Patrick Stella
email: patrick.stella@agroparistech.fr

Erwan Personne

Effects of conventional, extensive and semi-intensive green roofs on building conductive heat fluxes and surface temperatures in winter in Paris

Keywords: Green roofs, building insulation, urban heat island, conductive heat fluxes, winter conditions

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

-INTRODUCTION

Urban areas and land-use changes lead to severe environmental issues due to urbanization. On the one hand, urban areas are strong contributors to greenhouse gas (GHG) emissions at the global scale and therefore to global warming. It is estimated that buildings are responsible for 19% of all global 2010 energy-related GHG emissions, mainly indirect ones from electricity use. Yet, buildings account for 32% of total global final energy use, space heating and cooling representing more than one third of total building final energy consumption [START_REF]Climate Change 2014: Mitigation of Climate Change. Contribution of Working Group III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF]. On the other hand, at the local scale the land-use modifications due to urbanization alter the urban microclimate and induce the so-called "urban heat island" (UHI) effect [START_REF] Balchin | A micro-climatological investigation of Bath and the surrounding district[END_REF][START_REF] Oke | The energetic basis of the urban heat island[END_REF] which reflects the fact that cities are warmer than their surroundings. It has numerous impacts on building energy consumption (e.g., [START_REF] Akbari | Cool surfaces and shade trees to reduce energy use and improve air quality in urban areas[END_REF][START_REF] Li | Interaction between urban microclimate and electric air-conditioning energy consumption during high temperature season[END_REF]), citizen comfort and health (e.g., [START_REF] Vandentorren | Mortality in 13 French cities during the August 2003 heat wave[END_REF][START_REF] Duneier | Ethnography, the ecological fallacy, and the 1995 Chicago heat wave[END_REF][START_REF] Harlan | Climate change and health in cities: impacts of heat and air pollution and potential co-benefits from mitigation and adaptation[END_REF]), and urban air quality (e.g., [START_REF] Akbari | Cool surfaces and shade trees to reduce energy use and improve air quality in urban areas[END_REF][START_REF] Rosenfeld | Cool communities: Strategies for heat island mitigation and smog reduction[END_REF][START_REF] Sarrat | Impact of urban heat island on regional atmospheric pollution[END_REF]). This phenomenon originates from the alteration of radiative budget and energy balance, in which the heat released by anthropogenic activities is a strong contributor to UHI effect. Indeed, anthropogenic heat release occurs mainly through building cooling and heating (e.g., [START_REF] Pigeon | Anthropogenic heat release in an old European agglomeration (Toulouse, France)[END_REF]) and is responsible for an increase in urban air temperatures between 0.2 and 2.5°C (e.g., [START_REF] Wen | Influence of air conditioners utilization on urban thermal environment[END_REF][START_REF] De Munck | How much can air conditioning increase air temperatures for a city like Paris, France?[END_REF]). Within this context, the building insulation assessment allows (i) to mitigate UHI effect (through the decrease of anthropogenic heat release) and its related deleterious effects at the local scale, (ii) subsequent monetary savings, and (iii) to reduce the GHG emissions at the global scale [START_REF]Climate Change 2014: Mitigation of Climate Change. Contribution of Working Group III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF].

Among the numerous techniques to improve building insulation (e.g., insulation layer depth, new insulation materials) (e.g., [START_REF] Asdrubali | A review of unconventional sustainable building insulation materials[END_REF][START_REF] Schiavoni | Insulation materials for the building sector: A review and comparative analysis[END_REF][START_REF] Prakash | A review of heat dissipating passive cooling techniques for residential buildings at tropical region[END_REF][START_REF] Mirabella | Strategies to improve the energy performance of buildings: a review of their life cycle impact[END_REF]), green roofs currently receive strong attention.

Additionally to building insulation, they contribute to the assessment of urban air quality (e.g., [START_REF] Yang | Quantifying air pollution removal by green roofs in Chicago[END_REF][START_REF] Li | Green roofs against pollution and climate change. A review[END_REF]), the water retention to prevent runoff events (e.g., [START_REF] Li | Green roofs against pollution and climate change. A review[END_REF][START_REF] Dunnett | Influence of vegetation composition on runoff in two simulated green roof experiments[END_REF][START_REF] Cristiano | The role of green roofs in urban Water-Energy-Food-Ecosystem nexus: A review[END_REF]), sound reduction and insulation, ecological preservation (e.g., [START_REF] Cristiano | The role of green roofs in urban Water-Energy-Food-Ecosystem nexus: A review[END_REF][START_REF] Berardi | State-of-the-art analysis of the environmental benefits of green roofs[END_REF]), and last but not least direct UHI mitigation owing to their capability to refresh air throughout evapotranspiration (e.g., [START_REF] Peng | Green-roof effects on neighborhood microclimate and human thermal sensation[END_REF][START_REF] De Munck | Evaluating the impacts of greening scenarios on thermal comfort and energy and water consumptions for adapting Paris city to climate change[END_REF]).

Green roofs i.e., vegetated systems covering a building rooftop, are typically composed by some layers aiming to protect the building envelope (i.e., waterproofing membrane, root barrier, drainage layer), a growing medium (or substrate) layer, and a vegetation layer. It is distinguished the extensive, semi-intensive, and intensive green roofs, each characterized by the substrate thickness and plant communities [START_REF] Raji | The impact of greening systems on building energy performance: A literature review[END_REF]. Owing to plant shading, evaporative cooling, and additional insulation, green roofs help to reduce directly building energy consumption but also indirectly through UHI mitigation. However, the thermal performance of green roofs is highly variable according to the substrate and vegetation characteristics, building characteristics, and local climatic conditions. (e.g., [START_REF] Niachou | Analysis of the green roof thermal properties and investigation of its energy performance[END_REF][START_REF] Wong | Investigation of thermal benefits of rooftop garden in the tropical environment[END_REF][START_REF] Jaffal | A comprehensive study of the impact of green roofs on building energy performance[END_REF][START_REF] Cook | Towards a performance-based approach for multifunctional green roofs: An interdisciplinary review[END_REF][START_REF] Maiolo | The role of the extensive green roofs on decreasing building energy consumption in the Mediterranean climate[END_REF]).

Many studies focused their attention on the impacts of green roofs on building insulation and building thermal performance, as well as on UHI mitigation, under summer conditions (e.g., [START_REF] De Munck | Evaluating the impacts of greening scenarios on thermal comfort and energy and water consumptions for adapting Paris city to climate change[END_REF][START_REF] Jaffal | A comprehensive study of the impact of green roofs on building energy performance[END_REF][START_REF] Kumar | Performance evaluation of green roof and shading for thermal protection of buildings[END_REF][START_REF] Castleton | Green roofs, building energy savings and the potential for retrofit[END_REF][START_REF] Qiu | Effects of evapotranspiration on mitigation of urban temperature by vegetation and urban agriculture[END_REF][START_REF] Eksi | Effect of substrate depth, vegetation type, and season on green roof thermal properties[END_REF][START_REF] Yang | Evaluation of thermal performance of green roofs via field measurements and hygrothermal simulations[END_REF]). Overall, it is consensual that green roofs reduce surface and air temperatures, diminish heat fluxes entering inside the building, and therefore decrease energy consumption for cooling. However, the impact of green roofs under winter conditions received less attention, and is still under debate. For instance, Santamouris et al. [START_REF] Santamouris | Investigating and analysing the energy and environmental performance of an experimental green roof system installed in a nursery school building in Athens, Greece[END_REF] did not found any significant effect of green roof on heating load variation under winter Mediterranean climate, while Jaffal et al. [START_REF] Jaffal | A comprehensive study of the impact of green roofs on building energy performance[END_REF] reported positive impact of extensive green roofs under temperate oceanic climatic conditions. Yet, during heating period, Coma et al. [START_REF] Coma | Thermal assessment of extensive green roofs as passive tool for energy savings in buildings[END_REF] reported a negative impact of extensive green roof on building energy consumption under continental climate conditions, whereas under similar climatic conditions Lundholm et al. [START_REF] Lundholm | Snow depth and vegetation type affect green roof thermal performance in winter[END_REF] found lower net heat losses from extensive green compared to conventional roofs. Moreover, experimental studies carried out under winter conditions mainly considered only one kind of green roofs, often extensive ones [START_REF] Coma | Thermal assessment of extensive green roofs as passive tool for energy savings in buildings[END_REF][START_REF] Lundholm | Snow depth and vegetation type affect green roof thermal performance in winter[END_REF][START_REF] Lazzarin | Experimental measurements and numerical modelling of a green roof[END_REF][START_REF] Teemusk | Greenroof potential to reduce temperature fluctuations of a roof membrane: A case study from Estonia[END_REF][START_REF] Getter | Seasonal heat flux properties of an extensive green roof in Midwestern U.S. climate[END_REF][START_REF] Orazio | Green roof yearly performance: A case study in a highly insulated building under temperate climate[END_REF][START_REF] Zhao | Assessment of a green roof performance for sustainable buildings under winter weather conditions[END_REF][START_REF] Silva | Green roofs energy performance in Mediterranean climate[END_REF][START_REF] Squier | Heat flux and seasonal thermal performance of an extensive green roof[END_REF][START_REF] Collins | Thermal behavior of green roofs under Nordic winter conditions[END_REF], without comparing the extensive, semi-intensive, and intensive green roofs together.

Last, considering that winter UHI would be positive concerning building energy consumption, in particular by reducing the indoor and outdoor temperature difference, surface (and consequently air) cooling by green roofs would be deleterious. This issue has not been explored yet.

The objectives of this study are (i) to analyze how green roofs affect winter thermal fluxes reaching the building envelope, (ii) to determine their impact on winter surface UHI, and (iii) to compare extensive and semi-intensive green roofs under winter oceanic climatic conditions, from an experimental approach.

-MATERIALS AND METHODS

-Site description

The experiment was carried out from 1 st November 2018 to 28 th January 2019 in Paris, France.

The climate is oceanic and characterized by overall cool winter. The experiment was performed on the building rooftop of AgroParisTech, higher education and research institute located in the 5 th district of Paris (48°50'24"N, 2°20'55"E). It is a Haussmannian building built during the 19 th century. The rooftop area is about 900 m 2 and is exposed to direct solar radiation without shadowing effect from other buildings. The indoor spaces consisted in office rooms. Since the indoor uses are similar for the three roofs, the indoor conditions (especially indoor temperatures) did not differ between each roofs. The heating system only consisted in individual heater alimented by a common boiler, and no central temperature heating regulating system existed.

The experiment was performed on three different areas: the reference roof (REF) and the extensive (EXT) and semi-intensive (SI) green roofs. The reference is a conventional roof with a black bituminous waterproof membrane installed directly on the concrete slab. The green roofs are installed directly on the bituminous waterproof membrane and consisted in a root barrier, a substrate layer (13 cm for EXT, 27 cm for SI), and a vegetation layer (sparse Sedum for EXT; dense grass for SI) (Figure 1). For the EXT green roof, plant coverage accounted for 22% while it was close to 100% for the SI green roof.

-Instrumentation and measurements

For each roof, the mean surface temperature was measured by an infrared radiothermometer (IR120, Campbell Scientific Ltd, UK) installed on a mast at 1.05 m height above the REF roof the substrate (installed at 6.5 cm and 13 cm depth for EXT and SI green roofs, respectively) and outside the building envelope (i.e., installed directly on the bituminous membrane for the REF roof and at the substrate-building envelope interface for both green roofs) were also measured by heat flux plates (HFP01, Hukseflux, NL). Finally, water content reflectometers (CS655, Campbell Scientific Ltd, UK), installed at 6.5 cm and 13 cm depth for EXT and SI green roofs, respectively, provided measurements of volumetric substrate water content (Figure 1d). All the sensors were connected to a datalogger (CR1000X, Campbell Scientific Ltd, UK) coupled with two multiplexers (AM16/32, Campbell Scientific Ltd, UK). Measurements were averaged online over 30 min periods.

Additionally, standard meteorological conditions were measured at 2.5 m height above the roof by several sensors installed on a mast located on the roof at around 50 m from the experimental area: air temperature and relative humidity (HMP45C, Vaisala, FI), and rainfall (TE525WS, Campbell Scientific Ltd, UK). They were measured, averaged, and recorded every 30 min on a datalogger (CR1000, Campbell Scientific Ltd, UK).

The ranges and accuracies of the sensors used during the experiment are given Table 1. -3.5 to +1% according the intensity of rainfall

-RESULTS AND DISCUSSION

-Overview of weather conditions

During our experiment, weather conditions were representative of the winter climate in Paris.

Both daily and day-to-day variability of the weather conditions are reported through the halfhourly and daily statistics, respectively, in Figure 2 and Table 2. The cumulated solar radiation over the experimental period was 257 MJ.m -2 over the 83 days of the experiment (Table 2). The incident solar radiation followed a typical trend by increasing during the morning to reach its maximum at noon, on average around 150 W.m -2 , before decreasing during the afternoon to its minimum and nocturnal value at 0 W.m -2 (Figure 2a). Its intensity was weak: half-hourly solar radiation was 145 ± 88 W.m -2 on average and varied overall between 75 W.m -2 and 198 W.m - 2 , as indicated by the 1 st and 3 rd quartiles respectively. However, some sunny days occurred, as indicated by the maximum half-hourly solar radiation (520 W.m -2 ). At the daily scale, mean solar radiation was only 127 ± 61 W.m -2 and daily mean solar intensity usually ranged between 78 W.m -2 and 183 W.m -2 with a minimum and maximum at 51 W.m -2 and 256 W.m -2 , respectively (Table 2). Air temperature and relative humidity exhibited an inverse correlation.

Air temperature increased from its minimum, on average 6.3°C just before the sunrise, to its maximum in early afternoon, on average 8.7°C, and then decreased progressively during the afternoon and the night. Conversely, air relative humidity decreased from its maximum at the sunrise, around 87% on average, to its minimum, around 77% on average, reached in early afternoon, and finally increased in the afternoon and during nighttime (Figure 2b). Few particularly dry and warm days for the season occurred with maximum half-hourly and daily mean temperatures of 19.5°C and 15.3°C, respectively, and minimum half-hourly and daily mean air relative humidity of 40% and 65%, respectively. Some cold events also occurred, with negative minimum half-hourly and mean daily air temperatures. Overall, climatic conditions were cold and wet as indicated by mean and median values, around 7.4-7.8°C and 84-86% for air temperature and relative humidity respectively, both at the half-hourly and daily scales.

(Table 2). Yet, rainfall events regularly occurred: 51 days with at least one rainfall event were recorded corresponding to 337 half-hourly events over the whole experimental period. On average, rainfall events were 0.46 mm and 3.02 mm at the half-hourly and daily scales, respectively, but some exceptional and strong episodes appeared, 2.6 mm at the half-hourly scale and 14.2 mm at the daily scale on maximum. Over the three months of the experiment the cumulated rainfall was 154.2 mm (Table 2). As a consequence, green roof did not suffer from water limitation, mean substrate water content (0.29 m 3 .m -3 and 0.14 m 3 .m -3 for EXT and SI green roofs, respectively) being close to their maximum (0.46 m 3 .m -3 and 0.19 m 3 .m -3 EXT and SI green roofs, respectively). Yet, the regular rainfall events allowed a quite stable substrate water content over the period for both EXT and SI green roofs, as indicated by the weak range of variations between 1 st and 3 rd quartiles of the substrate water contents (0.25 to 0.32 for EXT green roof; 0.13 to 0.16 for SI green roof) (Table 2). As a consequence, the effect of soil water content variations on temperatures and conductive heat fluxes could be excluded.

-Albedo values of conventional and green roofs

The albedos for conventional and green roofs were determined by linear regressions between the measured incident and reflected shortwave solar radiations, albedos being given by the slopes of the regression lines. The SI green roof exhibited the largest albedo (0.16) while the EXT green roof had the weakest one (0.12). The REF roof showed an intermediate albedo (0.15) but also the most variable as indicated by the coefficient of determination of the relationships (R 2 = 0.82). Albedo variability was the weakest for the SI green roof (R 2 = 0.95) and intermediate for EXT green roof (R 2 = 0.90) (Figure 3).

Whereas Radhi et al. [START_REF] Radhi | On the colours and properties of building surface materials to mitigate urban heat islands in highly productive solar regions[END_REF] reported largest albedo (0.23) for bituminous membrane, the albedo for the REF roof determined experimentally is consistent with the typical range, 0.1-0.2, found for bitumen roofs [START_REF] Saadatian | A review of energy aspects of green roofs[END_REF]. For the green roofs, although previous studies did not distinguished between their different kinds i.e., extensive, semi-intensive or intensive, Lazzarin et al. [START_REF] Lazzarin | Experimental measurements and numerical modelling of a green roof[END_REF] reported green roof albedo of 0.23 and Takebayashi and Moriyama [START_REF] Takebayashi | Surface heat budget on green roof and high reflection roof for mitigation of urban heat island[END_REF] determined soil and grass albedos of 0.225 and 0.21, respectively, for dry surface conditions. The albedos for EXT and SI green roofs are lower than these studies, but consistent with the results obtained by D'Orazio et al. [START_REF] Orazio | Green roof yearly performance: A case study in a highly insulated building under temperate climate[END_REF] who measured green roof albedo of 0.13. On the one hand, these weak albedos for green roofs compared to those reported in previous studies can be explained first by the season of the experiment. While previous studies were carried out under spring and summer conditions implying green and fully developed canopy cover, the winter season implies brown and less developed vegetation. It induces that the vegetation albedo is lower, and the soil (typically a dark material) albedo contributes more to total green roof albedo. It also probably explains the lowest albedo for the EXT green roof than for the SI green roof: since vegetation is sparser, the lower soil albedo contributes more to the total EXT green roof albedo than for the SI green roof. On the other hand, wetter conditions during winter than during spring and summer induces darker color of the soil surface and bituminous membrane leading to lower albedos. This issue could also explain the large variability of albedo for the REF roof and EXT green roof which would reflect the wetting and drying of the surface alternately during the experiment.

-Impact of green roofs on building insulation

In order to evaluate the impact of the green roofs on building insulation, temperatures and conductive heat fluxes at the building surface (corresponding to surface temperature for the REF roof and temperature at the substrate-building envelope interface for green roofs) have been analyzed and compared between REF, EXT and SI roofs.

For the REF roof, the building surface temperature exhibited the largest fluctuation and followed the dynamic of solar radiation by increasing during the morning to reach its maximum around noon, on average around 13°C, before decreasing during the afternoon to its minimum and nocturnal value at 3.5-5°C (Figure 2c). Over the whole experimental period median building surface temperature was 6.4°C, typically ranged from 2.6°C to 9.5°C (1 st and 3 rd quartiles), and up to -7.2 to 19.8 for the minimum and maximum values (Figure 4a). The conductive heat fluxes reaching the building envelope strongly varied across the day. During nighttime, mean half hourly fluxes were negative indicating heat losses from the building, around -5 W.m -2 . It increased from sunrise to reach its maximum (around 7 W.m -2 ) in late morning, and decreased until middle afternoon to its minimum (-11 W.m -2 ). Finally it increased until sunset to reach its nocturnal value. The building therefore lost heat most of the time, excepted during a short period during the morning for which building gained heat as indicated by the positive values (Figure 2d).

Considering only the whole period the EXT green roof exhibited similar median building surface temperature (7°C) and 1 st and 3 rd quartiles (3.6°C and 9.4°C, respectively) to those observed for the REF roof (Figure 4a). However building surface temperature for EXT green roof exhibited less fluctuations than REF roof as indicated by both their minimum and maximum values (0.4°C and 15.7°C, respectively; Figure 4a) and its daily pattern (Figure 2c).

Indeed the minimal mean half hourly building surface temperature was 6°C and occurred only in late morning. It reached its maximum, around 8°C, in late afternoon and continuously decreased during nighttime (Figure 2c). Hence, the building surface for EXT green roof was warmer than for REF roof most of the time, excepted between 9:00 and 15:00 for which it was cooler (Figure 2c). The temperature difference between REF and EXT roofs typically ranged between -3.1°C (1 st quartile) and 1.2°C (3 rd quartile), overall at -0.8°C (median) indicating that the building surface was slightly warmer for the EXT green roof than for the REF roof.

However, the building surface for the EXT green roof could be warmer (typically during nighttime) or cooler (typically during daytime) by several degrees as indicated by the minimum (-9.8°C) and maximum (7.7°C) values of the temperature difference (Figure 4b). On average over the whole period, the conductive heat fluxes reaching the building envelope were always negative, indicating heat losses from the building. However, they exhibited less diurnal fluctuations than for the REF roof. The mean half hourly conductive heat flux was -5 W.m -2 during nighttime, increased during the morning to peak at -1.5 W.m -2 in early afternoon, and decreased to its nighttime value during the afternoon (Figure 2d).

The mean half hourly conductive heat fluxes reaching the building envelope and building surface temperatures for the SI green roof did not exhibited diurnal fluctuations: the former remained at around -4.5 W.m -2 and the latter at around 9°C (Figures 2d and2c, respectively).

The building surface temperature exhibited also less fluctuation as shown by the 1 st and 3 rd quartiles (6.9°C and 11.4°C, respectively) and minimum and maximum (3.1°C and 14.2°C, respectively) temperatures (Figure 4a). Over the whole period the SI green roof also exhibited the warmest building surface envelope with a median temperature of 8.9°C (Figure 4a). The building surface for the SI green roof was most of the time warmer than for the REF roof: the temperature difference over the whole period was -2.6°C (median), typically varied between -0.2°C to -5.4°C (1 st and 3 rd quartiles), and could reach -14.1°C (minimum). However, it could also be cooler than for the REF roof, although it remained exceptional and typically during daytime, as indicated by the maximum value of the building enveloped temperature difference reaching 7.6°C (Figure 4b).

Hence, huge differences has been observed concerning the impact of the green roofs on the conductive heat fluxes reaching the building envelope. Nevertheless, these fluxes are also closely linked with indoor temperatures, depending themselves to building users. In this study, indoor environments consisted in offices and it has been hypothesis that indoor temperatures were identical for each green roofs. However, since office users could control the room heating, some differences could occur between indoors temperatures. The conductive heat flux (G) depends on the building envelope surface temperature difference between outdoor and indoor (Tout and Tin, respectively) and on the thermal transmittance (Uvalue in W.m -2 .K -1 ) (i.e., G = Uvalue

x (Tout -Tin)). Considering a Uvalue of 3.33 W.m -2 .K -1 (reported by Fokaides and Kalogirou [START_REF] Fokaides | Application of infrared thermography for the determination of the overall heat transfer coefficient (U-value) in building envelopes[END_REF] for a flat and non-insulated roof made of reinforced concrete, which is comparable to the studied building), a change of 1°C of the indoor building surface envelope would lead to a change of 3.33 W.m -2 of the conductive heat flux reaching the external building envelope, which would not affect the trends observed during the experiment. In addition, the nocturnal conductive heat fluxes similar for each roofs (Figure 2d), corresponding to heat losses from the building, suggested that the indoor temperatures were identical for REF, EXT, and SI roofs. Are also indicated the results from the Wilcoxon statistical test (**** = p-value < 0.0001).

Although they were only interested in EXT green roofs during winter conditions, previous studies also found that the green roofs exhibited weak fluctuations of the building surface envelope temperatures under Mediterranean [START_REF] Coma | Thermal assessment of extensive green roofs as passive tool for energy savings in buildings[END_REF] and continental [START_REF] Squier | Heat flux and seasonal thermal performance of an extensive green roof[END_REF] climates, and less than for the conventional roofs. For instance Teemusk and Mander [START_REF] Teemusk | Greenroof potential to reduce temperature fluctuations of a roof membrane: A case study from Estonia[END_REF] reported during the winter period for continental climate monthly variation of daily temperature amplitudes between around 1°C and 2.5°C approximately for a green roof while they reached up to 5°C for the conventional roof. Similarly Getter et al. [START_REF] Getter | Seasonal heat flux properties of an extensive green roof in Midwestern U.S. climate[END_REF] reported for a Midwestern U.S. climate building envelope temperature varying between -4°C during nighttime and 10°C during daytime for a conventional roof whereas it only ranged from around -1°C to 2°C for the green roof. These largest temperature fluctuations for the conventional roof are linked with the absence of substrate and vegetation layer, allowing a direct exposure of the building envelope to solar radiation. It induces that the building envelope for the conventional roof rapidly warms during daytime due to the fast increase of incident radiation, but also cools faster due to fast radiative losses. On the contrary, the building envelope with a green roof is not directly exposed to the solar radiation. The energy received at the green roof surface needs to be transferred through the conductive heat flux within the substrate which is less efficient than the direct exposure to sun radiation. This issue explains that the maximum building envelope temperature for the EXT green roof only occurred in late afternoon while it occurred around noon for the REF roof. Yet, the green roof substrate also prevents direct building envelope cooling from radiative losses.

The presence of substrate is therefore of a key importance for the assessment of building insulation during winter, and its thickness the main factor controlling its efficiency since vegetation is sparse and weakly evapotranspirates under such conditions, whatever the kind of green roof (i.e., EXT or SI) considered. Indeed, the grass transpiration under winter conditions only accounts for a small part of the total evapotranspiration, between 5-20% [START_REF] Nelson | Ecosystem transpiration and evaporation: Insights from three water flux partitioning methods across FLUXNET sites[END_REF]. Hence, although the vegetation densities are quite different between EXT and SI green roofs (with a percentage of plant cover of approximatively 20-25% for EXT and 100% for SI, which is partly composed of yellow and inactive leaves), it could be expected that the total evapotranspiration is mainly driven by soil evaporation during winter conditions, which depends on atmospheric conditions (identical for the two roofs) and substrate water content (always close to the field capacity during the experiment). That would lead to a similar evapotranspiration for EXT and SI green roof under winter condition despite the differences in terms of LAI and canopy cover, as confirmed by Silva et al. [START_REF] Silva | Green roofs energy performance in Mediterranean climate[END_REF].

As illustrated in Figure 5 the conductive heat fluxes within the substrate varied between -20 W.m -2 and 35 W.m -2 for the EXT green roof (Figure 5a) while it only ranged from -10 W.m -2 to 5 W.m -2 for the SI green roof (Figure 5b) for the same period from 2 November to 17

November 2018. This trend is probably due to heat storage within the substrate which is all the more important that the substrate is deep. Yet it is observed a time lag between (i) the temperature difference between substrate surface and substrate-building envelope interface and

(ii) the conductive heat flux within the substrate for the SI green roof (Figure 5b), which did not occurred for the EXT green roof (Figure 5a). Hence the thick substrate of the SI green roof provides large thermal inertia. As a consequence the conductive heat fluxes reaching the building envelope exhibit less fluctuation with than without a green roof, and this effect increases with the substrate thickness. That issue was also found by D'Orazio et al. [START_REF] Orazio | Green roof yearly performance: A case study in a highly insulated building under temperate climate[END_REF] who compared EXT green roof with conventional ones.

-Impact of green roofs on winter surface urban heat island

The impact of the green roofs on winter surface UHI has been explored by comparing the surface temperatures for the REF, EXT, and SI roofs. The surface temperatures for the REF roof are those presented in the previous section and will not be described in this section.

Considering the whole period the EXT green roof had median surface temperature of 5.6°C (2°C and 8.4°C for the 1 st and 3 rd quartiles, respectively), minimum and maximum surface temperatures reaching -3.9°C and 18.2°C, respectively. The SI green roof had slightly warmer surface temperature with median value equal to 6.2°C and 1 st and 3 rd quartiles of 3°C and 8.9°C, respectively. The minimum and maximum surface temperature were similar than for the EXT green roof. Considering the whole period, the surface temperature of the SI green roof did not differ significantly to the surface temperature of the REF roof while the EXT green roof is cooler, as indicated by the results of the Wilcoxon statistical tests (Figure 6a). Indeed, the halfhourly difference between surface temperatures of the REF and SI roofs typically ranged between -1.1°C and 0.9°C (1 st and 3 rd quartiles) with a median value of 0.2°C whereas this difference varied between -0.5°C and 1.6°C (1 st and 3 rd quartiles) with a median value of 0.8°C for the EXT green roof (Figure 6b).

Huge differences occurred between surface temperature of the REF, EXT and SI roofs at the daily scale (Figure 2c). The surface temperatures for both the EXT and SI green roofs exhibited a similar daily evolution, following the same dynamics than the REF roof: they increased during the morning to reach their maximum around noon before decreasing during the afternoon to their minimum and nocturnal value. However, maximum mean half-hourly surface temperatures only peaked to 8.4°C and 8.9°C for the EXT and SI green roofs, respectively, while it reached 13°C for the REF roof. The nocturnal surface temperatures are similar between the EXT and REF roofs, around 3.5-5°C, but the SI green roof exhibited systematic slightly warmer surface, around 4.5-6°C (Figure 2c).

Overall, EXT and SI green roofs only had a slight negative (i.e., cooling) effect on winter surface urban heat island as reported by Teemusk and Mander [START_REF] Teemusk | Greenroof potential to reduce temperature fluctuations of a roof membrane: A case study from Estonia[END_REF]. However, under diurnal conditions the presence of green roofs resulted in surface cooling. Although this cooling for the SI green roof could be due its higher albedo compared to the REF roof, it would not be consistent for the EXT green roof (its albedo being lower than those of the REF roof, warmer surface would be expected). Hence, it would be probably due to low but significant evapotranspiration even during winter conditions [START_REF] Lazzarin | Experimental measurements and numerical modelling of a green roof[END_REF][START_REF] Silva | Green roofs energy performance in Mediterranean climate[END_REF], which is comparable for EXT and SI green roofs according to Silva et al. [START_REF] Silva | Green roofs energy performance in Mediterranean climate[END_REF]. Nevertheless, the SI green roof exhibited systematically warmer surface temperature, by around 1°C, inducing that it had a positive (i.e., warming) effect during the nocturnal conditions compared to the REF roof. It is probably due to its thicker substrate allowing a larger heat capacity and thermal inertia and therefore a slower surface cooling during nighttime.

-CONCLUSIONS

This study attempted to investigate the impact of two kinds of green roofs i.e., extensive (EXT) and semi-intensive (SI) green roofs, on both building insulation and surface urban heat island effect under winter conditions. To this aim we compared measurements of surface and building envelope temperatures as well as conductive heat fluxes reaching the external building envelope with those measured on a conventional bituminous roof under identical climatic conditions.

While the SI green roof provides a building envelope surface warming compared to the conventional bituminous roof, the EXT green roof has no clear effect on average. However, at the daily scale, although the conventional roof benefits to diurnal solar radiation, and therefore surface heating, it is also exposed to important radiative heat losses during nighttime. On the contrary, green roofs provide an additional insulation layer which diminishes the daily fluctuations of building envelope temperature under winter conditions whatever the kind of green roof. As a consequence, the conductive heat fluxes reaching the external building envelope also suffer to more daily fluctuations for the conventional bituminous roof than for the green roofs. These effects are all the more important that the substrate is thick, owing to its larger heat capacity and thermal inertia. Although it has not been quantified, it could be hypothesized that the reduction of daily fluctuations of conductive heat fluxes would also reduce the building energy consumption by reducing heating loads.

On average the green roofs only have an insignificant or slight effect on surface UHI. However they provide a surface cooling during daytime due to even low but significant evapotranspiration. During nighttime the EXT green roof exhibits similar surface temperature than the conventional roof, while the SI green roof is warmer.

Therefore green roofs can be suitable urban greening solutions since they do not have negative effect on surface urban heat island during winter while they provide an efficient cooling under summer conditions, and provide building insulation [START_REF] De Munck | Evaluating the impacts of greening scenarios on thermal comfort and energy and water consumptions for adapting Paris city to climate change[END_REF][START_REF] Jaffal | A comprehensive study of the impact of green roofs on building energy performance[END_REF][START_REF] Kumar | Performance evaluation of green roof and shading for thermal protection of buildings[END_REF][START_REF] Qiu | Effects of evapotranspiration on mitigation of urban temperature by vegetation and urban agriculture[END_REF].

Figure 1 :

 1 Figure 1: (a) The reference roof and vegetation covering the (b) extensive and (c) semi-

and 2 m

 2 height above EXT and SI green roofs. The surface measured by the sensor (half-angle of view of 20°) was therefore 0.4 m 2 for the REF roof and 1.6 m 2 for the EXT and SI green roofs. Additionally, each component of the radiative budget (i.e., incoming and outgoing shortand longwave radiations) were measured with a net radiometer (CNR4, Kipp & Zonen, NL) above the SI green roof, while only the outgoing (i.e., reflected) shortwave radiation was measured with a pyranometer (CMP11, Kipp & Zonen, NL) above the REF and EXT roofs. Moreover, five thermocouples (Type T, TC Direct, FR) were installed to provide temperature measurements: on the black bituminous membrane for the REF roof and on the substrate and at a b c d the substrate-building envelope interface for both green roofs. Conductive heat fluxes inside

Figure 2 :

 2 Figure 2: Half hourly means of (a) solar radiation, (b) air temperature (black line) and relative

Figure 3 :

 3 Figure 3: Relationships between incident and reflected solar radiation for the reference (REF;

Figure 4 :

 4 Figure 4: Boxplot statistics of (a) surface temperatures of the reference (REF) roof and

Figure 5 :

 5 Figure 5: Time series of half-hourly conductive heat fluxes within the substrate and

Figure 6 :

 6 Figure 6: Boxplot statistics of (a) the surface temperatures of the reference (REF), extensive

Table 1 :

 1 List of sensors, their range and accuracy used during the experiment

	Measured variable	Sensor	Range	Accuracy
	Mean surface temperature	Infrared radiothermometer IR120	-25 to +60°C	±0.2°C
	Temperature	Thermocouple Type T	-50 to +150°C	±0.1°C
	Air temperature and	Thermo-hygrometer	-39 to +60°C	±0.2°C
	relative humidity	HMP45C	0.8 to 100%	±1%
	Shortwave radiation	Net radiometer CNR4 Pyranometer CMP11	0 to 2000 W.m 2 0 to 4000 W.m 2	±10% <2%
	Longwave radiation Net radiometer CNR4	[-]	<10%
				-15 to +5%
	Conductive heat flux Heat flux plate HFP01	±2000 W.m -2	according to the
				material in contact
	Substrate water content	Reflectometer CS655	0 to 100%	±3%

Table 2 :

 2 Half-hourly and daily means (± standard deviations), medians, minimums, maximums, and 1 st and 3 rd quartiles of solar radiation, air temperature, air relative humidity, rainfall events, and substrate water content for extensive and intensive green roofs. The number of rainfall events (n) and cumulated rainfall over the experimental period are also given

				Solar radiation	Air temperature	Air relative humidity	Rainfall events	Substrate water content
				W.m -2	°C	%	mm	m 3 .m -3
								Extensive	Semi-intensive
			Mean ± SD 145±88	7.4±3.9	84±10	0.46±0.37 0.29±0.03 0.14±0.03
	Half-hourly	Min 1 st quartile Median 3 rd quartile Max	0 75 115 198 520	-2.4 4.8 7.7 10.3 19.5	40 77 86 92 99	0.2 0.2 0.2 0.6 2.6	0.15 0.25 0.29 0.32 0.46	0.03 0.13 0.15 0.16 0.19
			n	-	-	-	337	-	-
			Mean ± SD 127±61	7.4±3.7	84±8	3.02±3.20 0.29±0.06 0.14±0.03
			Min	51	-0.3	65	0.2	0.16	0.03
	Daily	1 st quartile Median 3 rd quartile	78 106 183	4.9 7.8 10.2	79 84 90	0.8 2.0 4.2	0.25 0.28 0.33	0.13 0.15 0.16
			Max	256	15.3	97	14.2	0.43	0.17
			n	-	-	-	51	-	-
	Whole	period	Sum	257 MJ.m -2	-	-	154.2	-	-

Acknowledgements

This work was supported by a grant from Vinci Construction France. The authors also acknowledge Hacene Bouhoun-Ali for its help in the experimental set-up and data acquisition.

Declaration of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Alejandro Rivera
email: alejandro.rivera@epfl.ch

High-dimensional Monochromatic Random Waves approximate the Bargmann-Fock field

Fix d ∈ N and let n ∈ N be a large integer. Let f d,n be the restriction to R d × {0} n-d of the Monochromatic Random Wave, i.e. the unique a.s. smooth, centered stationary Gaussian field whose spectral measure is the uniform measure on the unit sphere in R n . Then, we show that as n → +∞, f d,n approximates a Gaussian field g d,n whose spectral measure is a Gaussian measure with adequately chosen variance. As an application of this result, we deduce that for -> 0 small enough and n ∈ N large enough, the excursion sets {f d,n + > 0} percolate with probability one.

Introduction 1.Setting and main results

The n-dimensional Monochromatic Wave is the a.s. smooth centered stationary Gaussian field f n on R n with covariance function defined by

E[f n (x)f n (0)] = 1 |S n-1 | S n-1 e i x,ω dσ S n-1 (ω)
where dσ S n-1 is the standard area measure on the unit sphere S n-1 and |S n-1 | = S n-1 dσ S n-1 is its volume. The geometric and topological properties of the level sets {f + = 0} and excursion sets {f + > 0} of a.s. smooth stationary Gaussian fields f , such as f n , have been the subject of much interest in the past decade (we refer to Section 1.1 of [START_REF] Muirhead | The sharp phase transition for level set percolation of smooth planar Gaussian fields[END_REF] for an overview of the percolation results and to [START_REF] Anantharaman | Topologie des hypersurfaces nodales de fonctions aléatoires gaussiennes[END_REF] for a discussion of the counting of connected components of a given topology) . It has been notoriously difficult to describe its large scale connectivity properties. The only example we know of is [START_REF] Muirhead | The phase transition for planar gaussian percolation models without fkg[END_REF] (see Theorem 1.3, Theorem 1.7 and Example 1.4 therein). This is due in great part to the sign changes of the covariance function of f n . Indeed, by [START_REF] Pitt | Positively correlated normal variables are associated[END_REF], this implies that f n does not satisfy the Fortuin-Kasteleyn-Ginibre (or FKG) inequality, which is crucial to most tools used to study the percolation of excursion sets of Gaussian fields (see for instance the first point of Defintion 2.1 of [START_REF] Beffara | Percolation of random nodal lines[END_REF]).

Fix d ∈ N and n ∈ N such that n > d + 1 and let g d,n be the a.s. smooth centered stationary Gaussian field on R d with covariance function defined by E[g d,n (x)g d,n (0)] = e -2x 2 n-d-1 .

In contrast with f d,n , by [START_REF] Pitt | Positively correlated normal variables are associated[END_REF], the Gaussian field g d,n does satisfy the FKG inequality and its percolation properties have been studied before (see for instance [START_REF] Beffara | Percolation of random nodal lines[END_REF], [START_REF] Rivera | The critical threshold for Bargmann-Fock percolation[END_REF] and [START_REF] Hugo Duminil-Copin | Existence of an unbounded nodal hypersurface for smooth gaussian fields in dimension d ≥ 3[END_REF]). It is in fact, up to rescaling equal to the well known Bargmann-Fock field.

The main result of the present text is that, when restricted to R d × {0} n-d , when n → +∞ with d fixed, the field f n becomes in fact quite similar to g d,n in the following sense.

Proposition 1.1. For each n ∈ N, n > d + 1, there exists a coupling (f d,n , g d,n ) of f n restricted to R d × {0} n-d and g d,n with the following property. For each k ∈ N, there exists a constant C = C(k, d) < +∞ such that for each n ∈ N large enough, for each r ≥ 2 and for each t > 0,

P f d,n -g d,n C k ([0,r] d ) ≥ C(1 + t)n -1 ln(1 + r) ≤ e -t 2 /2 .
The key idea behind Proposition 1.1 is the following classical fact of high dimensional geometry:

As n → +∞ with d ∈ N fixed, the standard Gaussian measure on R n and the uniform measure on the sphere of radius √ nS n-1 , when projected onto R d × {0} n-d , become close to each other.

See for instance Section 1.1 of [START_REF] Ledoux | The concentration of measure phenomenon[END_REF] for a discussion of this phenomenon. We do not attempt to make it more precise here as the proof does not explicitely involve approximation of measures. However, with this in mind, we observe that the spectral measures 1 of the Gaussian field f n , when projected onto R d , should approximate the spectral measures of g d,n . This is a good hint that the two fields should be close in some sense. All that remains is to quantify their resemblence, which is what we do in Proposition 1.1.

As an application of Proposition 1.1, we extend a recent result from [START_REF] Hugo Duminil-Copin | Existence of an unbounded nodal hypersurface for smooth gaussian fields in dimension d ≥ 3[END_REF] on the percolation of excursion sets of g d,n to the percolation of excursion sets of f d,n . Theorem 1.2. There exists c < 0 such that for every > c , there exists an integer n 0 = n 0 ( ) ∈ N such that for each n ≥ n 0 , with probability one, the set {f n + ≥ 0}∩R 3 ×{0} n-3 contains an unbounded connected component.

As a corollary to Theorem 1.2, we obtain the following result.

Corollary 1.3. There exist n 0 ≥ 3 and > 0 such that for each n ≥ n 0 and for any ∈ [-, ], a.s. there exists an unbounded component in {f n = } ∩ R 3 .

Proof. Let = | c |/2 > 0 be as in Theorem 1.2 and let ∈ (0, ). Then, since f n and -f n have the same law, by Theorem 1.2, a.s., {f n > } and {f n < } both have an unbounded connected component. But these sets must then be separated by an unbounded connected component of their common boundary which is {f n = } (see for instance Lemma A.9 of [START_REF] Rivera | Quasi-independence for nodal lines[END_REF] and Lemma D.1 from [START_REF] Hugo Duminil-Copin | Existence of an unbounded nodal hypersurface for smooth gaussian fields in dimension d ≥ 3[END_REF]).

We prove Proposition 1.1 in Section 2 and Theorem 1.2 in Section 3. The proof of Theorem 1.2 extends a supercriticality result for the percolation of one model to another via an approximation procedure. This strategy may be compared to the proof of Theorem 3.3 of [START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF] in which the positivity of the critical level is established for super level sets of the discrete Gaussian Free Field in high dimensions via comparison with Bernoulli percolation.

Acknowledgements

I wrote this manuscript while occupying a research position in the Random Geometry chair of the mathematics department of the EPFL in Lausanne, Switzerland. I am thankful to Juhan Aru, head of this chair, for hiring me and providing me with ideal conditions to work on this question.

The idea behind this manuscript came while I was working in collaboration with Hugo Duminil-Copin, Pierre-François Rodriguez and Hugo Vanneuville on the project [START_REF] Hugo Duminil-Copin | Existence of an unbounded nodal hypersurface for smooth gaussian fields in dimension d ≥ 3[END_REF] cited below. I am thankful to them for many stimulating discussions during that period.

Proof of Proposition 1.1

In this section, we prove Proposition 1.1. Throughout this section we fix d ∈ N. For each n ≥ d + 1 and x ∈ R d we define

(2.1) d,n (x) = C d,n R d (1 -|ξ| 2 ) n-d-1 4 e i x,ξ 1 [|ξ| 2 ≤1] dξ where C d,n := 1 (2π) d/2 |S n-d-1 | 1/2 |S n-1 | 1/2 .
Then, using the d-dimensional convolution Fourier formula u * u(ξ) = (2π) d û(ξ) 2 where û(ξ

) := 1 (2π) d R d e -i x,ξ u(x)dx, we obtain, for each x ∈ R d , (2.2) d,n * d,n (x) = |S n-d-1 | |S n-1 | |ξ|≤1 (1 -|ξ| 2 ) n-d-1 2 e i x,ξ dξ = 1 |S n-1 | S n-1 e i x,ω dσ S n-1 (ω)
where dσ S n-1 is the surface area measure on S n-1 . Thus, d,n * d,n is the restriction to R d of the covariance function of the n-dimensional Monochromatic Random Wave.

Let n > d + 1 and x ∈ R d . Define

(2.3) γ d,n (x) := C d,n 2 (2π)(n -d -1) d/2 e -x 2 n-d-1
We note for later use that, since

|S n-1 | = nπ n/2 Γ( n 2 +1) , (2.4) γ d,n * γ d,n (x) = σ 2 d,n e -x 2 2(n-d-1) where σ 2 d,n = C 2 d,n 2 (2π)(n -d -1) d/2 = 1+O(n -1 )
where the implied constant in O(n -1 ) depends on d.

Let W be the L

Lemma 2.1. For each m ∈ N and β ∈ N d there exists C = C(m, β, d) < +∞ such that for each n > d + 1 and each x ∈ R d , ∂ β ( d,n (x) -γ d,n (x)) ≤ CC d,n n m-1-d/2 |x| -2m .
Proof. Throughout the proof, the constants implied by the symbol O(•) are assumed to be uniform in n. Let us take n > d + 1. In order to estimate d,n -γ d,n , we study its Fourier transform, defined, for each ξ ∈ R d , as

h d,n (ξ) := 1 (2π) d R d e -i x,ξ ( d,n (x) -γ d,n (x))dx .
From the definition of d,n and γ d,n , it follows that for each ξ ∈ R d ,

h d,n (ξ) = C d,n (1 -|ξ| 2 ) A d,n 1 [|ξ|≤1] -e -A d,n |ξ| 2 where A d,n := n -d -1 4 .
We derive the following estimate for h d,n .

Claim 2.2. For each α ∈ N d , there exists C = C(α, d) < +∞ such that for all large enough n and ξ ∈ R d ,

(2.5) |∂ α h d,n (ξ)| ≤ CA |α|/2 -1 d,n 1 + A d,n |ξ| 2 |α|/2 +1 e -A d,n |ξ| 2 . Proof. For each A > 0 and t ∈ [0, 1], let v A (t) = (1-t) A -e -At . We fix k ∈ N and estimate the k-th derivative of v A provided that A > 0 is large enough that v A ∈ C k ([0, 1]). In this case, writing r k (A) = k-1 j=0 (A -j) -A k , we get (-1) k v (k) A (t) = k-1 j=0 (A -j) (1 -t) A-k -A k e -At = O(A k )((1 -t) A-k -e -At ) + r k (A)e -At = O(A k )((exp(-(A -k)t + O(At 2 )) -e -At ) + r k (A)e -At = O(A k )e -At O(t + At 2 ) + r k (A)e -At . If k > 0, then r k (A) = O(A k-1
), and r 0 (A) = 0. All in all, for some

C 1 = C 1 (k) < +∞, uniformly for A large enough and t ∈ [0, 1], (2.6) |v (k) A (t)| ≤ C 1 (r k (A) + A k t + A k+1 t 2 )e -At . Recall that A d,n = n-d-1 4 . As long as |ξ| ≤ 1, h d,n (|ξ| 2 ) = C d,n v A d,n (|ξ| 2 )
. By the chain rule, for each α ∈ N d , for n large enough, we obtain, for all ξ ∈ R d with |ξ| ≤ 1,

C -1 d,n |∂ α h d,n (ξ)| = O   |α|/2 j=0 v |α|/2 +j A d,n (|ξ| 2 ) × |ξ| 2j   (2.6) = O (1) A |α|/2 -1 d,n |α|/2 +1 j=0 A d,n |ξ| 2 j × e -A d,n |ξ| 2 = O(1)A |α|/2 -1 d,n 1 + A d,n |ξ| 2 |α|/2 +1 e -A d,n |ξ| 2 .
This proves the estimate for |ξ| ≤ 1. If |ξ| > 1, then, h d,n (ξ) = -C d,n e -A d,n |ξ| 2 so the estimate follows from direct computation.

The statement follows from (2.5) by noticing that for each n ∈ N and

x ∈ R d , d,n (x) -γ d,n (x) = R d e i x,ξ h d,n (ξ)dξ
so that, for each m ∈ N, and β ∈ N, uniformly in x ∈ R d and n ∈ N large enough,

|x| 2m ∂ β ( d,n (x) -γ d,n (x)) = R d e i x,ξ ∆ m ((iξ) β h n (ξ))dξ ≤ R d ∆ m ((iξ) β h n (ξ)) dξ = O(1) |β| k=0 R d |ξ| |β|-k |γ|≤2m-k |∂ γ h n (ξ)|dξ = O(C d,n A m-1 d,n ) R d 1 + (A d,n |ξ| 2 ) m+1 e -A d,n |ξ| 2 dξ = O C d,n A m-1-d/2 d,n = O C d,n n m-1-d/2 .
In the above equations, ∆ denotes the Laplace operator on R d .

P φ d,n -ψ d,n C k ([0,r] d ) ≥ C(1 + t)n -1 ln(1 + r) ≤ e -t 2 /2 . Proof. Let θ d,n = d,n -γ d,n and η d,n = θ d,n * W = φ d,n -ψ d,
∈ R d , E ∂ α η d,n (0) 2 = R d ∂ α θ d,n (y) 2 dy =O C 2 d,n n -2-d |y|≤n 1/2 dy + |y|>n -1/2 n 2m |y| -4m dy = O C 2 d,n n -2-d/2 . Recall that |S n-1 | = nπ n/2 Γ( n 2 +1) so that C 2 d,n = O(n d/2
) and

E ∂ α η d,n (0) 2 = O(n -2 ) .
We conclude by applying Lemma A.1.

Proof of Proposition

P g d,n -ψ d,n C k ([0,r] d ) ≥ C(1 + t)n -1 ln(1 + r) ≤ e -t 2 /2 .
For each multiindex β ∈ N d , the variance In this section we prove Theorem 1.2. For the proof of this theorem, in addition to Proposition 1.1, we will use the following ingredients. We will use a truncation estimate (Lemma 3.1), comparing the field to a finite range field. This will be established in Subsection 3.1. Then, in Subsection 3.2 we will define a renormalized percolation based on the percolation of the set {f d,n + ≥ 0}. Using a result on the percolation of the level sets of g d,n (see Theorem 3.3) and Proposition 1.1, we will see that for small enough > 0, this renormalized percolation is supercritical (see Proposition 3.5) and that the renormalized percolation induces percolation in the original model (see Lemma 3.6). This will allow us to quickly reach the conclusion of the proof Theorem 1.2. The proof of Proposition 3.5 involves a variation on a classical planar renormalization argument. We relegate it to Subsection 3.3.

∂ β g d,n (0) is bounded uniformly in n. Moreover g d,n -ψ d,n = (1 -σ d,n )g d,n . Now, g d,n (

A truncation estimate

In order to prove Theorem 1.2, we will need to estimate the correlation of percolation events supported far from each other. We will do this by comparing the field to a field with finite dependence in Lemma 3.1. Estimates of this kind, often called truncation estimates, are quite common in the study of level sets of Gaussian fields see for instance Proposition 3.11 of [START_REF] Muirhead | The sharp phase transition for level set percolation of smooth planar Gaussian fields[END_REF].

Lemma 3.1 (Truncation estimate). Let χ : R → R + be a smooth function satisfying

χ(t) = 1 if |t| ≤ 1 and χ(t) = 0 if |t| ≥ 2. Let δ > 0. For each n ≥ d and s ≥ n (1+δ)/2 , let s d,n = χ(|x| 2 /s 2 ) d,n (x)
|∂ β s d,n (x)| ≤ C 1 n -d/4-|β|/2 (1 + |x| 2 /n) |β|/2 e -|x| 2 /n + n -1-d/4 (1 + |x| 2 /n) -m .
Notice that for each t > 0 large enough, (1 + t) |β|/2 e -t ≤ (1 + t) -m . Since s d,n (x) vanishes for all |x| ≤ n (1+δ)/2 ≤ s, we have, for each β ∈ N d such that |β| ≤ 1, for each n ∈ N large enough and x ∈ R d ,

|∂ β s d,n (x)| ≤ 2C 1 (1 + |x| 2 /n) -m 1 [|x|≥n (1+δ)/2 ] .
Of course the estimate is quite crude, but it is enough for our needs. Indeed, using the fact that |x| ≥ n (1+δ)/2 ⇒ |x| 2 /n ≥ |x|, we get, finally, for each β ∈ N d such that |β| ≤ 1, for each x ∈ R d and each n ∈ N large enough, that

(3.1) |∂ β s d,n (x)| ≤ C 1 (1 + |x|) -m . Now, since s d,n (x) vanishes for |x| ≥ s, E ∂ β (f d,n (0) -f s d,n (0)) 2 = (∂ β s d,n * ∂ β s d,n )(0) ≤ |y|≥s ∂ β s n (y) 2 dy ≤ (3.1) 2C 2 1 |y|≥s (1 + |y|) -2m dy = O s d-2m
as long as 2m > d. Since we can take m as large as we want, we deduce that there exists

C 2 = C 2 (m, δ, d) < +∞ such that for all β ∈ N d satisfying |β| ≤ 1, for each large enough n ∈ N, E ∂ β (f d,n (0) -f s d,n (0)) 2 ≤ C 2 s -2m
. We conclude thanks to Lemma A.1.

Defining a renormalized percolation and conclusion

In this section we prove Theorem 1.2. The proof will be a renormalization argument. First we will use Proposition 1.1 to show that a certain percolation event on a bounded box occurs with very high probability. Then, we combine this information with Lemma 3.1 and standard gluing constructions to build, with high probability, a path of boxes on which this percolation event happens (see Proposition 3.5 below). Finally, we conclude by showing that this implies the existence of a continuous path passing through the boxes of this discrete path (see Lemma 3.6 below). The construction of the path of boxes, i.e. the proof of Proposition 3.5, is postponed until Subsection 3.3.

We now define the percolation event that will serve as a building block for our later constructions. Definition 3.2 (Open renormalized vertices). For each 0 < r < R and , ∈ R, let AlmUni , (r, R) be the set of functions u ∈ C 0 (R d ) with the following properties:

For each x ∈ R 2 such that x + [-R, R] 2 ⊂ [-3R, 3R] 2 , the set {u + ≥ R -3/2 } contains a connected set of diameter at least r in (x + [-R, R] 2 ) × {0} d-2 .
Any two connected sets of diameter at least r in {u

+ ≥ 0} ∩ [-3R, 3R] 2 × {0} d-2 are connected by a continuous path in {u + ≥ 0} ∩ [-3R, 3R] 3 × {0} d-3 .
Given a fixed set of parameters r, R, , , we will study a renormalized vertex percolation on Z 2 where a vertex x will be called open if the event f d,n

•-x n 1/2 ∈ AlmUni , (r, R) is satisfied.
We will use the following result about the percolation of the sets {g d,n + ≥ 0}.

of [5]]

There exist < 0 and C > 0 such that if > , ε > 0, then, there exist δ 0 ∈ (0, 1) and R min < +∞ such that for each n ∈ N, n ≥ d + 1 and each R 0 > R min ,

P g d,n • n 1/2 ∈ AlmUni , (δ 0 R 0 , R 0 ) ≥ 1 -ε .
From Theorem 3.3, Proposition 1.1 and Lemma 3.1 we immediately deduce the following analogous result for f d,n (and f s d,n from Lemma 3.1).

Corollary 3.4 (Almost Uniqueness for f d,n ). Let < 0 be as in Theorem 3.3. Let > .

For each ε > 0 and ν > 0, there exist n 0 ∈ N, δ 0 ∈ (0, 1) and R min < +∞ such that for each n ∈ N, n ≥ n 0 > d + 1 and each R 0 ∈ [R min , e n ],

P f d,n • n 1/2 ∈ AlmUni -ν, +ν (δ 0 R 0 , R 0 ) ≥ 1 -ε .
Using this, by following a standard renormalization argument, we prove the following result.

Proposition 3.5 (Local uniqueness for renormalized percolation). Let > . For each ν > 0 there exist n 0 ∈ N, k 0 ∈ N and 0 < r 0 < R 0 < +∞ such that for each n ∈ N, n ≥ n 0 > d + 1 and k ≥ k 0 , the following event holds with probability at least

1 -e -(3/2) k . Let S 1 , S 2 ⊂ R 0 Z 2 ∩ [-10 k R 0 , 10 k R 0 ] 2 be
two subsets of diameter at least 10 k/2 R 0 each of which is connected in the graph R 0 Z 2 . Then, there exists a path in the graph Z 2 ∩ [-10 k R 0 , 10 k R 0 ] 2 which connects S 1 and S 2 such that for each vertex x of this path,

f d,n • -x n 1/2 ∈ AlmUni -ν, +ν (r 0 , R 0 ) .
The proof of Proposition 3.5 can be found in Subsection 3.3 below. This proposition will allow us to build long paths of boxes centered at open vertices of R 0 Z 2 . In the following lemma, we see that such paths induce continuous paths in {f d,n + -ν ≥ 0}. Lemma 3.6 (Renormalized paths induce continuous paths). Fix 0 < r 0 < R 0 < +∞ and , ∈ R, ≤ . Let be a (possibly infinite) path in R 0 Z 2 . Assume that for each vertex x visited by , the following event is satisfied:

(3.2) f d,n • -x n 1/2 ∈ AlmUni , (r 0 , R 0 ) .
Then, there exists a continuous path in {f d,n + ≥ 0} which visits

(x + [-3R 0 , 3R 0 ] 2 ) × {0} d-2 for each vertex x of .
Proof. Notice that for each two consecutive vertices x and y visited by , there exists a translate of the box

[-R 0 , R 0 ] 2 contained in (x + [-3R 0 , 3R 0 ] 2 ) ∩ (y + [-3R 0 , 3R 0 ] 2 )
. By the first point of Definition 3.2, in each of these boxes the set

{f d,n + ≥ R -3/2 0
} must contain a connected set S x,y of diameter at least r 0 . By the second point of Definition 3.2, if z is a third consecutive vertex of , S x,y and S y,z must be connected in {f d,n ≥ 0} ∩ (y + [-3R 0 , 3R 0 ] 3 ) × {0} d-3 . Hence, by induction, all such sets S x,y are connected and the conclusion follows.

Proof of Theorem 1.2. Let > and ν > 0. Fix n 0 ∈ N, k 0 ∈ N and 0 < r 0 < R 0 as in Proposition 3.5, fix also n ≥ n 0 and for each k ≥ k 0 , let Cluster k be the event described in the statement of the proposition. Then, with probability one, there exists k 1 ≥ k 0 such that Cluster k holds for each k ≥ k 1 . But this implies that there exists an infinite injective path in R 0 Z 2 ϕ such that for each vertex x of , the event (3.2) holds. By Lemma 3.6, this implies that there exists a continuous path in {f d,n + -ν ≥ 0} which visits every box (x + [3R 0 , 3R 0 ] 2 ) × {0} d-2 centered at a vertex x of . In particular, this path must be unbounded. Since this is true for every > and ν > 0, we conclude that for every > , there exists n 0 ∈ N such that for every n ≥ n 0 , {f d,n + > 0} contains an unbounded connected component with probability one.

Planar renormalization argument: proof of Proposition 3.5

We now prove Proposition 3.5. To do so, we first introduce the parameters we will use in the proof. Let < and ν 0 > 0. We also introduce ε 0 > 0 a parameter to be defined later as a function of a universal constant (see C 0 defined in Lemma 3.7 below). Let n 0 ∈ N, δ 0 ∈ (0, 1) and R min < +∞ be the parameters from Corollary 3.4 (with ν = ν 0 , ε = ε 0 /2 and δ = 1/4) and let R 0 ∈ [R min , e n ] to be fixed later. For each k ∈ N, let R k = 3 k R 0 .

Having defined our parameters, we now introduce a renormalized percolation model. For each ν > 0, we say that x ∈ R 0 Z 2 is ν-open if the following event is satisfied: Let G be the group of linear transformations of the plane generated by π 2 -rotations and the reflexion around the horizontal axis. Given T ∈ G, let Cross k,ν (x, T ) be the image of Cross k,ν (x) under the transformation

f d,n • -x n 1/2 ∈ AlmUni -ν, +ν (δ 0 R 0 , R 0 ) .

Note that being

f d,n → f d,n (T -1 •).
With our choice of parameters, by union bound, (3.3) P [Cross 0,ν 0 (0)] ≥ 1 -ε 0 .

A standard construction from planar percolation yields the following estimate (see for instance Section 3.4 of [START_REF] Bollobás | Percolation[END_REF]).

Lemma 3.7. There exists a universal constant C 0 < +∞ for which the following holds for each k ∈ N and ν > 0:

P [¬Cross k,ν (0)] ≤ C 0 sup x,T P [¬Cross k,ν (0) ∩ ¬Cross k,ν (x, T )] .
Where the sup is taken over pairs

x ∈ R k Z 2 and T ∈ G such that dist(x + T ([0, 2R k ] × [0, R k ]), [0, 2R k ] × [0, R k ]) ≥ aR k .
We now provide a proof of Proposition 3.5 from this estimate, combined with (3.3) and Lemma 3.1. The proof consists of a standard renormalization argument. In Section 3.4 of [START_REF] Bollobás | Percolation[END_REF] one can find an application of this technique to Bernoulli percolation. It has also been applied many times in the context of percolation of Gaussian fields, as in [START_REF] Rivera | The critical threshold for Bargmann-Fock percolation[END_REF], [START_REF] Muirhead | The sharp phase transition for level set percolation of smooth planar Gaussian fields[END_REF] for instance.

Proof of Proposition 3.5. Notice that for each 0 < r < R and ν > 0 and any two

f, g ∈ C(R d ), if δ = sup x∈[-3R,3R] 3 ×{0} d-3 |f (x) -g(x)|, (3.4) f ∈ AlmUni -ν, +ν (r, R) ⇒ g ∈ AlmUni -ν-δ, +ν+δ (r, R) .
Let us now introduce a parameter t 0 > 0 and an increasing sequence of positive real numbers (s k ) k to be fixed later. Recall the fields f s d,n introduced in Lemma 3.1. This lemma implies that there exists C 1 < +∞ depending only on R 0 such that, for k larger than some k 0 = k 0 (n) ∈ N, with probability at least 1 -e -(k+t 0 ) 2 /2 , the following holds (3.5) sup

x∈[-0,10R k+1 ] 3 ×{0} d-3 |f s k d,n (x) -f d,n (x)| ≤ C 1 log(1 + 3 k R 0 )(1 + k + t 0 )s -1 k =: δ k .
Let ν k = ν 0 + k j=1 δ j . At this point, we fix R 0 ∈ [R min , e n ] as a function of n and ν 0 so that it becomes possible to define s k as a function of t 0 , R 0 and ν 0 so that (3.6) sup

k ν k -ν 0 = k≥1 δ k ≤ ν 0 and n ≤ s 0 ≤ s k ≤ n 1/2 R k .
From (3.4) and (3.5) we immediately deduce that, if

x ∈ R k Z 2 and T ∈ G are such that dist(x + T ([0, 2R k ] × [0, R k ]), [0, 2R k ] × [0, R k ]) ≥ R k , then, as long as k ≥ k 0 , for each ν > 0, (3.7) P [¬Cross k,ν (0) ∩ ¬Cross k,ν (x, T )] ≤ P [¬Cross k,ν-δ k (0)] 2 + e -(t 0 +k) 2 /2 .
Together with Lemma 3.7, (3.7) implies that, setting p k := P [¬Cross k,ν k (0)] we have, for all k ∈ N,

(3.8) p k+1 ≤ C 0 p 2 k + e -(t 0 +k) 2 /2 .
On the other hand, by (3.3), p 0 ≤ ε 0 . Choosing ε 0 small enough and t 0 large enough, we deduce that for all k ≥ 0, p k ≤ e -2 k +O(k) ≤ e -(7/4) k .

Since by (3.6) ν k ≤ 2ν 0 , we deduce that (3.9) P [Cross k,2ν 0 (0)] ≥ P [Cross k,ν k (0)] ≥ 1 -e -(7/4) k .

To complete the proof of Proposition 3.5, observe that there exist a universal c 2 < +∞ and a covering of [-10 k R 0 , 10 k R 0 ] 2 with O(R c 2 k ) rectangles of size 2R k ×R k such that any R 0 Z 2path of (euclidean) diameter at least 10 k/2 R 0 must cross at least one of these rectangles width-wise. On the other hand, for k large enough, by union bound and (3.9) the events of the form Cross k,2ν 0 (x, T ) associated to these rectangles all happen simultaneously with probability at least 1 -e -(3/2) k as long as k is large enough. And if they do all happen and the covering is well chosen, the crossings must intersect any path of diameter at least 10 k/2 R 0 and must also be mutually connected by paths of 2ν 0 -open edges. Since ν 0 > 0 is arbitrary, we have reached the desired conclusion.

A An upper bound on the C k norm of Gaussian fields

In this section we recall a classical tail bound for the C k norm of Gaussian fields which we use repeatedly throughout the text. A second application of the Borell-TIS inequality completes the proof.

Lemma 2 . 3 .

 23 Let W be the L 2 (R d ) white noise. For each n ∈ N, n > d+1 let φ d,n = d,n * W and ψ d,n = γ d,n * W . Then, for each k ∈ N, there exists a constant C = C(k, d) < +∞ such that for each n ∈ N large enough, for each r ≥ 2 and each t > 0,

  0) has unit variance and by (2.4), σ d,n = 1 + O(n -1 ) so estimate (2.7) thus follows from Lemma A.1. 3 Application of Proposition 1.1: Proof of Theorem 1.2

.

  Write f s d,n = s d,n * W where W is the L 2 white noise on R d . Then, for each m ∈ N large enough there exists C = C(m, δ, d) < +∞ such that for each x ∈ R d , for each n ∈ N large enough and for each t > 0, s ≥ n (1+δ)/2 and r ≥ 1,P sup x∈[0,r] d |f d,n (x) -f s d,n (x)| ≥ C(1 + t)s -m ln(1 + r) ≤ e -t 2 /2 . Proof. Fix δ > 0. Let n ∈ N, n ≥ d + 1 and s ≥ n (1+δ)/2 . Let s d,n : x → d,n (x) -s d,n (x). Thus, f d,n -f s d,n = s d,n * W . The derivatives of s d,n up to any order are bounded uniformly in n by derivatives of d,n up to the same order. Therefore, by Lemma 2.1, for each m ∈ N, there exists C 1 = C 1 (m) < +∞ such that for each x ∈ R d each n ∈ N, large enough and each β ∈ N d satisfying |β| ≤ 1,

Theorem 3 . 3 (

 33 Almost Uniqueness for g d,n ). [Proposition 5.1 and Lemma 5.

  ν-open implies being ν open for each ν ≥ ν. We endow R 0 Z 2 with its usual graph structure and define a path in R 0 Z 2 to be a sequence of vertices each of which is connected to the next by an edge of the graph. Given a rectangle of the form [a, b] × [c, d], we call a left-right crossing of this rectangle a path in [a, b] × [c, d] ∩ R 0 Z 2 which connects a vertex x = (x 1 , x 2 ) whose first coordinate x 1 is minimal among the first coordinates of the vertices of [a, b] × [c, d] ∩ R 0 Z 2 with a vertex whose first coordinate is maximal among these. Given x ∈ R k Z 2 and ν > 0 let Cross k,ν (x) := There exists a left-right crossing of the rectangle x + [0, 2R k ] × [0, R k ] made of ν-open vertices .

Lemma A. 1 (

 1 Upper bound on C k norm). Let f be a Gaussian field on R d . Fix k ∈ N. Let κ(x) = E [f (0)f (x)] be the covariance of f . Assume that κ ∈ C 2k+2 (R d ). Then, there exists C = C(d, k) < +∞ such that the following hold. Let σ 2 be such that for eachβ ∈ N d with |β| ≤ k + 1, |∂ 2β κ(0)| ≤ σ 2 .Then, for each t > 0 and r ≥ 1,P sup |β|≤k sup x∈[0,r] d ∂ β f (x) ≥ C(1 + t)σ ln(1 + r) ≤ e -t 2 /2 .Proof. First, by homogeneity, we may assume that σ = 1. Next, by Kolmogorov's theorem (see Appendix A.9 of[START_REF] Nazarov | Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions[END_REF]), we see thatE sup |β|≤k sup x∈[0,1] d ∂ β f (x) = O(1). By the Borell-TIS inequality (see Theorem 2.8 of [2]), the quantity sup |β|≤k sup x∈[0,1] d ∂ β f (x) has Gaussian tails. A union bound then shows that E sup |β|≤k sup x∈[0,r] d ∂ β f (x) = O( d ln(1 + r)) .

  2 white noise on R d . By (2.2) and (2.4), the fields d,n * W and γ d,n * W are equal to f n | R d ×{0} n-d and close to g d,n respectively. In order to prove Proposition 1.1 we first prove that d,n and γ d,n are close in Lemma 2.1. Then, in Lemma 2.3 below, we deduce that the two fields d,n * W and γ d,n * W are close to each other. The proof of Proposition 1.1, given at the end of this section, follows readily from Lemma 2.3.

  n . Then, for each n ∈ N large enough η d,n is an a.s. smooth centered stationary Gaussian field on R d . Moreover, also by Lemma 2.1 applied for some m ∈ N such that 4m > d, for each α ∈ N d , uniformly for each n ∈ N large enough and x

I.e. the Fourier transform of their covariance functions.