Keywords: Infiltration tank, expansive clay, suction, water content, monitoring, soil heave

A large-scale infiltration tank was developed to study the water transfer in compacted expansive clay. Volumetric water content sensors were buried in a soil column for water content monitoring during infiltration. In addition to water content, soil suction and temperature at various locations and the heave at the soil surface were also monitored. Emphasis was put in minimizing the effect of sensors installation on water transfer and soil deformation. The results obtained for 338-days of infiltration were presented in terms of changes of suction, volumetric water content, temperature and the soil heave. Based on the recorded data, the performance and limitation of different suction and volumetric water content sensors and the adopted test procedure were analyzed. The recorded data on soil suction and volumetric water content were finally analyzed for determining the unsaturated hydraulic properties of soil such as the water retention curve and the unsaturated hydraulic conductivity. Note also that the results constitute useful data for further physical analysis or numerical models calibration.

Introduction

The water transfer in compacted expansive soil is a key issue in geotechnical and geoenvironmental engineering: in the safety assessment of geological radioactive disposal [START_REF] Alonso | Monitoring Large-Scale Tests for Nuclear Waste Disposal[END_REF] and surface waste disposal [START_REF] Chapuis | The 2000 R.M. Hardy Lecture: Full-scale hydraulic performance of soil-bentonite and compacted clay liners[END_REF][START_REF] Albright | Field performance of a compacted clay landfill final cover at a humid site[END_REF][START_REF] Delage | Geoenvironmental Testing[END_REF]; in the analysis of the impact of seasonal changes on the behavior of embankments and dams [START_REF] Ridley | Soil matrix suction: Some examples of its measurement and application in geotechnical engineering[END_REF]; in the analysis of soil-structure interaction caused by swell/shrinkage of expansive soils [START_REF] Abduljauwad | Laboratory and field studies of response of structures to heave of expansive clay[END_REF]; in the assessment of surficial stability of compacted clay slopes (Day and Axten, 1989;[START_REF] Aubeny | Shallow slides in compacted high plasticity clay slopes[END_REF]. From an experimental point of view, the water transfer analysis requires measurement of soil suction or water content. [START_REF] Tarantino | Field Measurement of Suction, Water Content, and Water Permeability[END_REF] present a review of techniques for field study on water transfer in unsaturated soils and show that several techniques are available for field measurement of suction, water content and unsaturated hydraulic conductivity. Nevertheless, as the boundary conditions in the field cannot be controlled and are difficult to be measured [START_REF] Cui | Field simulation of in-situ water content and temperature changes due to ground-atmospheric conditions[END_REF][START_REF] Cui | Water Balance and Evapotranspiration Monitoring in Geotechnical and Geoenvironmental Engineering[END_REF] additional laboratory tests are often required for field studies.

In laboratory, the unsaturated hydraulic conductivity of soil can be measured by various methods among which unsteady methods such as the instantaneous profile method [START_REF] Daniel | Measurement of hydraulic conductivity of unsaturated soils with thermocouple psychrometers[END_REF] are the most suitable for clayey soils [START_REF] Benson | Measuring unsaturated hydraulic conductivity in the laboratory and field[END_REF]. After [START_REF] Masrouri | Laboratory Hydraulic Testing in Unsaturated Soils[END_REF], unsaturated hydraulic conductivity of expansive soils is often determined in constantvolume conditions. This condition is verified in the case of engineered clay barriers used in the geological radioactive waste disposal. Indeed, the host geological barrier constrains the swelling of engineered clay barriers upon wetting [START_REF] Robinet | Détermination de la perméabilité non saturée des matériaux argileux a faible porosité[END_REF][START_REF] Loiseau | The gradient effect on the water flow through a compacted swelling soil[END_REF][START_REF] Kröhn | Results and interpretation of bentonite resaturation experiments with liquid water and water vapour[END_REF]. Nevertheless, in most cases, swelling takes place during infiltration [START_REF] Kim | Modelling of soil deformation and water flow in a swelling soil[END_REF][START_REF] Lemaire | Imbibition test in a clay powder (MX-80 bentonite)[END_REF]. With soil swelling, the advance rate of wet front at freeswell conditions was significantly higher than at constant-volume conditions [START_REF] Yong | A Study of Particle Interaction Energies in Wetting of Unsaturated Expansive Clays[END_REF]Cui et al., 2008b). This shows the importance of the coupling between water transfer and soil volume change during infiltration in compacted expansive soils. Further examination of laboratory infiltration studies reveals that, in the determination of unsaturated hydraulic conductivity by the instantaneous profile method, only the soil suction was continuously monitored [START_REF] Daniel | Measurement of hydraulic conductivity of unsaturated soils with thermocouple psychrometers[END_REF][START_REF] Chiu | Unsaturated hydraulic conductivity of compacted sand-kaolin mixtures[END_REF][START_REF] Bruckler | Testing an infiltration method for estimating soil hydraulic properties in the laboratory[END_REF][START_REF] Loiseau | The gradient effect on the water flow through a compacted swelling soil[END_REF], the water content evolution being deduced from the water retention curve determined separately. In the works of [START_REF] Robinet | Détermination de la perméabilité non saturée des matériaux argileux a faible porosité[END_REF] and [START_REF] Kröhn | Results and interpretation of bentonite resaturation experiments with liquid water and water vapour[END_REF], the infiltration test was stopped after a predefined period and the soil sample was dismantled for the direct determination of water content by oven-drying. Obviously, the number of water content profiles was limited by the number of tests performed. In the works of [START_REF] Kim | Modelling of soil deformation and water flow in a swelling soil[END_REF] and [START_REF] Lemaire | Imbibition test in a clay powder (MX-80 bentonite)[END_REF], the dual-energy gamma-ray technique was used for monitoring the variations of water content profile along the clay sample. Compared to the method employed by [START_REF] Robinet | Détermination de la perméabilité non saturée des matériaux argileux a faible porosité[END_REF] and [START_REF] Kröhn | Results and interpretation of bentonite resaturation experiments with liquid water and water vapour[END_REF], this method has the advantage of being non-destructive; but it needs sophisticated installation and is time consuming [START_REF] Masrouri | Laboratory Hydraulic Testing in Unsaturated Soils[END_REF].

Laboratory infiltration tests with both continuous suction and water content monitoring require usually medium to large-scale equipment [START_REF] Meerdink | Unsaturated hydraulic conductivity of two compacted barrier soils[END_REF][START_REF] Stormont | Capillary barrier effect from underlying coarser soil layer[END_REF][START_REF] Choo | Water flow through cover soils using modeling and experimental methods[END_REF][START_REF] Yang | A soil column apparatus for laboratory infiltration study[END_REF] where suction is monitored by tensiometers or psychrometers while volumetric water content is monitored by TDR probes.

Note that for the suction measurement, good contact should be ensured between the sensors and the soil, and for the measurement of water content using TDR probes, full insertion of the wave guides into the soil is needed. After [START_REF] Yang | A soil column apparatus for laboratory infiltration study[END_REF], embedding the full length TDR's waveguides in the soil allows a more accurate water content measurement.

In the present work, a laboratory infiltration tank was developed for the studies on water transfer in compacted expansive clays. Compacted Romainville expansive clay was tested. During water infiltration, the measurements of suction, volumetric water content and temperature were made at various locations. The soil heave was also monitored using 15 LVDTs (linear variable differential transformers) on the soil surface. A wide range of suction (from 0 to 100 MPa) was covered by three types of sensors: (1) sensors measuring the relative humidity in air-filled pores; (2) psychrometers for the total suction; and (3) high-capacity tensiometers for the matric suction. Volumetric water content was monitored using dielectric methods. These sensors were fully buried in the soil allowing free displacement of the sensors with soil deformation. A large soil volume was considered (800 × 1000 mm section with an initial height of 1000 mm). This is mainly justified by the significant volume of the sensors; a small soil volume would lead to unrepresentative measurements because of influence of the sensors on both water transfer and soil deformation. Moreover, the large soil section allowed minimizing the effect of the friction between the sidewall of the tank and the soil on the total soil heave in the central part. An infiltration period of 338 days was performed; the results are presented in terms of variations of suction, volumetric water content, temperature, and displacement observed on the soil surface. It is worth mentioning that even though the boundary conditions of the test performed do not correspond exactly to the field conditions in the applications cited previously, the first analysis on soil suction and volumetric water content allowed determining the unsaturated hydraulic properties such as water retention curve and unsaturated hydraulic conductivity on a significant soil body. Furthermore, the data recorded can be quite useful for further theoretical and numerical analysis covering at the same time changes in suction, volumetric water content, temperature and displacement.

Experimental setup

The schematic views of the experimental setup are presented in Fig. 1 and a picture is presented in Fig. 2. The soil was compacted in an acrylic transparent tank with a 20-mm-thick wall, 800-mm internal width and 1000-mm internal length, 1550-mm height (Fig. 1a). The tank is constituted of four separated acrylic plates joined together by epoxy glue on the four corners. The bottom of the tank is fixed on a PVC plate (30-mm thick). On the top, an acrylic cover is used. The system is strengthened by four vertical metallic stiffeners on the four corners and confined by various lateral metallic stiffeners in order to provide a good resistance against the lateral stress developed during wetting of the soil. The joints between the plates are finally sealed by silicon glue to prevent any leakage.

The details of the sensors used are presented in Table 1 and their locations are shown in Fig. 1. The distribution of the displacement sensors (LV) is shown in Fig. 1b. Five ThetaProbe sensors (TP) were buried in the soil (Fig. 1c). On the opposite side, five ECH 2 O-TE sensors (EC) were buried at the same levels. These sensors that allow the measurement of volumetric water content were installed every 200 mm along the height of the tank. Fig. 1c shows the 10 temperature sensors (PT) located every 100 mm along the height. The locations of highcapacity tensiometers (TS) and relative humidity sensors (RS) are indicated in Fig. 1d (section B-B). Each type of sensor was installed every 200 mm on both sides, allowing one measurement every 100 mm along the height from each type of sensor. The RS sensor T3111 is a programmable probe that is equipped with temperature and relative humidity sensors. The RS probe contains a microprocessor-based control circuitry in a durable plastic case with connection terminals and sensors in a filter with stainless steel mesh. The same distribution pattern was used for the installation of the psychrometers (PS) as shown in Fig. 1e (section C-C).

The details of the plugs designed for the installation of each type of sensor are shown in Fig. 3. The supports for ThetaProbe, ECH 2 O-TE and PT sensors (Fig. 3a) ensure the watertightness with the passages of cables. The RS sensor measures the relative humidity in a small tank (Fig. 3b) communicating with the soil through a porous metallic body. As for the tensiometers (Fig. 3c), they were kept in direct contact with the soil. The water flow system for the infiltration tank included the rainfall distributor and the drainage system. For the rainfall distributor, four perforated tubes with very small holes were installed on the cover of the tank. These tubes were connected to a tap of the local water system in the laboratory through a volume meter (± 50×10 -6 m 3 of accuracy) and a flow meter (±1.5×10 -3 m 3 /s of accuracy). Within this rainfall distributor, water can flow to the soil surface in a satisfactorily uniform pattern and at a controlled flow rate. For the drainage system, the soil was compacted on a layer of gravel (d = 2 -5 mm) of 30-mm thick separated by a layer of geotextile of 1-mm thick. Two outlets were perforated close to the bottom of the tank for water and air drainage.

Material and experimental procedures

The Romainville clay taken from a quarry in Cormeilles -en -Parisis, 25 km north-west from Paris, was studied. A general view of the geological profile of the site was presented by [START_REF] Arnould | Discontinuity networks in mudstones: A geological approach[END_REF]. Its geotechnical properties (after [START_REF] Audiguier | Caractérisation au laboratoire de la sensibilité au retrai-gonflement des sols argileux[END_REF][START_REF] Laribi | Assessing shrink/swell properties of two argillaceous soils from the Paris Basin: A comparison of cation exchange determination methods[END_REF] are presented in Table 2. According to the Casagrande's classification criterion, the soil is a high plasticity clay. It belongs to CH following the USCS soil classification system.

The soil transported to the laboratory was air-dried, crushed and sieved through a 2-mm sieve. It was then stocked in a sealed box for several days in order to homogenize its water content prior to the compaction. The water content at which the clay was compacted was 5.7%. For the compaction of the soil, 56.86 kg of the air-dried soil was first poured into the tank and compacted manually to have a layer of 50-mm thick. That corresponds to a dry density of 1.35 Mg/m 3 that was similar to the in-situ dry density [START_REF] Cui | Water retention properties and volume change behaviour of natural Romainville clay[END_REF]. The void ratio and the degree of saturation after compaction were 1.07 and 15.6% respectively. This procedure of compaction by layer was repeated until reaching the total height of 1000 mm. Note that this procedure allows obtaining a satisfactory homogeneity in the soil in terms of dry density.

During compaction, the installation of sensors buried in the soil was performed between the compaction of soil layers. The ECH 2 O-TE and PT sensors were installed on the compacted soil layer prior to pouring the subsequent soil layer. The installation procedure of ThetaProbe is shown in Fig. 4. After the compaction of the soil layer at the defined level, a hole having similar dimension of the sensor was created manually. The sensor was then placed horizontally in the hole by inserting the 4 steel guides inside the soil. The hole was finally filled and manually compacted with a calculated quantity of soil in order to ensure the same dry density of the soil (1.35 Mg/m 3 ). This procedure aimed at minimizing the effect of the sensors installation on the soil density.

The dielectric sensors (ThetaProbe and ECH 2 O-TE) were calibrated prior to use. For this purpose, Romainville clay was compacted at various water contents to a dry density of 1.35 Mg/m 3 . After the compaction, the sensors were then inserted into the compacted sample for the measurement. In Fig. 5, the volumetric water content (θ) of the compacted samples (determined using the gravimetric water contents by oven-drying and the global dry density) is plotted versus the square root of dielectric constant ( ε ) provided by the ThetaProbe sensor. The following best fitted curve was found:

θ = 2.5483( ε ) 2 -4.0981 ε + 0.0099 [1]
This equation was established based on the measurements made in the range of volumetric water content from 0 to 50%. In Fig. 6, the volumetric water content (θ) of the compacted samples (also determined using the gravimetric water contents by oven-drying and the global dry density) is plotted versus the raw value (RAW) provided by the ECH 2 O-TE sensor. A linear relationship can be proposed: [START_REF] Decagon | ECH2O-TE/EC-TM Water Content, EC and Temperature Sensors[END_REF] proposed a similar linear equation for mineral soils.

θ = 0.1049 (RAW) -65.47 [2] Note that
Chronologically, if the time for completion of the compaction is considered as t = 0, the relative humidity sensors and the displacement transducers (LV) were installed at t = 19 days. The system was then sealed to reach its equilibrium state for 25 days (t = 45 days). Infiltration was then started by rainfall on the soil surface at a rate of 0.6x10 -3 m 3 /min. The rainfall was stopped at t = 48 days, providing a total volume of water to the soil of 0.09 m 3 . From t = 48 days to t = 73 days, no more water was added. As it was difficult to control the water level above the soil surface when fixing the rainfall intensity at 0.6x10 -3 m 3 /mn, a permanent layer of water (10 -40 mm) was kept on the soil surface from t = 73 days. Subsequently, when the level of the water layer was lower than 10 mm, 0.02 m 3 of water was added, elevating the water level by 25 mm. The total volume of water added (Q inlet ) was plotted versus elapsed time as shown in Fig. 7a.

Experimental results

The results are presented in terms of changes in relative humidity (RH) (Fig. 7b), volumetric water content measured by ThetaProbe sensors (θ TP ) (Fig. 7c) and ECH 2 O-TE sensors (θ EC ) (Fig. 7d) and heave at soil surface (Fig. 7e). Note that the relative humidity sensors and the displacement transducer (LV) were installed at t = 19 days and the system was then sealed for 25 days (until t = 45 days) to reach its equilibrium state. At the end of the equilibration period, the relative humidity measured was 50±1% except that measured by the RS10 sensor. For RS10 which was situated close to the soil surface, the relative humidity was equal to 56% after the installation and then decreased progressively to 53% at t = 45 days. Actually, during the installation of the displacement transducers (at t = 19 days), the cover of the tank was opened, and the soil on the surface would be humidified by the air in the laboratory. After covering the tank, the relative humidity at the soil surface decreased approaching the value at lower soil layers. The volumetric water content at the end of the equilibration period was 12±1 % for θ TP and 9±1% for θ EC . It was significantly higher than the global volumetric water content (7.7%) estimated by considering a gravimetric water content of 5.7% and a dry density of 1.35 Mg/m 3 . Fig. 7b shows that after the start of infiltration at t = 45 days, the relative humidity measured at a depth of 50 mm (RS10) started to increase at t = 46 days and reached 96% after one day (t = 47 days). The relative humidity measured at a depth of 150 mm (RS9) started to increase at t = 47 days and reached 90% at t = 50 days. During the period t = 48 -73 days where no more water was added, the relative humidity measured at these two depths remained high while that of RS8 (at 250-mm depth) and RS7 (at 350-mm depth) increased significantly. The relative humidity measured at deeper levels remained unchanged during this period. For the second phase of infiltration (from t =73 days), the relative humidity measured by the other sensors increased progressively as a function of the distance from the sensor to the surface. In addition, the rate of the relative humidity increase was lower at deeper levels: for instance, 20 days were needed for RS6 (at 450-mm depth) to have the relative humidity increases from 50 to 96% (t = 90 -110 days) while 150 days were needed for RS2 at 850-mm depth (t = 150 -300 days).

The observations made on the changes in volumetric water content from the ThetaProbe sensors were similar to those from the relative humidity sensors (see Fig. 7c). The start of infiltration at t = 45 days induced a quick increase of the volumetric water content measured at 50-mm depth (TP5), from 11 to 52% after 2 days. During the period of t = 48 -73 days where no more water was added, this value decreased slightly to 46% while that measured at 250-mm depth (TP4) increased progressively to 20%. For the second phase of infiltration (from t =73 days), the volumetric water content measured at 50-mm depth (TP5) started first to increase quickly from 46 to 52% after 1 day. It continued to increase during the infiltration and reached 77% at t = 300 days. At t = 76 days, the volumetric water content measured at 250-mm depth (TP4) started to increase quickly (from 21%) and reached 47% after 2 days. It continued to increase and stabilized at 52% (t = 100 days). For the other ThetaProbe sensors, the larger the depth, the later the increase of volumetric water content. In addition, the final value of volumetric water content at a deeper sensor was lower. For instance, at t = 300 days, it was equal to 77% at 50-mm depth (TP5) against 47% at 650-mm depth (TP2). Note that the estimated volumetric water content at saturated state was 49% (corresponding to a density of soil particles of 2.67 Mg/m 3 and a dry density of 1.35 Mg/m 3 ). This high volumetric water content obtained at 50-mm depth (TP5) would be explained by a significant swelling of the soil near the surface; a volumetric water content of 77% (at saturated state) corresponds to a dry density of 0.58 Mg/m 3 . On the other hand, as the ThetaProbe was calibrated only in the range of volumetric water content from 0 to 50% (Fig. 5), this high volumetric water content would be also affected by the lack of accuracy.

The ECH 2 O-TE sensors were installed at the same levels as for the ThetaProbe sensors. Fig. 7d shows that the increase of the volumetric water content measured by the ECH 2 O-TE sensors during infiltration was similar to that by the ThetaProbe sensors. For instance, the volumetric water content measured at 50-mm depth (EC5) increased quickly at t = 45 days from 9% and reached its maximum at 49% after 2 days. That was similar to the response of ThetaProbe TP5 embedded at the same level. Nevertheless, significant difference can be observed between the two types of sensors after reaching high volumetric water contents. For the ThetaProbe sensors, from t = 73 days (second stage of infiltration), the volumetric water content increased continuously and stabilized at about 50% (except that measured at 50-mm depth, TP5). On the contrary, for ECH 2 O-TE sensors, the volumetric water content reached first a high value (between 60-70%) and then decreased significantly.

As far as the heaves are concerned (Fig. 7e), after the start of infiltration, at t = 45 days, it increased quickly and reached 29±1 mm after 2 days. During the period of t = 47 -73 days where no more water was added, the heaves continued to increase but at a significantly lower rate until it reached 34±2 mm at t = 73 days. During the second phase of infiltration (from t = 73 days) where a layer of water was permanently maintained on the soil surface, the heaves increased quickly in the beginning and trended to stabilize at the end of infiltration. The measurement was stopped at t = 283 days and the final heaves measured were in the range of 105-127 mm.

For further analysis, the results presented in Fig. 7b were used to plot the profiles of relative humidity at various times (Fig. 8). In this figure, the elevation h was calculated from the bottom of the soil column (from the level of the geotextile). Prior to the infiltration (t = 45 days), the relative humidity in the soil column was equal to 50%. When water was applied to the soil surface, it increased quickly at the highest layer: at t = 50 days, it was higher than 90% for h > 840 mm. At t = 100 days, the relative humidity was close to 100% for h > 640 mm while it remained unchanged at h < 440 mm. This transient zone (h = 440 -640 mm), where the soil started to be wetted but did not reach saturation, enlarged at higher values of time t (h = 140 -440 mm at t = 150 days; h = 40 -340 mm at t = 200 days). Note that in this figure, the relative humidity at the bottom was assumed to be constant at 50% which was the relative humidity of the air in the laboratory. Actually, as described previously (Fig. 1), the soil was compacted on a layer of gravel which was in contact with two air outlets. The bottom of the compacted soil column can be then assumed to be in contact with air in the laboratory. Thus both the relative humidity and the air pressure in the soil at the bottom corresponded to the conditions of the air in the laboratory.

The evolution of the volumetric water content measured by ThetaProbe sensors was plotted versus time in Fig. 9. In this figure, the elevation h corresponds to the initial position of the ThetaProbe sensors (the position would change due to soil swelling during infiltration). As mentioned previously, the volumetric water content at saturated state at a dry density of 1.35 Mg/m 3 was estimated at 49%. During infiltration, the swelling corresponded to a decrease of dry density and thus resulted in an increase of volumetric water content at saturated state. For the purpose of simplicity, in this analysis the soil was assumed to be saturated when the volumetric water content measured by ThetaProbe exceeded 49%. Fig. 9 shows that the saturated zone (θ TP > 49%) was advancing from the soil surface quickly in the beginning of the infiltration, but at a decreasing rate. For instance, at t = 100 days, the soil was saturated for h > 740 mm; and then h > 540 mm at t = 150 days; h > 340 mm at t = 200 days. Note also that at t = 300 days the soil at h = 140 mm still remained unsaturated.

Based on the heave data recorded by the LV sensors (15 points, see Fig. 1b) and the measurements performed using a ruler (accuracy: 1 ± mm) through the acrylic transparent walls of the tank, the heave of the soil surface at t = 283 days was plotted in a 3-D graph (Fig. 10). The maximum heave (120 -130 mm) was observed in the central part of the surface while the minimum heave (60 -80 mm) was observed on the four corners. The heave measured beside the walls remained lower than 100 mm.

The sensors measuring the soil suction at relatively low range (psychrometers and tensiometers) were unfortunately available only for the last months of the test. The results are presented in Fig. 11 at the same time with the data of temperature. In Fig. 11a, the total suctions measured by psychrometers are plotted. The total suction measured at h = 40 mm (PS1) was equal to 8 MPa (higher limit for the used psychrometer) at t = 304 days. It decreased to 2.21 MPa at t = 338 days. The total suction measured at h = 140 mm (PS2) decreased from 0.85 MPa (t = 285 days) to 0.16 MPa (t = 338 days). The total suction measured at h = 240 mm (PS3) fluctuated around 0.1 MPa. At higher elevation, the total suction was lower than the lower limit of psychrometers (0.05 MPa) and no results are then available.

The results obtained from the tensiometers are presented in Fig. 11b. After the installation of tensiometers, approximately 15 days were required to reach equilibrium. At h = 240 mm (TS3), the matric suction was equal to 50 kPa. It is lower but in the same order of magnitude of the total suction measured by the psychrometer PS3 at the same level. Note that the total suction is the sum of matric suction and osmotic suction and this latter was not measured. Lower matric suctions were recorded at higher elevation, in concordance with advancing of the infiltration front.

The variations of temperature are presented in Fig. 11c for the PT sensors within the soil and in Fig. 11d for the RS sensors beside the walls (see also Fig. 1). It can be observed that the diurnal variation of the RS sensors (± 1°C) was more pronounced than that of PT sensors (±0.1°C except for PT10).

The suction profile at t = 300 days measured by the three types of sensors was plotted in Fig. 12. The total suction at h = 0 was assumed to be equal to that imposed by the conditions of the ambient air (RH = 50% and T = 20°C, corresponding to s = 94 MPa, see [START_REF] Tang | Controlling suction by the vapour equilibrium technique at different temperatures and its application in determining the water retention properties of MX80 clay[END_REF]. At h = 40 mm, s = 18.5 MPa corresponding to a relative humidity of 87%. At higher elevation, the relative humidity was higher than 96% (s < 6 MPa). In this suction range, the error of measurement by RS sensor becomes significant (see [START_REF] Delage | The relationship between suction and swelling properties in a heavily compacted unsaturated clay[END_REF] and for this reason, these measurements were not considered. Three values of suction are available at t = 300 days for the psychrometers, corresponding to h = 40, 140 and 240 mm. The range of the total suction measured by psychrometers was 0.05 -8 MPa (see Table 1). For the tensiometers, the values of matric suction at h = 240, 340 and 440 mm are available. The results plotted in Fig. 12 show a satisfactory agreement between the measurements by the three types of sensors since all the results can be almost fitted with only one curve. To a certain extent, this also indicates that the total suction is approximately equal to the matric suction in the studied Romainville clay, the osmotic suction being negligible.

The volumetric water content (measured by the ThetaProbe) was correlated with the suction measured by the RS sensors (for s > 6 MPa), psychrometers (for s = 0.05 -8 MPa) and tensiometers (Fig. 13). One data set (volumetric water content and suction) every 30 days was taken from the measurements at each level. A satisfactory linear correlation was obtained in a ) log(s -θ plot. This relationship, which is independent of the sensors positions and of time, correspond to the wetting path of the soil water retention curve.

The unsaturated hydraulic conductivity was determined using the instantaneous profile method which is based on the generalized Darcy's law. The hydraulic gradient (i) at a moment t and a location z was calculated as the slope of the suction profile at t and z. This suction profile was plotted from the measurement of suction changes at various depths. The volume of water passing through the section at a location z during the time period from t to t + dt was calculated from the difference between the profiles of volumetric water content at t and t + dt. These profiles were plotted based on the measurements of volumetric water content at various depths. Finally, the unsaturated hydraulic conductivity (k) at the location z and at the moment t was calculated from the corresponding water fluxes and hydraulic gradient. The calculated unsaturated hydraulic conductivity is usually plotted versus the suction s at the corresponding location z and moment t. More details about the instantaneous profile method can be found in [START_REF] Daniel | Measurement of hydraulic conductivity of unsaturated soils with thermocouple psychrometers[END_REF], [START_REF] Chiu | Unsaturated hydraulic conductivity of compacted sand-kaolin mixtures[END_REF], [START_REF] Bruckler | Testing an infiltration method for estimating soil hydraulic properties in the laboratory[END_REF], Cui et al. (2008b). Fig. 14 presents the variations of hydraulic conductivity versus total suction. Note that the measurements by the relative humidity sensors allowed plotting the results for the high suction range (s = 4 -100 MPa) while that of psychrometers allowed plotting the results for low suction range (s = 1 -4 MPa). A good agreement between the two methods was obtained. The results show that the unsaturated hydraulic conductivity was close to 10 -14 m/s at high suction (100 MPa) and it increased to 10 -11 m/s at low suction (1 MPa).

Discussion

In this work, an infiltration test was performed by wetting compacted Romainville clay from the top surface and monitoring the change of relative humidity, volumetric water content, suction and temperature at various locations and the heave on the soil surface. The changes in relative humidity with time (Fig. 7b) are similar to that obtained by [START_REF] Loiseau | The gradient effect on the water flow through a compacted swelling soil[END_REF] when wetting a compacted expansive soil under confined conditions: the relative humidity close to the wetting end increased quickly while the rate of increase was lower at larger distances from the wetting end. This pattern is equally similar to that observed on the volumetric water content measured by ThetaProbe sensors (Fig. 7c) and to that obtained by [START_REF] Kröhn | Results and interpretation of bentonite resaturation experiments with liquid water and water vapour[END_REF].

The results of the volumetric water content measured by ThetaProbe sensors are in agreement with the results of suction. That confirms the validity of the sensor on one hand and that of the experimental procedure applied on the other hand. Actually, the performance of the ThetaProbe sensors in field conditions was confirmed in various works [START_REF] Lukangu | Use of sub-hourly soil water content measured with a frequency-domain reflectometer to schedule irrigation of cabbages[END_REF][START_REF] Robinson | Measurement of relative permittivity in sandy soils using TDR, capacitance and theta probes: comparison, including the effects of bulk soil electrical conductivity[END_REF][START_REF] Verhoef | The diurnal course of soil moisture as measured by various dielectric sensors: Effects of soil temperature and the implications for evaporation estimates[END_REF][START_REF] Smethurst | Seasonal changes in pore water pressure in a grass-covered cut slope in London Clay[END_REF][START_REF] Zhan | Instrumentation of an unsaturated expansive soil slope[END_REF]. The results obtained in this study show that the compaction procedure used allowed the sensors to be installed with negligible effect on soil density. In addition, comparing to the large dimension of the infiltration tank, the volume of the ThetaProbe sensors buried in the soil was negligible, and thus the sensors did not significantly influence the water transfer. The initial volumetric water contents measured by ThetaProbe and ECH 2 O-TE sensors were similar; that confirms the homogeneity of the compacted soil in terms of both density and water content.

As shown in Fig. 7, when the volumetric water content became higher than 50%, the responses of ECH 2 O-TE sensors were different from that of ThetaProbe sensors. This can be explained by the limited measurement range provided by the ECH 2 O-TE sensors. Indeed, after [START_REF] Decagon | ECH2O-TE/EC-TM Water Content, EC and Temperature Sensors[END_REF] and [START_REF] Kizito | Frequency, electrical conductivity and temperature analysis of a low-cost capacitance soil moisture sensor[END_REF], the ECH 2 O-TE sensor provides accurate measurements in the range of volumetric water content from 0 to 50%. For higher values, more calibrations are required.

The variations of heave with time recorded (Fig. 7e) are similar to that observed by [START_REF] Abduljauwad | Laboratory and field studies of response of structures to heave of expansive clay[END_REF] and [START_REF] Kim | Modelling of soil deformation and water flow in a swelling soil[END_REF]: wetting induced a quick increase of heave and the rate decreased then progressively with time. The 3-D presentation of heave at t = 283 days (Fig. 10) shows a significant boundary effect: the heave measured in the middle zone of the surface (120 -130 mm) was twice of that on the corners (60 -80 mm). The significant boundary effect can be attributed to the friction between the walls of the infiltration tank and the soil; this friction decreased the soil swelling as the vertical stress does in oedometer condition. Indeed, most works have showed that the maximum swelling strain recorded on expansive soils upon wetting in oedometer decreased drastically when the vertical stress increased. As a consequence, the measurements provided by the suction sensors near the edge of the tank affected by the edge friction and corresponding constraint on the porosity, and do not represent the suction changes in the central part of the tank.

The boundary effect can be also observed on the temperature measurement (Fig. 11): the diurnal variation of T measured within the soil column (±0.1 °C) was significantly lower than that measured on the walls (±1°C). In addition, T measured close to the soil surface (PT10) fluctuated with higher amplitude than other PT sensors.

The time needed to reach suction equilibrium was quite long for the tensiometers: 15 days (see Fig. 11). This long time is related to the low permeability of compacted expansive soils [START_REF] Singh | Estimation of unsaturated hydraulic conductivity using soil suction measurements obtained by an insertion tensiometer[END_REF][START_REF] Oliveira | Suction equilibration time for a high capacity tensiometer[END_REF]. In addition to the limited range of suction for the tensiometers (lower than 1.5 MPa), this long equilibrium time is another drawback of the tensiometers when used to monitor the soil suction in expansive soils. This is probably the reason why tensiometers have been not commonly used in the studies on expansive soils. In addition, as stated by Cui et al. (2008a), the high-capacity tensiometer is slightly sensitive to temperature changes (the pressure given increases with the temperature rise at a rate of 1 kPa/°C). For this reason, the accuracy of the tensiometers used in the conditions of the present test can be estimated at 1 kPa. That corresponds equally to the level of positive pore water pressure in the flooding condition (100 mm of water). Obviously, this water level can not be monitored by the high-capacity tensiometers used.

Besides the instantaneous profile method used for determining the unsaturated hydraulic conductivity, there are other methods for analyzing such experimental data. In the work of [START_REF] Robinet | Détermination de la perméabilité non saturée des matériaux argileux a faible porosité[END_REF], the unsaturated hydraulic conductivity was evaluated using a simple theoretical approach considering the presence of water in two forms, adsorbed and capillary. Water pressure and gas pressure were used as variables. [START_REF] Kim | Modelling of soil deformation and water flow in a swelling soil[END_REF] used a one-dimensional numerical model to analyze the soil deformation and unsaturated transient water flow in swelling soil based on a laboratory infiltration test. The [START_REF] Wind | Capillary conductivity data estimating by a simple method[END_REF] method, previously described to estimate soil hydraulic properties under evaporation conditions, was applied to infiltration conditions by [START_REF] Bruckler | Testing an infiltration method for estimating soil hydraulic properties in the laboratory[END_REF]. [START_REF] Lemaire | Imbibition test in a clay powder (MX-80 bentonite)[END_REF] analyzed the water transport in expansive clay (from an infiltration test) using the common diffusion equation involving Boltzmann variable. A parameter of hydraulic diffusivity was deduced using the Matano's [START_REF] Pel | Determination of moisture diffusivity in porous media using moisture concentration profiles[END_REF] method. After [START_REF] Lemaire | Imbibition test in a clay powder (MX-80 bentonite)[END_REF], two fluid transport mechanisms were involved in the test performed: imbibition of the microporosity in the scale of clay grains and capillary imbibition in the mesoporosity around the clay grains.

The results of unsaturated hydraulic conductivity are in agreement with that obtained in previous works [START_REF] Daniel | Measurement of hydraulic conductivity of unsaturated soils with thermocouple psychrometers[END_REF][START_REF] Chiu | Unsaturated hydraulic conductivity of compacted sand-kaolin mixtures[END_REF][START_REF] Bruckler | Testing an infiltration method for estimating soil hydraulic properties in the laboratory[END_REF]Cui et al. 2008b): the unsaturated hydraulic conductivity increased with infiltration. After Cui et al. (2008b), similar trend is usually observed on low-plasticity soils. For compacted expansive clays, Cui et al. (2008b) obtained also an increase in hydraulic conductivity with suction decrease when wetting the soil under free swell conditions whereas an inverse trend was observed at constant-volume conditions due to pore clogging by clay exfoliation.

The results obtained in this study are in agreement with existing data, confirming that the size of the tank is large enough for neglecting the effect of embedded sensors on the water transfer and soil swelling. On the other hand, the large size of the section allows developing studies on the soil-atmosphere interaction when rainfall, solar radiation, air circulation with controlled relative humidity are applied on the soil surface. The development of such a system can be based on the works reported by several authors as [START_REF] Kohsiek | A Rapid-Circulation Evaporation Tank for Measuring Bulk Stomatal-Resistance[END_REF], van de Griend and Owe (1994), [START_REF] Aluwihare | Measurement of evaporation on bare soil and estimating surface resistance[END_REF]. Note however that most studies on soilatmosphere interaction are usually conducted with soil and/or climate monitoring in field condition [START_REF] Tsaparas | Infiltration characteristics of two instrumented residual soil slopes[END_REF][START_REF] Fityus | Expansive soil test site near Newcastle[END_REF][START_REF] Smethurst | Seasonal changes in pore water pressure in a grass-covered cut slope in London Clay[END_REF][START_REF] Zhan | Instrumentation of an unsaturated expansive soil slope[END_REF][START_REF] Cui | Water Balance and Evapotranspiration Monitoring in Geotechnical and Geoenvironmental Engineering[END_REF].

Conclusion

A large-scale infiltration tank was developed in order to study the water infiltration in compacted expansive Romainville clay. The soil sample was instrumented by various suction sensors (high capacity tensiometer, psychrometer, and relative humidity sensor), volumetric water content sensors (ThetaProbe and ECH 2 O-TE), temperature sensors and displacement sensors. A specific installation was adopted for each type of sensor. Emphasis was put in minimizing the effect of sensors installation on water transfer and soil deformation. An infiltration test was performed with monitoring for 338 days. The following conclusions can be drawn.

-Monitoring total suction changes by measurement of relative humidity at various locations allowed tracking the water transfer in the soil. As these measurements are only valid for the high suction range they should be completed with the measurements by other types of sensors for lower range of suction (psychrometer, tensiometer). The results obtained show a good agreement between the three types of suction sensors. Note however that long equilibrium time was required when using high-capacity tensiometers for suction monitoring in expansive soils as Romainville clay.

-Sensors for monitoring the volumetric water content were buried inside the soil column, which allowed the free displacement of the sensors with the soil during infiltration. The good agreement between water content measurement and suction measurement confirms that the dimensions of the soil sample were large enough to minimize the effect of the sensors on water transfer.

-Heave measured at various locations on the soil surface showed a significant effect of the tank-soil friction on the soil swelling, the heave measured in the middle of the surface being twice that on the corners.

-With the experimental procedure applied, the response of the ThetaProbe sensor has been found to be in agreement with that of suction measurement while the response of the ECH 2 O-TE showed some anomalies.

-The variations of hydraulic conductivity versus total suction were determined from the suction and volumetric water content measurements. The results were in good agreement with other works on compacted expansive soils: the hydraulic conductivity was increasing with suction decreases. For the studied Romainville clay, the unsaturated hydraulic conductivity was close to 10 -14 m/s at high suction (100 MPa) and it increased to 10 -11 m/s at low suction (1 MPa).

Fig. 1 .FigFig. 14 .

 114 Fig. 1. Schematic views of the experimental setup

Table 1 . Sensors used

 1

	Name Type	Manufacturer Model	Parameter measured	Range	Accuracy	Number Position Reference
	TS	High-capacity tensiometer ENPC		Matric suction	0 -1500 kPa	_	6	Wall	Cui et al. (2008a)
	PS	Psychrometer	Wescor	PST-55	Total suction	0.05 -8 MPa	± 0.03 MPa 8	Wall	Wescor (2004)
	RS	Transmitter	Elcowa	T3111	-Relative humidity	0 -100%	±2.5%	10	Wall	Comet (2008)
					-Temperature	-30°C to +150°C	±0.4°C
	TP	ThetaProbe	Delta-T	ML2x	Volumetric water content 0 -100%	±1.0%	5	Soil	Delta-T (1999)
	EC	Capacity	Decagon	ECH 2 O-TE -Volumetric water content	0 -100%	±3.0%	5	Soil	Decagon (2007)
					-Electrical conductivity	0 -50 dS/m	±10.0%
					-Temperature	-40 to +50°C	±1.0°C
	PT	Resistance temperature	Correge	PT1000	Temperature	0 -100 °C	±0.3°C	10	Soil	Correge (2008)
		detectors
	LV	Linear variable differential	FGP	DX20EL	Displacement	0-50 mm	±0.125 mm 15	Surface FGP (2004)
		transformer

Table 2 . Geotechnical properties of Romainville green clay (after Audiguier et al., 2007; Laribi et al., 2008)

 2

	Property	Romainville green clay
	Carbonate content (%)	15-20
	Organic matter content (%) 0.12
	Specific surface area (m 2 /g) 98
	Blue methylene value	6.5
	Liquid limit (%)	75
	Plastic limit (%)	40
	Plasticity index (%)	35
	< 2 µm (%)	84
	Specific gravity	2.67

Acknowledgments

This work was carried out within the framework of the research project "Aléa et Risque Sécheresse" supported by the MAIF Foundation.


We present results on the Anderson localization in a quasi one-dimensional metallic wire in the presence of magnetic impurities. We focus within the same numerical analysis on both the universal localized and metallic regimes, and we study the evolution of these universal properties as the strength of the magnetic disorder is varied. For this purpose, we use a numerical Landauer approach, and derive the scattering matrix of the wire from electron's Green's function obtained from a recursive algorithm.

Interplay between disorder and quantum interferences leads to one of the most remarkable phenomenon in condensed matter : the Anderson localization of waves. The possibility to probe directly the properties of this localization with cold atoms [1,2] have greatly renewed the interest on this fascinating physics. In this paper, motivated by the transport properties of metallic spin glass wires [3,4,5], we study the electronic localization in the presence of both a usual scalar random potential and frozen random magnetic moments.

One dimensional disordered electronic systems are always localized. Following the scaling theory [6] this implies that by increasing the length L x of the wire for a fixed amplitude of disorder, its typical conductance ultimately reaches vanishingly small values. The localization length ξ separates metallic regime for small length L x ≪ ξ from the asymptotic insulating regime. In the present paper, we focus on several universal properties of both metallic and insulating regime of these wires in the simultaneous presence of two kind of disorder, originating for example from frozen magnetic impurities in a metal. The first type corresponds to scalar potentials induced by the impurities, for which the system has time reversal symmetry (TRS) and spin rotation degeneracy. In this class the Hamiltonian belongs to the so-called Gaussian Orthogonal Ensemble (GOE) of the Random Matrix Theory classification [7] (RMT). If impurities do have a spin, the TRS is broken as well as spin rotation invariance. The Hamiltonian is then a unitary matrix, which corresponds in RMT to the Gaussian Unitary Ensemble (GUE) with the breaking of Kramers degeneracy [8]. However, for the experimentally relevant case of a magnetic potential weaker than the scalar potential, the system is neither described by the GUE class, nor by the GOE class, but extrapolates in between.

In this paper, we study numerically the scaling of transport properties of wires with various relative strength of these two disorders. The phase coherence length L φ which phenomenologically accounts for inelastic scattering [START_REF] Akkermans | Mesoscopic Physics of electrons and photons[END_REF] is assumed larger than the wire's length L x . We describe the disordered wire using a tight-binding Anderson lattice model with two kinds of disorder potentials :

H = t <i,j>,s c † j,s c i,s + i,s v i c † i,s c i,s + J i,s,s ′ S i . σ s,s ′ c † i,s c i,s ′ , (1)
The first scalar disorder potential V = {v i } i is diagonal in electron-spin space. v i is a random number uniformly distributed in the interval [-W/2, W/2]. s, s ′ label the SU (2) spin of electrons and the S i correspond to the frozen classical spin of impurities, with random orientations uncorrelated from an impurity site to another. Varying the amplitude of magnetic disorder J allows to extrapolate from GOE to GUE. For a given realization of disorder, the Landauer conductance of this model on a 2D lattice of size (in units of lattice spacing) L x × L y is evaluated numerically by a recursive Green's function technique. We stay near the band center, avoiding the presence of fluctuating states studied in [START_REF] Deych | [END_REF]. Universal properties are identified by varying the transverse length L y from 10 to 80, with the aspect ratio L x /L y taken from 1 to 6000. Typical number of disorder averages is 5000, but for L y = 10 we sampled the conductance distribution for 50000 different configurations of disorder.

Localization length. The localization length ξ is extracted from the scaling of the typical conductance in the localized regime according to [11,12] :

exp log g = exp (-2L x /ξ) , (2)
where g = G/(e 2 /h) is the dimensionless conductance and • represents the average over scalar disorder V . Numerical plot of the logarithm of this equation is given in the inset of figure 1. The linear behavior of log g is highlighted in the insulating regime and fitted to provide the localization length for each value of the transverse length L y . Note that we checked that the very slow convergence

of the Lyapunov exponent γ(L x ) = 1 2Lx log 1 + 1 g(Lx)
towards ξ -1 (L y ) provided comparable results. The evolution of ξ with the transverse length is expected to follow [11]:

ξ = (βL y + 2 -β)l e , (3)
with l e the mean free path and β = 1 corresponds to the orthogonal universality class GOE while β = 2 for GUE. Note that this change in β is accompanied by an artificial doubling of the number of transverse modes N y ≡ L y → 2N y due to the breaking of Kramers degeneracy [11]. Comparison of numerical localization lengths for different J with (3) is shown in fig 1. Excellent agreement is found for J = 0 (GOE class, β = 1). In the case J = 0 we observe a crossover between GOE and GUE for intermediate values of magnetic disorder, while a good agreement with the GUE class is reached for J ≥ 0.2.

From these results, we already notice that the localization regime is reached for much longer wires in the GUE case than for GOE. As shown below, this allows for an easier numerical investigation of the universal metallic regime in the GUE case : magnetic impurities help in finding the universal conductance fluctuations ! Insulating regime. In the insulating regime L x ≥ ξ, we expect a Log-normal conductance statistical distribution [11]. However in the region g = 1 and for g 1, we find a non-analytical behavior of P (g) in agreement with [13,14,15,16] as shown for instance in fig. 2. In this figure we plot the distribution P (g) for similar values of g for GOE (J = 0) and GUE (J = 0.2). The shapes of these distributions are highly similar if g ≪ 1, showing that both distributions tend to become Log-normal with the same cumulants. In the intermediate regime, shapes are symmetry dependent. Moreover the non-analyticity appears for different values of conductance (close to 1) and the rate of the exponential decay [13] in the metallic g (J = 0) = 0.21 and g (J = 0.2) = 0.18. BOTTOM: PDF of conductance for g < 1(J = 0.2) and g > 1(J = 0) and Gaussian interpolations. Ly = 10. regime seems to differ from one ensemble to the other (see for instance curves (a) or (b)). Finally, the bottom curve of figure 2 represents the distribution of conductance for just above and below the threshold g = 1. Plain lines represent gaussian interpolations with a mean and a variance given by the first and the second cumulant of each numerical conductance distribution. For g > 1, the gaussian interpolation approximates very well the full distribution. On the other hand, as soon as g < 1, the gaussian law only approximates the tail g ≥ 1 of the distribution of conductance . This behavior is in agreement with the sudden appearance of the non-analyticity for distributions with average conductance inferior to 1 [16]. This conductance distribution converges to the Lognormal only deep in the insulating regime, the convergence being very slow (much slower than in the metallic regime). This qualitative result is confirmed by the study of moments : in the insulating regime the second cumulant is expected to follow [17]: Our numerical results are in agreement with this scaling (figure 3) with however very slow convergence towards this law : corrections are measurable even if the system is deeply in the localized state. More precisely, we find (see fig. 3) that for the deep insulating regime (log g) 2 c = -1.88 log g slope -1.88, with a slight discrepancy with (4). Finally in the inset of figure 3, we show the third cumulant of log g as a function of the first one. The linear behavior is in agreement with the single parameter scaling. We find that contrary to the second cumulant the coefficient of proportionality between the skewness and the average depends on the symmetry of disorder, which is not expected.

(log g -log g ) 2 = (log g) 2 c = -2 log g , (4)
Metallic regime. We now focus on the universal metallic regime described by weak localization. By definition weak localization corresponds to metallic diffusion, expected for lengths of wire l e ≪ L x ≪ ξ. For this regime to be reached, we thus need to increase the number of transverse modes L y and thus ξ for all other parameters fixed (see (3)). Moreover, for a fixed geometry, this regime will be easier to reach in the GUE class than in the GOE. Fig. 2 shows that the conductance distribution is in good approximation Gaussian with a variance described [START_REF] Akkermans | Mesoscopic Physics of electrons and photons[END_REF] by

δg 2 = g 2 c = 1 4 F (0) + 3 4 F x 4 3 + 1 4 F x √ 2 + 1 4 F x 2 3 , (5)
where x = L x /L m and the scaling function F (x) depends only on dimension [START_REF] Akkermans | Mesoscopic Physics of electrons and photons[END_REF]18]. In the bottom plot of figure 4, these conductance fluctuations are plotted as a function of longitudinal length L x for different values of J. A single parameter fit by (5) provides the determination of the magnetic dephasing length L m as a function of the magnetic disorder J. The determination of L m allows for very fine comparison with weak localization theory in this regime, allowing for example the study of conductance correlation between different disorder configurations (see [18,19]). The inset of Fig. 4 shows the scaling form of these fluctuations (as a function of L x /L m (J)) in excellent agreement with the theory (5). Moreover, for long wires (and large values of J) conductance fluctuations are no longer L x dependent and equal to 1/15. This is the so-called Universal Conductance Fluctuations (UCF) regime which is precisely identified numerically in the present work. The top plot of figure 4 confirms analytical results from [15] both qualitatively in the shape of the curves and quantitatively in the values of fluctuations in both universality classes. In our study, values of UCF are reached with a maximal error of 1% for GOE and 3% for GUE with respect to the analytical value of the UCF in the regime independent of g (i.e with much higher precision than e.g [20] and [14]) . The other information provided by this curve is the condition for having a universal behavior, i.e independence on the ge- ometry or on the disorder of the sample. As the localized regime is harder to reach for GUE, the universal metallic regime is easier to obtain, the plateau of UCF is widen. One needs then larger transverse lengths (recall that ξ increases with L y ) to get the UCF regime for GOE, this is exactly what we see on figure 4 where the UCF plateau is reached only for L y = 80 for GOE whereas it is reach for L y = 80 and L y = 40 for GUE. Finally we consider the third cumulant of the distribution of conductance. According to the analytical study of [15], this cumulant decays to zero in a universal way as g increases. Here we find a dependance of this decrease on the symmetry class : for GOE g 3 c goes to zero in a monotonous way whereas it decreases, changes its sign and then goes to zero in GUE case. For g > 4 numerical errors are dominant, then this part of the curve is irrelevant. Note that this fast vanishing of the third cumulant confirms the faster convergence of the whole distribution towards the gaussian, compared to what happens in the insulating regime. Based on our numerical results, we cannot confirm nor refute the expected law g 3 c ∝ 1/ g n , with n = 2 in GOE and n = 3 GUE [21,22].

To conclude we have conducted extensive numerical studies of electronic transport in the presence of random frozen magnetic moments. Comparing and extending previous analytical and numerical studies, we have identified the insulating and metallic regimes described by the universality classes GOE and GUE. We have paid special attention to the dependance on this symmetry of cumulants of the distribution of conductance in both metallic and insulating universal regimes. In particular, we have identified with high accuracy the domain of universal conductance fluctuations, and determined its extension in the present model.

We thank X. Waintal for useful discussions. This work was supported by the ANR grants QuSpins and Mesoglass. All numerical calculations were performed on the computing facilities of the ENS-Lyon calculation center (PSMN).

Note : during the very last step of completion of this paper, we became aware of a preprint by Z. Qiao et al. [23] which performed a similar numerical Landauer study of 1D transport for various universality classes, and focused mostly on the metallic regime. While both studies agree on the finding of UCF (although we have higher accuracy for β = 1), we did not find signs of a second universal plateau for (δg) 2 in our study.

  FIG.1: Evolution of localization length as a function of transverse length. le is the mean free path of the diffusive sample. Different behavior of the localization length if J = 0 or J = 0. Inset : Scaling of the typical conductance log g = -2Lxξ .

  FIG.2: TOP: Comparison of Probability density functions (PDF) of conductance for J = 0 (plain curves) and J = 0.2 (dashed curves). Plots are performed for different values of average conductance. (a): g (J = 0) = 0.84 and g (J = 0.2) = 0.79. (b): g (J = 0) = 0.67 and g (J = 0.2) = 0.62. (c): g (J = 0) = 0.45 and g (J = 0.2) = 0.42. (d): g (J = 0) = 0.21 and g (J = 0.2) = 0.18. BOTTOM: PDF of conductance for g < 1(J = 0.2) and g > 1(J = 0) and Gaussian interpolations. Ly = 10.

FIG. 3 :

 3 FIG.3: Plot of the variance of log g as a function of the mean for the orthogonal and unitary case. In inset is shown the third cumulant of log g as a function of the mean.

FIG. 4 :

 4 FIG. 4: TOP : second cumulant of g as a function of first cumulant of g showing the universal behavior in the metallic regime. BOTTOM : variance of g as a function of longitudinal size. UCF are shown. Different curves correspond to different values of magnetic disorder J. In inset variance of g is plotted as a function of Lx/Lm. Transverse length Ly = 40.

FIG. 5 :

 5 FIG.5: Plot of g 3 c as a function of g in the metallic regime. In insets are shown convergence curves for Ly = 10 and J = 0 or J = 0.2.

  Universal metallic and insulating properties of one dimensional Anderson Localization : a numerical Landauer study Guillaume Paulin and David Carpentier CNRS UMR 5672 -Laboratoire de Physique de l'Ecole Normale Supérieure de Lyon, 46, Allée d'Italie, 69007 Lyon, France (Dated: October 22, 2009)


Keywords: Node coloring, Distributed algorithms, SINR interference model, time complexity, Radio networks

A Note on Node Coloring in the SINR Model

Résumé : A V-coloring of a graph G is a coloring of the nodes of G with V colors in such a way any two neighboring nodes have different colors. We prove that there exists a O(∆ log n) time distributed algorithm computing a O(∆)-coloring for unit disc graphs under the signal-tointerference-plus-noise ratio (SINR)-based physical model (∆ is the maximum degree of the graph). We also show that, for a well defined constant d, a d-hop O(∆)-coloring allows us to schedule an interference free MAC protocol under the physical SINR constraints. For instance this allows us to prove that any point-to-point message passing algorithm with running time τ can be simulated in the SINR model in O(∆(log n + τ )) time using messages of well chosen size. All our algorithms are proved to be correct with high probability.

Introduction

Motivation We are interested in structuring wireless multi-hop radio networks formed of autonomous nodes communicating via radio and having no fixed built-in communication infrastructure. In such networks, the nodes must organize themselves distributively and establish some kind of structure allowing them to communicate and to perform their intended task efficiently.

Structuring radio networks in a local manner is of great practical and theoretical importance [START_REF] Kuhn | Initializing newly deployed ad hoc and sensor networks[END_REF]. Among the most useful structures allowing radio devices to self-organize their communications efficiently is coloring [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF][START_REF] Moscibroda | Coloring Unstructured Radio Networks[END_REF]. Many kinds of coloring may exist for different purposes. In fact, there is an impressive amount of work concerning network coloring (with several variants and applications) witnessing of the quintessential of this task, e.g., [BKM + 06, SGP08, SGP05]. For instance, assigning different colors to neighboring vertices and associating colors with different time-slots in a time-division multiple access (TDMA) scheme is nothing else than a medium access control (MAC) protocol without direct interferences.

Many distributed algorithms computing good colorings can be found in the literature. Although a lot of advance has been made, the existing distributed algorithms still suffer from some weaknesses. For instance, some existing distributed algorithms are proved to be efficient only through simulations or for some particular network topologies. More importantly, some other distributed algorithms assume rather simplified models of radio communication which allows a good theoretical understanding of the algorithmic issues but do not lead to practical implementations in real radio networks under different deployment scenarios. In particular, the best time efficient coloring algorithm [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF][START_REF] Moscibroda | Coloring Unstructured Radio Networks[END_REF] is described and analyzed in the graph-based model. In the graph based model and some of its variant, e.g., protocol model, the interference experienced by a node is modeled as local binary function. Typically in the classical graph based model, a node experiences an interference if more than two of its neighbors are sending at the same time slot. This model allows us to concentrate on the algorithmic issues and to give rigorous analysis of designed algorithms. However, it abstracts away the interferences experienced by a node in a realistic environment, thus the obtained algorithms cannot be guaranteed to work under more sophisticated interference constraints.

The aim of this paper is to study distributed coloring algorithms under the more realistic signal-to-interference-plus-noise ratio (SINR)-based physical model [START_REF] Gupta | The capacity of wireless networks[END_REF]. In this model, a node experience an interference if the ratio of the received signal strength and the sum of the interference caused by all nodes sending simultaneously in the network, plus noise is less then a hardwaredefined threshold β, where the signal fades with the distance to the power of some path-loss exponent α.

Contribution

In this paper, we adapt the best state-of-the-art coloring algorithm given initially by [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF][START_REF] Moscibroda | Coloring Unstructured Radio Networks[END_REF] to compute with high probability a O(∆)-coloring in O(∆ log n) time under the SINR physical constraints. The number of colors produced by the algorithm is optimal up to a constant factor, and the time complexity is optimal up to a log n factor. To the extent of our knowledge, this the first time where an almost optimal coloring algorithm is given in the SINR model.

As stated previously, one important application of coloring algorithms is to design interference free MAC protocols. However, it is well known that a O(∆)-coloring is not sufficient to design such interference free protocols even under the simple graph based model. In this paper we show that under the SINR model a MAC layer without direct interference can be scheduled in O(∆) time using a d-hop O(∆)-coloring for a well chosen constant d. A d-hop V-coloring is a coloring of nodes with at most V colors such that every node has a different color than the nodes in its d-hop Euclidian neighborhood 1 . As a direct application, we obtain upper bounds on the simulation of any point-to-point message passing algorithm in the SINR model.

Model and Definitions

We assume that nodes are placed arbitrarily in the 1 dimensional Euclidean space 2 , i.e., the plane. Given two nodes u and v, we denote by d(u, v) the distance between nodes u and v. We consider the signal-to-interference-plus-noise ratio model [START_REF] Gupta | The capacity of wireless networks[END_REF]. In this model, a node u successfully receives a message from a sender v if and only if the following condition holds:

P v d(u, v) α N + w∈V \{v} P w d(u, w) α ≥ β
where P v is the power level of the transmission of node v, α > 2 is the path-loss exponent, which depends on external conditions of the medium, β ≥ 1 denotes the minimum signal to interference ratio required for a message to be successfully received, N is the ambient noise, and w∈V \{v} P w d(u, w) α is the total amount of interference experienced by receiver u and caused by all simultaneously transmitting nodes in the network.

We assume that all nodes are transmitting with the same power P , i.e., uniform power assignment scheme. Let R T be the transmission range of nodes. Let B v be the transmission region of node v that is the disc of radius R T around v. From the definition of the SINR model, we have that R T ≤ ( P N β ) 1/α . We assume that the ambient noise level N is upper bounded by a fraction of the maximum tolerable interference level for a successful broadcast. More precisely, we assume that R T = ( P 2N β ) 1/α , i.e., N = P 2βR α T . Thus, the network can be viewed as a 'unit' disc graph G = (V, E) where two nodes are neighbors (i.e., they can potentially communicate) if they are at distance R T .

Given the transmission range R T of nodes, we define φ(R) to be size of the largest independent set (with respect to R T ) in the disc of radius R around any node of the graph 3 .

For every node v, let I v be the disc of radius R I around v where R I is defined as following (ρ > 1 is a constant to be fixed later):

R I = 96 ρ β • α -1 α -2 1/(α-2) • 2R T Note that R I ≥ 2R T .
For any constants ρ > 1 and c 2 ≥ c 1 ≥ 5, we define the following variables:

λ = 1 - 1 ρ • 1 e φ(RI )/φ(RI +RT ) • 1 - φ(R I ) φ(R I + R T ) 2 • ∆ • 1 - 1 φ(R I + R T ) φ(RI ) λ ′ = 1 - 1 ρ • 1 e • φ(R I + R T ) • 1 - 1 φ(R I + R T )∆ • 1 - 1 φ(R I + R T ) φ(RI +RT )
1 Here, the d-hop neighborhood of a given node v in the plane is the set of nodes at distance at most d • Rv where Rv is the transmission range of v that is the distance up to which a node can hear v's transmissions in absence of interferneces, i.e., in a clear environment.

2 We remark that our results can be proved to hold under any bounded independence graph. The assumption that nodes are placed in the plane is only made for the sake of clarity.

3 An independent set with respect to a given transmission range R T is a set of nodes mutually at distance d > R T . Notice that φ(R) can be roughly bounded as following:

φ(R) ≤ π(R + R T /2) 2 π(R T /2) 2 = " 2R R T + 1 « 2 . Notice
also that in our proofs, knowing only an upper bound on φ(R) affects our bound only by a constant, i.e., knowing the exact value of φ(R) is not required to prove our results.

σ = 2c 2 λ ′ , γ = c 1 • φ(R I + R T ) λ q ℓ = 1 φ(R I + R T ) , q s = 1 φ(R I + R T )∆
By a routine computation, one can easily verify that σ ≥ 2γ. Finally, we shall use any constant variables η and µ verifying: η ≥ 2γφ(2R T ) + σ + 1 and µ ≥ γ.

The coloring algorithm of [MW08, MW05]

The algorithm of [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF] is decomposed in three parts according to the states of nodes; see figures 1, 2 and 3 4 . It is important to remark that we do not use the same constant used in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF]. In fact, the value of constants γ, σ, η and µ used by the algorithm below are those defined in the previous section. As argued in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF], the choice of those constants is fundamental to prove the correctness of the coloring.

To make the paper self-contained, we recall the general idea of the algorithm. Each node can be in three state classes A, R or C. Upon wake up, a node enters state A 0 and executes the algorithm of Fig. 1. Whenever a node enters state R it executes the algorithm of Fig. 2. Whenever a node enters state C i it executes the algorithm of Fig. 3. As explained in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF][START_REF] Moscibroda | Coloring Unstructured Radio Networks[END_REF], the general idea of the algorithm is as follows. First, the algorithm attempts to compute an independent set of the graph: nodes in state A 0 compete in order to be in the independent set. Once a node becomes a leader it enters state C 0 . The algorithm also computes a clustering of the graph, i.e., each node is associated with one node in the independent set, each node of the independent set is the leader of its cluster. Then, every leader attempts to assign a different color to each node in its cluster. Nodes in state R are nodes waiting to receive a color from their leader. Once a node receives a color from its leader it must verify that no neighbors belonging to another cluster have been assigned the same color. Nodes verifying their color are in state A i with i > 0.

P v := ∅; ζ i := 1 if i = 0 ∆ if i > 0 ; A suc := R if i = 0 A i+1 if i > 0 ; 1 for ⌈η∆ ln n⌉ time slots do 2 for each w ∈ P v do d v (w) := d v (w) + 1; 3 if M i A (w, c w ) received then P v := P v ∪ {w}; d v (w) := c w ; 4 if M i C (w) received then state := A suc ; L(v) := w; 5 c v := χ(P v ),
χ(P v ) ≤ 0; while state = A i do 7 c v := c v + 1; 8 for each w ∈ P v do d v (w) := d v (w) + 1; 9 if c v ≥ ⌈σ∆ ln n⌉ then state := C i ; 10 transmit M i A (v, c v ) with probability q s ; 11 if M i C (w) received then state := A suc ; L(v) := w; 12 if M i A (w, c w ) received then 13 P v := P v ∪ {w}; d v (w) := c w ; 14 if |c v -c w | ≤ ⌈γζ i ln n⌉ then c v := χ(P v )
15 4 Note that we have used the same notations used in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF].

while state = R do 1 transmit M R (v, L(v)) with probability q s ; 2 if M 0 C (L(v), v, tc v ) received then
1 if i > 0 then 2 repeat transmit M i C (v)
with probability q s until protocol stopped; Given a time slot and two nodes u and v, we denote p v the sending probability of node v at that time slot and Ψ v u = p v /d(u, v) α the probabilistic interference caused by v to node u. For every node u, we define the probabilistic interference induced by nodes outside some region R in a given time slot as following :

3 else if i = 0 then 4 tc := 0; Q := ∅; 5 repeat 6 if M R (w, v) received and w / ∈ Q then add w to Q; 7 if Q is empty then 8 transmit M 0 C (v)
Ψ v / ∈R u = P • v / ∈R Ψ v u
The algorithm analysis follows the same scheme than the analysis made in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF][START_REF] Moscibroda | Coloring Unstructured Radio Networks[END_REF]. The correctness of the algorithm is based on the fact that at any time of the execution of the algorithm the set of node in state C i forms an independent set. This is announced in Theorem 1 below. In order to prove the theorem, we need to prove some intermediary properties expressed in Lemmas 1 and 25 . These two lemmas provide a bound on the time needed for a node in some particular state to communicate without interferences. To prove Lemmas 1 and 2 in the SINR model, we need to bound the interference experienced by a node at any step of the algorithm execution. This is the aim of Lemma 3 below. In the following, we start proving Lemma 3 and then come back to Lemmas 1 and 2.

Theorem 1 For all i, with high probability the color class C i forms an independent set throughout the execution of the algorithm.

Lemma 1 Assume C 0 forms an independent set. Consider nodes u, u ′ , and v such that d(u, v) ≤ R T , d(u ′ , v) ≤ R T , and u ∈ V \ C 0 and u ∈ C 0 . Let J and J ′ be time intervals of length γ∆ ln n

R 1 R 3 R A Bx R 2 R + 2 R T Bv v x
Figure 4: The rings around node x and γ ln n, respectively. The probability P no (resp. P ′ no ) that v does not get a message from u (resp. u ′ ) during an interval J (resp. J ′ ) is upper bounded by 1/n c1 .

Lemma 2 Assume C 0 forms an independent set. Consider a node v ∈ A i for an arbitrary i. Further, let J be a time interval of length |J| = σ 2 ∆ ln n. With probability at least 1 -1/n c2 , there is a time slot t ∈ J such that, at least one node w ∈ B v ∩ A i sends successfully.

Lemma 3 Assume C 0 forms an independent set. Then, for every node u, the probabilistic interference

Ψ v / ∈Iu u verifies: Ψ v / ∈Iu u ≤ P/(2ρβR α T ).
Proof of Lemma 3 Consider a fixed time slot and let p w the sending probability of node w.

From the algorithm, if w ∈ C 0 then p w = q ℓ otherwise p w = q s . Thus, the sum of transmitting probabilities over all nodes in a given region B v can be bounded as following:

w∈Bv p w = w∈Bv ∩C0 q ℓ + w∈Bv∩V \C0 q s ≤ w∈Bv ∩C0 1 φ(R I + R T ) + w∈Bv∩V \C0 1 φ(R I + R T )∆ ≤ 2 (1)
Given a node u, let R ℓ = {v ∈ V such that ℓR I ≤ d(u, v) ≤ (ℓ + 1)R I }. Consider an independent set I of maximum size in R ℓ . It is clear that ∪ z∈I B z (the union of the broadcast regions of nodes in I) covers entirely the ring R ℓ (otherwise we can add a new node to I leading to a contradiction since the size of I is maximum). Since nodes in I are independent, the discs of radius R T /2 around nodes in I are mutually disjoint (otherwise the independence of I is violated). Note that these discs are located inside the extended region R + ℓ defined by R + ℓ = {v ∈ V such that ℓR I -R T /2 ≤ d(u, v) ≤ (ℓ + 1)R I + R T /2} (Fig. 4 gives an example). Thus,

|I| ≤ Area R ℓ + Area (Disc(R T /2))
. Thus, the probabilistic interference caused by nodes inside R ℓ can be bounded as following:

Ψ R ℓ u = P • v∈R ℓ Ψ v u ≤ max v∈R ℓ Area R ℓ + Area (Disc(R T /2)) • P w∈Bv∩R ℓ p w (ℓR I ) α ≤ By Eq. 1 Area R ℓ + Area (Disc(R T /2)) • 2P ℓ α R α I = π ((ℓ + 1)R I + R T /2) 2 -(ℓR I -R T /2) 2 π(R T /2) 2 • 2P ℓ α R α I = 4(2ℓ + 1) R 2 I + R I R T R 2 T • 2P ℓ α R α I ≤ 1 ℓ α-1 • 48P R 2 I R 2 T • R α I Thus, Ψ v / ∈Iu u = ∞ ℓ=1 Ψ R ℓ u ≤ 48P R 2 I R 2 T • R α I • ∞ ℓ=1 1 ℓ α-1 ≤ P • 48R 2-α I R 2 T • α -1 α -2 ≤ P 2ρβR α T
The last inequality holds from the definition of R I .

Proof of Lemma 1 Consider a given time slot of the algorithm. Suppose that node u is the only node inside I v that transmits at that time slot. Then, we can show that node v is likely to hear the message sent by u, i.e., the SINR condition is likely to be verified. In fact, from the Markov inequality and using Lemma 3, we have that the probability that the interferences caused by nodes outside I v exceeds ρ • Ψ w / ∈Iv v is at most 1/ρ. Thus, given that u is the only sending node in I v , with probability at least 1 -1/ρ, the SINR at node v can be bounded as following:

P d(u, v) α ρ • Ψ w / ∈Iv v + N ≥ P d(u, v) α P 2βR α T + P 2βR α T ≥ β (2)
Thus the probability P((u -→ v)) that node v hears a message sent by u at a given time slot can be bounded as following:

P((u -→ v)) ≥ 1 - 1 ρ • p u • w∈Iv\{u} (1 -p w ) = 1 - 1 ρ • p u • w∈Iv∩C0 (1 -q ℓ ) • w∈Iv∩V \C0 (1 -q s ) ≥ 1 - 1 ρ • p u • 1 - 1 φ(R I + R T ) φ(RI ) • 1 - 1 φ(R I + R T ) • ∆ φ(RI )•∆ ≥ 1 - 1 ρ • p u • 1 - 1 φ(R I + R T ) φ(RI ) • 1 - φ(R I ) φ(R I + R T ) 2 • ∆ e -φ(RI )/φ(RI +RT ) = λ • p u
Thus, if u ∈ I v ∩ C 0 , i.e., p u = q s = 1/(φ(R I + R T )∆), then we get:

P no ≤ (1 -P((u -→ v))) |J| = 1 - λ φ(R I + R T )∆ c 1 φ(R I + R T ) λ ∆ ln(n) ≤ n -c1
Similarly, for node u ′ ∈ I v ∩ C 0 , we get

P no ≤ (1 -P((u -→ v))) |J ′ | = 1 - λ φ(R I + R T ) c 1 φ(R I + R T ) λ ln(n) ≤ n -c1
Proof of Lemma 2 Let v ∈ A a node and w ∈ B v ∩ A a neighbor of v.

Consider a node u such that u is a neighbor of w. We remark that if w is the only sending node in the region I u around u, then u is likely to hear the message sent by w, i.e., the SINR condition is likely to be verified for u. Thus, if w is the only sending node in ∪ u∈Bw I u , then the message sent by w is likely to be received by w's neighbors.

More precisely, let (w -→ ⋆) be the event "all w's neighbors hear a message from w". Using Lemma 3, we have that Ψ

x / ∈∪u∈B w Iu w ≤ P/(2ρβR α T ). Thus, provided that w is the only sending node in ∪ u∈Bw I u , and using the Markov inequality (as for Equation 2), it is not difficult to see that the SINR condition holds for all w's neighbor with probability at least 1 -1/ρ, i.e., if w is the only sending node in ∪ u∈Bw I u then for every node u ∈ B w the SINR condition holds and u hears w's message with probability at least 1 -1/ρ. Thus, we have:

P ((w -→ ⋆)) ≥ (1 - 1 ρ ) • p w • r∈( S u∈Bw Iu)\{w} (1 -p r ) = (1 - 1 ρ ) • p w • r∈(( S u∈Bw Iu)\{w})∩C0 (1 -q ℓ ) • r∈(( S u∈Bw Iu)\{w})∩(V \C0) (1 -q s ) ≥ (1 - 1 ρ ) • 1 φ(R I + R T )∆ • 1 - 1 φ(R I + R T ) φ(RI +RT ) • 1 - 1 φ(R I + R T )∆ φ(RI +RT )•∆ ≥ (1 - 1 ρ ) • 1 φ(R I + R T )∆ • 1 - 1 φ(R I + R T ) φ(RI +RT ) • 1 - 1 φ(R I + R T )∆ • e -1
≥ λ ′ ∆ Thus the probability that w fails sending successfully during the interval J is at most:

(1 -P ((w -→ ⋆))) |J| = 1 - λ ′ ∆ c 2 λ ′ ∆ ln(n) ≤ n -c2
Having Lemma 1 and Lemma 2 at hand, Theorem 1 is straightforwardly proved using the same arguments than those in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF]'s proof. In fact, apart from using Lemma 1 and Lemma 2, the reasoning in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF] simply needs that σ > 2γ which is actually verified by our constants.

Having proved Theorem 1, Lemmas 1 and 2, the analysis of the correctness and time complexity of the coloring algorithm is basically the same than the analysis made in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF] 6 . Therefore we can state the following theorem (For a full proof we refer the reader to the proof of Theorem 5 in [START_REF] Moscibroda | Coloring unstructured radio networks[END_REF]).

Theorem 2 With high probability, the algorithm produces a correct coloring with at most φ(2R T )∆ colors within at most c∆ ln n time slots after nodes wake up, where c is a constant verifying c = O(φ(2R T ) 2 • σ).

MAC layer in the SINR model

In this section, we show how to use the coloring algorithm to design a generic MAC Layer allowing to simulate classical interference free message passing algorithms in the SINR model.

Let us consider the classical graph-based point-to-point message passing model. In other words, we consider a distributed model where every two neighboring nodes are connected by a private channel allowing them to communicate in a bi-directional manner without any interferences, i.e., no interference can prevent a message sent by a node to arrive to the second node. For simplicity let us assume that in such a model, any algorithm proceeds into rounds. In each round, a node can receive messages, do some local computation and send messages. We consider two classes of algorithms: the uniform model and the general model. In the former, each node is allowed to send the same message to all its neighbors at a given round, e.g. broadcast algorithms. In the latter, a node can send a different message to each neighbor.

We first consider a uniform algorithm. Our goal is to transform the algorithm so we can run it in the SINR physical model and obtain the same output. The idea is to simulate each round of the algorithm and to attempt to deliver each original message to destination while avoiding interference. For that purpose, we shall use the previous coloring algorithm as a MAC layer by associating each color with a time slot where a node can transmit. However, given a node u having two neighbors v and w with the same color c v = c w , if v and w both transmit a message during the time slot corresponding to color c v , then an interference occurs at node u, i.e., node u does not receive any message. This is a well known problem which is commonly solved by using a distance 2 coloring as a MAC layer, that is a coloring such that each node have a different color than his 2-hop neighbors. Unfortunately, such a coloring do not allow us to avoid interferences in the SINR physical model.

In the following we show how to schedule a MAC layer without direct interferences in the SINR model. In fact, suppose we have constructed a (d + 1)-hop V-coloring for some parameter d, that is a coloring using at most V colors such that each node have a different color than his (d + 1)-hop neighbors (by t-hop neighborhood, we mean nodes at distance t • R T in the Euclidean plane). We associate each color c with a time slot t c where nodes having color c can transmit in time slot t c . Using this simple protocol, we obtain the following:

Theorem 3 For d = 32 • α -1 α -2 • β 1/α
, a (d+ 1)-hop V-coloring defines a scheduled MAC layer protocol allowing every node to successfully send a message to its neighbors within at most V time.

Proof. Consider a time slot t c and a node u having a neighbor v with color c v = c. Let us compute the amount of interference Φ u\v experienced by u and caused by nodes w ∈ V \ {v} during time slot t c .

Let H ℓ,d = {w ∈ V such that ℓdR T ≤ d(u, v) ≤ (ℓ + 1)dR T }. The d hop neighborhood of any node w ∈ H ℓ must be located in an extended region defined as following:

H + ℓ,d = {w ∈ V such that (ℓ -1/2)dR T ≤ d(u, v) ≤ (ℓ + 3/2)dR T }. Let Disc(r) any disc of radius r.
Let us denote by φ ′ (ℓ, d) the maximum number of nodes with the same color in some region H ℓ,d . Since the d/2 hop neighborhoods of any two nodes w and w ′ with the same color c are disjoint, we can bound φ ′ (ℓ, d) as following:

φ ′ (ℓ, d) ≤ Area(H + ℓ,d ) Area(Disc(dR T /2)) = π ((ℓ + 3/2)dR T ) 2 -π ((ℓ -1/2)dR T ) 2 πd 2 R 2 T /4 = 4 • (ℓ + 3/2) 2 -(ℓ -1/2) 2 ≤ 4(4ℓ -2) ≤ 16ℓ
At time slot t c corresponding to color c, we have that: To summarize, a simple idea to obtain a d-hop O(∆)-coloring of G is to compute a 1-hop coloring algorithm of G d on top of G. One simple idea to achieve that is to initially increase the transmission power of every node from P to d • P . Since ∆(G d ) = O(∆), the time needed to output the coloring is still O(∆ log n).

Φ u\v = w∈V \{v} P d(w, u) α = w∈V \{v} s.t., cw=c P d(w, u) α = P • ∞ ℓ=1 w∈H ℓ,d 1 (ℓ • d • R T ) α ≤ P • ∞ ℓ=1 φ ′ (ℓ, d) (ℓ • d • R T ) α ≤ 16P d α R α T • ∞ ℓ=1 1 ℓ α-1 ≤ 16P d α R α T • α -1 α -2 ≤ P
From the previous discussion, we get the following:

Corollary 1 Consider a uniform (resp. general) point-to-point message passing algorithm A (resp. A ′ ) running on an n-node graph in τ time and using messages of size at most s bits.

With high probability,

• Algorithm A can be simulated in the SINR physical model in O(∆ • (log n + τ )) time using messages of size at most O(s log n) bits.

• Algorithm A ′ can be simulated in the SINR physical model in O(∆ • (log n + τ )) (resp. O(∆ • log n + ∆ 2 • τ )) time using messages of size at most O(s∆ log n) (resp. O(s log n)) bits.

The latter corollary can be of special interest, for instance, for some random geometric graphs with polylogaritmic expected maximum degree.

Figure 1 :

 1 Figure 1: Coloring Algorithm: code for node v in state A i

3 4 Figure 2 :

 42 Figure 2: Coloring Algorithm: code for node v in state R

Remove w from Q; 14 until protocol stopped ; 15 Figure 3 :

 14153 Figure 3: Coloring Algorithm: code for node v in state C i

  Thus, node u receives the message sent by v. In other words, a message sent by node v at time slot t cv is correctly received by all its neighbors and the theorem is proved. Now, suppose that we have an algorithm A computing a 1-hop (c∆)-coloring of a given (unit disc) graph G where c is a constant. It is not difficult to adapt algorithm A to compute a d-hop coloring of G as following. In fact, we remark that a 1-hop coloring of G d (where G d is the graph obtained from G by adding an edge between any tow nodes at distance at mostd • R T in the Euclidean plane) is also a d-hop ∆(G d )-coloring of G where ∆(G d ) is the maximum degree of G d . The maximum degree of G d can be upper bounded as following ∆(G d ) ≤ φ(d•R T )∆ ≤ (2d+ 1) 2 ∆.Thus, running algorithm A on G d produces a d-hop (c • (2d + 1) 2 • ∆)-coloring of G.

RR n°7058

These two lemmas corresponds to Lemmas 2, 3 and 4 proved in[START_REF] Moscibroda | Coloring unstructured radio networks[END_REF] under the simple graph based interference model.

The analysis made in[START_REF] Moscibroda | Coloring unstructured radio networks[END_REF] needs that η ≥ 2γφ(2R T ) + σ + 1 and µ ≥ σ, which is actually verified by our constants.