Nao Harada

Alban Ferrier

Diana Serrano

Mauro Persechino

Emrick Briand

Romain Bachelet

Ian Vickridge

Jean Jacques Ganem

Philippe Goldner

Alexandre Tallaire

Chemically Vapour Deposited Eu 3+ :Y 2 O 3 thin films as a material platform for quantum technologies

Rare earth ions hosted in solids are good candidates for quantum technologies due to their chemical stability and optical and spin transitions exhibiting long coherence lifetimes. While bulk oxide crystals are usually the preferred host material, the development of a scalable silicon-compatible thin film platform would be desirable. In this paper, we report on the growth of Y 2(1-x) Eu 2x O 3 thin films on silicon in the full range of Eu 3+ concentration by direct liquid injection Chemical Vapour Deposition. Our sub-micron polycrystalline films with a strong-(111) texture were grown for all compositions into the bixbyite cubic phase. The variation of growth rates with temperature and flow indicated that deposition occurred through a mass-transport controlled regime. Optical assessment of the Eu-doped thin films showed inhomogeneous linewidths as narrow as 50 GHz and fluorescence lifetimes of 1ms for the lowest concentrations. Finally, a spectral hole was successfully burned in a 200 nm-thin film with a 2 % Eu doping leading to a homogeneous linewidth of 11 MHz. These values are still below those reported for bulk single crystals indicating that additional decoherence mechanisms exist in such nanometric films, which might be alleviated by further improvement of the crystalline quality. Nevertheless, these results pave the way to the use of CVD-grown Eu:Y 2 O 3 thin films as a platform for integrated quantum devices.

1-Introduction

Quantum technologies (QT) are witnessing an increasing interest with the promise to offer new systems and devices with unrivalled performance: ultra-sensitive sensors, quantum computers, information processing, highly-secured communication networks etc 1 . Often, these achievements are conditioned by the availability of custom-made materials with exceptional control over the purity as well as defect positioning at the atomic scale. Adopting a stable solid-state, scalable and efficient platform for hosting the required qubits would be advantageous for these devices. Among the various candidates, rare-earth ions (REI) are outstanding as they exhibit extremely narrow optical homogeneous linewidths at low temperature, in the range of a few kHz to less than 100 Hz in bulk single crystals 2 . This behaviour stems from their particular electronic configuration in which 4f electrons are shielded from the environment by outer electronic shells 3 . Optical coherence lifetimes of up to T 2 = 2.6 ms (homogeneous linewidth  h = 1/πT 2 = 122 Hz) have been reported for Eu 3+ :Y 2 SiO 5 at cryogenic temperatures 2 while 151 Eu 3+ spin transitions that can be optically addressed, have shown coherence lifetimes up to 10's of ms 4 or even hours 5 . REI can be chosen with optical transitions in a wide range of wavelengths, including some that might be relevant for optical fibre telecommunications. In addition, they can be hosted in a variety of matrices and are easily interchangeable due to their chemical similarity. All these properties have for example allowed demonstrating quantum storage 6 and quantum state teleportation 7 with such systems.

Single crystals produced by crystal pulling techniques such as Czochralski-grown Y 2 SiO 5 , are the preferred host for manipulation of large ion ensembles since they provide a well-controlled environment 8,9 . These bulk materials are however costly to produce especially when isotopic enrichment is needed. Due to their size, they also do not provide a platform as appropriated as nano-materials when it comes to integrating REI emitters into other nanoscale systems like high finesse and small volume optical cavities 10 . Such systems can enhance RE interactions with light, a pre-requisite to detect the weak emission from single ions 11,12 . Manipulating single emitters in nanomaterials could indeed potentially unleash quantum processing and quantum computing applications 13 . Recently, we have shown that optical homogeneous linewidths as narrow as 30 kHz 14 can be obtained in specially prepared yttrium oxide (Y 2 O 3 ) nanocrystals doped with Eu 3+ ions, the lowest ever measured for nanoscale solids 15,16 . While this value remains above that of bulk crystals, integration of these nanoparticles into an optical fibre cavity could be successfully obtained 10 .

Single crystalline thin films constitute an alternative approach. They could possibly allow one to benefit from the flexibility of multi-layers of variable thicknesses with spatially localized emitters and compatibility with scalable platforms like silicon [17][18][19][20] . Integration into a device or coupling to other quantum materials in a hybrid system 21,[START_REF] Kurizki | Proc. Natl. Acad. Sci[END_REF] would thus be greatly facilitated. In the context of QT and high resolution optical spectroscopy, single crystalline films with high purity and low defect content are preferred to achieve an optimized environment for RE ions in which strain, structural defects and electrical or magnetic noise are as limited as possible, possibly leading to extended coherence properties. Indeed, in polycrystalline or even amorphous films, the remaining disorder is expected to induce additional inhomogeneous broadening of the optical linewidths. Epitaxial thin films made by Molecular Beam Epitaxy (MBE) or lifted-off from bulk crystals have been recently investigated and showed promising results 18,[START_REF] Dutta | [END_REF]24 . Chemical methods have also been explored. Atomic Layer Deposition (ALD) has been used to deposit thin optically active Eu or Er-doped Y 2 O 3 films with precise thickness control in the 10-200 nm range and high conformity 25 . Regarding optical properties of the emitters, encouraging results have been recently obtained with inhomogeneous linewidth of 200 GHz and 1.5 ms lifetime for the infrared transition of Erdoped ultra-thin films (10 nm) 26 . However, ALD is limited to a restricted low-temperature window that does not easily allow for the synthesis of epitaxial oxide films. High temperature post-treatments (950-1050 °C) can partially improve the structural order, but they lead to the formation of parasitic interfacial phases 27 as well as unwanted dopant diffusion 28 which may be a limiting factor especially on silicon substrates. Chemical Vapour Deposition (CVD) offers an interesting approach for the deposition of high quality epitaxial RE oxide films due to the possibility to grow under a more flexible range of conditions and higher temperatures 29 . Eu 3+ :Y 2 O 3 films have already been deposited using standard metal-organic CVD on sapphire, and inhomogeneous and homogeneous optical linewidths have been reported for films that are several microns thick and high temperature annealed 30 .

In this work, we used a variant of the CVD technique involving Direct Liquid Injection (DLI) with metal-organic precursors. This technique allow to partially overcomes the issue of low volatility of heavy element precursors as REI ones [31][32][33] and helps having a better control over their constant feeding inside the reaction chamber. Europium ions doped in Y 2 O 3 were chosen among other REI in this study because they exhibit ultra-narrow optical homogeneous linewidths and long coherent times while the matrix is a simple cubic binary oxide. Moreover, using this growth technique, almost any other REI or even oxide matrix could be prepared as long as suitable precursors in terms of temperature stability and volatility are outsourced. Deposition was performed on silicon for compatibility with standard wafer processing and silicon-on-insulator waveguide technologies 34 . A range of deposition conditions were explored, and highly textured polycrystalline thin films were obtained with a good control over a wide composition range. We provide an assessment of their optical properties and evaluate their relevance to the development of scalable platforms for quantum technologies.

2-Experimental details

To deposit the films, we used a home-made DLI-CVD set-up composed of a double injection and evaporation head (KEMSTREAM Vapbox1500) and a horizontal tube furnace (Carbolite) (Fig. 1). High purity (3N-99.9 %) solid precursors from the -diketonate family (Fig. 1a, Y(TMHD) 3 and Eu(TMHD) 3 ) purchased from Strem Chemicals were dissolved in mesitylene (purity of 99.9%) to obtain concentrations in the range of 0.1-0.01 mmol/L and stored in pressurized tanks. The liquid solution was mixed to a carrier gas (300 sccm of Ar) and brought to the evaporation head maintained at 220 °C where it was sprayed into tiny droplets and flash-evaporated (Fig. 1b). Composition of the films was adjusted by monitoring the opening time and frequency of the two injectors (one for Y and one for Eu). The injectors can be controlled independently allowing a precise control of the incoming molar flow. The gas mixture was then carried through a heated line towards a tubular quartz tube furnace at 750 °C in which Si wafer pieces (10 x 10 mm²) were placed. An additional oxygen flow (150 sccm) was introduced inside the tube. The gases were pumped with a rotary vane pump allowing the pressure to settle at around 10 mbar. Unreacted precursors and chemicals were collected in a nitrogen-cooled trap placed downstream the chamber. Prior to deposition, asreceived silicon (100) substrates were cleaned using the standard Radio Corporation of America (RCA) protocol. This includes: first a (1:1:5) solution of (NH 4 OH:H 2 O 2 :H 2 O) then a 5 % HF solution and finally a (1:1:5) solution of (HCl:H 2 O 2 :H 2 O). Each step is followed by a deionised water rinsing. This procedure allows creating a 4 nm-thin, reproducible and stable chemical oxide at the surface of the silicon wafer. Spectroscopic ellipsometry (Woollam iSE) in the range 400-1000 nm was used to evaluate film's thickness. The ellipsometric angle  and phase difference  (see supplementary information file, Fig. S1) were recorded at an incidence angle of 65°. The optical refractive index of yttria was obtained using a Cauchy model.

The crystallinity of the films was evaluated using X-ray Diffraction (XRD) in a PanalyticalXPert Pro diffractometer with a Ge monochromator. Additional high-resolution (HR) XRD characterisations were performed using a high-brilliance Rigaku SmartLab diffractometer with a 9 kW copper rotating anode and a 2 bounces Ge(220) monochromator.

Chemical composition of the films was obtained by Rutherford Backscattering Spectroscopy (RBS). RBS was performed with a Van de Graaff accelerator, 4 He + particles at 1.8 MeV, 100 nA and a scattering detection angle of 165 °. The areal density of atoms was determined using a reference Si sample implanted with a controlled dose of 209 Bi ions (see supplementary information file, Fig. S3). The spectra were acquired for a 10 µC dose of incident ions for each sample. For Eu concentration below 1 % the doping level could not be precisely assessed by RBS and the nominal doping value was used.

Field Effect Gun Scanning Electron Microscope (FEG-SEM) images of the surface of the films were recorded using a ZEISS Leo 15030 microscope. Micro photoluminescence (µPL) measurements were performed using a Renishaw InVia apparatus with a 50 × objective and a 532 nm laser as the excitation source. Decay measurements at room temperature were also recorded using a tunable optical parametric oscillator pumped by a Nd:YAG Q-switched laser (Ekspla NT342B-SH) with 6 ns pulse length, an ACTON SP2300i monochromator and a photomultiplier tube. The detection time constant was between 50 µs and 200 µs and data was recorded on a digital oscilloscope (LeCroy 332A).

Inhomogeneous linewidths of the 7 F 0  5 D 0 transition were measured in reflection mode using a closed-cycle cryostat (Janis CTI-Cryogenics Model CCS-150) with the temperature monitored by a LakeShore model 330 controller. The 7 F 0  5 D 0 transition inhomogeneous linewidths were probed by monitoring the fluorescence intensity of the 5 D 0 → 7 F 2 transition at 612 nm as a function of the excitation wavelength around 580.883 nm. Excitation was provided by a single mode dye laser (Coherent 899-21, 1 MHz linewidth) pumped by a semi-conductor laser (Coherent Verdi G10). Red fluorescence was collected by a 75 mm diameter lens and focused on a photomultiplier tube after suitable filtering to eliminate scattered laser light.

For spectral hole burning (SHB) experiments, samples were cooled down to liquid helium temperatures in a bath cryostat (Janis SVT 200). A continuous wave narrow linewidth (250 kHz) dye laser (Sirah Matisse DS) provided the excitation around 580 nm ( 7 F 0  5 D 0 transition), which was modulated in time and frequency using an acousto-optic modulator driven by an arbitrary waveform generator (Agilent N8242A). Holes were probed by monitoring the red fluorescence ( 5 D 0  7 F 2 ) at 612 nm, which was collected using a high numerical aperture lens mounted on a piezo-driven mount and located inside the cryostat. Fluorescence was filtered using low-pass and interferential filters and sent to a high sensitivity photomultiplier tube (Hamamatsu R10699). Further details on the SHB setup used and the procedure are given in the supplementary information file (Fig. S5).

3-Results

3.1-Structural and morphological characterization of the Y2O3 thin films

Undoped Y 2 O 3 thin films were first grown on (100) silicon wafers in order to investigate the deposition process by DLI-CVD. By adjusting the deposition time while keeping all other deposition parameters constant, films with thicknesses from 90 to 650 nm as evaluated by ellipsometry (see supplementary information file, Fig. S1) were obtained and observed by SEM. The resulting morphologies are presented in Fig. 2a-c. The films show a polycrystalline structure with an increasing but uniformly distributed grain size from about 20 to 80 nm when increasing thicknesses from 50 to 650 nm and with an obvious saturation of grain coarsening above 400 nm (Fig. 2d). This behaviour is typical from a columnar growth mechanism 35 where nucleated grains competitively grow depending on their faster growth direction until they reach a steady-state.

In Fig. 3a, growth rates are plotted as a function of temperature. Below 500 °C no growth is observed due to the well-known limited reactivity of the -diketonate precursors with molecular oxygen [START_REF] Aspinall | Rare Earth Oxide Thin Films[END_REF] . The use of more powerful oxidizing agents such as ozone would be a possible way to increase reactivity . At higher temperatures, three regions can be distinguished indicating the different CVD growth modes [START_REF] Pierson | [END_REF] . From the slope of the growth rate dependence with temperature, one can extract the activation energy of the reaction from an Arrhenius law: ln(𝐺𝑅) = 𝐴 -𝐸 𝑎 𝑅𝑇 where GR is the growth rate, A is a constant, E a is the activation energy and R is the universal gas constant. In region I (500-650 °C), growth rates strongly depend on the temperature with an estimated activation energy of 1.2 kJ/mol. This is characteristic of a kinetic regime where surface reactions are the limiting process. In region II (650-1000 °C), growth rates are less dependent on temperature and activation energy reduces to 0.14 kJ/mol. This indicates that mass transport, i.e. the ability to carry the reactants towards the substrate is the main limiting step. Finally, in region III (T > 1000 °C) the trend is inverted and growth rates continuously decrease with temperature. This can possibly indicate pre-reactions of the precursors that occur before reaching the substrate and that limit the deposition process. This sets the upper temperature bound for this particular CVD system and chemistry.

To confirm that in region II growth rates can be controlled by the supply of the metallic cation precursor, a series of growth runs, in which the Y(TMHD) 3 flow was varied from 3 to 14 µmol/min, was carried out at a constant temperature of 750 °C. A linear dependence was found from 0.2 to 0.9 µm/h without any apparent saturation (Fig. 3b). This confirms that masstransport is the controlling step for growth at this temperature. Although this trend may seem trivial, it indicates that the design of the reactor and the growth conditions (total flow, pressure, concentration etc.) are suitable, avoiding pre-reactions between precursors or uncontrolled turbulent flow inside the chamber. In order to get more insight into the structural properties of the films, HR-XRD analysis was carried out on a 100 nm-thick pure Y 2 O 3 film grown on Si (100). The -2 scan presented in Fig. 4a shows that the cubic phase of Y 2 O 3 was obtained. The position of the peaks is consistent with a fully-relaxed film. The absence of the (400) diffraction peak and the presence of only (hhh) reflections indicate that the film is (111)-textured. The -scan of Fig. 4b acquired for the (222) lattice plane indicates a relatively high out-of-plane mosaicity of about 8° which shows that an important crystalline disorder exists. Preferential orientation along (111) has already been reported for Y 2 O 3 polycrystalline films obtained by various techniques [START_REF] Wu | [END_REF]39 .Such texturing is a consequence of the minimization of free surface energy of the grains as well as a competitive growth between grains of different orientations as expected from a columnar growth model. We note that, although all films produced in this study displayed preferential texture, full (111)-texture was only obtained when a thin chemical silicon oxide layer (around 4 nm) was intentionally created by using an RCA cleaning procedure ending with an oxidation step. With as-received silicon substrates, a significant fraction of (100)-oriented grains was found (see supplementary information file, Fig. S2). This suggests that surface preparation of the substrate plays an important role in the texture formation. To assess the in-plane orientation of the films, a pole figure was also acquired on an asymmetric (400) reflection (Fig. 5). The presence of a concentric circle (Fig. 5a) rather than clear diffraction spots in the map underlines that the film was not in-plane oriented. Although no epitaxial growth was obtained here, the full (111)-texture and thus the preferential out-of-plane orientation as well as the relatively narrow diffraction peaks provide positive indicators for the optical properties, even though they do not correspond to specific incorporation sites for the REI. By using appropriate substrates, surface treatments and growth conditions the possibility to epitaxially grow the films on Si will be attempted as already reported for films grown by Molecular Beam Epitaxy (MBE) 40,41 , Pulsed Laser Deposition (PLD) 42 and e-beam evaporation 43 for example.

Eu doping of the Y2O3 thin films

We then evaluated the ability to produce Y 2(1-x) Eu 2x O 3 films with a controlled doping and in a wide range of compositions (x = 0 to 1). To this end, a variable amount of Eu(TMHD) 3 was added together with Y(TMHD) 3 using the double injection head of the DLI-CVD set-up. The films were grown at 750°C in the mass-transport regime with a 6 µmol/min flow which is a good trade-off between the deposition's duration and thickness homogeneity on the substrate. To perform RBS measurements, thinner films of 60 nm were grown. Indeed, for too high thicknesses Eu and Y peaks widen and may overlap, further complicating their quantification. The composition of the films was determined using non-channelling conditions from the Eu/(Eu+Y) ratio of each related peak in the RBS spectrum (see supplementary information file, Fig. S3). It indicates that incorporation efficiency of Eu and Y in the films was similar. This could be expected since the use of identical ligand molecules in the precursor (TMHD) and the interchangeability of rare-earth elements that possess similar radii and properties are likely to lead to similar chemistry. We note that a systematic cleaning procedure of the injectors between each deposition run using mesitylene solvent was crucial to limit cross-contaminations. This approach allowed us to address a wide Eu range from 0 to 99 % as deduced from RBS analysis. For the highest doping (99 %), background yttrium contamination could not be entirely suppressed despite careful cleaning of the lines preventing reaching a completely pure Eu 2 O 3 film. The crystal structure was then further analysed by XRD (Fig. 6a). The (222) peak position varied in between the reported values for Y 2 O 3 (a = 10.60 Å) to Eu 2 O 3 (a = 10.87 Å) cubic crystalline structures in a linear fashion (Fig. 6b).

Full solid solution was therefore achieved in agreement with what is expected for cation substitution by the Vegard's law 44 . Lanthanide sesquioxides present three major structural polymorphs: cubic bixbyite, trigonal, and monoclinic. Under ambient conditions, the thermodynamically stable phase of Eu 2 O 3 is monoclinic unlike Y 2 O 3 which is cubic. Interestingly we note that even for the highest Eu contents, the crystal structure remained in the cubic phase. This might be related to a stabilizing effect of yttrium on the cubic structure or to an influence of the cubic substrate underneath.

4-Optical properties of the Eu:Y 2 O 3 thin films

The optical properties of the films grown with different Eu concentrations (thickness around 200 nm) were assessed and benchmarked against Eu:Y 2 O 3 transparent ceramics and nanoparticles already investigated in our lab. The µPL spectra of films with a similar thickness and Eu content from 1 to 99 % as measured by RBS were recorded using a non-resonant 532 nm excitation laser (Fig. 7a). All spectra were normalized to the samples' thicknesses. For all compositions, a narrow emission band around 612 nm, corresponding to the 5 D 0  7 F 2 Eu 3+ transition in Y 2 O 3 and characteristic of the cubic phase, is observed (Fig. 7b). Furthermore, although weaker, the fluorescence of europium ions in Eu 2 O 3 cubic host is detected. The integrated intensity of this main emission was found to increase up to concentrations of about 4 % of Eu 3+ (Fig. 7c) and then decreases due to concentration quenching, i.e. a reduction of the emission related to interaction between Eu and defects or impurities (Fig. 7c) 47 . In addition, migration of the excitation energy over the Eu 3+ sub-lattice increases the probability of quenching by impurities at higher concentrations. For bulk crystals, the critical concentration was found to be around 10 % 48 .

The lifetime of the 5 D 0 level was then assessed by probing the red fluorescent transition ( 5 D 0  7 F 2 ) following a resonant excitation of the 7 F 0  5 D 2 level at 465 nm (Fig. 7a). The curves in Fig. 6d show a single exponential decay from which the lifetime T 1 can be extracted. T 1 exhibits a maximum around 1 ms for the lowest concentrations and then reduces drastically down to about 36 µs for the highest Eu concentration (Fig. 7e). This is in good agreement with the concentration quenching previously observed in the fluorescence spectra. We note that at a nominal concentration of 1 % the signal was weak and the error in measuring the lifetime was much larger. Nevertheless this maximum lifetime value is comparable to that measured on transparent ceramics with similar Eu 3+ doping 49 which is an encouraging indicator of performance of this thin film material. Inhomogeneous linewidths (Γ inh ) were also evaluated as a function of Eu 3+ nominal concentration at a temperature of 10 K. To this end, emission of the 5 D 0  7 F 2 transition was monitored near 612 nm while the laser excitation frequency was scanned around a central frequency to resonantly probe the 7 F 0  5 D 0 transition (Fig. 6a). The results presented in Fig. 8a indicate that, as the Eu content increased, Γ inh broadened from a minimum of 50 GHz (at 0.25 % Eu) to a maximum of 325 GHz (at 50 % Eu). This behaviour is consistent with the fact that the 50 % doping level corresponds to the maximum statistical disorder in the crystal structure where one half of the cations are Y 3+ and the other half are Eu 3+ . A broadening rate of about 6 GHz per percent of Eu is thus deduced. A blueshift of the line as a function of increasing Eu concentration is also observed, in agreement with the larger Eu ionic radius compared to Y one 50,51 . The Γ inh measured here is comparable although slightly narrower than that reported for ALD-grown and annealed thin films (around 80 GHz at 2 % Eu) 26 as well as CVD-grown and high-temperature annealed films on sapphire (around 90 GHz at 2 % Eu) 30 . This relatively low value of inhomogeneous linewidth suggests that RE ions experience a fairly uniform crystalline environment. Nevertheless, at low Eu concentrations, Γ inh of our films are significantly broader than that measured in high-quality nanoparticles annealed at 1200 °C (10 GHz at 0.3 %) 15 or transparent ceramics 49 of the same composition. This larger contribution can be attributed to the presence of intrinsic defects such as dislocations or point defects (impurities, vacancies) in the films, indicating that the insertion of Eu 3+ ions is not the main source of broadening at low doping levels. The Lorentzian rather than Gaussian shape of the  inh line rather points towards impurities-induced broadening 52 . We indeed know that carbon from the precursors or silicon from the substrate might be present in the films. The use of annealing post-treatment at high temperature (> 1100 °C) could possibly help in relieving this issue as demonstrated in Eu:Y 2 O 3 nanoparticles 53 . In order to get more insight into the coherent properties of RE ions in our CVD-grown Eu:Y 2 O 3 thin films, we performed a SHB experiment at 2 K on a 200 nm-thick film with 2 % Eu doping. Spectral Hole Burning (SHB) is an optical method that can probe homogeneous linewidths and therefore the effect of dynamical fluctuations in the material on RE ion transitions 3 . In our experiments, the spectral hole is induced by a modification of the population in the long-lived hyperfine level of the ground state of europium. While at thermal equilibrium all the hyperfine levels are equally populated, a narrow laser excitation can change this population repartition by depleting the pumped hyperfine level (see supplementary information file, Fig. S6). This change of population is directly observed by a decrease of the red fluorescence emission. Spectral hole width is as broad as twice the homogeneous linewidth provided that it is not increased by other effects like too high excitation power and laser jitter or drift.

The spectral hole burning sequence used in this work was composed of 3 different parts (Fig. 9a). A burn pulse of 20 ms at a fixed frequency (), a shorter probe pulse at a variable frequency ( + ) and a series of strong pulses scanned over 200 MHz to erase the spectral hole by distributing the population equally between the three hyperfine levels of Eu. The measurement was performed by monitoring the fluorescence intensity after the weak probe pulse. Contribution from laser frequency instability and other hole broadening effects was determined using a Eu:Y 2 SiO 5 single crystal in which the homogeneous linewidth is below 1 kHz. Here a minimal hole linewidth of 2 MHz was recorded, corresponding to the spectral resolution of our setup.

Using the sequence outlined above, we observed a clear spectral hole burning phenomenon with a FWHM of about 22 MHz (Fig. 9b), which gives a homogeneous linewidth of about  hom = 11 MHz. This is similar to the value obtained on Eu 3+ :Y 2 O 3 high temperature annealed CVD films of higher thickness 30 and deposited on sapphire. Nevertheless, homogeneous linewidths lower by almost two order of magnitude were measured on nanoscale systems (Eu:Y 2 O 3 nanoparticles 15,16 ). The origin of this broadening might be connected with local disorder that leads to additional dephasing mechanisms known as Two Level System (TLS) 30 . As discussed above, the large inhomogeneous linewidths observed in the films is a clear indication of such a disorder. Another homogeneous broadening mechanism suggested in nanoparticles is fluctuating electric fields due, for example, to charge noise on surfaces or grain boundaries 15,16 . Decreasing these processes could be achieved by reducing defects and increasing grain or by combining high-temperature deposition and annealing, as well as other post-treatments 14 . This has proved efficient in obtaining linewidths below 100 kHz in Eu 3+ :Y 2 O 3 nanoparticles 15,53 . Epitaxially depositing the film could also potentially help reducing dephasing phenomena.

5-Conclusion

Using DLI-CVD we successfully grew Y 2(1-x) Eu 2x O 3 oxide thin films on (100)-oriented silicon wafers in the full range of Eu content (0 < x < 1). The films obtained through a columnar growth mode were polycrystalline with full (111)-texture. High structural order was achieved for deposition temperatures in the range 650-1050 °C which corresponds to a mass-transport dominated regime. A good control of the solid solution composition was obtained with lattice parameters matching the expected values from a relaxed structure.

The optical properties of such thin films on Si were then assessed and benchmarked against other systems (nanoparticles and ceramics) within the specific context of quantum technologies. Optical spectroscopy showed that fluorescence from the 5 D 0  7 F 2 transition of Eu 3+ ions had strongest emission intensity and longest decay time (1 ms) for low concentrations (below 4 %). At high concentrations, quenching of the luminescence occurred. The lowest inhomogeneous linewidth was found to be around 50 GHz which is a relatively low value for such a nanoscale system. Notably, a 22 MHz-wide hole was spectrally burned in a 200 nm-thin film on silicon, corresponding to a homogeneous linewidth of 11 MHz. Future efforts to reduce  hom should be directed to the improvement of the structural quality of our Y 2 O 3 thin films and in particular reduce the presence of grain boundaries as well as impurities and defects that can possibly act as a source of decoherence. Nevertheless, the first structural and optical assessment indicates that CVD-grown Eu:Y 2 O 3 thin films on silicon offer interesting perspectives towards their integration into quantum devices.

Fig. 1 .

 1 Fig. 1. (a) Schematics of the DLI-CVD set-up used in this study and that uses specially developed liquid injectors. The reaction chamber consists of a horizontal quartz tube furnace. (b) Chemical representation of Y(TMHD) 3 diketonate precursor used.

Fig. 2 .

 2 Fig. 2. (a) -(c) Film morphologies observed by SEM for different thicknesses (t) from 90 to 650 nm showing polycrystalline grains. (d) Grain size directly estimated from SEM images as a function of film thickness.

Fig. 3 .

 3 Fig. 3. (a) Growth rate dependence on temperature showing three distinctive regions. Activation energies are extracted from the slopes of the lines and are indicated. (b) Y 2 O 3 film growth rate dependence as a function of the incoming precursor flow at 750°C. This confirms the mass transport regime for region II. The dotted line is a linear fit of the data.

Fig. 4 .

 4 Fig. 4. HR-XRD analysis. (a) -2 scans showing that only the (111)-related diffraction peaks of cubic Y 2 O 3 are observed. (b)  scan around the (222) diffraction peak showing an 8° out-of-plane mosaicity. The solid red line is a Lorentzian fit from which FWHM is extracted.

Fig. 5 .

 5 Fig. 5. (a) Pole figure on the asymmetric (400) reflection from which no preferential in-plane orientation is evidenced. (b) and (c) Line scans at particular values of and.

Fig. 6 .

 6 Fig. 6. Y 2(1-x) Eu 2x O 3 thin films. (a) Position of the (222) X-ray diffraction peak for different Eu contents measured by RBS. Vertical dotted lines are the expected values of the two rare-earth oxides in the literature 45,46 . (b) Deduced lattice parameter from the peak position using Bragg's formula. The expected trend for a full solid solution between Y 2 O 3 and Eu 2 O 3 cubic phases are indicated by the red line.

Fig. 7 .

 7 Fig. 7. Fluorescence and lifetime measurements at room temperature as a function of Eu content estimated by RBS. (a) Energy level of Eu 3+ in Y 2 O 3 with the 612 nm transition that is monitored. (b) µPL spectra acquired with a non-resonant 532 nm excitation. (c) Area of the 612 nm PL emission. (d) Fluorescence dynamics of the 5 D 0 level obtained by resonantly exciting 5 D 2 at 465 nm. (e) Lifetime values determined by fitting the experimental fluorescence decay curve with a single exponential.

Fig. 8 .

 8 Fig. 8. Inhomogeneous linewidth measurement (a) Emission of the 612 nm line as a function of excitation laser frequency to probe the 7 F 0  5 D 0 transition for various Eu concentrations at 10 K. The solid line is a Lorentzian fit of the experimental data points. (b) Variation of the inhomogeneous linewidth with Eu content.

Fig. 9 .

 9 Fig. 9. (a) SHB sequence used. (b) SHB spectrum obtained when varying the detuning frequency between the burn and probe pulses. A distinct fluorescence loss is observed indicating an efficient hole burning. The solid line is a Lorentzian fit of the data.

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 820391 (Square). This research work has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 712721 (NanOQTech). We acknowledge the financial support of the Paris network on quantum technologies (SIRTEQ), and the French EMIR-A network for provision of irradiation beam time using the SAFIR facility. The authors would also like to thank Hervé Guillon from Kemstream as well as Dominique De Barros for advice and technical support throughout this project.

Data availability

The data that supports the findings of this study are available within the article and its supplementary material.

Supplementary material

See the supplementary material for a more complete description about thickness measurement by Spectroscopic Ellipsometry (SE), effect of silicon surface preparation, composition analysis by RBS, hole burning experimental setup and energy levels of europium.


Keywords: Gaussian multiplicative chaos, tail expansion, reflection coefficient, DOZZ formula

 to higher dimensions and other kernels of the type (1.2) because their argument is based on a functional relation, which is obtained 1 We will only consider this case in this note though there

THE TAIL EXPANSION OF GAUSSIAN MULTIPLICATIVE CHAOS AND THE LIOUVILLE REFLECTION COEFFICIENT

By Rémi Rhodes * and Vincent Vargas * Université Aix-Marseille and École Normale Supérieure de Paris

In this short note, we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d GFF on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main interest of our work consists of two points. First, our derivation is based on a simple method which we believe is universal in the sense that it can be generalized to all dimensions and to all log-correlated fields. Second, in the 2d case we consider, the value of the constant in front of the inverse power is (up to explicit terms) nothing but the Liouville reflection coefficient taken at a special value. The explicit computation of the constant was performed in the recent rigorous derivation with A. Kupiainen of the DOZZ formula [START_REF] Kupiainen | Local Conformal Structure of Liouville Quantum Gravity[END_REF][START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF]; to our knowledge, it is the first time one derives rigorously an explicit value for such a constant in the tail expansion of a GMC measure. We have deliberately kept this paper short to emphasize the method so that it becomes an easily accessible toolbox for computing tails in GMC theory.

Introduction.

Gaussian multiplicative chaos (GMC) measures are widespread in probability and statistical physics; indeed they appear in a wide variety of contexts and in particular in the fields of finance, number theory, Liouville quantum gravity and turbulence (see [START_REF] Rhodes | Gaussian multiplicative chaos and applications: a review[END_REF] for references). In view of the broad applications of GMC theory, it is very natural to study these measures in great detail. The foundations of GMC theory were laid in Kahane's 1985 seminal work [START_REF] Kahane | Sur le chaos multiplicatif[END_REF]. If Ω is some open subset of R d then the theory of GMC enables one to define random measures of the form

(1.1) M γ (dx) = e γX(x)-γ 2 2 E[X(x) 2 ] dx
where dx denotes the Lebesgue measure and X is a log-correlated Gaussian field, i.e. a centered Gaussian field with covariance

(1.2) E[X(x)X(y)] = ln 1 |x -y| + f (x, y), x, y ∈ Ω
with f some smooth and bounded function. Of course, definition (1.1) is only formal since X is not defined pointwise hence the measure M γ can only be defined via a regularization procedure. More specifically, the measure M γ is defined via the limit in probability of the sequence

(1.3) M γ, (dx) = e γX (x)-γ 2 2 E[X (x) 2 ] dx
where (X ) >0 is a reasonable family of smooth Gaussian fields converging towards X when goes to 0: see Berestycki's very simple and elegant approach [START_REF] Berestycki | An elementary approach to Gaussian multiplicative chaos[END_REF] for an introduction to GMC and an account on the above issues of convergence. Kahane [START_REF] Kahane | Sur le chaos multiplicatif[END_REF] proved in 1985 that the measure M γ is different from 0 if and only if γ 2 < 2d 1 ; moreover, a standard result in GMC theory is the following condition on existence of moments (see [15, section 2]): if O is some nonempty and bounded open subset then

(1.4) E[M γ (O) p ] < ∞ ⇐⇒ p < 2d γ 2 .
In view of (1.4), it is natural to seek the exact tail behaviour of M γ (O). This was achieved in the beautiful work by Barral and Jin [START_REF] Barral | On exact scaling log-infinitely divisible cascades[END_REF] in the 1d case and for a covariance kernel of the form E[X(x)X(y)] = ln 1 |x-y| on the interval [0, 1]. More specifically, the work of Barral and Jin established that

(1.5) P(M γ [0, 1] > t) = C t 2 γ 2 + o( 1
t 2 γ 2 )
where the constant C is given by

(1.6) C = 2γ 2 2 -γ 2 E[M γ [0, 1] 2 γ 2 -1 M γ [0, 1 2 ] -M γ [0, 1 2 ] 2 γ 2 ] ln 2 .
from a specific geometric representation (called cone construction) of the Gaussian field with kernel E[X(x)X(y)] = ln 1 |x-y| in dimension 1. Moreover, the approach of Barral and Jin does not provide an explicit value for the constant C . Let us also mention the appendix of Leblé-Serfaty-Zeitouni [START_REF] Leblé | Large deviations for the two-dimensional two-component plasma[END_REF] in a slightly different framework, which contains a description of the tail of the modulus of complex GMC (for β ∈ (0, √ 2))

M β (D) := lim →0 -β 2 2 D e iβX (x) dx
with X a full plane Gaussian Free Field in dimension 2 and D a bounded open subset of the plane. This type of GMC possesses moments of all orders, which are related to Coulomb gas computations. This makes possible to compute the tail with a value of the constant given by a variational formula (whose solution is not explicit).

Let O denote an open subset of R2 with a C 1 boundary 2 . The aim of this work is to introduce a simple method to compute the tail of M γ (O) together with an exact value for the constant in the leading order term. We will consider the case of the unit disk D equipped with a Gaussian Free Field (GFF) X with vanishing mean over the unit circle (eventually augmented by an independent Gaussian perturbation; this covers for instance the important case of the GFF with Neumann boundary condition). Indeed, though we believe our method can be generalized to all dimensions and all kernels, we stick to the 2d GFF setting to keep this note rather short. Let us stress furthermore that these generalizations may eventually raise serious additional technicalities.

We further mention that some material in this note is inspired by the work of Duplantier-Miller-Sheffield [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF] and also by the tail estimates which appear in our recent proof with Kupiainen of the DOZZ formula [START_REF] Kupiainen | Local Conformal Structure of Liouville Quantum Gravity[END_REF][START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF]. In fact, our main result (Theorem 2.2 below) can be seen as a strengthening of the convergence results which underly the construction of the so-called quantum sphere in [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF]. However, this note is mostly self-contained: it requires no a priori knowledge of the paper [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF] and we recall the definition of the reflection coefficient of Liouville conformal field theory (LCFT hereafter) before stating the main result.

1.1. The Liouville reflection coefficient. From now on we restrict to the case of dimension 2. Consider γ ∈ (0, 2) and define Q = γ 2 + 2 γ . In this subsection, we introduce the (unit volume) reflection coefficient of LCFT. It was defined in [START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF] where it plays an important role in the proof of the DOZZ formula. This coefficient shows up when one analyzes the large values of GMC measures with a singularity, i.e. when one looks at variables of the type

B(v,r) 1 |x -v| γα M γ (d 2 x).
For this integral to exist, the exponent must satisfy α < Q [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]. It turns out that when α is large enough (in fact α > γ 2 ), large values of this random variable mostly come from the singularity at the point v and the reflection coefficient quantifies this statement (more details later). Therefore it can intuitively be understood as a coefficient of mass localization in GMC theory. Our proof for the tail is based on a localization trick (see subsection 3.1), which allows us to express the tail of GMC in terms of singular integrals as above. This is the reason why the reflection coefficient is instrumental in identifying the constant in the tail of GMC measures.

In order to introduce the reflection coefficient, we first recall basic material introduced in [START_REF] Sheffield | Conformal weldings of random surfaces: SLE and the quantum gravity zipper[END_REF]. For all α < Q, we define the process B α s

(1.7) B α s = B α -s if s < 0 Bα s if s > 0
where (B α s ) s 0 , ( Bα s ) s 0 are two independent standard Brownian motions with negative drift α -Q and conditioned to stay negative. We also consider an independent centered Gaussian field Y defined for s ∈ R and θ ∈ [0, 2π] with covariance

(1.8) E[Y (s, θ)Y (s , θ )]
= ln e -s ∨ e -s |e -s e iθ -e -s e iθ | and associated GMC measure

N γ (dsdθ) = e γY (s,θ)-γ 2 2 E[Y (s,θ) 2 ] dsdθ.
We introduce the integrated chaos measure with respect to Y (1.9)

Z s = 2π 0 e γY (s,θ)-γ 2 2 E[Y (s,θ) 2 ] dθ.
This is a slight abuse of notation since the process Z s is not a function (for γ √ 2) but rather a generalized function; in this setup, Z s ds is a stationary 1d random measure, meaning that this random measure has same law as its pushforward by any translation. Now, we define the (unit volume) reflection coefficient R(α) for all α ∈ ( γ 2 , Q) by the following formula

(1.10) R(α) = E ∞ -∞ e γB α s Z s ds 2 γ (Q-α)
.

Notice that the condition α ∈ ( γ 2 , Q) ensures that 2 γ (Q -α) < 4 γ 2 hence one can show that R(α) is well defined for all α ∈ ( γ 2 , Q). Indeed, for all α < Q and p ∈ (0, 4 γ 2 ) the following holds (see [START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF]Lemma 2.8])

(1.11) E ∞ -∞ e γB α s Z s ds p < ∞.
Finally, one of the main results of [START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF] is an integrability result for R. Indeed, one has the following remarkable explicit formula for R

(1.12) R(α) = - (πl( γ 2 4 )) 2 γ (Q-α) 2 γ (Q -α) Γ(-γ 2 (Q -α)) Γ( γ 2 (Q -α))Γ( 2 γ (Q -α))
where Γ is the standard Gamma function and l(x) = Γ(x)/Γ(1 -x).

Now that we have introduced R, we can announce the organization of the paper. First, we state our main result in section 2 and then proceed with the proof in section 3. The final section of the paper discusses how the method can be extended to more general situations.

imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018 2. Setup and Main Results. We consider the open unit disk D ⊂ C equipped with a full plane Gaussian Free Field (GFF) X with vanishing mean over the unit circle. This is a centered Gaussian field with covariance

(2.1) ∀x, y ∈ D, E[X(x)X(y)] = ln 1 |x -y| .
Remark 2.1. Recall that the full plane GFF is a centered Gaussian distribution X defined up to additive constant with covariance ln 1 |x-y| . Thus it makes sense to consider X := X -

1 2πi |z|=1 X(z) dz z ,
for which the covariance is easily identified to be given by

E[X(x)X(y)] = ln 1 |x -y| + 1 {|x| 1} ln |x| + 1 {|y| 1} ln |y|.
When restricted to D, this gives (2.1).

For γ ∈ (0, 2), we consider the GMC measure

(2.2) M γ (d 2 z) := e γX(z)-γ 2 2 E[X(z) 2 ] d 2 z
where d 2 z is the standard Lebesgue measure in the plane (recall that (2.2) is defined as the limit of (1.3) when goes to 0). In the sequel, we adopt the following standard convention: for β ∈ R, we denote o(t β ) (respectively O(t β )) a quantity of the form t t β where t goes to 0 (resp. remains bounded) as t goes to infinity. Denote by |B| the Lebesgue measure of the Borel measurable set B and by Ā the closure of the set A.

Our main result is:

Theorem 2.2. Set (2.3) p 0 := (2 -γ 2 2 ) 2 γ 4 + 2 γ 2 + (2 -γ 2 2 ) γ 2 .
For any δ ∈ (0,

1+p 0 -4 γ 2 2+p 0 ) and for any open set O ⊂ D with a C 1 boundary, we have (2.4) P(M γ (O) > t) = 2 γ (Q -γ) 2 γ (Q -γ) + 1 R(γ) t 4 γ 2 |O| + o(t -4 γ 2 -δ )
where R(γ) is the (unit volume) reflection coefficient of LCFT evaluated at γ. It has explicit expression

(2.5) R(γ) = - (πl( γ 2 4 )) 2 γ (Q-γ) 2 γ (Q -γ) Γ(-γ 2 (Q -γ)) Γ( γ 2 (Q -γ))Γ( 2 γ (Q -γ)) with Q = 2 γ + γ 2
and the function l is given by a ratio of Gamma functions l(x) = Γ(x)/Γ(1 -x).

The origin of the value p 0 will be more transparent in Remark 3.7 below. Also, notice that 1 + p 0 -4 γ 2 > 0 so that our statement is not trivially empty.

Remark 2.3. In a previous version of this note, we stated (2.4) with a better bound on δ than the present condition; unfortunately this was due to a slight mistake in the proof. Also, let us mention that we do not know what is the optimal bound on δ such that (2.4) holds.

More than the result, our method of proof makes us believe in a higher level of generality for this result. For general GMC measures in 2d with respect to a Gaussian field with covariance (1.2): we expect that the ln 1 |x-y| part is responsible for the appearance of the reflection coefficient term whereas the perturbation term f (x, y) in (1.2) produces an integral which depends only on the on-diagonal values of f .

As an illustration of this discussion we claim:

Corollary 2.4. Assume the GMC measure (2.
2) is defined with respect to the Gaussian distribution X with covariance (1.2) for some function f that is locally Hölder and positive definite. For any open set O ⊂ Ō ⊂ D with a C 1 boundary, we have

(2.6) P(M γ (O) > t) = O e 4 γ (Q-γ)f (v,v) d 2 v 2 γ (Q -γ) 2 γ (Q -γ) + 1 R(γ) t 4 γ 2 + o(t -4 γ 2 )
where R(γ) is still the (unit volume) reflection coefficient of LCFT evaluated at γ. Actually we even believe that the above structure of the tail of GMC measures is universal in the sense that in all dimensions there should be an analogue of the unit volume reflection coefficient, with a probabilistic representation similar to (1.10), such that the tail is given by (2.6) (see section 4 for further discussion). A specific feature of the 2d case is that there exists an explicit analytic expression (2.5) established in [START_REF] Kupiainen | Local Conformal Structure of Liouville Quantum Gravity[END_REF][START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF] for the expectation formula (1.10).

Let us comment on the physics literature. To our knowledge, an explicit tail expansion for GMC was derived (at the level of physics rigor) in the papers [START_REF] Fyodorov | Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential[END_REF][START_REF] Fyodorov | Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of 1/f Noises generated by Gaussian Free Fields[END_REF] in the 1d case for two specific logcorrelated models: the circular case in [START_REF] Fyodorov | Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential[END_REF] and the unit interval case in [START_REF] Fyodorov | Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of 1/f Noises generated by Gaussian Free Fields[END_REF] (with exact logarithmic correlations, i.e. the case where f = 0 in (1.2)). Their derivation is based on exact integrability results for GMC hence their results are much stronger than just tail expansions for these specific GMC measures; however, these works do not adress the derivation of tail expansions for 1d GMC associated to general logarithmic kernels or more importantly for GMC in higher dimensions. We refer to the section 4 below for more on this.

Finally, let us mention that the our main result Theorem 2.2 (together with its Corollary 2.4) reinforces/strengthens the convergence results used to define the unit volume quantum sphere [6, definition 4.21] stated by Duplantier-Miller-Sheffield. Indeed the definition of quantum sphere in [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF] is the following: consider the random field h defined on the cylinder R × [0, 2π] by the sum of two independent processes h 1 (s) + Y (s, θ) such that • The radial part h 1 is 2 γ ln U where U is chosen according to the a Bessel excursion measure with index δ = 4 -8 γ 2 (and reparametrized to have quadratic variation ds).

• The non radial part Y has the law given by (3.4).

The (unit volume or area) quantum sphere is obtained by conditioning the measure e γh dsdθ to have volume 1. By definition, 2 γ ln U is distributed according to B γ + 2 γ ln e where (e ) 2 (which is the maximum of the excursion) is distributed according to the (infinite) measure u δ 2 -2 du (recall that δ = 4 -8 γ 2 ). Recalling the definition (1.9) of Z s , we define

ρ(γ) = ∞ -∞ e γB γ s Z s ds.
Therefore the law of the (unit volume or area) quantum sphere µ h is obtained as the following limit when goes to 0

(2.7) E[F (µ h )] := lim →0 1 C E F (e ) 2 e γB γ s N γ (dsdθ) 1 (e ) 2 ρ(γ)∈[1,1+ ]
where

C = E[1 (e ) 2 ρ(γ)∈[1,1+ ]
] and F is an arbitrary bounded measurable functional defined on the space of Radon measures equipped with the weak-topology. The asymptotics of this quantity is easily identified with a simple change of variables

E[F ((e ) 2 e γB γ s N γ (dsdθ))1 (e ) 2 ρ(γ)∈[1,1+ ] ] = ∞ 0 E[F (ue γB γ s N γ (dsdθ))1 uρ(γ)∈[1,1+ ] ]u δ 2 -2 du = ∞ 0 E F (y e γB γ s N γ (dsdθ) ρ(γ) )ρ(γ) 1-δ 2 1 y∈[1,1+ ] y δ 2 -2 dy.
It is then straightforward to get the equivalent as → 0

E[F ((e ) 2 e γB γ s N γ (dsdθ))1 (e ) 2 ρ(γ)∈[1,1+ ] ] ≈ ∞ 0 1 v∈[1,1+ ] v δ 2 -2 dv E F ( e γB γ s N γ (dsdθ) ρ(γ) )ρ(γ) 2 γ (Q-γ)
where a ≈ b means that the ratio a/b converges to 1 and we have used that 1 -δ 2 = 2 γ (Q -γ). Therefore, definition (2.7) is equivalent to the following Definition 2.6. The law of the unit area quantum sphere µ h (in the space of quantum surfaces with two marked points -∞ and ∞4 ) is given by

(2.8) E[F (µ h )] = 1 R(γ) E F ( e γB γ s N γ (dsdθ) ρ(γ) )ρ(γ) 2 γ (Q-γ)
where R(γ) is the unit volume reflection coefficient and F is an arbitrary bounded measurable functional defined on the space of Radon measures equipped with the weak-topology.

In view of expression (2.8), it is quite natural to interpret R(γ) as the partition function of the unit area quantum sphere. Hence our paper suggests (and proves to the extent of Corollary 2.4) that the unit area quantum sphere is a universal feature of tails of GMC measures. Also, the above simple computation shows the equivalence between the reflection coefficient (or 2 point correlation function) constructed in [START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF] and the unit area quantum sphere (with 2 marked points). Let us mention that this equivalence [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]↔ [START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF] has been established in [START_REF] Aru | Two perspectives of the 2D unit area quantum sphere and their equivalence[END_REF] for quantum surfaces with 3 marked points.

Proof of Theorem 2.2.

Conventions and notations. We will denote |•| the norm in C of the standard Euclidean metric. For all r > 0 we will denote by B(x, r) (resp. C(x, r)) the Euclidean ball (resp. circle) of center x and radius r. We will write X Y when the random variables X and Y are independent.

3.1. Localization trick. The difficulty in computing right tails of random measures usually lies in the fact that the right tail should depend on the macroscopic shape of the random measure. An important observation in our argument is that tails of GMC measures depend only very softly on their global shape. It is based on the Girsanov transform which we apply in a way so as to localize the computations for the right tail of GMC measures around a fixed point. Girsanov's transform is a standard tool in the study of Gaussian processes (or even GMC measures) but, to the best of our knowledge, it has never been used to localize large mass effects of GMC measures. Anyway, this simple observation reduces drastically the difficulty of computing the tails.

P(M γ (O) > t) = O E[ 1 M γ (v, O) 1 {Mγ (v,O)>t} ]d 2 v (3.1)
where we have set

M γ (v, O) := O 1 |z -v| γ 2 M γ (d 2 z).
Proof. First write

P(M γ (O) > t) = E[ M γ (O) M γ (O) 1 {Mγ (O)>t} ] = O E[e γX(v)-γ 2 2 E[X(v) 2 ] 1 M γ (O) 1 {Mγ (O)>t} ] d 2 v. (3.2)
Girsanov's transform then asserts that weighting the probability law by e γX(v)-γ 2 2 E[X(v) 2 ] amounts to shifting the law of X by γ ln 1 |•-v| and our claim follows. Of course, as the field X is distributionvalued, it cannot be evaluated pointwise: the last equality in (3.2) thus requires a cut-off regularization of X(v) and then a harmless passage to the limit in the regularization parameter to be rigorously established (use the fact that sup >0 E[ 1 Mγ, (O) ] < +∞ where stands for the regularization parameter).

The important point is that the asymptotic behavior of the quantity

E[ 1 Mγ (v,O) 1 {Mγ (v,O)>t}
] is completely dominated by the behavior close to v because of the singularity 1 |z-v| γ 2 . So we will establish the following estimate, which will be enough to complete our argument.

× O → R such that ∀v ∈ O, E[ 1 M γ (v, O) 1 {Mγ (v,O)>t} ] = 2 γ (Q -γ) 2 γ (Q -γ) + 1
P(M γ (O) > t) = 2 γ (Q -γ) 2 γ (Q -γ) + 1 R(γ) t 4 γ 2 + 1 t 4 γ 2 +δ O (t, v) d 2 v.
Notice that since O has a C 1 boundary and sup v∈O,t

1 dist(v, O c ) α | (t, v)| < +∞, the term O (t, v) d 2 v
is indeed well defined. In fact, the term O (t, v) d 2 v converges to 0 as t goes to infinity by using the dominated convergence theorem and the fact that for all v one has lim t→∞ | (t, v)| = 0. This yields Theorem 2.2. The remaining part of this section is thus devoted to the proof of this lemma.

Remark 3.3. Let us stress here that the assumption that O has a C 1 boundary ensures that

O 1 dist(v,O c ) α d 2 v < ∞ for all α < 1. Therefore the term O (t, v) d 2 v
is well defined and converges to 0. This is the only place where the C 1 assumption is used and perhaps one could relax a bit the C 1 assumption by a finer analysis of the (t, v) term.

3.2.

GFF with vanishing mean over a circle. Given v ∈ C, we consider the GFF X v,r with vanishing mean over the circle C(v, r), namely a centered Gaussian random distribution with covariance for x, y ∈ D Let us denote by (X v,r u ) u 0 the circle average of this field

(3.3) E[X v,r (x)X v,r (y)
X v,r u = 1 2πi |w-v|=re -u X v,r (w) dw w -v .
A simple computation shows that (X v,r u ) u 0 is a standard Brownian motion starting from 0 independent of the sigma algebra σ{X v,r (z); z ∈ B(v, r) c }.

3.3.

Polar decomposition and the reflection coefficient. The asymptotic expansion in Lemma 3.2 will be determined by the behavior of the integral M γ (v, O) around the singularity at v. As already noticed in [START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF], an important ingredient in the analysis is the reflection coefficient. We consider the polar decomposition of the chaos measure. We have the following equality in the sense of distributions

X v,r (v + re -s e iθ ) = B s + Y (s, θ)
where B s is Brownian Motion starting form the origin at s = 0 and Y (s, θ) the independent centered Gaussian field with covariance

(3.4) E[Y (s, θ)Y (s , θ )] = ln e -s ∨ e -s |e -s e iθ -e -s e iθ | .
Then we get by the change of variables z = v + re -s e iθ5

(3.5)

B(v,r) e γX v,r (z)-γ 2 2 E[X v,r (z) 2 ] |z -v| γ 2 d 2 z law = r 2-γ 2 ∞ 0 e γ(Bs-(Q-γ)s) Z s ds
The following decomposition lemma due to Williams (see [START_REF] Williams | Path Decomposition and Continuity of Local Time for One-Dimensional Diffusions, I[END_REF]) will be useful in the study of M γ (v, B(v, r)): Lemma 3.4. Let (B s -νs) s 0 be a Brownian motion with negative drift, i.e. ν > 0 and let M = sup s (B s -νs). Then conditionally on M the law of the path (B s -νs) s 0 is given by the joining of two independent paths:

• A Brownian motion ((B 1 s + νs)) s τ M with positive drift ν > 0 run until its hitting time τ M of M . • (M + B 2
s -νs) s 0 where B 2 s -νs is a Brownian motion with negative drift conditioned to stay negative.

By the joining of two paths (X s ) s T (with T > 0) and ( Xs ) s 0 , we mean the path (X s 1 s T + Xs-T 1 s>T ) s 0 . Moreover, one has the following time reversal property for all C > 0 (where τ C denotes the hitting time of C)

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

Remark 3.5. As a consequence of the above lemma, one can also deduce that the process

( B L -C +s -ν(L -C + s) + C) s 0 is equal in distribution to ( B s -νs) s 0 .
We may apply Lemma 3.4 to (3.5). Let M = sup s 0 (B s -(Q -γ)s) and L -M be the last time (B γ s ) s 0 hits -M (recall that B γ s was defined in (1.7)). Then where we used stationarity of the process Z s (and independence of Z s and B s ). The distribution of M is well known (see section 3.5.C in the textbook [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] for instance):

(3.7) P(e γM > t) = 1 t 2(Q-γ) γ , t 1.
Now, notice that R(γ) appears as the coefficient of the tail of the random variable e γM ∞ -∞ e γB γ s Z s ds

∀η > 0, P e γM ∞ -∞ e γB γ s Z s ds > t = R(γ)t -2 γ (Q-γ) + o(t -2 γ (Q-γ)-(1-η) )
Indeed, setting Z := ∞ -∞ e γB γ s Z s ds, we have because of (3.7)

P e γM Z > t =P e γM Z > t, Z t + P e γM Z > t, Z > t =E[Z 2 γ (Q-γ) 1 Z t ]t -2 γ (Q-γ) + P e γM Z > t, Z > t = R(γ)t -2 γ (Q-γ) + E[Z 2 γ (Q-γ) 1 Z>t ]t -2 γ (Q-γ) + P e γM Z > t, Z > t
Now, using (1.11) and Markov's inequality, the two correction terms are easily seen to be o(t

-2 γ (Q-γ)-(1-η) ).
imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018

3.4. Decomposition of the GMC around v. We fix v ∈ O and set r = dist(v, ∂O). Our purpose now is to replace the field X in the definition of M γ (v, B(v, r)) by the field X v,r in order to use the polar decomposition of the GMC measure in B(v, r) detailed in the previous subsection. It suffices to subtract the mean value of X along the circle C(v, r). Therefore we introduce

N v,r = 1 2πi |w-v|=r X(w) dw w -v .
This is a centered Gaussian variable with variance

E[N 2 v,r ] = -ln r.
We introduce the field X = X -N v,r , which has the same law as X v,r . One can notice that

E[ X(x)N v,r ] = 0
for all |x -v| r hence X is independent from N v,r in this region. Therefore, using the decomposition in distribution obtained in subsection 3.3

(3.8) M γ (v, B(v, r)) = r 2-γ 2 e γM E r,v I γ (M )
where we have set

E r,v := e γNv,r-γ 2 2 E[N 2 v,r ] , I γ (M ) := ∞ -L -M e γB γ s Z s ds with the convention that I γ (∞) = ∞
-∞ e γB γ s Z s ds. We will constantly use in the sequel the fact that x → I γ (x) is an increasing function.

Getting rid of the non-singularity. Now we argue that the behaviour of

E[ 1 Mγ (v,O) 1 {Mγ (v,O)>t} ] is completely determined by that of E[ 1 Mγ (v,B(v,r)) 1 {Mγ (v,B(v,r))>t}
]. We will show this claim by giving an upper/lower bound of

E[ 1 Mγ (v,O) 1 {Mγ (v,O)>t} ] up to o(t -4 γ 2 -δ ) for any δ ∈ (0, 1+p 0 -4 γ 2 2+p 0 ). We introduce the notation A := M γ (v, B(v, r) c ∩ O).
A key lemma in the argument is the following. Lemma 3.6. For all p ∈ (0, 4 γ 2 ) there is some constant

C p > 0, ∀v ∈ O, E[A p ] C p ×      1 if p ∈ (0, 4 γ 2 -1) (ln 1 r ) p∨1 if p = 4 γ 2 -1 r ψ(p) if 4 γ 2 -1 < p < 4 γ 2 with ψ(p) = (2 -γ 2 2 )p -γ 2 2 p 2 and r = dist(v, O c ).
Remark 3.7. Notice that ψ(p) > 0 for p ∈ (0, 4/γ 2 -1). The value of p 0 in (2.3) is determined in such a way that ψ(p 0 ) = -1. This ensures that ψ(p) ∈ (-1, 0) for p ∈ (4/γ 2 -1, p 0 ). Our proof will show that the exponent α appearing in Lemma 3.2 will actually be given by ψ(p) for some p ∈ (4/γ 2 -1, p 0 ).

Proof. If p > 0 observe that ψ(p) > 0 if and only if p < 2 γ (Q -γ) = 4 γ 2 -1.
In this case, the result is a simple consequence of Lemma A.1 in [START_REF] David | Liouville Quantum Gravity on the Riemann sphere[END_REF]. Now we study the case p

4 γ 2 -1. Set B n := {x ∈ D; 2 -n-1 |x-v| 2 -n } and A n = M γ (0, B n ) p .
Assume p > 1 (if p 1 use sub-additivity of the mapping x → x p in the next argument). Using in turn Minkowski's inequality and invariance under translations we have

E[A p ] 1/p -ln r ln 2 n=0 E[A p n ] 1/p + E[M γ (O) p ] 1/p .
It is standard fact in GMC theory that A n scales as

E[A p n ] = 2 -nψ(p) E[A p 1 ]
(see the review [START_REF] Rhodes | Gaussian multiplicative chaos and applications: a review[END_REF] for instance). We deduce that

E[A p ] 1/p C p -ln r ln 2 n=0 2 -nψ(p)/p .
Since ψ(p) 0, we get our claim. Now we first give the upper bound. Fix δ ∈ (0,

p 0 +1- 4 γ 2 2+p 0 ). We can find η > δ such that (3.9) (1 -η)(1 + p 0 ) > 4 γ 2 + δ, 1 + p 0 > 4 γ 2 + η.
Hence we can choose 4 γ 2 < p < p 0 + 1 such that (1 -η)p > 4 γ 2 + δ.

E[ 1 M γ (v, O) 1 {Mγ (v,O)>t} ] E[ 1 M γ (v, B(v, r)) + A 1 {Mγ (v,B(v,r))>t-t 1-η } ] + E[ 1 A 1 {A>t 1-η } ] E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t-t 1-η } ] + E[A p-1 ]t -(1-η)p . (3.10)
Notice that the expectation E[A p-1 ] can be analyzed with Lemma 3.6 and gives E[A p-1 ] C p r ψ(p-1) with ψ(p -1) ∈ (-1, 0).

For the lower bound we first restrict to the set {A < t 1-η } to get where we have used the inequality (1 + u) -1 1 -u. The last of the three terms

E[ 1 M γ (v, B(v, r)) + A 1 {Mγ (v,B(v,r))+A>t} ] E[ 1 M γ (v, B(v, r)) + t 1-η 1 {Mγ (v,B(v,r))>t,A<t 1-η } ] =E[ 1 M γ (v, B(v, r)) + t 1-η 1 {Mγ (v,B(v,r))>t} ] -E[ 1 M γ (v, B(v, r)) + t 1-η 1 {Mγ (v,B(v,r))>t,A>t 1-η } ] E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t} ] -t 1-η E[ 1 M γ (v, B(v, r)) 2 1 {Mγ (v,B(v,r))>t} ] -E[ 1 M γ (v, B(v, r)) + t 1-η 1 {Mγ (v,B(v,r))>t,A>t 1-η } ] (1 - 1 t η )E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t} ] -E[ 1 M γ (v, B(v, r)) + t 1-η 1 {Mγ (v,B(v,r))>t,A>t 1-η } ] (3.
E[ 1 M γ (v, B(v, r)) + t 1-η 1 {Mγ (v,B(v,r))>t,A>t 1-η } ]
is less than t -(1-η) P(A > t 1-η ) and by the Markov inequality,

t -(1-η) P(A > t 1-η ) t -(1-η)p E[A p-1 ] C t -(1-η)p r ψ(p-1)
with ψ(p -1) ∈ (-1, 0). The second will be treated in the same time as the first (next subsection) and will be proved to be O(t

-4 γ 2 -η ), hence o(t -4 γ 2 -δ ).
3.6. Behaviour near the singularity. In view of the bounds (3.10) and (3.11), it remains to study the term

E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t} ].
We will proceed by using decomposition (3.8) and establishing lower/upper bounds.

Lower bound. In the study of the lower bound, let us fix η such that η > δ and η satisfies (3.9) in such a way that we can find 1 < p < 1 + p 0 with p(1 -η) > 4 γ 2 + δ. We will introduce the event {γM > η ln t}. On this event, one has I γ (M ) I γ ( η γ ln t) and therefore

E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t} ] = E[ 1 r 2-γ 2 E r,v e γM I γ (M ) 1 {r 2-γ 2 Er,ve γM Iγ (M )>t} ] E[ 1 r 2-γ 2 E r,v e γM I γ (∞) 1 {r 2-γ 2 Er,ve γM Iγ ( η γ ln t)>t} 1 {γM >η ln t} ] = E[ 1 r 2-γ 2 E r,v e γM I γ (∞) 1 {r 2-γ 2 Er,ve γM Iγ ( η γ ln t)>t} ] -E[ 1 r 2-γ 2 E r,v e γM I γ (∞) 1 {r 2-γ 2 Er,ve γM Iγ ( η γ ln t)>t} 1 {γM <η ln t} ]
The correction term (second expectation in final line above) is r -α o(t

-4 γ 2 -δ ) with α < 1.
Indeed on the event {γM < η ln t}, the event

{r 2-γ 2 E r,v e γM I γ ( η γ ln t) > t} is contained in the event {r 2-γ 2 E r,v I γ (∞) > t 1-η }. Hence the correction term is less than E[(r 2-γ 2 E r,v ) p-1 ]t -(1-η)p E[I γ (∞) p-1 ] = r ψ(p-1) t -(1-η)p E[I γ (∞) p-1 ]
, where the function ψ has been defined in Lemma 3.6.

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Then we average with respect to M to get (recall that the distribution of M is given by (3.7))

E[ 1 r 2-γ 2 E r,v e γM I γ (∞) 1 {r 2-γ 2 Er,ve γM Iγ ( η γ ln t)>t} ] E[ 1 r 2-γ 2 E r,v e γM I γ (∞) 1 {r 2-γ 2 Er,ve γM Iγ ( η γ ln t)>t} 1 {r 2-γ 2 Er,vIγ (∞) t} ] = 2(Q -γ) 2(Q -γ) + γ E[ r 2-γ 2 E r,v 2 γ (Q-γ) I γ ( η γ ln t) 4 γ 2 I γ (∞) 1 {r 2-γ 2 Er,vIγ (∞) t} ]t -4 γ 2 = 2(Q -γ) 2(Q -γ) + γ E I γ ( η γ ln t) 4 γ 2 I γ (∞) t -4 γ 2 - 2(Q -γ) 2(Q -γ) + γ E[ r 2-γ 2 E r,v 2 γ (Q-γ) I γ ( η γ ln t) 4 γ 2 I γ (∞) 1 {r 2-γ 2 Er,vIγ (∞)>t} ]t -4 γ 2 .
Above we have used the fact

E[ r 2-γ 2 E r,v 2 γ (Q-γ) ] = 1. One has E[ r 2-γ 2 E r,v 2 γ (Q-γ) I γ ( η γ ln t) 4 γ 2 I γ (∞) 1 {r 2-γ 2 Er,vIγ (∞)>t} ]t -4 γ 2 E[ r 2-γ 2 E r,v 4 γ 2 -1+η I γ (∞) 4 γ 2 -1+η ]t -4 γ 2 -η = r ψ( 4 γ 2 -1+η) E[I γ (∞) 4 γ 2 -1+η ]t -4 γ 2 -η
where 4 γ 2 -1 + η < p 0 by condition (3.9). Hence it remains to show that E Iγ ( η γ ln t) + η γ ln t is independent from everything and distributed like (B γ s ) s 0 . We can then write

∞ -∞ e γB γ s Z s ds = I γ ( η γ ln t) + t -η B with B = 0 -∞ e γ Bγ s Z s-L -η γ ln t ds. We set m = 4 γ 2 .
Then we have (use the triangle inequality to get the third line below)

R(γ) -E[I γ ( η γ ln t) m /I γ (∞)] =E[ I γ ( η γ ln t) + t -η B (I γ ( η γ ln t) + t -η B) 1/m m ] -E[ I γ ( η γ ln t) (I γ ( η γ ln t) + t -η B) 1/m m ] E[ I γ ( η γ ln t) (I γ ( η γ ln t) + t -η B) 1/m m ] 1/m + E[ t -η B (I γ ( η γ ln t) + t -η B) 1/m m ] 1/m m -E[ I γ ( η γ ln t) (I γ ( η γ ln t) + t -η B)
R(γ) -E[I γ ( η γ ln t) m /I γ (∞)] Ct -η E[ I γ ( η γ ln t) (I γ ( η γ ln t) + t -η B) 1/m m ] 1-1/m E[ B (I γ ( η γ ln t) + t -η B) 1/m m ] 1/m .
In the above expression the first expectation is less than R(γ) 1-γ 2 /4 . We aim to bound the second one. Let us fix η ∈ (0, 1). Then

E[ B (I γ ( η γ ln t) + t -η B) 1/m m ] 1/m E[ B m I γ (0) 1-η (t -η B) η ] 1/m t η η m E[B m-η I γ (0) -1+η ] 1/m .
Notice that the last expectation is finite (use Hölder inequality and the fact that B has finite moments of order q < m and I γ (0) has finite negative moments of all order). This proves our claim for the lower bound provided that we choose η small enough so as to make η(1 -η m ) > δ.

Upper bound. For the upper bound, we use again the decomposition (3.8)

E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t} ] = E[ 1 r 2-γ 2 E r,v e γM I γ (M ) 1 {r 2-γ 2 Er,ve γM Iγ (M )>t} ]
We want to replace the term I γ (M ) in the fraction by I γ (∞). Hence we write

E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t} ] = E[ 1 r 2-γ 2 E r,v e γM I γ (∞) 1 {r 2-γ 2 Er,ve γM Iγ (M )>t} ] + C(t)
where C(t) stands for the cost for this replacement Now we observe that, under the condition δ ∈ (0,

C(t) := E[ 1 r 2-γ 2 E r,v e γM 1 I γ (M ) - 1 I γ (∞) 1 {r 2-γ
p 0 +1- 4 γ 2
2+p 0 ) one can find η ∈ (0, 1) and p ∈ (0, 1+p 0 ) such that (3.13) p 2+p 0 . Notice that the η introduced in the proof of the upper bound is not the same as the η introduced in the proof of the lower bound. We are going to evaluate the cost term on different possible event,

i.e we introduce (3.14)

C i (t) := E[ " 1 A i ]
where " stands for the integrand inside the expectation in (3.12) and the event A i (i = 1, 2) ranges respectively among the two events {M > η γ ln t} and {M η γ ln t}.

Let us start with C 1 (t), which can be estimated by

C 1 (t) =E[ 1 r 2-γ 2 E r,v e γM I γ (∞) -I γ (M ) I γ (M )I γ (∞) 1 {r 2-γ 2 Er,ve γM Iγ (M )>t,M > η γ ln t} ] E[ 1 e 2γM B I γ (M ) 2 F t e γM I γ (M ) 1 {M > η γ ln t} ]
where we have introduced the function

(3.15) F (u) = E[ 1 r 2-γ 2 E r,v 1 {r 2-γ 2 Er,v>u} ].
The above expression for the cost will be analyzed according to different possible regimes of the function F , depending on the possible values of its argument t e γM Iγ (M ) . For this, observe that 1

< 2γQ(1 - √ 2
Q ), hence we can choose 0 < q < 2 and 0 < a < 1 such that

(3.16) 1 < aqγQ and 0 < a < 1 - √ 2 Q .
Now we restrict the cost C 1 (t) further to the events {tr -aγQ e γM I γ (M )} and then {tr -aγQ > e γM I γ (M )}, producing two quantities that we respectively call C 1 1 (t) and C 2 1 (t). Concerning C 1 1 (t), on the event {tr -aγQ e γM I γ (M )}, we can use the rough estimate F (u) r -2 (obtained by using the fact that indicator functions are bounded by 1 in (3.15)) to deduce that for any 0 < q < 2 We have the obvious bound for p ∈ (0, 4 γ 2 )

C 1 1 (t) r -2 E[ B (e γM I γ (M )) 2 1 {M > η γ ln t} 1 {tr -aγQ <e γM Iγ (M )} ] =r qaγQ-2 t -q E[ B (e γM I γ (M )) 2-q 1 {M > η γ ln t} 1 {tr -aγQ <e γM Iγ (M )} ] Cr qaγQ-2 t -q E[ 1 (e γM ) 2-q 1 {M > η γ ln t} ]. ( 3
E[A p ] E[S p ]E[A p 0 ].
The assumption of f being Hölder on Ō ensures that E[S p ] is finite by standard arguments for the supremum of Gaussian processes (see lecture 6 in [START_REF] Biskup | Extrema of the two-dimensional Discrete Gaussian Free Field[END_REF] for example). Hence A satisfies the same bounds as Lemma 3.6 and the second term in (3.21) is again of the form (r ) ψ(p-1) o(t

-4 γ 2 -δ
). The first term in (3.21) concerning the behaviour near the singularity at v is bounded as in subsection 3.6 which states that

E[ 1 M 0 γ (v, B(v, r )) 1 {M 0 γ (v,B(v,r ))>t-t 1-η } ] 2(Q -γ) 2(Q -γ) + γ R(γ)t -4 γ 2 + (r ) -α o(t -4 γ 2 -δ )
for some α ∈ (0, 1). By independence of S r (v) and conditioning on S r (v), we deduce

E[ 1 M 0 γ (v, B(v, r )) 1 {S r (v)M 0 γ (v,B(v,r ))>t-t 1-η } ] 2(Q -γ) 2(Q -γ) + γ R(γ)t -4 γ 2 E[(S r (v)) 4 γ 2 ]+(r ) -α o(t -4 γ 2 -δ ).
Integrating this relation over O we deduce from (3.18) that lim sup

t→∞ t 4 γ 2 P(M γ (O) > t) O E[(S r (v)) 4 γ 2 ] dv 2(Q -γ) 2(Q -γ) + γ R(γ).
This bound is valid for arbitrary (recall that r = min(r, )). So we want to let → 0. An easy application of the dominated convergence theorem ensures that lim

→0 O E[(S r (v)) 4 γ 2 ] dv = O E[(e γG(v)-γ 2 2 E[G(v) 2 ] ) 4 γ 2 ] dv = O e 4 γ (Q-γ)f (v,v) dv. Hence lim sup t→∞ t 4 γ 2 P(M γ (O) > t) O e 4 γ (Q-γ)f (v,v) dv 2(Q -γ) 2(Q -γ) + γ R(γ).
This shows the upper bound. The lower bound is established in the same way (using (3.20)).

4. Extensions to other cases. In this section, we explain how to generalize our results to the other cases. Though the claims of this section cannot be taken for granted (due to the fact that it sweeps under the rug potential technical difficulties), it provides nonetheless arguments which we believe are convincing to tackle the other cases. with respect to the measure d 2 v. The advantage of this localization trick is that the tail of (4.1) is completely concentrated around v, hence is determined by a local analysis which is more transparent than a direct study of M γ (O). For r small, the variable X has roughly covariance of the form ln 1 |x-y| + f (v, v) and therefore X ≈ X + N (in law) where X has ln 1 |x-y| covariance and N is a centered Gaussian independent from everything and with variance f (v, v). Combining this decomposition with our estimates for X, one expects that Gathering the above considerations, we expect the following generalization of Theorem 2.2 for kernels of type (1.2)

(4.3) P(M γ (O) > t) = O e 4 γ (Q-γ)f (v,v) d 2 v 2 γ (Q -γ) 2 γ (Q -γ) + 1 R(γ) 1 t 4 γ 2 + o(t -4 γ 2 ).
We leave open the determination of bounds on the o(t

-4 γ 2 ) term.
4.2. The other dimensions. In higher dimensions, we expect the method to work as well by decomposing the log-correlated field into a radial part (Brownian motion) and an independent radial part around each localization point v. The constant in the expansion will then be given by explicit terms times the d-dimensional analog Rd (γ) of the reflection coefficient R(γ) defined by (1.10). The question is then to know if we can compute explicitly this expectation depending on d; presently, getting explicit formulas in dimension d 3 seems out of reach since one can not rely on the powerful framework of 2d conformal field theory for the GFF.

Like in dimension 2, we also have an explicit expression for R1 (γ) in dimension 1. Indeed, one has the following expression for R1 (γ) (the so-called boundary unit volume reflection coefficient in the terminology of Liouville field theory):

R1 (γ) = E[ ∞ -∞ e γB 1,γ s e γY (s)-γ 2 E[Y (s) 2 ] 2 Z s ds 2 γ (Q 1 -γ) ]
where B 1,γ s is defined like B γ s in (1.7) with Q replaced by Q 1 = γ 2 + 1 γ and Y s is the restriction to the real line of the centered Gaussian field with covariance (3.4). This yields the asymptotic

P(M γ (O) > t) = O e 2 γ (Q 1 -γ)f (v,v) d 2 v (1 -γ 2 2 )
R1 (γ)

t 2 γ 2 + o(t -2 γ 2 ).
The recent integrability results of Rémy for GMC on the circle [START_REF] Rémy | The Fyodorov-Bouchaud formula and Liouville Conformal Field theory[END_REF] allows us to compute explicitly the tail of M γ (O) in the case when X is the circular logarithmic noise and O = (0, 2π); as a matter of fact, the result of Rémy is much more precise since it gives the precise density of the total mass of GMC on the circle (this density was conjectured in the physics literature in 2008 by Fyodorov-Bouchaud [START_REF] Fyodorov | Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential[END_REF]; see also a similar conjecture in [START_REF] Fyodorov | Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of 1/f Noises generated by Gaussian Free Fields[END_REF] for the case of the unit interval). This leads to the following explicit expression for R1 (γ)

R1 (γ) = (2π) 2 γ (Q 1 -γ) (1 -γ 2 2 )Γ(1 -γ 2 2 ) 2 γ 2
.

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CONTENTS 1

 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Liouville reflection coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Setup and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conventions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Localization trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 GFF with vanishing mean over a circle . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Polar decomposition and the reflection coefficient . . . . . . . . . . . . . . . . . . . 3.4 Decomposition of the GMC around v . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Getting rid of the non-singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Behaviour near the singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Extensions to other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The 2d case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The other dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Author's addresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Remark 2 . 5 .

 25 One can apply Corollary 2.4 with f (x, y) = ln 1 |1-xȳ| to get the right tail of GMC based on the GFF X with Neumann boundary condition in D, which has covariance E[X(x)X(y)] = ln 1 |x-y||1-xȳ| . Indeed f (x, y) = ln 1 |1-xȳ| is positive definite 3 and locally Hölder in D.

Lemma 3 . 1 .

 31 (Localization trick) Let O be an open set. We have

Lemma 3 . 2 .- 4 γ 2 2+p 0 )

 32420 For any δ ∈ (0, 1+p 0 and for all open sets O ⊂ D with C 1 boundary, there exists some function : R +

  r where we use the notation |z| + = |z| if |z| 1 and |z| + = 1 if |z| 1.

  11) imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018

  -η)) to get the desired lower bound. By Remark 3.5, the process Bγ s defined for s 0 by the relation Bγs = B γ s-L -η γ ln t

2 4 γ 2

 242 Er,ve γM Iγ (M )>t} ].(3.12) Now we establish that this cost satisfies C(t) = r -α o(t --δ ) with α < 1. By Remark 3.5, conditionally on M , the process Bγ s defined for s 0 by the relation Bγ s = B γ s-L -M + M is independent from everything and distributed like (B γ s ) s 0 . We can then write ∞ -∞ e γB γ s Z s ds = I γ (M ) + e -γM B Bγ s Z s-L -M ds.

4. 1 .

 1 The 2d case. Consider now the general case in 2d of a log-correlated kernel of the type (1.2). Along the same lines as for the proof of Theorem (2.2) the localization trick allows us to trade the study of the tail of the random variable M γ (O) for the study of the tail of the singular integral (for some r > 0)(4.1) M γ (v, B(v, r)) := B(v,r) e γ 2 f (v,z) e γX(z)-γ 2 2 E[X(z) 2 ] |z -v| γ 2 d 2 z for each point v ∈ O (f appears in (1.2)). The full tail of M γ (O) is obtained by integration of the term E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t} ]imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018

(4. 2 ) t 4 γ 2 2 γ (Q -γ) 2 γ (Q -γ) + 1 R(γ)e 4 γ

 2422214 E[ 1 M γ (v, B(v, r)) 1 {Mγ (v,B(v,r))>t} ] → t→∞ (Q-γ)f (v,v) .

  . Plugging this estimate in the above expression yields imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018

			m
			1/m	].
	This expression is of the form (α t + β t ) m -α m t with βt αt us consider another constant C such that (1 + x) m -1	c for some c > 0 and all t Cx for 0 x c. We deduce that 0. Let
	(α t + β t ) m -α m t	Cα m-1

t β t

  Indeed this condition is equivalent to the set of conditions (1+ p 0 )(1 -η) + 4 γ 2 η > δ + 4 γ 2 and η(1 + 4 γ 2 ) > δ + 4 γ 2 ,which are in turn equivalent to η < 1 -

					δ 1+p 0 -4 γ 2	and η(1 + 4 γ 2 ) > δ + 4 γ 2 . This is
	possible if and only if (1 -	δ 1+p 0 -4 γ 2	)(1 + 4 γ 2 ) > δ + 4 γ 2 , which produces our condition δ <	1+p 0 -4 γ 2
	(1 -η) +	4 γ 2 η > δ +	4 γ 2 and η(1 +	4 γ 2 ) > δ +	4 γ 2 .

Our techniques could perhaps handle more general cases but for the sake of simplicity we consider only the case of a smooth boundary: see Remark

3.3. imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018

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The fact that f is positive definite can be seen after a series expansion of the ln.imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018

see[START_REF] Duplantier | Sheffield: Liouville quantum gravity as mating of trees[END_REF] for the definition of quantum surfaces. Here, we choose a parameterisation of the unit area quantum sphere such that the radial part is maximal at 0. imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018

Observe that the measures involved in this relation are defined through a limiting procedure. Thus one needs to apply the change of variables in the regularized measures and then pass to the limit to obtain the relation.imsart-aop ver. 2014/10/16 file: Tail-short_revisionAOP_finalversion.tex date: Thursday 20 th December, 2018

Acknowledgements. We thank the excellent restaurant Le Pelagos on the island of Porquerolles where this work was initiated.

Research supported in part by ANR grant Liouville (ANR-15-CE40-0013). MSC 2010

Above we have used the fact that E[ B Iγ (M ) q-2 |M ] can be bounded by constant. Indeed, we use Hölder's inequality with conjugate exponents m, m and B M as well as I γ (0) M to get the fact that

provided that m is chosen smaller than 4 γ 2 (recall that I γ (0) has negative moments of all order: see [START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ Formula[END_REF]). Now we can use the explicit exponential law for M to get that (3.17) is less than

, which is as expected thanks to condition (3.13).

Concerning C 2 1 (t), by using the Girsanov transform with the term e -γNv,r-

, we get for some standard Gaussian random variable Z and for ur -aγQ > 1

Then, using this estimate, C 2 1 (t) is seen to be less than

which is less than Cr

γ 2 )η using the explicit exponential distribution (3.7) of M as in (3.17). Again, conditions (3.16) and (3.13) ensure respectively that the r-exponent and texponent behave as expected. This concludes the case of C 1 (t).

Now we analyze

where we have used that almost surely

1. Hence condition (3.13) ensures that the cost term is r ψ(p-1) o(t

To conclude it suffices to bound the term where o(t

The last line is obtained by independence of E r,v , M , I γ (∞) and the explicit tail of the exponential distribution. Our claim follows.

Proof of Corollary 2. [START_REF] Biskup | Extrema of the two-dimensional Discrete Gaussian Free Field[END_REF]. In what follows, the Gaussian process X 0 stands for the GFF inside D with vanishing mean on the unit circle, i.e. with covariance structure given by (2.1) whereas G is an independent centered Gaussian field in D such that

with f the function appearing in (1.2). Then we may assume that X = X 0 + G. Now we follow and adapt the proof of Theorem 2.2. For this, we set

. The localization trick then yields

and Girsanov's transform asserts that weighting the probability law by

where we have set

Let us set r = dist(v, ∂O) and r = min(r, ), where > 0 is a regularization parameter which will be sent to 0 in the end. Sticking to the notations of subsection 3.5, we set

)) + A and similarly for the corresponding items with index 0. Our next step will be to use the computations already done for items with index 0 in the proof of Theorem 2.2 and compare with items M γ (v, B(v, r )) and A. The difference between these quantities involves the process G, which will be estimated in terms of the quantities

Then we can reproduce the argument (3.10) for the upper bound