dt) stands for a lognormal multiplicative cascade at generation n, the parameters of which are adjusted to be in the critical situation (see [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF] for a precise definition).

It is now well established that the right-hand side of (37) is bounded uniformly with respect to n. The reader may consult [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF][START_REF] Barral | Critical Mandelbrot's cascades[END_REF][START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF] for instance about this topic. By the Fatou lemma and Theorem 5, we have

E M ′ ([0, 1] d ) α lim inf n→∞ E √ nM √ 2d n ln 2 ([0, 1] d ) α
This shows that the measure M ′ possesses moments of order q for 0 q < 1. We already know that it possesses moments of negative order [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale[END_REF].

It remains to prove that sup

t 1 E 1 √ tM √ 2d t (A) q < +∞.
We write the proof in dimension 1 (generalization to higher dimensions is straightforward) and we assume that the kernel k of (6) vanishes outside the interval [-1, 1]. Our proof is based on an argument in [START_REF] Molchan | Scaling exponents and multifractal dimensions for independent random cascades[END_REF], which we adapt here to get bounds that are uniform in We set

Y i = 1 8 inf x∈[ i 4 , 2i+1 8 ] e √ 2X ln 8 (x)-ln 8 , N i = 8 2i+1 8
i 4 e √ 2(X t+ln 8 -X ln 8 )(x)-t dx.

for i = 0, . . . , 3. A straightforward computation of covariances shows that ((X t+ln 8 -X ln 8 )(x)) x∈R has same distribution as (X t (8x)) x∈R . It is plain to deduce that

• the random variables (Y i ) i are independent of (N i ) i ,

• the random variables (Y i ) i are identically distributed,

• the random variables (N i ) i are identically distributed with common law M

√ 2d t ([0, 1]). Let us define ϕ t (s) = E[e -s √ tM √ 2d t ([0,1]) ].
Since the mapping x → e -s √ tx is convex, we can apply Lemma 10 and obtain

ϕ t (s) E[e -s √ tM √ 2d t+ln 8 ([0,1]) ].
Therefore:

ϕ t (s) E[e -s √ tM √ 2d t+ln 8 ([0,1]) ] E[e -s √ t 3 i=0 Y i N i ] =E[ 3 i=0 ϕ t (sY i ))] E[ϕ t (sY 1 ) 4 ].
Let us denote by F the distribution function of Y 1 . We have:

ϕ t (s) ∞ 0 ϕ t (sx) 4 F (dx) = s -1/2 0 ϕ t (sx) 4 F (dx) + ∞ s -1/2 ϕ t (sx) 4 F (dx) F (s -1/2 ) + ϕ t (s 1/2 ) 4 .
It is plain to check that E[(Y 1 ) -q ] < +∞ for all q > 0. Fix q 0 > 0. By the Markov inequality, we deduce:

ϕ t (s) s -q 0 2 E[(Y 1 ) -q 0 ] + ϕ t (s 1/2 ) 4 . Therefore ϕ t (s) s 2q-q 0 2 E[(Y 1 ) -q 0 ] + ϕ t (s 1/2 ) 2 s -2q + ϕ t (s 1/2 ) 2 . ( 38
)
for all q < q 0 /4. Let us admit for a while the following lemma Lemma 18. The family of functions

(ϕ t (•)) t uniformly converges on R + as t → ∞ towards ϕ(s) = E[e -sM ′ ([0,1]) ].
Since P(M ′ ([0, 1]) = 0) = 0, we deduce lim s→∞ ϕ(s) = 0. Therefore there exist s 0 > 0 and t 0 > 0 such that

∀s > s 0 , ∀t > t 0 s 2q-q 0 2 E[(Y 1 ) -q 0 ] + ϕ t (s 1/2 ) 2 1 √ 2 .
Plugging this estimate into (38) yields:

∀s > s 0 , ∀t > t 0 ϕ t (s) 1 √ 2 s -2q + ϕ t (s 1/2 ) 2 .
Then, by induction, we check that:

ϕ t (s 2 n ) 1 √ 2 a n+1 s -q2 n+1 + ϕ 2 n+1 t (s 1/2 ) ( 39
)
where a 1 = 1 and a n+1 = a 2 n + 1. Let us choose Q 1+ √ 5 2 so as to have Q 2 -Q -1 0. It is then plain to check by induction that Q 2 n Q + a n . Let us choose s = s 0 + Q 1/q . Then for all x > s, there exists n ∈ N such that:

s 2 n x < s 2 n+1 .
In other words,

2 n ln x ln s < 2 n+1 .
Thus we obtain from (39)

ϕ t (x) ϕ(s 2 n ) 1 √ 2 Q 2 n+1 s -q2 n+1 + ϕ 2 n+1 t (s 1/2 ) 1 √ 2 (x -α + x -β )
where α = q -ln Q ln s > 0 and β = -ln ϕt(s 1/2 ) ln s > 0. Since lim t→∞ ϕ t (s) = E[e -sM ′ ([0,1]) ], we can choose β arbitrarily close to β 0 = -ln ϕ(s 1/2 ) ln s > 0. To sum up, we have proved that, for all β < β 0 , there exists x 0 = s 0 + Q 1/q and t 0 > 0 such that

∀x > x 0 , ∀t > t 0 ϕ t (x) 1 √ 2 (x -α + x -β ). ( 40
)
It is straightforward to deduce sup

t 1 E 1 √ tM √ 2d t (A) q
< +∞ for all 0 < q < min(α, β 0 ). Put in other words, we have proved the result "only for small q". But, remembering that

ϕ t (x) E[ϕ t (xY 1 ) 4 ]
and E[Y -q 1 ] < +∞ for all q > 0, we can deduce from [START_REF] Kostov | Loop amplitudes for nonrational string theories[END_REF] the result for arbitrary q by induction.

Proof of Lemma 18. Define the family of functions

f t : y ∈ [0, 1[ → ϕ t (tan πy 2
).

Since P √ tM

√ 2
t ([0, 1]) = 0 = 0, it is plain to deduce that f t can be continuously extended to [0, 1] by setting f t (1) = 0 for all t. In the same way we define a continuous function on

[0, 1] by f : y ∈ [0, 1[ → ϕ(tan πy 2 )
and f (1) = 0 (possible because P M ′ ([0, 1]) = 0 = 0). The family (f t ) t pointwise converges as t → ∞ towards f . Furthermore, the functions f t are non increasing for all t. It is then standard to deduce the uniform convergence. The lemma follows.

C. Proof of Section 5

Proof of Proposition 7. Consider a solution M of the ⋆-equation [START_REF] Biggins | Fixed points of the smoothing transform; the boundary case[END_REF] with ω ε given by ( 11) and γ 2 = 2d. Consider an exponent 0 < q < 1, an open bounded set A and λ < 1. From equation ( 9) (with ε = λ) and the Jensen inequality we have:

E[(M(λA) q ] = E λA e γX ln 1 λ (r)-γ 2 2 E[X ln 1 λ (r) 2 ] λ d M λ (dr) q λ dq E e γqX ln 1 λ (0)-γ 2 q 2 E[X ln 1 λ (0) 2 ] M(A) q = λ ξ(q) E M(A) q .
Conversely, we use Lemma 10 and equation ( 9):

E[M(λA) q ] = E λA e γX ln 1 λ (r)-γ 2 2 E[X ln 1 λ (r) 2 ] M λ (dr) q E λA e γ √ a λ Z-γ 2 2 a λ M λ (dr) q 29
where Z is a standard Gaussian random variable and a λ = inf x,y∈λA K ln 1 λ (y -x). Therefore:

E[M(λA) q ] λ dq E e γq √ a λ Z-qγ 2 2 a λ E M(A) q .
We complete the proof by noticing that a λ C + ln 1 λ . Remark 19. Actually, using the ⋆-equation to compute the power-law spectrum is not necessary: it can be computed with similar arguments for any derivative martingale associated to a log-correlated Gaussian field.

Proof of Theorem 8. Without loss of generality, we assume that k vanishes outside the ball B(0, 1). Let K be a compact set included in the ball B(0, 1) with Lebesgue Hausdorff dimension dim Leb (K) < 1 (the case dim Leb (K) = 1 is obvious). Let q ∈ [0, 1[ be such that ξ(q) > d dim Leb (K) with ξ(q) < d. For ε > 0, there is a covering of K by a countable family of balls (B(x n , r n )) n such that n r ξ(q) n < ε.

By using in turn the stationarity and the power law spectrum of the measure, we have

E n M ′ (B(x n , r n )) q = n E M ′ (B(0, r n )) q C q n r ξ(q) n C q ε.
Using the Markov inequality, we deduce

P n M ′ (B(x n , r n )) q C q √ ε 1 - √ ε.
Thus, with probability 1 -√ ε, there is a covering of balls of K such that

n M ′ (B(x n , r n )) q C q √ ε. So q dim M ′ (K) almost surely. Therefore d dim Leb (K) dim M (K).
Conversely, consider q ∈ [0, 1[ such that ξ(q) < d dim Leb (K). By the Frostman Lemma, there is a probability measure γ supported by K such that B(0,1) 2 1 |x -y| ξ(q) γ(dx)γ(dy) < +∞. [START_REF] Kostov | Strings with discrete target space[END_REF] For q ∈ [0, 1[, let us define the random measure γ as the almost sure limit of the following family of positive random measures:

γ(dx) = lim t→∞ e q √ 2dXt(x)-q 2 dE[(Xt(x)) 2 ] γ(dx). ( 42
)
The limit is non trivial: Kahane [START_REF] Kahane | Sur le chaos multiplicatif[END_REF] (see also a more general proof in [START_REF] Rhodes | ⋆-scale invariant random measures[END_REF]) proved that, for a Radon measure γ(dx) satisfying (41) with a power exponent κ (instead of ξ(q) in ( 41)) the associated chaos

γ(dx) = lim t→∞ e sXt(x)-s 2 2 E[(Xt(x)) 2 ] γ(dx)
is non degenerate (i.e. the martingale is regular) provided that κ -s 2 2 > 0. In our context, this condition reads q 2 d < ξ(q), that is q < 1.

From the Frostman lemma again, we just have to prove that the quantity

B(0,1) 2 1 M ′ (B(x, |y -x|)) q γ(dx) γ(dy)
is finite almost surely, which will follow from

E B(0,1) 2 1 M ′ (B(x, |y -x|)) q γ(dx) γ(dy) < +∞. (43)
Actually, by using the Fatou lemma in [START_REF] Kostov | Multicritical phases of the O(n) model on a random lattice[END_REF], it is even sufficient to prove that:

lim inf t→∞ B(0,1) 2 E e q √ 2dXt(x)x+q √ 2dXt(y)-2q 2 dE[(Xt(x)) 2 ] √ tM √ 2d t (B(x, |y -x|)) q γ(dx)γ(dy) < +∞. ( 44
)
From now on, we will focus on computing the above integral [START_REF] Lalley | A conditional limit theorem for frontier of a branching Brownian motion[END_REF]. There is a way of making the computations with minimal effort: we change the process X t with the perfect scaling process introduced in [START_REF] Rhodes | Multidimensional multifractal random measures[END_REF]. We just have to justify that this change of processes is mathematically rigorous. So let us admit for a while the following lemma:

Lemma 20. If (44) is finite for the process (X t (x)) t, 0,x∈R d with correlations given by

E[X t (x), X s (y)] = S g e -min(s,t) (| x -y, s |)σ(ds) (45)
where S stands for the sphere of R d , σ the uniform measure on the sphere and the function g u (for 0 < u 1) is given by [START_REF] Lalley | A conditional limit theorem for frontier of a branching Brownian motion[END_REF] is finite for the process (X t (x)) t, 0,x∈R d with correlations given by (6).

g u (r) = ln + 2 r if r u ln 2 u + 1 -r u if r < u then (
So, from now on, we assume that the correlations of (X t (x)) t, 0,x∈R d are the new correlations specified in Lemma 20 (see [START_REF] Rhodes | Multidimensional multifractal random measures[END_REF] for further details). Notice also that the measure M γ also involves this new process. Such a family of kernels possesses useful scaling properties, namely that for |x| 1 and 0 < λ < 1, K t+h (e -h x) = K t (x) + h. In particular, we have the following scaling relation for all 0 < l < 1 and all 0 < λ < 1:

(X t+h (e -h x)) x∈B(0,1) , (M √ 2d t+h (e -h A)) A⊂B(0,1T ) law = (X t (x) + Ω h ) x∈B(0,1) , (e √ 2dΩ h -2dh M √ 2d t (A)) A⊂B(0,1) . ( 46
)
where Ω h is a centered Gaussian random variable with variance h and independent of the couple (X t x ) x∈B(0,1) , (M √ 2d t (A)) A⊂B(0,1) .

We will use the above relation throughout the proof.

By stationarity of the process X t , we can translate the integrand in the quantity (44) to get We first estimate I 1 (t). For |x -y| > e -t , by using [START_REF] Molchan | Scaling exponents and multifractal dimensions for independent random cascades[END_REF] = E e q √ 2dΩ h -2(q 2 -q)dh E e q √ 2dX t-h (0)+q

B(0,1) 2 E e q √ 2dXt(x)+q √ 2dXt(y)-2q 2 dE[(Xt(x)) 2 ] √ tM √ 2d t (B(x, |y -x|)) q γ(dx)γ(dy) B(0,1) 2 E e q √ 2dXt(0)+q √ 2dXt(y-x)-q 2 dE[(Xt(0)) 2 ] √ tM √ 2d t (B(0, |y -x|)) q γ(dx)γ(dy).
√ 2dX t-h ( y-x |y-x| )-2q 2 dE[(X t-h (0)) 2 ]
√ tM

√ 2d
t-h (B(0, 1))

q C |y -x| ξ(q) E 1 √ tM √ 2d
t-h (B(0, 1/2))

q .

In the last line, we have used a Girsanov transform to get rid of the numerator. Therefore, if we set c = sup u 1 E

1 M √ 2d u (B(0,1/2)) q , then I 1 (t) C( sup u 1 E 1 √ uM √ 2d u (B(0, 1/2)) q + c √ t )
|x-y|>e -t C |y -x| ξ(q) γ(dx)γ(dy). [START_REF] Motoo | Proof of the iterated logarithm through diffusion equation[END_REF] From Corollary 6, the quantity sup

u 1 E 1 √ uM √ 2d u
(B(0, 1/2)) q is finite.

Remark 21. Corollary 6 only deals with Gaussian fields with correlations given by [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF]. From Kahane's convexity inequality, this quantity is also finite for every Gaussian fields with correlations given K(x, y) = 2d ln + 1 |y -x| + g(x, y)

for some bounded function g. In particular, it is finite for the field considered in Lemma 20.

To treat the term I 2 (t), we use quite a similar argument excepted that we use the E e q √ 2dΩt-2(q 2 -q)dt E e q √ 2dX 0 (0)+q √ 2dX 0 (e t (y-x))-q 2 2dE[(X 0 (0)) 2 ] √ tM 0 B(0, e t (y -x))

q γ(dx)γ(dy)

= |y-x| e -t
e tξ(q) t q/2 E e q 2 2dK 0 e t (y-x) B(0,e t (y-x)) e √ 2dX 0 (u)-dE[(X 1 (0)) 2 ]+q2dK 0 (e t (y-x)-u)+q2dK 0 (u) du q γ(dx)γ(dy).

By using the fact that K 0 is positive and bounded by ln 2, we have (for some positive constant C independent of t)

I 2 (t) C |y-x| e -t
e tξ(q) t q/2 E 1 B(0,e t (y-x)) e √ 2dX 0 (u)-dE[(X 1 (0)) 2 ] du q γ(dx)γ(dy).

Since E[X 0 (u)X 0 (0)] E[(X 0 (0)) 2 ], we can use Kahane's convexity inequalities to the convex mapping x → 1

x q . We deduce (for some positive constant C independent of t, which may change from line to line)

I 2 (t) C
|y-x| e -t e tξ(q) t -q/2 E 1 B(0,e t (y-x)) e √ 2dX 0 (0)-dE[(X 1 (0)) 2 ] du q γ(dx)γ(dy)

C

|y-x| e -t e tξ(q) 1 e dqt |y -x| dq γ(dx)γ(dy) Ct -q/2 |y-x| e -t 1 (y -x) ξ(q) γ(dx)γ(dy).

Hence lim

t I 2 (t) C lim sup t→∞ t -q/2 |y-x| e -t 1 (y -x) ξ(q) γ(dx)γ(dy) = 0.

The KPZ formula is proved.

Proof of Lemma 20. Let us denote by K t the ⋆-scale invariant kernel given by (6) associated to the process (X t (x)) t,x . We will use the superscript p to denote the corresponding quantities associated to the "perfect" kernel of [START_REF] Rhodes | Multidimensional multifractal random measures[END_REF]: we denote by K p t the kernel described in Lemma 20, by (X p t (x)) t,x (resp. M p, √ 2d t

) the associated Gaussian field (resp. approximate multiplicative chaos). It is plain to see there is a constant C > 0 such that ∀t > 0, ∀x ∈ R d , K t (x) -C K p t (x) K t (x) + C. Now we prove the Lemma. By using the Girsanov transform, we have: q e q 2 2dKt(x-y) γ(dx)γ(dy) e (2dq q e q 2 2dK p t (x-y) γ(dx)γ(dy).

B(0,1) 2 E e q √ 2dXt (
By the Girsanov transform again, we have q γ(dx)γ(dy).

The lemma follows.

Remark 22. To sum up, we have proved that establishing the KPZ formula for the perfect kernel is equivalent to establishing the KPZ formula for all ⋆-scale invariant kernels. Furthermore the above argument is obviously valid for any log-correlated Gaussian field (as it only involves the Girsanov transform) and for other values of γ: for γ 2 < 2d with the techniques developed in [START_REF] Rhodes | KPZ formula for log-infinitely divisible multifractal random measures[END_REF] or for γ 2 > 2d with the techniques developed in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF]. In particular, the KPZ formula established in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Rhodes | KPZ formula for log-infinitely divisible multifractal random measures[END_REF] in terms of Hausdorff dimensions are valid for the GFF. The reader may compare with [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF] where the KPZ formula is stated in terms of expected box counting dimensions. At criticality, things are a bit more subtle: the KPZ formula established in this paper is valid for all the derivative martingale for which you can establish the renormalization theorem 5. This theorem is necessary to be in position to apply Kahane's convexity inequalities.

t.

  We work with the ball A = [0, 1]. The first step consists in writing an appropriate decomposition for the measure √ tM t+ln 8 -X ln 8 )(x)-t dx.

  Now we split the latter integral according to scales larger or smaller than e -t and obtain 0, |y -x|))q γ(dx)γ(dy) |x-y|>e -t E e q √ 2dXt(0)+q √ 2dXt(y-x)-q 2 dE[(Xt(0)) 2 ] √ tM √ 2d t (B(0, |y -x|)) q γ(dx)γ(dy) + |x-y| e -t E e q √ 2dXt(0)+q √ 2dXt(y-x)-q 2 dE[(Xt(0)) 2 ] |y -x|)) q γ(dx)γ(dy) def = I 1 (t) + I 2 (t)

2 √

 2 scaling relation on h = t instead of -ln |y -x|, and the Girsanov transform again: 2dt e q √ 2dX 0 (0)+q √ 2dX 0 (e t (y-x))-q 2 2dE[(X 0 (0)) 2 ] tM 0 B(0, e t (y -x))q γ(dx)γ(dy)= |y-x| e -t

  x)+q √ 2dXt(y)-2q 2 dE[(Xt(x)) 2 ]x,|y-x|) e q2dKt(u-x)+q2dKt(u-y)

  with e -h = |y -x|, we deduce

	E	√ = E 2dXt(0)+q e q √ tM √ √ 2d 2dXt(y-x)-q 2 dE[(Xt(0)) 2 ] t q (B(0, |y -x|)) e q2 √ 2dΩ h -2q 2 dh e q √ 2dX t-h (0)+q √ e √ 2dΩ h -2dh √ tM 2dX t-h ( y-x |y-x| )-2q 2 dE[(X t-h (0)) 2 ] √ 2d t-h (B(0, 1))

q

Partially supported by grant ANR-08-BLAN-0311-CSD5 and by the MISTI MIT-France Seed Fund. Partially supported by grant ANR-11-JCJC. Partially supported by NSF grants DMS 064558, OISE 0730136 and DMS 1209044, and by the MISTI MIT-France Seed Fund. Partially supported by grant ANR-11-JCJC.