Sylvain Gnamien

Véronique Yoboué

Cathy Liousse

Sékou Keita

Julien Bahino

Silué Siélé

Lamine Diaby

Particulate pollution in Korhogo and Abidjan (Cote d'Ivoire) during the dry season

Keywords: Urban air quality, Aerosols, Africa cities, PASMU project

Particulate pollution in West African cities seriously impacts public human health. Assessing the levels of pollution to which populations are exposed is problematic, as to date very few countries in Africa have an air quality monitoring network in place. However, given the specific anthropogenic sources present in West African countries and the increase in their projected emissions in the coming years if no regulations are put in place, solutions must be found. This study evaluates with a simple existing methodology particulate air pollution in two West African cities (Korhogo and Abidjan in Cote d'Ivoire) which have different population practices and local meteorology.

From two measurement campaigns at about ten sites during the dry season and using the inverse distance-weighted interpolation method, maps of spatial variation in PM10 and PM2.5 concentrations are obtained at the scale of each of these cities, which provide a snapshot of the concentration levels to which populations are actually exposed. The results of this study showed that populations in low-and middle-income areas are exposed to higher concentrations than those in high-income areas. The analysis of the relative contribution of fine particles to coarse particles shows that PM2.5 contributes to PM10 at less than 40% in Korhogo but more than 50% in Abidjan. Daily average concentrations of PM2.5 are all above WHO standards, and are 2 to 8 times higher in Korhogo, while in Abidjan they are up to 4 times higher. The daily profiles of the mean hourly concentrations of the sites are similar across all sites in Korhogo, while in Abidjan, that of some sites show some differences.

This study, by the methodology used, offers an opportunity for different research teams to assess at lower cost the population's exposure levels to urban particulate pollution.

A C C E P T E D M A N U S C R I P T INTRODUCTION

Air pollution is a global public health problem and its extent in the cities of many developing countries is still poorly known [START_REF] Kirenga | The State of Ambient Air Quality in Two Ugandan Cities : A Pilot Cross-Sectional Spatial Assessment[END_REF]. Among all pollutants, particulate pollution is highly harmful to health, especially in low-and middle-income countries [START_REF] Kalisa | Chemical and Biological Components of Urban Aerosols in Africa[END_REF].

Recent studies have shown alarming levels of particulate pollutants in West African cities [START_REF] Dieme | Relationship between physicochemical characterization and toxicity of fine particulate matter (PM2.5) collected in Dakar city (Senegal)[END_REF][START_REF] Djossou | Mass concentration, optical depth and carbon composition of particulate matter in the major southern West African cities of Cotonou (Benin) and Abidjan (Côte d'Ivoire)[END_REF], due to various sources of pollution that are generally uncontrolled and relatively close to living areas Val et al., (2013) and Adon, ( 2019) highlighted the inflammatory nature of particles rich in organic carbon (OC) and small in size emitted by urban west African sources. Also, a few studies [START_REF] Cohen | The Global Burden of Disease Due to Outdoor Air Pollution[END_REF][START_REF] Liousse | Dynamicsaerosol-chemistry-cloud interactions in West Africa -Inflammatory risk regional maps[END_REF] have shown the importance of premature death estimations for cardiovascular, lung cancer and respiratory diseases causes, due to anthropogenic fine particles in West Africa.

Anthropogenic sources are harmful in developing countries because of the absence or ineffectiveness of regulations on pollutant emissions. The typical anthropogenic sources in sub-Saharan Africa are domestic fires (biomass and gas for cooking), road traffic (combustion emissions and dust resuspension), waste combustion, industrial activities (petrochemicals, agriculture, construction. More precisely, air quality in Cote d'Ivoire and in West Africa, is impacted by emissions from a very old vehicle fleet, on average 20 years according to the Ministry of Transports, (2017), using poor quality fuel and from the use of traditional stoves with wood and/or charcoal incomplete consumption. Furthermore, open burning of waste in urban areas of Africa is a very common phenomenon due to the ineffectiveness of municipal waste collection and management policies [START_REF] Keita | African Anthropogenic Emissions Inventory for gases and particles from 1990 to[END_REF]. In addition to these local sources, there are important imported sources of aerosols: (1) desert dust and natural savannah fire sources from prevailing

Harmattan wind (north wind) as well as (2) marine aerosols brought by monsoon flow (southwest wind).

A C C E P T E D M A N U S C R I P T

3 Unlike developed countries, most cities in the developing countries do not have the capacity to monitor urban air concentrations in order to effectively solve the problem of air pollution. In West Africa, Dakar (in Senegal), is the only city with an air quality monitoring network. Considering the anthropogenic source diversity and their strength which is expected to increase if no regulation is taken [START_REF] Liousse | Explosive growth in African combustion emissions from 2005 to 2030[END_REF], it is necessary to expand pollution studies in African cities.

Studies carried out in West Africa have shown high concentrations of PM2.5 and PM10 in several cities: Accra in Ghana [START_REF] Dionisio | Within-Neighborhood Patterns and Sources of Particle Pollution: Mobile Monitoring and Geographic Information System Analysis in Four Communities in Accra, Ghana[END_REF][START_REF] Arku | Characterizing air pollution in two lowincome neighborhoods in Accra, Ghana[END_REF]; Dakar in Senegal [START_REF] Dieme | Relationship between physicochemical characterization and toxicity of fine particulate matter (PM2.5) collected in Dakar city (Senegal)[END_REF]Val et al., 2013) and Bamako in Mali (Val et al., 2013 and[START_REF] Antonel | Measuring ambient particulate matter in three cities in Cameroon, Africa[END_REF][START_REF] Antonel | Measuring ambient particulate matter in three cities in Cameroon, Africa[END_REF]. In the European program DACCIWA (Dynamic-Aerosol-Chemistry-Cloud Interactions in West Africa), [START_REF] Djossou | Mass concentration, optical depth and carbon composition of particulate matter in the major southern West African cities of Cotonou (Benin) and Abidjan (Côte d'Ivoire)[END_REF]

METHODOLOGY Description of the experimental sites

In this study, two measurement campaigns were conducted successively in Korhogo (January It is important to note that local stations are therefore not included in this study as they only provide very local information [START_REF] Jeannée | Évaluation de l'exposition en zones urbaines à la pollution atmosphérique : méthodes existantes et application aux PM10 en France métropolitaine[END_REF]. In our study, urban background stations are mainly influenced by major west-African urban sources, i.e. the domestic fires, the waste burning and the road traffic (from combustion or non-combustion). In addition, dust resuspension of particulate matter is also important because of road conditions and urban developments. In this study, 14 non-permanent sites in Korhogo and 20 in Abidjan were sampled for 3 days (2 working days and 1 weekend day). The 2 cities and their sampling sites are described below.

3

Korhogo sampling sites

Korhogo is the third largest city in Cote d'Ivoire in terms of population. A large part of the city's area is not built. Korhogo is a city still marked by agricultural activities with a few rare agricultural product factories for processing and packaging. The principal sources of anthropogenic emissions in Korhogo are traffic sources with intense activities of heavy vehicles, national bus line but also motorcycle traffic, which is a particularity of Korhogo. Furthermore, in this city, only main roads are paved and represent, 10% of the urban roads [START_REF] Ageroute | Etendue du Réseau Routier Ivoirien[END_REF]. This implies important road resuspension of aerosols which is also a major source of PM emissions. Another important anthropogenic source is the domestic fire with the use of firewood for economic reasons or for convenience as because these resources are still available in the surrounding villages. In addition to the local anthropogenic sources present in Korhogo, the harmattan wind is a major contributor of PM in the dry season with desert dust, biomass burning and aerosol transport. Fig. 2 shows the geographical distribution of the 14 measurement sites as well as the boundaries of the city of Korhogo. Also, the specificities of each sites are summarized in Table S1 (in Supplementary files),

where are mentioned site localizations as well as roads infrastructure and possible sources of

A C C E P T E D M A N U S C R I P T

6 pollution around. Additional information like that geographic positions of Korhogo measurement sites is given in Table S2 (in Supplementary files). Sites K2, K10, K11 and K14 are surrounded by paved roads (i.e. under the influence of the traffic source) and far from other urban sources. The other sites are located in residential areas where wood is used extensively for cooking, with the exception of K1 site which is in a peri-urban area but close to a major traffic road.

Abidjan sampling sites

Abidjan is the economic capital regrouping the main industrial activities with a population of 4,707,404 inhabitants for an area of 2,119 km 2 according to the General Census of Population and Housing (RGPH, 2014).

Abidjan is an Autonomous District with different levels of urbanization and includes populations of different social classes. Its principal anthropic sources are also traffic, domestic fires and waste burning sources with different practices depending on the communes. For instance, wood is intensively used in the commune of Yopougon whereas the use of gas dominates in Cocody.

Less road dust particles compared to Korhogo are expected since more than 50% of roads are paved according to MCLAU, (2015), (MCLAU for Ministère de la Construction, du Logement, de l'Assainissement et de l'Urbanisme). However, this value also depends on the communes. In Cocody, Le Plateau and Marcory, a higher percentage of paved roads are observed compared to Abobo and Yopougon, where a large part of the city population lives.

Fig. 3 shows the geographical distribution of the measurement sites in Abidjan and boundaries of the sampled communes, which represent the study area of this city. In Supplementary files, Table S5 provides information on the 20 sites sampled in Abidjan. In this study, only the sub-prefecture of Bingerville (Abidjan peripheral municipality) is associated with the 10 municipalities of Abidjan, because Bingerville city is completely integrated into Abidjan. As shown in, in Supplementary files,

A C C E P T E D M A N U S C R I P T

7 in Table S5, measurement sites are located in neighborhoods inhabited by high-income populations (A1, A2, A11, A20, A4, A6, A8, A14, A15), middle (A18) and low (A3, A5, A7, A9, A10, A12, A13, A16, A17, A19). The latter sites are generally close to unpaved roads. In addition, Table S6 (in Supplementary files), give latitude and longitude of Abidjan measurement sites.

Meteorological data and surrounding pollution

To assess the impacts of weather parameters on PM concentration levels measured at each site during the 3-day sampling period, temperature, wind intensity and direction are analyzed at local As expected, Korhogo is under the influence of northern trajectories whereas Abidjan is under the influence of southern air masses, associated to air masses coming from the Nigerian coast.

In order to identify which aerosols could be advected into Korhogo and Abidjan using these of the study period. Associated with Figs. S1, this shows that Korhogo is under the influence of regional winds that carry desert dust and biomass burning aerosols coming from the north and south of the city. While at Abidjan these regional winds show that only marine aerosols are expected.

Material and Methods

Measurement campaigns and equipment

The first step of the methodology deals with intensive data collection through measurement campaigns conducted at sites in the two cities, as presented in previously. The measurement campaigns of PM10 and PM2.5 concentrations were carried out with 4 portables PM measuring devices (Aeroqual monitor S500 with PM10/PM2.5 sensor). Particulate pollution is measured over 3 days at each site, with 2 consecutive working days and 1 weekend day, because these 2 periods (working and weekends days) present different activities levels in urban areas. Indeed, studies have shown different levels of pollution at these 2 periods of the week [START_REF] Adame | Weekend-Weekday Effect Assessment for O3, NOx, CO and PM<sub>10<sub> in Andalusia, Spain (2003-2008[END_REF]; Doumbia et

A C C E P T E D M A N U S C R I P T

sites of each city because rotations are organized. During the campaigns, the measurements are not simultaneous on all sites of each city. Rotations are organized with at maximum 4 sites running in parallel. This is possible due to atmospheric stability detailed in the meteorological paragraph. This technique of measuring on non-permanent sites during short periods allows to maximize measurements on a large number of non-permanent sites with a small number of equipment.

In this study, optical PM10/PM2.5 sensor on S500 Aeroqual monitor is used to measure particulate pollution, because it is easy to use and take up very little space. This sensor covers a concentration range from 0 to 1000 µg m -3 , and simultaneously measures PM10 and PM2.5. These types of sensors are based on light scattering and have an accuracy that can be influenced by the physical and chemical characteristics of aerosols (Wang et al., 2015). The measuring device has been placed in a box to protect it from natural elements. The device is installed at a height of between 2 and 3 m. The time step chosen is 3 minutes, in order to allow fine sampling and to store the data over the entire measurement period. After data retrieval, the data logger of the equipment is formatted before being installed at another site.

Quality Control data:

The evaluation of the uncertainties associated with low-and middle-cost sensor measurements is the subject of several studies, as shown by Wang et al., (2015) who tested 3 low-cost sensors in combustion chambers with different results. Consequently, it is necessary to evaluate and correct optical sensor concentration measurements against a reference method [START_REF] Clements | Low-Cost Air Quality Monitoring Tools[END_REF]. The reference method here proposed, is the gravimetric method. At some of the selected sites, aerosol sampling lines are installed, including a fixed flow pump, a volume meter, a valve, an aerosol size selection system (cane or cut-off head). These lines allow sampling by filtration of aerosols on quartz filters, over the same periods the same sites than the low-cost sensor measurements. The filters are weighed on a submicron balance before and after exposure in ) and the DACCIWA program [START_REF] Djossou | Analyse de la pollution atmosphérique en zones urbaines en Côte d'Ivoire et au Bénin dans le cadre du programme DACCIWA : Pollution particulaire[END_REF].

Geometrical means of the measurements obtained with low-cost sensors are compared with those obtained by the gravimetric method (reference method). Geometrical means concentrations of optical sensors are consistently lower than the gravimetric concentrations of 2.7±1.1 and 3.5±0.9

for PM10 and PM2.5 respectively. The standard deviation of these coefficients is higher for PM10 than for PM2.5, the latter having more points of comparison. It is worth noting that these values are in agreement with those of Zuidema et al., (2019) who showed the need for a correction factor of 2.9 for an optical photometric sensor (GP2Y1010AU0F, Sharp Electronics, Osaka, Japan) compared to the gravimetric method. Similarly, [START_REF] Egondi | Measuring exposure levels of inhalable airborne particles (PM 2.5) in two socially deprived areas of Nairobi, Kenya[END_REF] also showed overestimated concentration values given by the DustTrak II 8532 portable samplers compared to a reference method. These factors (2.7 and 3.5) are then applied to PM10 and PM2.5 concentrations at all measurement sites.

Spatial mapping

Geographic Information System (GIS) provide a robust method of spatial analysis through interpolation. The most commonly used methods, kriging (geostatistical method) and reverse weighting of IDW (deterministic method), have already been compared in works. For example, Shukla et al., (2020) in the Delhi megacity showed that kriging and IDW methods are associated to errors close to 22% and 24% respectively. In addition, [START_REF] Bahino | A pilot study of gaseous pollutants' measurement (NO2, SO2, NH3, HNO3 and O3) in Abidjan, Côte d'Ivoire: contribution to an overview of gaseous pollution in African cities[END_REF]

A C C E P T E D M A N U S C R I P T

11 a comparison between kriging and IDW, and showed that IDW method is associated to lower uncertainty than kriging method. In addition, our measurement network (number of points and geographic locations) is developed on the basis of the network of [START_REF] Bahino | A pilot study of gaseous pollutants' measurement (NO2, SO2, NH3, HNO3 and O3) in Abidjan, Côte d'Ivoire: contribution to an overview of gaseous pollution in African cities[END_REF]; therefore, in this study, the IDW method is chosen and applied to the collected data.

ArcGIS version 10.3 software (ESRI Inc.) is used because it combines geostatistical data processing and map edition. The shapefiles for map editing are available for free on OpenStreetMap (https://www.openstreetmap.org). Briefly, IDW interpolation allows to predict the concentrations around a sampled point from the following formula presented in Eq. ( 1).

𝑍 𝑝 𝑗 = ∑ λ 𝑖 * 𝑍 𝑂 𝑖 𝑁 𝑖=1

(1)

where, 𝑍 𝑂 𝑖 is the observed value at location i; form geometrical mean of measured values at each site during the different measurement campaigns. These geometrical means, averages deviation, minimums and maximum are available, for Korhogo in Table S3 (for PM10) and Table S4 (for PM2.5) and for Abidjan in Table S7 (for PM10) and Table S8 (for PM2.5), in Supplementary files for each site.

𝑍 𝑃 𝑗 is the predicted value for location j;

N is the number of measured sample points surrounding the predicted location that will be used in the prediction;

𝜆 𝑖 are weighting factors applied at each measured value at location Oi. They decrease according to the separation distance.

The geostatistical analysis of the data enables the values to be evaluated in order to develop the spatial variation maps. Thus, the IDW method calculates values from the collected data for each point (unsampled and sampled) and gives average predicted values everywhere [START_REF] Rivera-González | An assessment of air pollutant[END_REF]Zamri et al., 2012;[START_REF] Bahino | A pilot study of gaseous pollutants' measurement (NO2, SO2, NH3, HNO3 and O3) in Abidjan, Côte d'Ivoire: contribution to an overview of gaseous pollution in African cities[END_REF]. (2)

A C C E P T E D M A N U S C R I P T

𝑀𝐴𝐸 = 1 𝑁 ∑ |𝑍 𝑃 𝑖 -𝑍 𝑂 𝑖 | 𝑁 𝑖=1
(3) where, 𝑍 𝑃 𝑖 and 𝑍 𝑂 𝑖 are respectively the predicted and observed value for a location i; N is the number of measured sample point.

Table 1 presents such values for the 2 cities as well as the coefficient of variation of error. In Abidjan, the MAE values are lower than the RMSE values due to the high uncertainties associated with some sites whereas no marked differences appear in Korhogo. Also, both coefficients are higher in Korhogo than in Abidjan: that could be due to the high concentration values found in Korhogo. As shown in Table 1, the coefficients of variation of error in Abidjan are higher than in Korhogo. These low values obtained in Korhogo may be due to a smaller study area than in Abidjan. This analysis highlights the impact of the distance between measurement sites on the uncertainties.

RESULTS AND DISCUSSIONS

A C C E P T E D M A N U S C R I P T

13

Spatial distribution of PM concentrations

Korhogo PM concentrations maps

As previously described, the PM10 and PM2.5 concentrations obtained by geographical averages allowed the mapping of spatial variations for the Korhogo study area. These maps are presented in K13 and K5 sites have also high PM10 concentrations because they are in neighborhoods with unpaved roads. Moreover, even if downtown roads are mostly paved, a high resuspension of dust by intense traffic is also observed. Indeed, as mentioned earlier, dust deposition on roads are also observed at Korhogo due to northerly air masses transporting desert dust aerosols at the period of this study (Fig. S1). Such aerosols significantly contribute to the pollution level of Korhogo.

Finally, Fig. 5(a) shows that in Korhogo, PM2.5 and PM10 spatial variation maps roughly present the same pattern of high and low concentrations despite some differences which are detailed in the following section.

A C C E P T E D M A N U S C R I P T

14

Abidjan PM concentrations maps

The spatial variation maps of PM10 and PM2.5 concentrations (obtained from the geographical averages of the measured concentration time series) in Abidjan study area are given respectively in Figs. 6(a) and 6(b). The average concentrations are between 38.1 µg m -3 and 160.4 µg m -3 for PM10 and 23.8 µg m -3 and 113.4 µg m -3 for PM2.5. For PM10, the highest concentrations are found at sites A7 (160.4 µg m -3 ), A18 (125.3 µg m -3 ) and A20 (120.7 µg m -3 ). The A7 and A20 sites are located nearby unpaved roads (Table S2) therefore, dust from road resuspension contribute to the increase of concentrations. In A18 site, the high concentrations may be explained by the proximity of important road traffic and industrial activities. In addition, A18 and A19 sites are located on the sea front, and are therefore subject to a high contribution of marine aerosols.

Sites included in areas with better quality infrastructures and with middle-or high-class populations, have generally lower PM10 concentrations (e.g. A2, A11) compared to sites in neighborhoods with poor quality infrastructures (e.g. A3, A5, A7, A8, A16, A18 and A20).

Compared to PM10, PM2.5 spatial variation (Fig. 6(b)) presents high concentrations over most of the study area, despite some very localized low concentrations in A1 (32.9 µg m -3 ), A2 (24.9 µg m - 3 ), A5 (30.8 µg m -3 ), A14 (23.8 µg m -3 ) and A19 (24.5 µg m -3 ). PM2.5 concentrations in northern Abidjan are higher than in southern Abidjan. Indeed, the wind rose of Abidjan shows a dominance of south, southwest winds, which could lead to a transport and an accumulation of pollutants in the northern area of the city, as for [START_REF] Bahino | A pilot study of gaseous pollutants' measurement (NO2, SO2, NH3, HNO3 and O3) in Abidjan, Côte d'Ivoire: contribution to an overview of gaseous pollution in African cities[END_REF]. Values for these sites (A7 with 113.4 µg m -3 , A9 with 91.7 µg m -3 , A12 with 73.2 µg m -3 and A20 with 91 µg m -3 ) are 2 to 6 times higher than the WHO daily standard (25 µg m -3 ). Moreover, these sites mainly appear in low income neighborhood.

Indeed, in such areas, high concentrations result from the combination of the use of wood for cooking, the proximity of public transport (small bus: gbaka, municipal taxis: wôrô-wôrô) with very old and poorly maintained vehicles and the burning of solid domestic waste due to collection

A C C E P T E D M A N U S C R I P T

15 failure. In contrary, the less densely populated areas (high-class districts) have lower concentrations (sites A1, A2, A14) than the medium and low-class districts. In these areas (sites A1, A2, A14), people have higher incomes and are more likely to use gas for cooking than wood and charcoal for cooking activities. Lower concentrations are observed in A5 and A19 sites, although located in popular districts. That may be due to less intensive combustion activities than usually observed in these districts for the A5 site and to the proximity of the sea front for A19 site.

Relative importance of fine particles

In order to highlight the relative contribution of particulate size and aerosol sources, spatial variation maps of the percentage of relative contribution of PM2.5 to PM10 are presented in Figs. 7 and 8 for Korhogo and Abidjan respectively. Low values represent a high contribution of PM2.5 to PM10, while high percentages represent a low contribution of PM2.5 to PM10. It is important to note that these figures show totally different values for the two cities. In Korhogo, more than 50% of PM10 are coarse particles, with the highest values in the north and south of the city and the lowest values in the city center, where most urban activities take place. In Abidjan, the situation is totally different: coarse particles in PM10 represent less than 40%, with the highest values in A4, A5, A14, A18 and A19 sites and the lowest values in A6, A9, A11, A12, A13 and A15.

In both cities, the lowest values associated with high PM2.5 are found particularly in densely populated sites or in economically very active neighborhoods with intense combustion sources and fine particle emissions. In addition, the percentages of coarse particles (PM2.5-10) relative to PM10 are high near unpaved roads and sometimes in areas with heavy traffic.

In Abidjan, low contributions of PM2.5 to PM10 are also observed in industrial areas (site A18).

In these areas, heavy truck traffic damages the roads and causes dust resuspension. In addition, a contribution from marine aerosols is possible due to the south-westerly (SW) wind direction. These In Korhogo, all the sites show the same daily average concentration profiles for PM10 and PM2.5 respectively in Figs. 9(a) and 9(b), with two very important daily peaks. One peak in the morning and one in the evening. These peaks are due to the mix of several local sources: mainly, household sweepings (twice a day), use of firewood (morning and evening), and road traffic (dominated by 2-wheelers). Focusing on these peaks, PM2.5 morning and evening peaks are of similar importance whereas PM10 peak is higher in the evening than in the morning. This could be due to favorable conditions for dust resuspension due to the higher dryness of the soil in the evening caused by sunshine during the day. According to [START_REF] Cheng | Influences of Traffic Emissions and Meteorological Conditions on Ambient PM10 and PM2.5 Levels at a Highway Toll Station[END_REF] and [START_REF] Gualtieri | Influence of road traffic, residential heating and meteorological conditions on PM10 concentrations during air pollution critical episodes[END_REF], meteorological parameters such as temperature and relative humidity influence particulate matter concentrations. Moreover, the low values observed during the day, twenty times lower than in morning could be due to a decrease of urban activities. However, it is interesting to note that such low average concentrations observed in Korhogo are still higher than the WHO daily standard.

A C C E P T E D M A N U S C R I P T

those of the evening. Traffic source are mainly responsible of these peaks. The difference between the morning and evening peaks can be explained by the fact that the flow of road traffic in the evening is more spread out in time compared to that of the morning. In -3 ) in Bafossam (Cameroon), which is further south, in a forest area, therefore less exposed to dust. Furthermore, the results obtained by [START_REF] Dionisio | Within-Neighborhood Patterns and Sources of Particle Pollution: Mobile Monitoring and Geographic Information System Analysis in Four Communities in Accra, Ghana[END_REF] showed that concentrations in low-income areas are higher than in high-income areas in Accra, which is in agreement with the results of this study. Finally, the ranges of concentrations obtained at Korhogo (in Tables S3 andS) and Abidjan (in Tables S7 andS8) measurement sites are on the same order of magnitude as those obtained in other studies carried out in Sub-Saharan Africa. Values from other studies conducted in Africa are presented in Table S9 and agree very well with the results of this study.

CONCLUSIONS

This study provides an assessment of air quality in the cities of Korhogo and Abidjan in Cote The spatial variation maps obtained through the application of GIS in Korhogo show that the high activity areas (market and national bus station) have higher PM10 and PM2.5 concentration levels than other areas. Our study also highlights that the relative contribution of PM2.5 to PM10 is very different in Korhogo and Abidjan, varying from 50 to 100% in Abidjan, while 20 to 50% in Korhogo. In Abidjan, a difference between the concentrations is observed in high-and low-income neighborhoods, with higher concentration levels in low-and middle-income neighborhoods. One likely reason is that, high-income areas have better services (roads, waste collection, ...), and also use less polluting cooking methods (e.g. gas). This result can be also explained by the status of the city: (1) in Korhogo, the higher presence of coarse particles is due to anthropogenic (road dust resuspension) and natural sources (desert dust transport); (2) Abidjan is an economic capital with a higher population density, road traffic and generally a higher activity density than in Korhogo.

The concentrations measured in this work are comparable to previous studies conducted in Abidjan and in Africa.

In conclusion, this study indicates that this simple and reproductive method provides a quality assessment of particulate pollution at the city level. Note that this method could be also used for other purposes such as performing a quick assessment of the impact of a new industrial unit or infrastructure on the surrounding air quality. Finally, the original results obtained on urban

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T

List of Table titles

Table 1. Interpolation error statistics, where RMSE is root mean absolute error in µg m -3 ; MAE is mean average error in µg m -3 and CV is mean of coefficient of variation of error in %.

  observed average concentrations (February 2015 to March 2016) in Abidjan (Cote d'Ivoire) and Cotonou (Benin), well above WHO standards. Similarly, in the POLCA program (POLlution of African Capitals), Doumbia et al., (2012) measured PM2.5 concentrations at an urban site in Dakar from June 2008 to May 2009, which were above the WHO standard. These various studies highlight overexposure of populations to atmospheric particles whose economic impacts were assessed by Stern and van Dijk, (2017), and mortality by[START_REF] Cohen | The Global Burden of Disease Due to Outdoor Air Pollution[END_REF] and in Global Burden of Disease Study 2010 byMurray et al., (2012). Also, within the framework of DACCIWA,[START_REF] Keita | African Anthropogenic Emissions Inventory for gases and particles from 1990 to[END_REF] have calculated emission factors (EF) for particulate pollutants based on sources and considering the specificities of African regions in order to improve emission inventories for these pollutants. These results showed very high EF values, highlighting the importance of particulate pollutant emissions from sources such as domestic fires and traffic, which are very important sources of pollution in West Africa. These different results are a challenge for us and we must act.The lack of air pollution monitoring networks, even in the most urbanized cities, due to the high cost of acquiring and maintaining air quality measurement equipment can be offset by the -and medium-cost sensors offering simple and inexpensive alternatives.Although accessible, these sensors cannot be deployed always and everywhere. To overcome this problem, an experimental existing methodology based on the use of low-cost sensors during intensive measurement campaigns and a geostatistical treatment tool is used to assess particulate air pollution in two cities of Cote d'Ivoire. Specifically, this study quantifies the level of PM10 and PM2.5 concentrations and their spatial and temporal distribution in these cities of different geography and morphology (Korhogo and Abidjan). This work is carried out within the framework of the project Air Pollution and Health in Urban Areas, Cote d'Ivoire (PASMU).

  rd to January 16 th ) and in Abidjan (January 20 th to February 10 th , 2019), whose economic activities, transport specificities and types of roads, cooking fuels, population practices and local meteorology differ. As shown in Fig.1, the population densities of these two cities are also very different, with a population of Abidjan 10 times that of Korhogo. It can also be noticed that population densities from south to center of Cote d'Ivoire are generally higher compared to areas further north. The population density of Korhogo remains high for a northern region due to agricultural industries. In each city, the measurement sites are chosen to allow representative mapping of the variability of pollution including different districts. Generally, measurement sites can be of 3 types: (1) Local stations, generally used to control classified establishments, are located in proximity of the polluted source to be monitored and therefore under the direct influence of this pollution source; (2) Urban background stations, used to monitor urban pollution, are located in an urban environment but influence of pollution sources, and (3) Background stations, are located in a periurban environment.

  and regional scales. These meteorological parameters are obtained from the data of the synoptic stations managed by the Agency for the Safety of Air Navigation in Africa and Madagascar (ASECNA) at Korhogo Airport (station No. 65536) and at Felix Houphouet-Boigny Airport in Abidjan (station No. 65578).Average daily temperatures (available on https://www.accuweather.com/fr/ci/cote-divoireweather) show a range of 32 to 36 °C in Korhogo and 31 to 35 °C in Abidjan. These daily averages do not vary by more than one degree over several consecutive days. This may suggest stability of the atmosphere.Wind roses presented in Fig.4highlight important north (N) and north-east (NE) winds in Korhogo (Fig.4 (a)) and south-west (SW) winds in Abidjan (Fig.4 (b)). The regional climate is defined by the position of the Intertropical Convergence Zone (ITCZ), which regulates the succession of weather seasons[START_REF] Nouaceur | Rainfall Variability and Trend Analysis of Rainfall in West Africa (Senegal, Mauritania, Burkina Faso)[END_REF]. During measurement campaign, the ITCZ is located above the latitude of Abidjan, favoring the arrival of air masses from the south to Abidjan and air masses from the north to Korhogo where high temperature and low relative humidity are typical for dry seasonal conditions. Such features are confirmed by the use of HYSPLIT trajectories (https://www.ready.noaa.gov/HYSPLIT_traj.php) as shown by Fig.S1(in . HYSPLIT trajectories represent the movement of air masses arriving at the study areas during the measurement campaigns. Fig.S1shows 14-and 21-days back-trajectories before the end dates of each measurement campaign respectively for Korhogo and Abidjan respectively. It gives an overview of the dominant air mass origins over the measurement period.

  trajectories, emphasis was put on MODIS active fire maps and Aerosol Optical Thickness data (AOD), obtained from https://neo.sci.gsfc.nasa.gov. Figs. S2 present actives fires (a and b) and Aerosol Optical Depth (c and d) in January (a and c) and February (b and d) 2019, representative

  the collected aerosol masses. This reference method has been used and validated in several research projects and programs such as the International Network to study Deposition and Atmospheric chemistry in Africa (INDAAF) program (ref.

  carried out a spatial distribution of gaseous pollutants (NO2, SO2, NH3, HNO3 and O3) in Abidjan (Cote d'Ivoire) and Cotonou (Benin) during the dry season within DACCIWA Program. The latter study included also

Figs. 5

 5 Figs. 5(a) and 5(b), respectively for the PM10 and PM2.5. Concentrations are very high, between

  line with our expectations and confirm the good representation of the measurement sites and the good performance of the spatial distribution, because they reflect the main sources of pollution that influence each city.Daily distribution of PM concentrationsDaily profiles of fifteen minutes average concentrations of PM are shown in Fig. 9. The average daily profiles of PM10 and PM2.5 concentrations in Korhogo are presented in Figs. 9(a) and 9(b) respectively. For Abidjan, Figs. 9(c) and 9(d) represent the sites with similar patterns of PM2.5 concentrations and the other sites respectively. Since PM10 and PM2.5 daily profiles are similar in Abidjan, only PM2.5 concentrations are presented here. Note that the daily profiles of PM10 concentrations in Abidjan are shown in Figs. S5(a) and S5(b).

  Fig. 9(d), some sites show no clear daily variation. For example, A18 and A20 sites show high concentrations throughout the day, characteristic of local economic activities. Indeed, the A8, A18 and A20 sites are respectively located in a low-income neighborhood (mainly using wood and charcoal for cooking), in a neighborhood under construction and in an industrial area respectively. The daily minimum average concentrations are 19.3 µg m -3 for PM10 and 14 µg m -3 for PM2.5. These low values are compensated by the high values observed during peaks (beyond 350 µg m -3 as well as PM10 and PM2.5), thus providing a daily average concentration well above WHO standards. To conclude, this data collection methodology provides daily profiles of PM concentrations, which are important in the analysis of population exposure levels.Comparison with other Sub-Saharan Africa citiesConcentrations of PM10 and PM2.5 in Korhogo and Abidjan are generally above the WHO daily exposure standard. As shown previously the local variability of these concentrations is strongly related to the predominant local sources.Concentrations vary on average from 212.2 µg m -3 to 534.7 µg m -3 for PM10 and from 57.4 µg m -3 to 230.4 µg m -3 for PM2.5 in Korhogo and in Abidjan, from 38.1 µg m -3 to 160.4 µg m -3 for PM10 and from 23.8 µg m -3 to 113.4 µg m -3 for PM2.5. The mean standard deviations of measured concentrations are of 89 µg m -3 and 23.5 µg m -3 for PM10 and of 41.5 µg m -3 and 17.9 µg m -3 for PM2.5 respectively in Korhogo and Abidjan. The standard deviation values are lower in Abidjan compared to Korhogo, which shows a greater variability in the latter. The maximum concentrations respectively more than 3 to 5 times higher than in Abidjan whereas the minimum are 2 to 3 times higher. As already mentioned, this may be explained by additional aerosol sources occurring in Korhogo such as dust aerosol coming from northern African areas and/or from unpaved roads and the 2-wheel emissions.

  d'Ivoire by proposing for the first time to our knowledge, spatial variation maps of PM10 and PM2.5 concentrations. It is part of the objectives of the PASMU project, which aims to characterize urban air pollution and its impact on health in Cote d'Ivoire. The use of low-cost sensors at 14 sites in Korhogo and 20 in Abidjan resulted in daily average PM10 concentrations ranging from 212.2 µg m -3 to 534.7 µg m -3 in Korhogo and from 38.1 µg m -3 to 160.4 µg m -3 in Abidjan. PM2.5 concentrations range from 57.4 µg m -3 to 230.4 µg m -3 and from 23.8 µg m -3 to 113.4 µg m -3 in Korhogo and Abidjan respectively.

Figure Captions Fig. 1 .

 Captions1 Figure Captions

Fig. 2 .

 2 Fig. 2. Localization of measurement sites in Korhogo (Cote d'Ivoire). Green diamond represents

Fig. 3 .

 3 Fig. 3. Localization of measurement sites in Abidjan (Cote d'Ivoire). Green diamond represents

Fig. 4 .

 4 Fig. 4. Wind rose from synoptic station measurements during the study periods in Korhogo (a) and

Fig. 5 .

 5 Fig. 5. Spatial variation of PM10 (a) and PM2.5 (b) concentrations in Abidjan (Cote d'Ivoire). Black

Fig. 6 .

 6 Fig. 6. Spatial variation of PM10 (a) and PM2.5 (b) concentrations in Abidjan (Cote d'Ivoire). Black

Fig. 7 .

 7 Fig. 7. Spatial variation of relative difference between PM10 and PM2.5 in percentage in Korhogo

Fig. 8 .

 8 Fig. 8. Spatial variation of relative difference between PM10 and PM2.5 in percentage in Abidjan

Fig. 9 .

 9 Fig. 9. Daily averages of PM10 and PM2.5 concentrations in Korhogo (a, b) and PM2.5

  12The Root Mean Square Error (RMSE)[START_REF] Mckeen | Assessment of an ensemble of seven real-time ozone forecasts over eastern North America during the summer of 2004[END_REF] Savage et al., 2013;[START_REF] Chai | Root mean square error (RMSE) or mean absolute error (MAE) -Arguments against avoiding RMSE in the literature[END_REF] and the Mean Absolute Error (MAE)(Taylor et al., (2013);[START_REF] Chatterjee | Background error covariance estimation for atmospheric CO2 data assimilation[END_REF]

	Jerez et al., (2013) are the two parameters widely used to evaluate the quality of the predictions of
	interpolation methods. In order to assess the quality of the prediction in the two cities, both RMSE
	and MAE values are calculated for each measurement campaign following the formulas present in
	Eqs. (2) and (3).
	𝑅𝑀𝑆𝐸 = √ 𝑁 1	∑ (𝑍 𝑃 𝑖 -𝑍 𝑂 𝑖 ) 𝑁 𝑖=1	2

Table 2

 2 presents the daily concentrations measured in African urban areas by this study, compared to literature results for Sub-Saharan Africa. PM2.5 concentrations measured in this study are lower than those measured with the individual exposures of Xu et al., (2019) in Abidjan. Personal exposure measurements are generally closer to the emission sources than our sites. The PM10 measurements performed by Ana et al., (2014) from January to March 2008 in Ibadan (Nigeria) are on the same order of magnitude as the concentrations measured at Korhogo in January 2019. These PM10 concentration levels are characteristic of the dry season in West Africa. For the same period of the year, Antonel and Chowdhury (2014) found lower PM10 concentrations (105 µg m

Table 2 .

 2 Comparison between PM10 and PM2.5 daily concentrations obtained in this study and other studies Sub-Saharan Africa.

	M A N U S C R I P T
	A C C E P T E D
	32

Table 1 .

 1

	Pollutants	PM 10	Korhogo	PM 2.5	PM 10	Abidjan	PM 2.5
	Root Mean Absolute Error (RMSE)	97.0			43.2	41.5		24.9
	Mean Average Error (MAE)	105.6			40.0	31.0		19.98
	% Coefficient of Variation of error	33.4 ± 12.6	36.7 ± 31.7	42.5 ± 41.8	39.7 ± 29.2

particulate pollution in Korhogo and Abidjan, relative contribution of fine particle to coarse and daily variations of PM concentrations will be very useful for urban modelling in West African cities.

ACKNOWLEDGEMENTS

This project is funded by the Education and Research Ministry of Cote d'Ivoire, as part of the Debt Reduction-Development Contracts (C2Ds) managed by Institut de Recherche pour le

DISCLAIMER

The authors decline any commercial link with the companies producing equipment or software used in this study.

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T

A C C E P T E D M A N U S C R I P T


Michael Goldman
email: goldman@u-paris.fr

Dario Trevisan
email: dario.trevisan@unipi.it

CONVERGENCE OF ASYMPTOTIC COSTS FOR RANDOM EUCLIDEAN MATCHING PROBLEMS

Keywords: 2010 Mathematics Subject Classification. 60D05, 90C05, 39B62, 60F25, 35J05 Matching problem, optimal transport, geometric probability. 1

published or not. The documents may come

Convergence of asymptotic costs for random Euclidean matching problems

Introduction

The aim of this paper is to extend the results of [START_REF] Dobrić | Asymptotics for transportation cost in high dimensions[END_REF][START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Dereich | Constructive quantization: approximation by empirical measures[END_REF] on the existence of the thermodynamic limit for some random optimal matching problems. Because of their relations to computer science, statistical physics and quantization of measures, optimal matching problems have been the subject of intense research both from the mathematical and physical communities. We refer for instance to [START_REF] Yukich | Probability theory of classical Euclidean optimization problems[END_REF][START_REF]Upper and lower bounds for stochastic processes: modern methods and classical problems[END_REF][START_REF] Caracciolo | Scaling hypothesis for the Euclidean bipartite matching problem[END_REF] for more details in particular regarding the vast literature.

Probably the simplest and most studied variant of these problems is the bipartite (or Euclidean bipartite) matching on the unit cube in d dimensions. Given p ≥ 1 and two independent families of i.i.d. random variables (X i ) i≥1 and (Y i ) i≥1 with common law the uniform (Lebesgue) measure on [0, 1] d , the problem is to understand the behavior for large n of

E 1 n min π n i=1 |X i -Y π(i) | p ,
where the minimum is taken over all permutations π of {1, . . . , n}. It is by now well-known, see [START_REF] Ajtai | On optimal matchings[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Bobkov | One-dimensional empirical measures, order statistics, and Kantorovich transport distances[END_REF][START_REF]On optimal matching of Gaussian samples[END_REF] that for 1 n 1 (see [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF] for some non-asymptotic bounds) Let us point out that while the case d ≥ 3, p ≥ d/2 is not explicitly covered in the literature, the proof of [START_REF]On optimal matching of Gaussian samples[END_REF] clearly extends to any p = 2 (see also [START_REF] Bobkov | A simple Fourier analytic proof of the AKT optimal matching theorem[END_REF]). Our main result is the following:

E 1 n min π n i=1 |X i -Y π(i) | p ∼        n -p 2 for d = 1
Theorem 1.1. For every d ≥ 3 and p ≥ 1, there exists a constant

f bi ∞ = f bi ∞ (p, d) > 0 such that lim n→∞ n p d E 1 n min π n i=1 |X i -Y π(i) | p = f bi ∞ . (1.2)
This extends earlier results of [START_REF] Dobrić | Asymptotics for transportation cost in high dimensions[END_REF][START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Dereich | Constructive quantization: approximation by empirical measures[END_REF] where the same conclusion was obtained under the more restrictive condition p < d/2. See also [START_REF] Talagrand | Matching random samples in many dimensions[END_REF] for bounds on f bi ∞ (1, d) as d becomes large. As in the previously quoted papers, our proof is based on a sub-additivity argument and makes use of the classical observation that, thanks to the Birkhoff-Von Neumann Theorem, the bi-partite matching problem is actually an optimal transport problem. Indeed, if µ = n i=1 δ X i and λ = n i=1 δ Y i are the associated empirical measures, then min

π n i=1 |X i -Y π(i) | p = W p p (µ, λ) ,
where W p denotes the Wasserstein distance of order p (see [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF]). However, the papers [START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Dereich | Constructive quantization: approximation by empirical measures[END_REF] rely then upon combinatorial arguments, and in fact their results apply to a larger class of random optimization problems, while [START_REF] Dobrić | Asymptotics for transportation cost in high dimensions[END_REF] strongly uses the dual formulation of optimal transport, which in the case p = 1 is quite specific, since it becomes a maximization over the set of 1-Lipschitz functions.

The optimal transport point of view allows us to treat the defect in sub-additivity as a defect in local distribution of mass rather than a defect in local distribution of points. More precisely, even if µ([0, 1] d ) = λ([0, 1] d ) it is in general not true that for a given partition of [0, 1] d in sub-cubes Q i , µ(Q i ) = λ(Q i ). Therefore, in order to use sub-additivity, one needs to relax the definition of the matching problem to take into account this disparity. In [START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Dereich | Constructive quantization: approximation by empirical measures[END_REF] this is done by requiring that as many points as possible are matched. Here we allow instead for varying weights. That is, for µ and λ containing potentially a different number of points, we consider the problem

E W p p µ µ([0, 1] d ) , λ λ([0, 1] d )
.

The main sub-additivity argument for this quantity is contained in Lemma 3.1. In order to estimate the error in sub-additivity, we then use in Lemma 3.4 a PDE ansatz similar to the one proposed in the context of the matching problem in [START_REF] Caracciolo | Scaling hypothesis for the Euclidean bipartite matching problem[END_REF] and then used in [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] (see also [START_REF]On optimal matching of Gaussian samples[END_REF][START_REF]Quantitative linearization results for the Monge-Ampère equation[END_REF][START_REF] Benedetto | Euclidean random matching in 2d for non-constant densities[END_REF]) to show that when d = 2

lim n→∞ n log n E 1 n W 2 2 (µ, λ) = 1 2π .
Notice however that our use of this linearization ansatz is quite different from [START_REF] Caracciolo | Scaling hypothesis for the Euclidean bipartite matching problem[END_REF][START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF]. Indeed, for us the main contribution to the transportation cost is given by the transportation at the smaller scales and linearization is only used to estimate the higher order error term. On the other hand, in [START_REF] Caracciolo | Scaling hypothesis for the Euclidean bipartite matching problem[END_REF][START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] (see also [START_REF] Ambrosio | Finer estimates on the 2-dimensional matching problem[END_REF]), the main contribution to the cost is given by the linearized i.e. H -1 , cost. This is somewhat in line with the prediction by [START_REF] Caracciolo | Scaling hypothesis for the Euclidean bipartite matching problem[END_REF] that for d ≥ 3, the first order contribution to the Wasserstein cost is not given by the linearized problem while higher order corrections are. In any case, we give in Proposition 5.3 an alternative argument to estimate the error term without relying on the PDE ansatz. There we use instead an elementary comparison argument with one-dimensional transport plans. We included both proofs since we believe that they could prove useful in other contexts.

Let us make a few more comments on the proof of Theorem 1.1. As in [START_REF] Dobrić | Asymptotics for transportation cost in high dimensions[END_REF][START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF], the proof of (1.2) is actually first done on a variant of the problem where the number of points follows a Poisson distribution instead of being deterministic. This is due to the fact that the restriction of a Poisson point process is again a Poisson point process. For this variant of the problem, rather than working on a fixed cube [0, 1] d with an increasing density of points, we prefer to make a blow-up at a scale L = n 1/d and consider in Theorem 5.1, a fixed Poisson point process of intensity 1 on R d but restricted to cubes Q L = [0, L] d with L 1 (hence the terminology thermodynamic limit). We believe that the sub-additivity argument is slightly clearer in these variables (a similar rescaling is actually implicitly used in [START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF]). This setting is somewhat reminiscent of [START_REF] Huesmann | Optimal transport from Lebesgue to Poisson[END_REF], where super-additivity is used to construct an optimal coupling between the Poisson point process and the Lebesgue measure on R d . In order to pass from the Poisson version of the problem to the deterministic one, we prove a general de-Poissonization result in Proposition 6.1 which can hopefully be useful in other contexts.

Besides the bipartite matching we also treat in Theorem 4.1 and Theorem 6.2 the case of the matching to the reference measure. We actually treat this problem first since the proof is a bit simpler. Indeed, while the general scheme of the proof is identical to the bipartite case, the PDE ansatz used in Lemma 3.4 works well for "regular" measures and a more delicate argument is required for the bipartite matching. Notice that by Jensen and triangle inequalities, (1.1) also holds for the matching to the reference measure.

We point out that in [START_REF] Dobrić | Asymptotics for transportation cost in high dimensions[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Dereich | Constructive quantization: approximation by empirical measures[END_REF], it is more generally proven that if the points X i and Y i have a common law µ supported in [0, 1] d instead of the Lebesgue measure (for measures with unbounded support a condition on the moments is required), then for 1 ≤ p < d/2 and d ≥ 3, lim sup

n→∞ n p d E 1 n min π n i=1 |X i -Y π(i) | p ≤ f bi ∞ [0,1] d dµ dx 1-p d .
However, when p > d/2 and without additional assumptions on µ, the asymptotic rate may be different and thus this inequality may fail, see e.g. [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF]. Positive results for specific densities can be obtained nonetheless. For instance, it is proven in [START_REF] Ledoux | On optimal matching of Gaussian samples III[END_REF] that for the standard Gaussian measure µ on R d , d ≥ 3, the asymptotic bound

n p d E 1 n min π n i=1 |X i -Y π(i) | p ∼ 1
holds true also for d/2 ≤ p < d.

Finally, we notice that usual results on concentration of measure allow us to improve from convergence of the expectations to strong convergence. However, we are able to cover only the case 1 ≤ p < d, see Remark 6.5.

The plan of the paper is the following. In Section 2, we fix some notation and recall basic moment and concentration bounds for Poisson random variables. In Section 3, we state and prove our two main lemmas namely the sub-additivity estimate Lemma 3.1 and the error estimate Lemma 3.4. In Section 4, we then prove the existence of the thermodynamic limit for the matching problem of a Poisson point process to the reference measure. The analog result for the bipartite matching between two Poisson point processes is obtained in Section 5. Finally, in Section 6 we pass from the Poissonized problem to the deterministic one and discuss stronger convergence results.

Notation and preliminary results

We use the notation |A| for the Lebesgue measure of a Borel set A ⊆ R d , and A f for the Lebesgue integral of a function f on A. For L > 0, we let Q L = [0, L] d . We denote by |p| the Euclidean norm of a vector p ∈ R N . For a function φ, we use the notation ∇φ for the gradient, ∇ • φ for the divergence, ∆φ for the Laplacian and ∇ 2 φ for the Hessian.

2.1. Optimal transport. In this section we introduce some notation for the Wasserstein distance and recall few simple properties that will be used throughout. Proofs can be found in any of the monographs [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF][START_REF] Peyré | Computational optimal transport: With applications to data science[END_REF] with expositions of theory of optimal transport, from different perspectives.

Given p ≥ 1, a Borel subset Ω ⊆ R d and two positive Borel measures µ, λ with µ(Ω) = λ(Ω) ∈ (0, ∞) and finite p-th moments, the Wasserstein distance of order p ≥ 1 between µ and λ is defined as the quantity

W p (µ, λ) = min π∈C(µ,λ) R d ×R d |x -y| p dπ(x, y) 1 p
, where C(µ, λ) is the set of couplings between µ and λ. Moreover, if µ(Ω) = λ(Ω) = 0, we define W p (µ, λ) = 0, while if µ(Ω) = λ(Ω), we let W p (µ, λ) = ∞.

Let us recall that since W p is a distance, we have the triangle inequality

W p (µ, ν) ≤ W p (µ, λ) + W p (ν, λ). (2.1)
We will also use the classical sub-additivity inequality

W p p i µ i , i λ i ≤ i W p p (µ i , λ i ), (2.2)
for a finite set of positive measures µ i , λ i . This follows from the observation that if

π i ∈ C(µ i , λ i ), then i π i ∈ C( i µ i , i λ i ).
Remark 2.1. In fact, our results deal with the transportation cost W p p (µ, λ) rather than W p (µ, λ). To keep notation simple, we write

W p Ω (µ, λ) = W p p (µ ¬ Ω, λ ¬ Ω).
Moreover, if a measure is absolutely continuous with respect to Lebesgue measure, we only write its density. For example,

W p Ω µ, µ(Ω) |Ω| ,
denotes the transportation cost between µ ¬ Ω to the uniform measure on Ω with total mass µ(Ω).

Occasionally we may write W Ω (µ, λ) instead of W p Ω (µ, λ) 1/p . This may lead to some ambiguity, but it should be clear from the context.

Poisson point processes.

As in [START_REF] Dobrić | Asymptotics for transportation cost in high dimensions[END_REF][START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF], we exploit invariance properties of Poisson point processes on R d with uniform intensity in order to obtain simpler sub-additivity estimates. We refer e.g. to [START_REF] Last | Lectures on the Poisson process[END_REF] for a general introduction to Poisson point processes. Here we only recall that a Poisson point process on R d with intensity one can be defined as a random variable taking values on locally finite atomic measures

µ = i δ X i such that, for every k ≥ 1, for any disjoint Borel sets A 1 , . . . , A k ⊆ R d , the random vari- ables µ(A 1 ), . . . , µ(A k ) are independent and µ(A i ) has a Poisson distribution of parameter |A i |, for every i = 1, . . . , k. In particular, if A ⊆ R d is Lebesgue negligible, then µ(A) = 0 almost surely.
Existence of a Poisson point process of intensity one is obtained via a superposition argument, noticing that on every bounded subset Ω ⊆ R d the law of µ ¬ Ω can be easily described: conditionally on µ(Ω) = n, the measure µ ¬ Ω has the same law as the random measure

n i=1 δ X i ,
where (X i ) n i=1 are independent random variables with uniform law on Ω. Uniqueness in law can be also obtained, so that translation invariance of Lebesgue measure entails that the process is stationary, i.e., any deterministic translation of the random measure µ leaves its law unchanged.

Let us finally recall the classical Cramér-Chernoff concentration bounds for Poisson random variables. Lemma 2.2. Let N be a Poisson random variable with parameter n 1. Then, for every t > 0

P[|N -n| ≥ t] ≤ 2 exp - t 2 2(t + n) . (2.3)
As a consequence, for every q ≥ 1,

E [|N -n| q ] q n q 2 .
(2.4)

Proof. For the concentration bound (2.3), see for instance [START_REF] Boucheron | Concentration inequalities[END_REF]. By the layer-cake representation,

E [|N -n| q ] ∞ 0 t q-1 exp - t 2 2(t + n) dt √ n 0 t q-1 dt + n √ n t q-1 exp -c t 2 n dt + ∞ n t q-1 exp(-ct)dt q n q 2 + n q 2 + n q-1 exp(-cn) q n q 2 .

The main lemmas

Our sub-additivity argument rests on a general but relatively simple lemma (which we only apply here for rectangles). Lemma 3.1. For every p ≥ 1, there exists a constant C > 0 depending only on p such that the following holds. For every Borel set Ω ⊂ R d , every Borel partition (Ω i ) i∈N of Ω, every measures µ, λ on Ω, and every ε ∈ (0, 1),

W p Ω µ, µ(Ω) λ(Ω) λ ≤ (1 + ε) i W p Ω i µ, µ(Ω i ) λ(Ω i ) λ + C ε p-1 W p Ω i µ(Ω i ) λ(Ω i ) χ Ω i λ, µ(Ω) λ(Ω) λ . (3.1)
Proof. We first use the triangle inequality (2.1) to get

W p Ω µ, µ(Ω) λ(Ω) λ ≤ W Ω µ, i µ(Ω i ) λ(Ω i ) χ Ω i λ + W Ω i µ(Ω i ) λ(Ω i ) χ Ω i λ, µ(Ω) λ(Ω) λ p .
The proof is then concluded by combining the elementary inequality

(a + b) p ≤ (1 + ε)a p + C ε p-1 b p ∀a, b > 0 and ε ∈ (0, 1), (3.2)
with the sub-additivity of W p Ω (2.2) in the form

W p Ω µ, i µ(Ω i ) λ(Ω i ) χ Ω i λ ≤ i W p Ω i µ, µ(Ω i ) λ(Ω i ) λ .
Remark 3.2. Alternatively, we could have also stated (3.1) in the slightly more symmetric form:

W p Ω µ µ(Ω) , λ λ(Ω) ≤ (1 + ε) i µ(Ω i ) µ(Ω) W p Ω i µ µ(Ω i ) , λ λ(Ω i ) + C ε p-1 W p Ω 1 µ(Ω) i µ(Ω i ) λ(Ω i ) χ Ω i λ, λ λ(Ω)
.

However, the sub-additivity argument turns out to be a little bit simpler using (3.1) instead.

Lemma 3.1 shows that in order to estimate the defect in sub-additivity, it is enough to bound the local defect of mass distribution. This will be done here through a PDE argument.

Definition 3.3. We say that a rectangle R = x + d i=1 [0, L i ] is of moderate aspect ratio if for every i, j, L i /L j ≤ 2. A partition R = {R i } of R is called admissible if for every i, R i is a rectangle of moderate aspect ratio and 3 -d |R| ≤ |R i | ≤ |R|.
Notice that in particular #R 1 for every admissible partition. Lemma 3.4. Let R be a rectangle of moderate aspect ratio, µ and λ be measures on R with equal mass, both absolutely continuous with respect to Lebesgue and such that inf R λ > 0. Then, for every p ≥ 1 We first argue that

W p R (µ, λ) diam p (R) (inf R λ) p-1 R |µ -λ| p . ( 3
W p R (µ, λ) 1 (inf R λ) p-1 R |∇φ| p . (3.5)
Let us point out that this estimate is well-known and has already been used in the context of the matching problem, see [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF][START_REF]Quantitative linearization results for the Monge-Ampère equation[END_REF]

W R (µ, λ) ≤ W R µ, 1 2 (µ + λ) + W R 1 2 (µ + λ), λ ≤ W R 1 2 µ, 1 2 λ + W R 1 2 (µ + λ), λ = 2 -1 p (W R (µ, λ) + W R (µ + λ, 2λ))
and thus W R (µ, λ) W R (µ + λ, 2λ). We now recall that by the Benamou-Brenier formula (see [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF]Th. 5.28])

W p R (µ + λ, 2λ) = min ρ,j 1 0 R 1 ρ p-1 |j| p : ∂ t ρ + ∇ • j = 0, ρ 0 = µ + λ, ρ 1 = 2λ .
Estimate (3.5) follows using

ρ t = (1 -t)µ + tλ + λ and j = ∇φ
as competitor and noticing that for t

∈ [0, 1], ρ t ≥ inf R λ. We now claim that R |∇φ| p diam p (R) R |µ -λ| p , (3.6)
which together with (3.5) would conclude the proof of (3.3). Estimate (3.6) is a direct consequence of Poincaré inequality and Calderón-Zygmund estimates for the Laplacian. However, since we did not find a precise reference for (global) Calderón-Zygmund estimates on rectangles with Neumann boundary conditions, we give here a short proof. By scaling, we may assume that diam(R) = 1. Furthermore, using even reflections along ∂R we may replace Neumann boundary conditions by periodic ones in (3.4) . By Bochner's formula and Hölder inequality,

R |∇ 2 φ| 2 = R |µ -λ| 2 R |µ -λ| p 2 p
, which concludes the proof of (3.6).

Remark 3.5. For p = 2, combining the energy identity

R |∇φ| 2 = R φ(λ -µ)
with Poincaré inequality, we see that (3.6) (and thus (3.3)) holds for any convex set R.

Although the situation for p ≥ 2 is more subtle, see [14, Prop. 2], we believe that (3.6) holds for any rectangle, not necessarily of moderate aspect ratio.

Matching to the reference measure

In this section, we consider the optimal matching problem between µ a Poisson point process on R d with intensity one and the Lebesgue measure. More precisely, for every L ≥ 1 we let

f ref (L) = E 1 |Q L | W p Q L (µ, κ) ,
where

Q L = [0, L] d and κ = µ(Q L ) |Q L |
is the generic constant for which this is well defined.

Theorem 4.1. For every d ≥ 3 and p ≥ 1, the limit

f ref ∞ = lim L→∞ f ref (L)
exists and is strictly positive. Moreover, there exists C > 0 depending on p and d such that for L ≥ 1,

f ref ∞ ≤ f ref (L) + C L d-2 2 . (4.1)
The proof follows the argument of [START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF] (see also [START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF]) and is mostly based on the following sub-additivity estimate. Proposition 4.2. For every d ≥ 3 and p ≥ 1, there exists a constant C > 0 such that for every L ≥ 1 and m ∈ N,

f ref (mL) ≤ f ref (L) + C L d-2 2 . (4.2)
Proof. We start by pointing out that since f ref (L) L p , it is not restrictive to assume that L 1 in the proof of (4.2).

Step 1. [The dyadic case] For the sake of clarity, we start with the simpler case m = 2 k for some k ≥ 1. We claim that

f ref (2L) ≤ f ref (L) + C L d-2 2 . (4.
3)

The desired estimate (4.2) would then follow iterating (4.3) and using that k≥0

1 2 k(d-2) < ∞ for d ≥ 3. In order to prove (4.3), we divide the cube Q 2L in 2 d sub-cubes Q i = x i + Q L and let κ i = µ(Q i ) |Q L | (and κ = µ(Q 2L ) |Q 2L |
). Notice that we are considering a partition up to a Lebesgue negligible remainder, which gives no contribution almost surely. By (3.1), for every ε ∈ (0, 1),

W p Q 2L (µ, κ) ≤ (1 + ε) i W p Q i (µ, κ i ) + C ε p-1 W p Q 2L i κ i χ Q i , κ .
Dividing by |Q 2L |, taking expectations and using the fact that by translation invariance

E[ 1 |Q L | W p Q i (µ, κ i )] = f ref (L), we get f ref (2L) ≤ (1 + ε) i |Q L | |Q 2L | f ref (L) + C ε p-1 E 1 |Q 2L | W p Q 2L i κ i χ Q i , κ = (1 + ε)f ref (L) + C ε p-1 E 1 |Q 2L | W p Q 2L i κ i χ Q i , κ . We now estimate E[ 1 |Q 2L | W p Q 2L ( i κ i χ Q i , κ)]. Using 1 |Q 2L | W p Q 2L ( i κ i χ Q i , κ) L p κ to- gether with P κ ≤ 1 2 ≤ exp(-cL d ),
which follows from the Cramér-Chernoff bounds (2.3), we may reduce ourselves to the event {κ ≥ 1 2 }. Under this condition, by (3.3), we have 1

|Q 2L | W p Q 2L i κ i χ Q i , κ L p L d Q 2L i |κ i -κ| p χ Q i L p |κ -1| p + i |κ i -1| p .
Recalling that µ(Q i ) are Poisson random variables of parameter |Q i | and that

κ i = µ(Q i ) |Q i | , we get from (2.4) E [|κ -1| p ] ∼ E [|κ i -1| p ] 1 L pd 2 . Thus E 1 |Q 2L | W p Q 2L i κ i χ Q i , κ 1
L p 2 (d-2) (4.4)
and we conclude that for ε ∈ (0, 1),

f ref (2L) ≤ (1 + ε)f ref (L) + C ε p-1 1 L p 2 (d-2) . (4.5)
Optimizing in ε by choosing ε = L -(d-2)/2 , and using that f ref (L) is bounded (by (1.1) and (2.3), see for instance [START_REF] Goldman | A large-scale regularity theory for the Monge-Ampere equation with rough data and application to the optimal matching problem[END_REF]Prop. 2.7] for details) we conclude the proof of (4.3). Let us point out that we used here boundedness of f ref (L) for simplicity but that as shown below it can also be obtained as a consequence of our proof.

Step 2.[The general case] We now consider the case when m ∈ N is not necessarily dyadic. We will partition Q mL into rectangles of almost dyadic size and thus need to deal with slightly more general configurations than dyadic cubes. Let us introduce some notation. Let R be a rectangle with moderate aspect ratio and R = {R i } be an admissible partition of R (recall Definition 3.3). Slightly abusing notation, we define

f ref (R) = E 1 |R| W p R (µ, κ) . (4.6)
Step 2.1. We claim that the following variant of (4.5) holds: for every rectangle R of moderate aspect ratio with |R| 1, every admissible partition R of R and every ε ∈ (0, 1), we have

f ref (R) ≤ (1 + ε) i |R i | |R| f ref (R i ) + C ε p-1 1 |R| p(d-2) 2d . (4.7)
Defining κ i = µ(R i ) |R i | and using (3.1) as above, we get

f ref (R) ≤ (1 + ε) i |R i | |R| f ref (R i ) + C ε p-1 E 1 |R| W p R i κ i χ R i , κ .
The estimate

E 1 |R| W p R i κ i χ R i , κ 1 |R| p(d-2) 2d (4.8)
is then obtained arguing exactly as for (4.4), using first the Crámer-Chernoff bound (2.3) to reduce to the event {κ ≥ 1 2 } and then (3.3) (recalling that diam(R) ∼ |R| 1 d since R has moderate aspect ratio) in combination with (2.4) and the fact that #R 1 since R is an admissible partition.

Step 2.2. Starting from the cube Q mL , let us construct a sequence of finer and finer partitions of Q mL by rectangles of moderate aspect ratios and side-length given by integer multiples of L. We let R 0 = {Q mL } and define R k inductively as follows. Let R ∈ R k . Up to translation we may assume that R = d i=1 (0, m i L) for some m i ∈ N. We then split each interval (0,

m i L) into (0, m i 2 L) ∪ ( m i 2 L, m i L).
It is readily seen that this induces an admissible partition of R. Let us point out that when m i = 1 for some i, the corresponding interval (0, m i 2 L) is empty. This procedure stops after a finite number of steps

K once R K = {Q L + z i , z i ∈ [0, m -1] d }. It is also readily seen that 2 K-1 < m ≤ 2 K and that for every k ∈ [0, K] and every R ∈ R k we have |R| ∼ (2 K-k L) d .
Let us prove by a downward induction that there exists Λ > 0 such that for every k ∈ [0, K] and every R ∈ R k ,

f ref (R) ≤ f ref (Q L ) + Λ(1 + f ref (Q L ))L -d-2 2 K j=K-k 2 -j d-2 2 . (4.9)
This is clearly true for k = K. Assume that it holds true for k

+ 1. Let R ∈ R k . Applying (4.7) with ε = (2 K-k L) -(d-2)/2 1, we get f ref (R) ≤ (1 + ε) R i ∈R k+1 ,R i ⊂R |R i | |R| f ref (R i ) + C ε 1 |R| p(d-2) 2d (4.9) ≤ (1 + ε)   f ref (Q L ) + Λ(1 + f ref (Q L ))L -d-2 2 K j=K-k+1 2 -j d-2 2   + C(2 K-k L) -d-2 2 ≤ f ref (Q L ) + Λ(1 + f ref (Q L ))L -d-2 2 ×   K j=K-k+1 2 -j d-2 2 + 2 -(K-k) d-2 2   C Λ + L -d-2 2 K j=K-k+1 2 -j d-2 2     . If L is large enough, then ( K j=K-k+1 2 -j d-2 2 )L -(d-2)/2 ≤ 1 2 .
Finally, choosing Λ ≥ 2C yields (4.9). Applying (4.9) to R = Q mL and using that j≥0 2 -j d-2 2 < ∞, we get

f ref (mL) ≤ f ref (L) + C(1 + f ref (L)) 1 L d-2 2
.

Since f ref (L) L p , writing that every L 1 may be written as L = mL for some m ∈ N and L ∈ [1, 2], we conclude that f ref (L) is bounded and thus (4.2) follows.

Remark 4.3. We point out that as a consequence of the proof of Proposition 4.2 we have for every rectangle R of moderate aspect ratio (recall definition (4.6))

f ref (R) 1. (4.10)
We can now prove Theorem 4.1

Proof of Theorem 4.1. The existence of a limit f ref ∞ is obtained from (4.2) arguing exactly as in [START_REF] Boutet De Monvel | Almost sure convergence of the minimum bipartite matching functional in Euclidean space[END_REF] using the continuity of L → f ref (L) (which can be obtained for instance by dominated convergence). The fact that f ref (L) 1 and thus f ref ∞ > 0 follows from (1.1) and (2.3). Finally, (4.1) follows from (4.2) by sending m → ∞ for fixed L.

Remark 4.4. It may be conjectured from [START_REF] Caracciolo | Scaling hypothesis for the Euclidean bipartite matching problem[END_REF] that

|f ref (L) -f ref ∞ | 1 L d-2 .
Let us notice that if we could replace the term ε -(p-1) in (4.5) by a constant then letting ε → 0 we would get the lower bound

f ref (L) -f ref ∞ - 1 L p(d-2) 2
, which, at least for p = 2, is in line with the conjectured rate. See also Remark 6.4 for rates in the case of a deterministic number of points.

Bi-partite matching

We now turn to the bi-partite matching. For µ and λ two independent Poisson point processes of intensity one, we want to study the asymptotic behavior as L → ∞ of

f bi (L) = E 1 |Q L | W p Q L (µ, κλ) , with κ = µ(Q L ) λ(Q L )
. Analogously to Theorem 4.1, we have:

Theorem 5.1. For every d ≥ 3 and p ≥ 1, the limit

f bi ∞ = lim L→∞ f bi (L)
exists and is strictly positive. Moreover, there exists C > 0 depending on p and d such that for L ≥ 1,

f bi ∞ ≤ f bi (L) + C L d-2 2p
.

The proof of Theorem 5.1 follows the same line of arguments as for Theorem 4.1. We only detail the estimate of the sub-additivity defect i.e. the counterpart of (4.8), since it is more delicate in the bipartite case. Proposition 5.2. Let R be a rectangle of moderate aspect ratio with |R| 1 and R = {R i } be an admissible partition of R (recall Definition 3.3). Defining

κ i = µ(R i ) λ(R i ) and κ = µ(R)
λ(R) , we have for every d ≥ 3 and p ≥ 1,

E 1 |R| W p R i κ i χ R i λ, κλ 1
|R| d-2 2d
.

(5.1)

Proof. As opposed to (4.8), since λ is atomic, we cannot directly use (3.3). The idea is to use as intermediate step the matching between λ and the reference measure. Let us point out that since

f ref (R i ) = E 1 |R i | W p R i λ, λ(R i ) |R i |
is of order one, we cannot apply naively the triangle inequality. The main observation is that since |κ i -κ| 1 with overwhelming probability, the amount of mass which actually needs to be transported is very small.

Let 1

θ > 0 to be optimized later on. For every i let

θ i := (κ -κ i ) + θ and assume that θ ≥ 2 max i |κ -κ i | (5.2) so that 3 2 θ ≥ θ i ≥ 1 2 θ > 0.
Notice that thanks to the Crámer-Chernoff bounds (2.3), (5.2) is satisfied with overwhelming probability as long as θ |R| -1/2 . Using the triangle inequality (2.1) we have

W p R i κ i χ R i λ, κλ W p R i κ i χ R i λ, i χ R i (κ i -θ)λ + θ λ(R i ) |R i | dx + W p R i χ R i (κ i -θ)λ + θ λ(R i ) |R i | dx , i χ R i (κ i -θ)λ + θ i λ(R i ) |R i | dx + W p R i χ R i (κ i -θ)λ + θ i λ(R i ) |R i | dx , κλ .
Notice that i θλ(R i ) = i θ i λ(R i ) by definition of θ i and the fact that κλ(R) = µ(R) = i κ i λ(R) so that the second term is well defined. We now estimate the three terms separately. The first and third are estimated in a similar way and we thus focus only on the first one. By sub-additivity (2.2) of W p R , we have

W p R i κ i χ R i λ, i χ R i (κ i -θ)λ + θ λ(R i ) |R i | dx ≤ θ i W p R i λ, λ(R i ) |R i | .
We turn to the middle term. Again by sub-additivity of W p R ,

W p R i χ R i (κ i -θ)λ + θ λ(R i ) |R i | dx , i χ R i (κ i -θ)λ + θ i λ(R i ) |R i | dx ≤ W p R i χ R i θ λ(R i ) |R i | , i χ R i θ i λ(R i ) |R i | .
Using (3.3) in the event {λ(R i )/|R i | ∼ 1} (which has overwhelming probability) and recalling that we assumed 3 2 θ ≥ θ i ≥ 1 2 θ, we have

W p R i χ R i θ λ(R i ) |R i | , i χ R i θ i λ(R i ) |R i | diam p (R) θ p-1 i |κ -κ i | p |R i |.
Putting these two estimates together, dividing by |R| and taking expectations we find

E 1 |R| W p R i κ i χ R i λ, κλ θ i f ref (R i ) + diam p (R) θ p-1 i E [|κ -κ i | p ] (4.10)&(2.4) θ + 1 θ p-1 1 |R| p (d-2) 2d . Optimizing in θ by choosing θ = |R| -d-2 2d
|R| -1/2 (so that (5.2) is satisfied) this yields (5.1).

Comparing (4.4) and (5.1), one may wonder if (5.1) is suboptimal and could be improved. Let us prove that it is not the case, at least if we consider the slightly more regular situation of the matching to a deterministic grid.

Q 2L = ∪ 2 d i=1 Q i where Q i are disjoint cubes of sidelength L and defining κ i = µ i λ(Q i ) , κ = i µ i λ(Q 2L ) , we have for d ≥ 3 and p ≥ 1, E 1 |Q 2L | W p Q 2L i κ i χ Q i λ, κλ ∼ 1 L d-2 2
.

Proof.

Step 1.[The lower bound] We start by proving that

E 1 |Q 2L | W p Q 2L i κ i χ Q i λ, κλ 1
L d-2 2
.

(

For this we notice that if π is any admissible coupling, then |x -y| ≥ 1 for every (x, y) in the support of π with x = y and thus 2

W p Q 2L i κ i χ Q i λ, κλ ≥ W 1 Q 2L i κ i χ Q i λ, κλ . Let Σ = ∪ i ∂Q i ∩ Q 2L and ε i = sign(κ i -κ). Using ξ(x) = d(x, Σ) i ε i χ Q i (x) as test function in the dual formulation of W 1 Q 2L , we obtain W 1 Q 2L i κ i χ Q i λ, κλ ≥ i X∈Z d ∩Q i d(X, Σ)|κ i -κ| L d+1 i |κ i -κ|. 2 recall that W 1 Q 2L denotes the 1-Wasserstein distance.
Taking expectations we find (5.3).

Step 2.[The upper bound] We turn to the corresponding upper bound,

E 1 |Q 2L | W p Q 2L i κ i χ Q i λ, κλ 1 L d-2 2
.

(5.4)

One could argue exactly as for (5.1) but we provide an alternative proof which uses a one-dimensional argument instead of (3.3). Notice that this argument could have also been used to give a different proof of (4.4).

Step 2.1.[The one-dimensional estimate] Let λ = X∈Z δ X and for κ 1 , κ 2 > 0 and L 1, define κ = κ 1 λ(0,L/2)+κ 2 λ(L/2,L) λ(0,L)

. We claim that if

|κ 1 -κ 2 | L -1 min(κ 1 , κ 2 ), then W p (0,L) κ 1 χ (0,L/2) λ + κ 2 χ (L/2,L) λ, κλ L 2 |κ 1 -κ 2 |. (5.5)
Let us assume without loss of generality that κ 1 ≥ κ 2 . The optimal transport map is essentially symmetric around L/2 and is given in (0, L/2) by sending a mass (k + 1) λ(L/2,L) λ(0,L) (κ 1 -κ 2 ) from position k to k + 1 (which is admissible since by hypothesis

k|κ 1 -κ 2 | min(κ 1 , κ 2 )) so that W p (0,L) (µ, λ) L/2 k=0 k|κ 1 -κ 2 | ∼ L 2 |κ 1 -κ 2 |.
This proves (5.5).

Step 2.2.[Proof of (5.4)] The proof is made recursively by layers. In the first step, we pass from 2 d cubes to 2 d-1 rectangles of the form x + (0, L) × (0, L/2) d-1 . For this we remark that for every cube Q i there is exactly one cube Q j such that Q j = Q i ± L 2 e 1 (where e i is the canonical basis of R d ). Let us focus on

Q 1 = (0, L/2) d and Q 2 = L 2 e 1 + Q 1 . Let R = Q 1 ∪ Q 2 = (0, L) × (0, L/2) d-1 . Define κ = κ 1 λ(Q 1 )+κ 2 λ(Q 2 ) λ(R) = µ 1 +µ 2 λ(R) . We claim that E 1 |R| W 2 R (κ 1 χ Q 1 λ + κ 2 χ Q 2 λ, κλ) 1 L d-2 2
.

For this we notice that in R, the measures κ 1 χ Q 1 λ + κ 2 χ Q 2 λ and κλ are constant in the directions orthogonal to e 1 and thus

1 |R| W p R (κ 1 χ Q 1 λ + κ 2 χ Q 2 λ, κλ) 1 L W p (0,L) κ 1 χ (0,L/2) λ + κ 2 χ (L/2,L) λ, κλ . Since κ i = µ i λ(Q i )
, by the Cramér-Chernoff bounds (2.3), we have

|κ i -1| = O(L -d/2
) with overwhelming probability. Hence, if d ≥ 3 we may apply (5.5) to get

E 1 |R| W p R (κ 1 χ Q 1 λ + κ 2 χ Q 2 λ, κλ) LE[|κ 1 -κ 2 |] 1 L d-2 2
which proves the claim. We finally iterate this this argument 2 d times. At every step k we have 2 d-k rectangles of the form x + (0, L) k × (0, L/2) d-k and each iteration has a cost of the same order (namely L -(d-2)/2 ). Using triangle inequality this concludes the proof of (5.4).

Remark 5.4. Notice that in the proof of (5.5), we locally move a mass of order L -(d-2)/2 which corresponds to the optimal choice of θ in the proof of (5.1).

De-Poissonization

In this section we discuss how to transfer limit results from matching problems with Poisson point process, i.e., with a random number of points, to those of with a deterministic number of points.

We use the following general result. Proof. Let 0 < δ < L d and introduce the event

A = |N L,i -L d | < δ for i = 1, . . . , k .
By independence and Poisson tail bounds (2.3),

P(A) ≥ 1 -2 exp - 1 2 δ 2 L d + δ k . (6.1)
We decompose

f (L) = E [f (L|N L )|A] P(A) + E [f (L|N L )|A c ] P(A c ).
If A holds, we use monotonicity of f (L|n) to argue that

L p f (1|a) ≤ E [f (L|N L )|A] ≤ L p f (1|b),
where a = L d + δ, b = L d -δ. Otherwise, we use ( 1) and ( 2) to obtain

0 ≤ E [f (L|N L )|A c ] ≤ L p .
Combining these inequalities, we have

L p f (1|a)P(A) ≤ f (L) ≤ L p (f (1|b) + (1 -P(A))) . (6.2)
For any n ≥ 1, let L = L(n) be such such that L d + L d/2 √ 2β log L = n, for some fixed β > p. Then, we have from (6.2)

with δ = L d/2 √ 2β log L, f (1|n) ≤ f (L) L p P(A) .
As n → ∞, we have

n p d L p = (1 + L -d 2 2β log L) p d = 1 + O log n n .
Moreover, by (6.1), To obtain the converse inequalities, we argue analogously choosing instead

P(A) ≥ 1 -2 exp - 1 2 δ 2 L d + δ k = 1 -O(L -β ) = 1 -O n -
L = L(n) such that L d -L d/2 √ 2β log L = n.
We now apply Proposition 6.1 to matching problems. Let us first consider the case of matching to the reference measure. Theorem 6.2. Let d ≥ 3, p ≥ 1, and (X i ) i≥1 be i.i.d. uniform random variables on [0, 1] d . Then,

lim n→∞ n p d E W p [0,1] d 1 n n i=1 δ X i , 1 = f ref ∞ , with f ref ∞ as in Theorem 4.1. Proof. Recalling the notation Q 1 = [0, 1] d , we introduce the function f (L|n) = L p E W p Q 1 1 n n i=1 δ X i , 1 , if n ≥ 1,
and f (L|0) = 0. It is clearly bounded and p-homogeneous in the sense of Proposition 6.1.

To show monotonicity, let 1 ≤ m ≤ n and use the identity

1 n n i=1 δ X i = n m -1 I⊆{1,...,n} |I|=m 1 m i∈I δ X i
in combination with the convexity of the transportation cost, to obtain

W p Q 1 n i=1 δ X i , 1 = W p Q 1     n m -1 I⊆{1,...,n} |I|=m 1 m i∈I δ X i , n m -1 I⊆{1,...,n} |I|=m 1     ≤ n m -1 I⊆{1,...,n} |I|=m W p Q 1 1 m i∈I δ X i , 1 .
Taking expectation yields f (1|n) ≤ f (1|m), since for I ⊆ {1, . . . , n} with |I| = m, {X i } i∈I have the same law as {X i } m 1=1 . Let µ be a Poisson point process of intensity one on R d and for L > 1 let N L = µ(Q L ) be a Poisson random variable of parameter L d . For n ≥ 1, we notice that

f (L|n) = E W p Q L µ µ(Q L ) , 1 |Q L | N L = n = L d n f ref (L|n), (6.3)
where we write, extending the notation from Section 4,

f ref (L|n) = E 1 |Q L | W p Q L (µ, κ) N L = n , with κ = µ(Q L )/|Q L | (and f ref (L|0) = 0). Notice that f ref (L) = E[f ref (L|N L )].
In order to apply Proposition 6.1 and conclude the proof, we argue that lim

L→∞ E f (L|N L ) -f ref (L|N L ) = 0. ( 6.4)
To this aim, we first bound E [f (L|N L )] uniformly from above as L → ∞. For this we combine the following two simple facts. First, since f (L|n)

L p , letting A L = {N L ≥ |Q L |/2}, we have E [f (L|N L )|A c L ] L p . Second, since by (2.3) P[A L ] 1, E[f (L|N L )|A L ] (6.3) = E L d N L f ref (L|N L ) A L E[f ref (L|N L )] = f ref (L) 1,
where in the last inequality we used that f ref is bounded as a consequence of Theorem 4.1. Therefore,

E[f (L|N L )] = E[f (L|N L )|A c L ]P[A c L ] + E[f (L|N L )|A L ]P[A L ] L p P[A c L ] + 1 (2.3) L p exp(-cL d ) + 1 1.
Using Hölder inequality with 1 q + 1 q = 1, and the fact that

f (L|N L ) L p , E f (L|N L ) -f ref (L|N L ) (6.3) = E N L |Q L | -1 f (L|N L ) ≤ E N L |Q L | -1 q 1 q E [f (L|N L ) q ] 1 q . (2.4) q L -d 2 L p(q-1) q E [f (L|N L )] 1 q q L -d 2 +
p(q-1) q

.

Choosing q close enough to 1, in particular 1 < q < 2p 2p-d if p > d 2 , we get (6.4). Arguing similarly, we obtain the corresponding result for the bi-partite matching on the unit cube, that is Theorem 1.1, which we restate for the reader's convenience. Theorem 6.3. Let d ≥ 3, p ≥ 1, and (X i ) i≥1 , (Y i ) i≥1 , be independent uniform random variables on [0, 1] d . Then,

lim n→∞ n p d E W p [0,1] d 1 n n i=1 δ X i , 1 n n i=1 δ Y i = f bi ∞ ,
with f bi ∞ as in Theorem 5.1. Proof. The proof is very similar to that of Theorem 6.2, but in this case we define the function

f (L|n 1 , n 2 ) = L p E W p [0,1] d 1 n 1 n 1 i=1 δ X i , 1 n 2 n 2 i=1 δ Y i
for n 1 , n 2 ≥ 1, and let f (L|n 1 , n 2 ) = 0 otherwise. It is clearly bounded and p-homogeneous.

Using the identities

1 n 1 n 1 i=1 δ X i = n 1 m 1 -1 n 2 m 2 -1 I 1 ⊆{1,...,n 1 } |I|=m 1 I 2 ⊆{1,...,n 2 } |I|=m 2 1 m 1 i∈I 1 δ X i 1 n 2 n 2 i=1 δ Y i = n 1 m 1 -1 n 2 m 2 -1 I 1 ⊆{1,...,n 1 } |I|=m 1 I 2 ⊆{1,...,n 2 } |I|=m 2 1 m 2 i∈I 2 δ Y i
we obtain monotonicity arguing analogously as in Theorem 6.2. The proof is then concluded as in Theorem 6.2 and we omit the details.

Remark 6.4 (Convergence rates). An inspection of the proof of Proposition 6.1 shows that one can transfer rates of convergence (even only one-sided) from the Poisson case to that of a fixed number of points. In the case of matching to the reference measure this leads to the inequality

f ref ∞ ≤ n p d E W p Q 1 1 n n i=1 δ X i , 1 + Cn 2-d 2d ,
while for the bi-partite matching we obtain

f bi ∞ ≤ n p d E W p Q 1 1 n n i=1 δ X i , 1 n n i=1 δ Y i + Cn 2-d 2dp ,
for some constant C ≥ 0. Besides being one-sided bounds, these are still far from the conjectured rates in [START_REF] Caracciolo | Scaling hypothesis for the Euclidean bipartite matching problem[END_REF], which for p = 2 read

n 2 d E W 2 Q 1 1 n n i=1 δ X i , 1 n n i=1 δ Y i = f bi ∞ + Cn 2-d d + o(n 2-d d ),
with an explicit constant C. Remark 6.5 (Strong convergence). If p < d, standard concentration of measure arguments allow to obtain strong convergence from convergence of the expected values (see also [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] for a similar argument in the case p = d = 2). Let us consider for example the case of bi-partite matching, and show that, both P-a.s. and in L 1 (P),

lim n→∞ n p d W p Q 1 1 n n i=1 δ X i , 1 n n i=1 δ Y i = f bi ∞ , (6.5)
with (X i ) i≥1 , (Y i ) i≥1 independent uniform random variables on Q 1 = [0, 1] d . For any n ≥ 1, the function This relies on the triangle inequality (2.1) and the fact that

(x i ) n i=1 → µ (x i ) n i=1 is n -min 1 2 , 1
p -Lipschitz if we endow the set of probability measures on [0, 1] d with the Wasserstein distance of order p. Indeed, for (x i ) n i=1 , (y i ) n i=1 ∈ [0, 1] d×n , then

W p µ (x i ) n i=1 , µ (y i ) n i=1 ≤ 1 n n i=1 |x i -y i | p 1 p ≤ n -min 1 2 , 1 p n i=1 |x i -y i | 2 1 2
.

Gaussian concentration for the uniform measure on the unit cube [20, Prop. 2.8] yields that if

Z n = n 1 d W p µ (X i ) n i=1 , µ (Y i ) n i=1
, then, for r > 0,

P (|Z n -E [Z n ]| ≥ r) ≤ 2 exp -cn 2 min 1 2 , 1 p -1 d r 2 ,
where c > 0 is an absolute constant. A standard application of Borel-Cantelli Lemma gives that, if 1 ≤ p < d and d ≥ 3, then lim n→∞ (Z n -E [Z n ]) = 0, P-a.s. (6.6)

Moreover, using the layer-cake formula, we obtain the inequality

E [|Z n -E [Z n ]| p ] n p d -min{ p 2 ,1} , (6.7)
which is infinitesimal as n → ∞. To conclude, it is sufficient to argue that lim n→∞ E [Z n ] = (f bi ∞ ) 1/p , since it yields by (6.7) and (6.6) that lim n→∞ Z n = (f bi ∞ ) 1/p in L p (P) and P-a.s. and hence (6.5). By Theorem 6.2, lim n→∞ E [Z p n ] = f bi ∞ . By Jensen's inequality, lim sup n→∞

E [Z n ] p ≤ lim n→∞ E [Z p n ] = f bi ∞ .
By the elementary inequality (3.2), we have, for any ε > 0,

Z p n ≤ (E [Z n ] + |Z n -E [Z n ] |) p ≤ (1 + ε) E [Z n ] p + C ε p-1 |Z n -E [Z n ] | p .
Taking expectation and letting n → ∞, using (6.7), we obtain

f bi ∞ = lim n→∞ E [Z p n ] ≤ (1 + ε) lim inf n→∞ E [Z n ] p .
Letting ε → 0 we conclude.

. 3 )

 3 Proof. Let φ be a solution of the Poisson equation with Neumann boundary conditions ∆φ = µ -λ in R and ν • ∇φ = 0 on ∂R.(3.4)

Proposition 5 . 3 .

 53 Let λ = X∈Z d δ X and for every L 1, (µ 1 , • • • , µ 2 d ) be 2 d independent Poisson random variables of parameter L d . Writing for L 1,

Proposition 6 . 1 .

 61 Let f : (0, ∞) × N k → [0, ∞), (L, n) → f (L|n), satisfy the following assumptions:(1) (p-homogeneity)f (L|n) = L p f (1|n), for every n ∈ N k , L > 0, (2) (boundedness) f (1|n) 1, for every n ∈ N k , (3) (monotonicity) f (1|n) ≤ f (1|m), for every m, n ∈ N k such that m i ≤ n i for i = 1, . . . , k. Define f (L) = E [f (L|N L )] where N L = (N L,i ) k i=1 are i.i.d.Poisson random variables with parameter L d . Then, lim inf n→∞ n p d f (1|(n, . . . , n)) = lim inf L→∞ f (L), and lim sup n→∞ n p d f (1|(n, . . . , n)) = lim sup L→∞ f (L).

  f (1|(n, . . . , n)) ≤ lim sup L→∞ f (L), and lim inf n→∞ n p d f (1|(n, . . . , n)) ≤ lim inf L→∞ f (L).

[0, 1 ] 1 d 1 pδ

 111 d×2n (x 1 , . . . , x n , y 1 , . . . , y n ) → n W p µ (x i ) n i=1 , µ (y i ) -Lipschitz with respect to the Euclidean distance, where we writeµ (x i ) x i , for x i ∈ [0, 1] d .

  . By Poincaré inequality[START_REF] Leoni | A first course in Sobolev spaces[END_REF] Prop. 12.29] (notice that thanks to the periodic boundary conditions we now have R ∇φ = 0), By interior Calderón-Zygmund estimates (see for instance[START_REF] Giaquinta | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs[END_REF] Th. 7.3]), periodicity and the fact that R has moderate aspect ratio, we get

	|∇φ| p	|∇ 2 φ| p .
	R	R

R |∇ 2 φ| p R |µ -λ| p + R |∇ 2 φ| 2 p 2

Acknowledgments

M.G was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR) and by the LYSM LIA AMU CNRS ECM INdAM. M.G. also thanks the Centro de Giorgi in Pisa for its hospitality. D.T. was partially supported by the University of Pisa, Project PRA 2018-49, and Gnampa project 2019 "Proprietà analitiche e geometriche di campi aleatori".