Zebo Zhou

Shanhui Mo

Jin Wu

Hassen Fourati
email: hassen.fourati@gipsa-lab.grenoble-inp.fr.

Behaviors classification based distance measuring system for pedestrians via a foot-mounted inertial sensor

Keywords: pedestrian navigation, step detection, step length estimation, behavior classification, state machine

In this paper, we develop a foot-mounted pedestrian navigation system prototype with the emphasis on distance measuring with an inertial measurement unit (IMU) which implies the characteristics of pedestrian gait cycle and thus can be used as a crucial step indicator for distance calculation. Conventional methods for step detection and step length estimation cannot adapt well to the general pedestrian applications since the parameters in these methods may vary for different persons and motions. In this paper, an adaptive time-and frequency-domains joint distance measuring method is proposed by utilizing the means of behaviors classification. Two key issues are studied: step detection and step length determination. For the step detection part, first behavior classification along with state transition strategy is designed to identify typical pedestrian behaviors including standing still, walking, running and irregular swing. Then a four-stage step detection method is proposed to adaptively determine both step frequency and threshold in a flexible window. Based on the behavior classification results, a two-segment functional based step length model is established to adapt the walking and running behaviors. Finally, real experiments are carried out to verify our proposed step detection method and step length model. The results show that our method outperforms the existing representative methods and it exhibits the merits of accuracy and adaptability for different persons in real time and significantly improves the accuracy of distance measuring.

I. INTRODUCTION

With the increasing demands of location based services (LBS), the pedestrian navigation system (PNS) has gained widespread attentions [START_REF] Mezentsev | Pedestrian dead reckoning a solution to navigation in GPS signal degraded areas[END_REF]. The global navigation satellite system (GNSS) can achieve satisfactory positioning accuracy for outdoor pedestrians. However, it is not necessarily the case for challenging environments, e.g. indoors, city canyons, signal jamming, etc. With the development of wireless technologies, WiFi, ZigBee, Bluetooth are widely utilized as potential solutions for providing one's location in GNSS-denied regions. However, multipath effects and electromagnetic interferences seriously cause signal attenuation, diffraction and refraction, thus degrading the quality of location information. Alternatively, an inertial measurement unit (IMU) consisting of triaxial gyroscopes and accelerometers senses the egomotion of the moving object and is free of external radio-frequency influences. Since MEMS-IMU is self-contained, environment-independent and low-cost, it has been equipped in almost all the existing smart wearable devices for gait analysis, monitoring, sport statistics etc. [START_REF] Godha | Foot mounted inertial system for pedestrian navigation[END_REF]. However, the inertial sensor inevitably suffers from gyroscope drift and accelerometer bias, leading to the divergence of navigation solutions in a short period [START_REF] Zhou | Integrated navigation system for a low-cost quadrotor aerial vehicle in the presence of rotor influences[END_REF]. Great deals of research efforts have been made to enhance the navigation results. The most representative one is the zero velocity update (ZUPT) strategy [START_REF] Nilsson | Performance characterisation of foot-mounted ZUPTaided INSs and other related systems[END_REF][START_REF] Skog | Zero-velocity detection an algorithm evaluation[END_REF]. However, to some extent, it can only minimize the divergence speed of solutions. Using auxiliary sensors is another efficient way to compensate for the inertial sensor drift and bias. Unfortunately, for a PNS, the volume, weight and power consumption should be cautiously considered, thus the lighter and more compact hardware system is preferred. From the aspect of dead reckoning, the inertial instruments potentially imply the characteristics of pedestrian gait cycle, thus they can be naturally used as a step indicator for dead reckoning navigation with a pre-determined step length [START_REF] Sun | Activity classification and dead reckoning for pedestrian navigation with wearable sensors[END_REF].

For the distance measuring with inertial sensors, two critical issues should be considered, i.e. step detection and step length estimation. For the step detection, a popular way is to use empirical acceleration thresholds in time domain [START_REF] Cho | MEMS based pedestrian navigation system[END_REF]. Likewise, a given threshold can be applied to the differential specific force series [START_REF] Fang | Design of a wireless assisted pedestrian dead reckoning system -the NavMote experience[END_REF]. [START_REF] Yang | Walking status detection for pedestrian navigation[END_REF] computes the variance and its duration information to judge one's step. The g-crossing algorithm counting from the rising-point of gravity is efficient and commonly employed as well [START_REF] Leppkoski | Pedestrian navigation based on inertial sensors, indoor map, and WLAN signals[END_REF][START_REF] Perttula | Distributed indoor positioning system with inertial measurements and map matching[END_REF]. Alternatively, many efforts have been put in the frequency domain. The most representative one is proposed by Levi and Judd [START_REF] Levi | Dead reckoning navigational system using accelerometer to measure foot impacts[END_REF]. It substantially implements the fast Fourier transformation (FFT) to extract the frequency subsequently used for peak detection. Unfortunately, pseudo-peak seriously degrades its success rate. Under such framework, similar algorithms e.g. zero-crossing and autocorrelation are also developed in [START_REF] Sun | Activity classification and dead reckoning for pedestrian navigation with wearable sensors[END_REF] and [START_REF] Gusenbauer | Selfcontained indoor positioning on off-the-shelf mobile devices[END_REF]. For the step length estimation, in the past few decades, the most common model is a bi-parametric linear model [START_REF] Levi | Dead reckoning navigational system using accelerometer to measure foot impacts[END_REF][START_REF] Sun | Activity classification and dead reckoning for pedestrian navigation with wearable sensors[END_REF]. It is simple and only related to the step frequency. Recent years, apart from the step frequency, other parameters are introduced into the refined step length model as well, for instance, the acceleration variance [START_REF] Zhang | A handheld inertial pedestrian navigation system with accurate step modes and device poses recognition[END_REF], acceleration boundary [START_REF] Krach | Cascaded estimation architecture for integration of foot-mounted inertial sensors[END_REF][START_REF] Li | A Bluetooth/PDR integration algorithm for an indoor positioning system[END_REF][START_REF] Ngoc-Huynh | Stepdetection and adaptive step-length estimation for pedestrian dead-reckoning at various walking speeds using a smartphone[END_REF], height [START_REF] Tian | A multi-mode dead reckoning system for pedestrian tracking using smartphones[END_REF] and leg length [START_REF] Lan | Using smart-phones and floor plans for indoor location tracking[END_REF]. These model parameters can be pre-calibrated analytically or trained with machine learning [START_REF] Sy | MEMS based pedestrian navigation system[END_REF]. In addition, from the aspect of inertial navigation, a real-time step length estimation strategy is put forward to directly compute each step length with the aid of ZUPT [START_REF] Fourati | Heterogeneous data fusion algorithm for pedestrian navigation via foot-mounted inertial measurement unit and complementary filter[END_REF]. However, it is more timeconsuming and has a higher computational complexity compared with other empirical step length models.

Above all, the aforementioned methods have two main drawbacks: (i) the thresholds and parameters in step detection are person-dependent and hence need to be tuned for different users and different motions. Otherwise, the success rate will be heavily degraded. (ii) Even if the parameters in these step length models are pre-determined, unique step length model still cannot adequately describe different motion behaviors, e.g. running and walking. Therefore, in this contribution, to overcome these drawbacks, an adaptive time-and frequency-domains joint distance measuring method is proposed by utilizing the means of behaviors classification. It consists of two critical issues: step detection and step length determination. Based on behavior classification, we propose a fourstage step detection strategy, where thresholds and step frequency are adaptively adjusted for different persons. For step length estimation, a two-segment functional model is established to accommodate different motion behaviors. This paper is arranged as follows: the distance measuring problem and the basic principle are described in Section 2. Pedestrian features and behaviors recognition are investigated in Section 3. Section 4 focuses on the step detection strategy based on behavior classification. A two-segment functional step length model is constructed in Section 5. Field tests are carried out to evaluate performances of our proposed method in Section 6. Finally, concluding remarks are drawn in Section 7.

II. Problem formulation and background

Step based pedestrian navigation system, in essence, is a typical dead reckoning system which recursively computes the one's position in real time by using two kinds of fundamental information during a motion cycle, i.e. moving distance and orientation. Usually, the orientation is acquired from the magnetometers, while the moving distance is indirectly measured by inertial sensors. The fundamental principle of pedestrian dead reckoning is generally interpreted in a defined plane by

x k y k = x 0 y 0 + k-1 i=1 d i sin θ i cos θ i (1)
c 2016 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls where (x k , y k ) denotes the two-dimensional coordinate after the k-th motion cycle; (x 0 , y 0 ) denotes the initial location. d i and θ i represent distance and moving orientation during the i-th motion period respectively. In the following, we will get an insight into how to accurately measure one's distance d with triaxial accelerometers periodically. A wearable distance measuring device is particularly designed and developed with several modules including builtin BOSCH MEMS-IMU, STM32, Bluetooth, Lithium battery [START_REF] Zhou | Behaviors recognition and step detection for pedestrian navigation via a foot-mounted inertial measurement unit[END_REF]. The chip model version 'nRF51822' integrates STM32 with Bluetooth module. Bluetooth low energy (BLE) 4.0 is applied to ensure the lowpower consumption for communications in real time. Also the collected IMU data is transmitted to cell-phone through the serial peripheral interface (SPI) for the implementation of the accumulated distance measuring procedure, which is mainly decomposed as two basic parts: step counting and step length determination.

The maximum system sampling rate is 250 Hz. The specifications of Lithium battery is 3.7 V 80mAh. The device prototype is mounted on the surface of one's shoes. Figure 1 shows the top and slide views of the developed system.

III. Pedestrian features and behaviors recognition

In previous works, gaits information is extracted and recognized as steps mainly through fast Fourier transformation (FFT) and peak detection with thresholds and zero-crossing [START_REF] Weimann | A pedestrian navigation system for urban and indoor environments[END_REF]. However, it is much more complicated in real applications accompanied with irregular motions (swings), strenuous exercise etc. These activities probably generate multiple pseudopeaks resulting in a risk of wrong step detection. Therefore the invariant frequency bandwidth would not be suitable for various activities. The activity classification performance is highly dependent on where the wearable device is mounted [START_REF] Maurer | Activity recognition and monitoring using multiple sensors on different body positions[END_REF][START_REF] Kawahara | Recognizing user context using mobile handsets with acceleration sensors[END_REF][START_REF] Troped | Prediction of activity mode with global positioning system and accelerometer data[END_REF]. In this section, four sorts of activities including standing still, walking, running and irregular motions are considered and to be classified. Features in time and frequency domains can be utilized to identify the activities. For instance, the first and second moments, maximum, discrete cosine transformation coefficients, FFT coefficients, direct component, energy and entropy etc. [START_REF] Bao | Activity recognition from user-annotated acceleration data[END_REF][START_REF] He | Activity recognition from acceleration data based on discrete consine transform and SVM[END_REF][START_REF] Wang | Recognizing transportation mode on mobile phone using probability fusion of extreme learning machines[END_REF]. Three of them i.e. variance, direct component (DC) and energy are jointly utilized to identify behaviors. For the j-th j = 1, 2, • • • , n sampled specific force in dataset i, its derived acceleration can be written such as

a i,j = l i,j -b a -g r ( 2
)
where l represents the specific force vector measured by accelerometer; g r represents the reference gravity in the local frame, i.e. [0 0 g] T ; b a denotes zero-bias term of accelerometer and can be approximated as

b a ≈ 1 N N s=1 l s -g r (3)
where N is the number of static samples. Noting that the zero-bias term of IMU is estimated in a horizontal plane in a static mode, then it is parallel to the local frame of gr. Therefore we can reliably calculate the bias term b a by (3) in the body frame. It should be noted that after computing accelerations with (2), we still need to remove the influences of high-frequency noise from them. For this reason, the Hamming-window based, linear-phase finite impulse response (FIR) filter is introduced to generate the filtered acceleration ã.

Then it is rather easy to compute the variance σ 2 ãi . The DC component denoted as f 0 can be determined by using FFT and accordingly the energy of dataset i is defined as [START_REF] Bao | Activity recognition from user-annotated acceleration data[END_REF]

E ãi = n j=1 |f j | n ( 4
)
where E represents the energy indicator; f denotes the frequency after the DC component is removed. Figure 2 shows the three features values for different motion behaviors. It can be observed that these features are very effective for distinguishing the four behaviors.

Based on the previous field tests, the thresholds are rather easy to be chosen empirically due to the features of the behaviors significantly differing from each other. Next we will investigate how to recognize the recognition, for instance, supported vector machine, neural network, K-means and decision tree [START_REF] Sun | Activity classification and dead reckoning for pedestrian navigation with wearable sensors[END_REF][START_REF] Deng | Crossperson activity recognition using reduced kernel extreme learning machine[END_REF].

In this paper, we propose the following real-time state machine based multi-mode behaviors recognition algorithm (see Figure 3). It is worthy to point out that the behavior recognition is manipulated by making use of the acceleration series within a two-second time window in real time.

IV. Time-and frequency-domain joint step detection

Based on the behaviors identified in Section 3, a time-and frequency-domains joint step detection method is proposed. It needs to be clarified that for the step detection, once behavior is standing still or irregular swing, it will not be in the step counting process. In other words, only walking and running modes will trigger the step counting procedure. If one is in the transition state, we will count the step till the walking or running behavior is detected. The time resolution of behavior recognition is set as two seconds. The step counting procedure mainly involves four stages: FIR for walking and running data, adaptive window length determination, extrema points searching and cross-points identification. It needs to be pointed out that all the parameters in step detection are assigned with different configurations settings to suit for walking and running behaviors. Stage 2: Adaptive window length determination Define f D as the dominant frequency which is determined with following process: first we process acceleration data of a short-period through FFT then remove its DC component. Furthermore, an empirical threshold is introduced to avoid the influences from the high-order harmonics. Finally, the dominant frequency f D is reliably acquired. To minimize the computation burdens for a stepping cycle, the window length is adaptively determined through

w ã = f A f D ( 5
)
where w ã represents the window length and varied with the dominant frequency.

Stage 3: Extrema points searching Within the determined window by ( 5), we search the acceleration maxima and minima. Then these points with extremum are the candidates for identifying a pedestrian motion cycle. However, in real applications, due to pseudo-peaks, we additionally use thresholds (T L , T S ) to construct the following criterion to screen out the maxima and minima points maxima:

[ã(p -1) < ã(p) > ã(p + 1)] & [ã > TL] minima: [ã(p -1) > ã(p) < ã(p + 1)] & [ã > TS]
(6) where p = 1, 2, ..., w ai .

Stage 4: Cross-points identification

Based on the extrema points confirmed above, we further introduce the 'cross-points' with the aid of three reference lines with values of R L , R M and R S (shown in Figure 4), since these cross-points c 2016 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls are crucial to exhibit details of moving pace. Note that the symbol R represents the reference line; the subscripts 'L', 'M', 'S' correspond to 'large', 'median', 'small' respectively. They can be adaptively adjusted according to the maximum and minimum accelerations in a sliding window, which is flexible. The window size is selfdetermined based on the results of FFT analysis and there is no need to define the size manually. Given the current window index i, the reference lines are produced according to the following rules (see Table 2).

(i) = R L (i -1) and R S (i) = R S (i -1)

Rule 2:

If the maxima or minima points exist, we have

R L (i) = max [ã(p)] or R S (i) = 2 × min [ã(p)] , p = 1, 2, • • • , w ai Rule 3: If R S (i) < R S (i -1), let R S (i) = (R S (i) + R S (i -1))/2 Rule 4: The reference value R M is calculated by R M (i) = [R L (i) + R S (i)]/4
Meanwhile, with the established rules, we invent a simple step confirmation procedure as below. For convenience, 'START' and 'END' flags are marked for indicating a whole step interval.

(i) Search cross-points till the first rising cross-point on RM is found and mark it as 'START'. (ii) Keep searching till the next descent cross-point on RS is found and mark it as 'END'. (iii) Confirm the whole step cycle ranging from 'START' to 'END'. (iv) Continue to seek a new rising cross-point on RM after the 'END' in (iii). Then mark it as a new 'START' and go to (ii).

V. Two-segment functional based step length model

A variety of step length models have been developed in the past few decades. These methods can be mainly categorized into the following three types: (i) Constant models. They usually contain a group of empirical personal parameters (e.g. height, gender) which require to be calibrated beforehand from person to person. It is time-invariant once the parameters are fixed, thus is rather simple and convenient to be applied. However, its accuracy cannot satisfy the high precision applications. (ii) Observation based models. Besides some empirical parameters mentioned in (i), the models also consist of inertial measurements, e.g. mean or minimum and maximum of accelerations. (iii) Indirect motion characteristics based models. Such models are usually constructed by some indirect motion information, typically e.g., step frequency. In general, models in (ii) and (iii) achieve comparative performances. To sum up, the representative models are given by Eqs. ( 7) ∼ [START_REF] Levi | Dead reckoning navigational system using accelerometer to measure foot impacts[END_REF],

d i = K 0 + K 1 × f Si ( 7
)
d i = K 0 + K 1 × f Si + K 2 × σ 2 ãi ( 8
)
d i = K 0 + K 1 × 1 f Si + K 2 × max(ã i ) ( 9
)
d i = K 0 × 4 max(ã i ) -min(ã i ) ( 10
)
d i = K 0 × h × f Si ( 11
)
d i = K 0 × 3 ãi (12)
where d i indicates the step length of the i-th step period; K 0 , K 1 , K 2 are coefficients to be calibrated beforehand; h denotes one's height; ãi denotes the mean value of ãi during the i-th step period; f Si indicates the step frequency during the i-th step period and is defined by

f Si = 1 ∆t i ( 13
)
in which ∆t i represents the duration of the i-th detected step. However, any individual model among them is not able to adequately describe different motion behaviors, e.g. walking and running. As a matter of fact, based on our tests, it is found that the models along with their parameters both need to be tuned according to different motions. Otherwise, the step length estimation will be inaccurate or even wrong. For instance, the step frequency contained models are more accurate in running scenarios, whereas they do not perform well in walking cases. On the contrary, the acceleration information shows its advantages on expressing the step length in running mode. For this reason, considering the merits and performances of all models above, the following two-segment functional model is constructed to estimate step length of running and walking separately:

     d W i = K W 0 + K W 1 × 1 f S i + K W 2 × 1 √ f S i + K W 3 × max(ã i ) (W alking) d R i = K R 0 × 3 ãR i (Running) (14)
c 2016 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls

VI. Experiments and analysis

In this section, pedestrian experiments are carried out to test the proposed method and fully evaluate its performance. The distance measuring system prototype is mounted on the left or right foot. The sampling rate is set as 200 Hz and the sensed data is transmitted to a smart phone through Bluetooth communication module in real time for latter distance estimation. Four pedestrian behaviors, including standing still, walking, running and irregular swing are all intentionally experienced during the pedestrian experiments. To examine the efficiency of behaviors recognition in Section 3, we present the related recognition results of one tested person (see Figure 5). It can be seen from this figure that the pedestrian samples contain these four sorts of behaviors and the 'true' behaviors are also marked there (top). The Green line indicates the 'on' and 'off' of transition state. When the behaviors change, the transition state 'on' is triggered and kept till a different activity occurs. In rare cases, transition state may be incorrectly triggered but wrong behaviour will not be determined, meanwhile the transition state quickly switches to the right behaviour mode. Thus, the identified behaviors by our method (see the bottom of Figure 5) are completely consistent with the true ones. Next we continue to validate the success rate of our step detection strategy based on the behaviors recognition results. 205 persons (78 are female and 127 are male) have been tested in the past few months. We classify the statistic results according to four different height ranges: (i) < 160 cm (42 persons); (ii) 161 ∼ 170 cm (85 persons); (iii) 171 ∼ 180 cm (58 persons); (iv) > 180 cm (20 persons). Considering the space limitation, here we only list the detailed personal results of four representative persons in Table 3 where the 'WD' and 'RD' denote the reference walking and running distance respectively. The reference distance is acquired according to the standard track length of UESTC stadium which has the length of 400 meters for the innermost lane for one lap and 200 meters for half lap. All persons are required to move along with the trajectory of innermost lane. During the tests, the walking speed ranges 1 ∼ 2 m/s and the running speed ranges 2 ∼ 4 m/s. The true total steps are manually counted by tested persons themselves.

Since the reference [START_REF] Levi | Dead reckoning navigational system using accelerometer to measure foot impacts[END_REF] has been extensively used in applications of step detection [START_REF] Perttula | Distributed indoor positioning system with inertial measurements and map matching[END_REF][START_REF] Leppkoski | Pedestrian navigation based on inertial sensors, indoor map, and WLAN signals[END_REF], we count the steps by using Levi and Judd's method in comparison with our proposed behaviour classification based timeand frequency-domains joint (BCTFD) step detection method as well. The statistical results for Levi and Judd's method and BCTFD method are respectively summarized in Tables 4 and5. It is noted that overcounted and missed steps are both counted as wrong steps for success rate statistics. It is observed from Table 4 that for Levi and Judd's method, in general, the success rate of walking is slightly higher than running. The reason is that step signals during running are more likely suffered from the residual pseudo-peaks which cannot be entirely eliminated by FIR filter with only one invariant frequency-band. In fact, the frequency-band should be different for running and walking modes. Levi and Judd's method. The relative accuracy of step detection for 205 persons is drawn with the boxplot for graphically depicting groups of tested persons through their quartiles (see Figure 6). Accordingly, the average accuracy results are given in Table 6. Overall, the performance of BCTFD method shows almost no difference between walking and running. Moreover, it exhibits better adaptability to all the tested persons in both walking and running behaviors. This is due to two aspects: behaviors classification and the step identification strategy which in principle ensures the steps to be reliably confirmed. In addition, observed from Figure 6, no relationship can be found between step detection accuracy and one's height. Furthermore, we examine the window flexibility, which is strongly related to time and memory consumptions. Figure 7 clearly shows the variations of window length for different behaviors.

To evaluate the efficiency of step length models mentioned in Section 5, the walking and running distances (see Table 3) of all models are computed for c 2016 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls 14) is denoted as M7. Table 7 presents the error statistics of all methods and their relative errors are also drawn in Figure 8. The boxplot of the relative distance error for all tested persons are presented in Figure 9. The corresponding average distance relative errors statistics are also given in Table 8. It can be concluded that: in running case, M1, M2, M3 and M5 show extremely poor performances, especially for person PD. Similarly in walking case, M4 and M6 provide the unreliable distances particular for person PB. Compared with M1 ∼ M6, with the behaviors classification based twosegment function, distance estimated by M7 achieves the best accuracy, reliability and adaptability. More specifically, recalling the parameters in step length models for M1 ∼ M7, the following points can be made: (i) M5 with the height parameter shows no evident advantage compared with those methods without height. In other words, height parameter is not indispensable in the step length model. (ii) For walking behavior, the step frequency is the most crucial parameter for step length estimation. For example, the step frequency parameter is absent in M4 and M6, thus they produce the worst results. (iii) For running behavior, the acceleration is very critical for capturing ones motion. Without acceleration parameter, M1 and M5 generate the lowest step length accuracy.

VII. Conclusion

In this paper, we develop a foot-mounted pedestrian distance measuring system. An adaptive time-and frequency-domains joint distance measuring method is proposed based on the behaviors classification. Behaviors recognition, step detection and step length estimation are all intensively investigated. Real experiments are carried out to demonstrate the vadility and efficiency of our proposed method compared with representative ones. The results show that our proposed method outperforms the existing representative methods and achieves the best accuracy, reliability and adaptability. Therefore, the proposed method is suitable and is of great benefit to pedestrian navigation applications. In the future, attention will be put on dead reckoning with the aid of auxiliary sensors [START_REF] Zhou | Equality constrained robust measurement fusion for adaptive Kalman-filter-based heterogeneous multi-sensor navigation[END_REF][START_REF] Zhou | Optimal batch distributed asynchronous multi-sensor fusion with feedback[END_REF], e.g. magnetometers, barometers etc. On the other hand, more pedestrian behavior modes will be taken into account, such as going upstairs, downstairs, climbing and etc.

Fig. 1 .

 1 Fig. 1. The prototype of foot-mounted distance measuring system

Fig. 2 .

 2 Fig. 2. Three features extracted from validation datasets

Fig. 3 .Fig. 4 .

 34 Fig. 3. Schematics of the designed state machine for behaviors recognition

Stage 1 :

 1 FIR for walking and running dataThe parameters setting for FIR should be particularly configured for walking and running. Denoting f A as sampling frequency, based on our previous sufficient tests (over 200 persons), the passband and stopband are set as 0∼4 Hz and 4.5∼f A /2 Hz respectively, while for running, these values are 0∼10 Hz and 10.5∼f A /2 Hz respectively.

Fig. 5 .

 5 Fig. 5. The accelerations and identified behaviors

Fig. 6 .

 6 Fig. 6. The relative accuracy statistical results of step detection for tested 205 persons

Fig. 7 .

 7 Fig. 7. The variation of window length corresponding with behavior transition

Fig. 8 .

 8 Fig. 8. The relative errors for all methods

Fig. 9 .

 9 Fig. 9. The relative distance error statistical results for 205 tested persons

Table 1 .

 1

	Transition state:
	Compute the three features: variance, DC component, energy.
	IF Condition 6 is satisfied (f 0 > T R f ), the current state turns to running
	mode.
	ELSE
	IF Condition 5 is satisfied (σ 2 ãi > T W σ &E ãi > T W E ), the current state
	turns to walking mode.
	ELSE
	IF Condition 4 is satisfied (f 0 > T I f &E ãi > T I E ), the current state
	turns to irregular mode.
	ELSE satisfying Condition 3 (complementary set of Condition 4),
	the current state turns to standing still mode.
	GO TO next cycle.
	Table 1 depicts the implementation mechanism of
	our designed state machine, in detail. Note that symbol
	'T ' represents the threshold and its superscripts 'R',
	'W' and 'I' identify the corresponding behavior modes
	running, walking and irregular swing respectively.

Implementations of state machine based multi-mode behaviors recognition algorithm (one-cycle)

Initial information:

Denote the maximum ã of previous window as ãp max .

Find the maximum of ã in the current window and denote it as ãc max

Original state:

IF Condition 1 is satisfied (|ã c max -ãp max | < T ã) the current

state is kept. GO TO next cycle. ELSE satisfying Condition 2 (complementary set of Condition 1) the current state turns to transition state. c 2016 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls

Table 2 .

 2 Rules of producing RL, RM and RS If no extrema point exists, R L and R S maintain the values in previous window, R L

	Rule 1:

Table 3 .

 3 Pedestrian information of tested four persons

	Person ID Gender	Height Weight WD RD (cm) (kg) (m) (m)
	PA	Male	170	55	1200 800
	PB	Male	180	70	1200 800
	PC	Female	159	42	1200 800
	PD	Female	158	61	1200 800
	Table 6. The average accuracy results for step detection (205
	persons)
	Behaviors Levi and Judd's method BCTFD method
	Walking		98.5 %		99.1 %
	Running		97.1 %		98.5 %

Table 5

 5 verifies such a point of view. Compared with statistics in Table4, our proposed BCTFD method outperforms

Table 4 .

 4 The step detection statistical results for Levi and Judd's method

	Behaviors Person ID True Detected Over-counted Missed Wrong Success rate
		PA	891	887	6	10	16	98.2%
	Walking	PB PC	747 877	741 881	0 10	6 6	6 16	99.2 % 98.2%
		PD	870	870	4	4	8	99.1%
		PA	384	380	0	4	4	99%
	Running	PB PC	349 470	338 460	0 0	11 10	11 10	96.8% 97.9%
		PD	401	393	0	8	8	98%

Table 5 .

 5 The step detection statistical results for the proposed BCTFD method

	Behaviors Person ID True Detected Over-counted Missed Wrong Success rate
		PA	891	891	0	0	0	100%
	Walking	PB PC	747 877	747 878	0 1	0 0	0 1	100% 99.8%
		PD	870	871	1	0	1	99.8%
		PA	384	384	0	0	0	100%
	Running	PB PC	349 470	350 474	1 4	0 0	1 4	99.7% 99.1%
		PD	401	404	3	0	3	99.2%

c 2016 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls

Table 7 .

 7 The statistics of distance errors for M1 ∼ M7 (unit: m)

	Person ID Behaviors	M1	M2	M3	M4	M5	M6	M7
	PA	Walking	53.44	49.78 22.61 52.82 44.41	38.46 20.07
		Running	55.87	35.54 34.17 37.79 57.56	31.11 20.46
	PB	Walking	86.17	53.46 15.76 117.6 64.14 157.06 14.06
		Running	72.52	70.14 70.73 43.29 64.83	49.75	19.7
	PC	Walking	44.44	26.36 27.29 19.81 41.82	28.39 30.74
		Running	53.72	51.15 51.15 50.91 53.22	50.05 46.95
	PD	Walking	35.59	85.46 31.82 56.42 16.35	62.81 16.21
		Running 124.63 128.34 80.53 65.86 122.89 55.73 50.44

Table 8 .

 8 The average distance relative errors for M1 ∼ M7 (205 persons)

	Behaviors	M1	M2	M3	M4	M5	M6	M7
	Walking	6.01 % 5.08 % 3.63 % 6.33% 5.07% 6.47% 3.31%
	Running	12.1%	9.9%	11.1% 8.32% 12.56% 7.83% 6.32%

c 2016 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls

Asian Journal of Control, Vol. 00, No. 0, pp.1-12, Month 2016

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Acknowledgement This work is supported by the National Natural Science Funds of China (Grant No. 41604025), the State Key Laboratory of Geodesy and Earth's Dynamics (Institute of Geodesy and Geophysics, CAS) with the Grant No. SKLGED2018-3-2-E, and the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2018J080).


LARGE DEVIATIONS FOR THE LARGEST EIGENVALUE OF THE SUM OF TWO RANDOM MATRICES ALICE GUIONNET AND MYLÈNE MAÏDA

Keywords: Random matrix, large deviations, extreme eigenvalues, free convolution

In this paper, we consider the addition of two matrices in generic position, namely A + U BU * , where U is drawn under the Haar measure on the unitary or the orthogonal group. We show that, under mild conditions on the empirical spectral measures of the deterministic matrices A and B, the law of the largest eigenvalue satisfies a large deviation principle, in the scale N, with an explicit rate function involving the limit of spherical integrals. We cover in particular all the cases when A and B have no outliers.

Introduction

Understanding the spectrum of the sum A + B of two Hermitian matrices knowing the spectra of A and B respectively is a classical and difficult problem. Since the pioneering works of [START_REF] Voiculescu | Limit laws for random matrices and free products[END_REF], we know that free probability provides efficient tools to describe, at least asymptotically, the spectrum of the sum of two large Hermitian matrices in generic position from one another. More precisely, if A N and B N are two deterministic N × N Hermitian matrices and U N is a unitary random matrix distributed according to the Haar measure, then, in the large N limit, A N and U N B N U * N are asymptotically free and the spectral distribution of

H N := A N + U N B N U *
N is given by the free convolution of the spectral distributions of A N and B N . This global law, that is the convergence of the spectral distribution of H N at macroscopic scale, has been studied in details by [START_REF] Speicher | Free convolution and the random sum of matrices[END_REF], [START_REF] Pastur | On the law of addition of random matrices[END_REF] among others. The local law, that is the comparison of the spectral distribution of H N with the free additive convolution of the spectral distributions of A N and B N below the macroscopic scale was then investigated by [START_REF] Kargin | A concentration inequality and a local law for the sum of two random matrices[END_REF] and [START_REF] Bao | Local law of addition of random matrices on optimal scale[END_REF]. In this paper, we will be interested in the behavior of the largest eigenvalue of H N . As a corollary of the results of [START_REF] Collins | The strong asymptotic freeness of Haar and deterministic matrices[END_REF] on strong asymptotic freeness, we know that if A N and B N have no outliers, then the largest eigenvalue of H N converges to the right edge of the support of the free convolution of the spectral distributions of A N and B N . In this work, we investigate the large deviations of this extreme eigenvalue.

In the framework of random matrix theory, there are very few large deviation results known about the spectrum, basically because the eigenvalues are complicated functions of the entries. A notable exception is given by the Gaussian invariant ensembles for which the joint law of the eigenvalues can be explicitly written as a Coulomb gas. Based on this explicit formula, large deviation principles for the spectral measure at global scale have been established by Ben Arous and [START_REF] Ben Arous | Large deviations for Wigner's law and Voiculescu's noncommutative entropy[END_REF] and for the largest eigenvalue by Ben Arous et al. [2001]. Another special case is given by the sum of a deterministic matrix and a Gaussian invariant ensemble. Then, the spectrum can be constructed as the realization at time one of a Hermitian (or symmetric) Brownian motion starting from a given deterministic matrix. This point of view was used by [START_REF] Guionnet | Large deviations asymptotics for spherical integrals[END_REF] to study the large deviations of the empirical measure, and the large deviations for the process of the largest eigenvalue starting from the origin were derived by Donati-Martin and Maïda [2012]. One of the application of this paper is to provide the large deviation for the largest eigenvalue of this sum by using another approach based on spherical integrals. Beyond these cases where specific tools are available, it was observed by [START_REF] Bordenave | A large deviation principle for Wigner matrices without Gaussian tails[END_REF] that deviations of the spectrum of Wigner matrices for which the distribution of the entries has a tail which is heavier than Gaussian are naturally created by big entries. This key remark allowed to obtain the large deviations for the empirical measure in [START_REF] Bordenave | A large deviation principle for Wigner matrices without Gaussian tails[END_REF] (see also [START_REF] Groux | Asymptotic freeness for rectangular random matrices and large deviations for sample convariance matrices with sub-Gaussian tails[END_REF] for the counterpart for covariance matrices) and for the largest eigenvalue in [Augeri, 2016b]. Large deviations for the spectrum of Wigner matrices with subgaussian entries is still completely open as far as the empirical measure is concerned. One can mention the deviations results of Augeri [2016a] for the moments of the spectral measure in several models. Concerning the deviations of the largest eigenvalue, beyond the works [Ben Arous et al., 2001, Donati-Martin and Maïda, 2012, Augeri, 2016b] already cited above, the following models have been so far studied : Gaussian ensembles plus a rank one perturbation by [START_REF] Maïda | Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles[END_REF], very thin covariance matrices by [START_REF] Fey | Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems[END_REF], finite rank perturbations of deterministic matrices or unitarily invariant ensembles by Benaych-Georges et al. [2012]. In a companion paper, [START_REF] Guionnet | Large deviations for the largest eigenvalue of Rademacher matrices[END_REF] have established a large deviation principle for the largest eigenvalue of Wigner matrices with entries having sharp sub-Gaussian tails, such as Rademacher matrices. They show that the speed and the rate function of this large deviation principle are the same as in the Gaussian case.

Acknowledgments The idea to tilt measures by the spherical integral came out magically from a discussion with M. Potters in UCLA in 2017 and we wish to thank him for this beautiful inspiration. We also benefited from many discussions with J. Husson and F. Augeri with whom one of the author is working on a companion project on Wigner matrices. Finally, we are very grateful for stimulating discussions with O. Zeitouni and N. Cook.

Statement of the results

Let (A N ) N ≥1 and (B N ) N ≥1 be two sequences of deterministic real diagonal matrices, with A N and B N of size N × N. We denote by λ

(A N ) 1 ≥ . . . ≥ λ (A N ) N and λ (B N ) 1 ≥ . . . ≥ λ (B N ) N
their respective eigenvalues in decreasing order, by

A N := max(|λ (A N ) 1 |, |λ (A N ) N |) and B N := max(|λ (B N ) 1 |, |λ (B N ) N |)
their respective spectral radius and by

μA N := 1 N N j=1 δ λ (A N ) j and μB N := 1 N N j=1 δ λ (B N ) j
their respective spectral measures. For β = 1 or 2, we denote by m β N the Haar measure on the orthogonal group O N if β = 1 and on the unitary group U N if β = 2. For any U a N × N matrix, we denote by H N (U ) := A N + U B N U * and by λ N max the largest eigenvalue of H N (U ). The goal of the present work is to establish a large deviation principle for the law of λ N max under the Haar measure m β N . This large deviation principle holds under mild assumptions that we now detail.

Assumption 1.

(H bulk ) The sequences of spectral empirical measures (μ A N ) N ≥1 and (μ B N ) N ≥1 converge weakly as N grows to infinity respectively to µ a and µ b , compactly supported on R. Moreover,

sup N ≥1 ( A N + B N ) < ∞.
(H edge ) The largest eigenvalues λ

(A N ) 1
and λ

(B N ) 1
converge as N grows to infinity to ρ a and ρ b respectively.

A key argument of the proof will be a tilt of the measure by a rank one spherical integral. Similar strategies are used in the companion paper [START_REF] Guionnet | Large deviations for the largest eigenvalue of Rademacher matrices[END_REF] to study some classes of sub-Gaussian Wigner matrices. The rank one spherical integral is defined as follows: for any θ ≥ 0 and M N an Hermitian matrix of size N,

I β N (θ, M N ) := e N θ(U M N U * ) 11 m β N (dU ) and J β N (θ, M N ) := 1 N log I β N (θ, M N ).
The rate function of our large deviation principle will crucially involve the limit of J β N (θ, H N ) as N grows to infinity, which we now describe. For µ a compactly supported probability measure on R, we denote by r(µ) the right edge of the support of µ and by G µ the Stieltjes transform of µ : for λ ≥ r(µ),

G µ (λ) := 1 λ -y µ(dy).
It is decreasing on the interval (r(µ), ∞). By taking the limit as λ decreases to r(µ), one can also define G µ (r(µ)) ∈ R + ∪ ∞. As G µ is bijective from (r(µ), ∞) to (0, G µ (r(µ))), one can define its inverse on this latter interval, that we denote by K µ . Then, for any z ∈ (0, G µ (r(µ))), we define

R µ (z) := K µ (z) - 1 z .
The function R µ is called the R-transform fo µ. One can check that R µ is increasing and that lim z→0 R µ (z) = λµ(dλ), so that it is bijective from (0, G µ (r(µ))) to λµ(dλ), r(µ)) -1 Gµ(r(µ)) . We denote by Q µ its inverse on this interval. We can now define, for β = 1 or 2, θ ≥ 0, µ a compactly supported probability measure and ρ ≥ r(µ):

J β µ (θ, ρ) :=    β 2 2θ β 0 R µ (u)du, if 0 ≤ 2θ β ≤ G µ (ρ), θρ -β 2 log θ -β 2 log(ρ -y)µ(dy) + β 2 log β 2 -1 , if 2θ β > G µ (ρ).
If µ 1 and µ 2 are two probability measures compactly supported on R, we denote by µ 1 ⊞ µ 2 the free convolution of µ 1 and µ 2 . It is uniquely determined as the unique probability measure with R-transform equal to the sum of the R-transforms of µ 1 and µ 2 (see [START_REF] Voiculescu | Limit laws for random matrices and free products[END_REF]). For any θ ≥ 0 and x ≥ r(µ a ⊞ µ b ), we denote by

I β (θ, x) := J β µa⊞µ b (θ, x) -J β µa (θ, ρ a ) -J β µ b (θ, ρ b ),
and

I β (x) := sup θ≥0 I β (θ, x), if x ≥ r(µ a ⊞ µ b ), +∞, otherwise. (2.1)
It is easy to check the following:

Lemma 1. Let µ a , µ b , ρ a and ρ b be given as in Assumption 1. For β = 1 or 2, the function I β is a good rate function. Moreover, for any x > ρ a + ρ b , I β (x) = +∞.

The proof will be given at the beginning of Section 4. We can now state the main results of this paper. The first result is the following large deviation upper bound:

Proposition 2. Under Assumption 1, for β = 1 or 2, for any x ∈ R, lim sup δ↓0 lim sup N →+∞ 1 N log m β N λ N max ∈ [x -δ, x + δ] ≤ -I β (x).
We will then derive the following large deviation lower bound:

Proposition 3. Assume that Assumption 1 holds and that µ a is not a Dirac mass at ρ a and µ b is not a Dirac mass at ρ b . Then, for β = 1 or 2, for any

x ∈ R such that G µa⊞µ b (x) ≤ min (G µa (ρ a ), G µ b (ρ b )) , (2.2) we have lim inf δ↓0 lim inf N →+∞ 1 N log m β N λ N max ∈ [x -δ, x + δ] ≥ -I β (x).
This leads to the following important corollary:

Theorem 4. Under Assumption 1 and if moreover,

G µa⊞µ b (r(µ a ⊞ µ b )) ≤ min (G µa (ρ a ), G µ b (ρ b )) , (NoOut)
then, for β = 1 or 2, the law of λ N max under m β N satisfies a large deviation principle in the scale N with good rate function I β .

One can in fact check (see Lemma 11 for more details) that the condition (NoOut) is automatically satisfied if there is no outliers, namely ρ a = r(µ a ) and ρ b = r(µ b ). This leads to the following corollary Corollary 5. Under the assumption (H bulk ), if A N and B N have no outliers, then for β = 1 or 2, the law of λ N max under m β N satisfies a large deviation principle in the scale N with good rate function I β .

Observe that in the case where one of the measures µ a or µ b is a Dirac mass at ρ a or ρ b respectively and the other matrix has no outliers, r(µ a ⊞µ b ) = ρ a +ρ b so that the above result still holds, but with a degenerate rate function which is infinite except at ρ a + ρ b . To get a taste of what happens in the case with outliers, we also consider in Appendix A the following model: let (U (1) , . . . , U (d) ) be independent random matrices with distribution m β N , independent of U and γ 1 , . . . , γ d be nonnegative real numbers. For any 1 ≤ i ≤ d, we denote by U (i) 1 the first column vector of U (i) and we set:

X N := A N + U B N U * + d i=1 γ i U (i) 1 (U (i) 1 ) * . (2.3)
We show in Theorem 14 that we still have a large deviation principle, for which the rate function will depend on the γ i 's. The rest of the paper will be organized as follows: in the next section, we will first prove a more general result than Proposition 2, that holds not only for m β N but also for a whole family of tilted measures. This will be helpful in the proof of Proposition 3, that will be developed in Section 5. Before getting there, we will study in Section 4 some properties of the rate function I β . The last section will be devoted to the proof of Theorem 4 and Corollary 5, with Lemma 11 as prerequisite. At the end of the paper, in Appendix A, we will study the deviations of the largest eigenvalue of X N for the deformed model (2.3).

Large deviation upper bound for tilted measures

For θ ≥ 0, β = 1 or 2, we define a tilted measure on

O N if β = 1 and U N if β = 2 as follows m β,θ N (dU ) := I β N (θ, A N + U B N U * ) I β N (θ, A N )I β N (θ, B N ) m β N (dU ).
It is easy to check that m β,θ N is a probability measure: indeed, for any U, we have that

I β N (θ, A N + U B N U * ) ≥ 0 and E m β N (I β N (θ, A N + U B N U * )) = I β N (θ, A N )I β N (θ, B N ).
For these tilted measures, we have the following weak large deviation upper bound : Proposition 6. Under Assumption 1, for β = 1 or 2, for any θ ≥ 0, for any

x < r(µ a ⊞ µ b ), lim sup δ↓0 lim sup N →+∞ 1 N log m β,θ N λ N max ∈ [x -δ, x + δ] = -∞, (3.1
)

and for any x ≥ r(µ a ⊞ µ b ), lim sup δ↓0 lim sup N →+∞ 1 N log m β,θ N λ N max ∈ [x -δ, x + δ] ≤ -I β (x) -I β (θ, x) . (3.2)
Remark 7. Applying this proposition with θ = 0 gives Proposition 2.

As we will see in Section 5, establishing an upper bound for any θ ≥ 0 will be useful in the proof of Proposition 3. To prove Proposition 6, and in particular its first statement, we will need to check that, under m β,θ N the spectral measure

μN := 1 N N j=1 δ λ (H N (U )) j of H N (U ) = A N + U B N U * concentrates around a deterministic probability measure ν β N much faster than e -N .
A natural choice for this deterministic equivalent of μN will be its expectation E m β N μN . More precisely, we equip the set P(R) of probability measures on R with the bounded Lipschitz distance d: for any Lipschitz function

f : R → R, we define f ∞ := sup x∈R |f (x)| and f Lip := sup x =y |f (x)-f (y)| |x-y|
, then for any µ and ν in P(R),

d(µ, ν) := sup f ∞≤1 f Lip ≤1 f dµ -f dν.
We then have the following concentration result: Lemma 8. Under Assumption (H bulk ), for β = 1 or 2 and any θ ≥ 0, lim sup

N →∞ 1 N log m β,θ N d(μ N , E m β N μN ) > N -1/4 = -∞.
Proof. Let β = 1 or 2 and θ ≥ 0 be fixed. For any Borel subset A of O N if β = 1 and U N if β = 2, we have:

m β,θ N (A) = 1 I β N (θ, A N )I β N (θ, B N ) A I β N (θ, A N + U B N U * )m β N (dU ) ≤ e 2N θK m β N (A), with K := sup N ≥1 ( A N + B N )
, which is assumed to be finite. Therefore it is enough to prove Lemma 8 for θ = 0, that is lim sup

N →∞ 1 N log m β N d(μ N , E m β N μN ) > N -1/4 = -∞.
For β = 2, Theorem 3.8 in [START_REF] Meckes | Concentration and convergence rates for spectral measures of random matrices[END_REF] states that there exists c, C > 0 such that

m 2 N d(μ N , E m 2 N μN ) > N -1/4 ≤ Ce -cN 3/2 , (3.3)
from which the lemma follows. A careful reading of [START_REF] Meckes | Concentration and convergence rates for spectral measures of random matrices[END_REF] shows that the exact same result as (3.3) also holds for β = 1.

We can now prove Proposition 6. In the sequel, we will denote by

ν β N := E m β N μN .
Proof of Proposition 6. The first claim (3.1) is a direct consequence of the previous lemma. Indeed, let x < r(µ a ⊞ µ b ) and

δ 0 := r(µa⊞µ b )-x 2
. Then, for any δ ≤ δ 0 , there exists ε(δ) > 0,

{λ N max ∈ [x -δ, x + δ]} ⊂ {d(μ N , µ a ⊞ µ b ) > ε(δ)}. ( 3
{λ N max ∈ [x -δ, x + δ]} ⊂ {d(μ N , ν β N ) > ε(δ)/2}
so that, by Lemma 8, for any δ ≤ δ 0 , lim sup

N →∞ 1 N log m β,θ N λ N max ∈ [x -δ, x + δ] = -∞.
We now prove (3.2). Let δ > 0 and x ≥ r(µ a ⊞µ b ) be fixed and define the following event:

E x N,δ := λ N max ∈ [x -δ, x + δ], d(μ N , ν β N ) ≤ N -1/4 . (3.5)
Then we have,

m β,θ N λ N max ∈ [x -δ, x + δ] ≤ m β,θ N (E x N,δ ) + m β,θ N (d(μ N , ν β N ) > N -1/4 ).
By Lemma 8, it is therefore enough to show that lim sup δ↓0 lim sup

N →∞ 1 N log m β,θ N E x N,δ ≤ -I β (x) -I β (θ, x) .
To lighten a bit the notations we write A, B and H for A N , B N and H N = A N + U B N U * respectively. For any θ, θ ′ ≥ 0, we have

m β,θ N (E x N,δ ) = 1 I β N (θ, A)I β N (θ, B) E m β N 1 E x N,δ I β N (θ, H) I β N (θ ′ , H) I β N (θ ′ , H) ≤ E m β N (I β N (θ ′ , H)) I β N (θ, A)I β N (θ, B) sup U ∈E x N,δ I β N (θ, A + U BU * ) I β N (θ ′ , A + U BU * ) = I β N (θ ′ , A)I β N (θ ′ , B) I β N (θ, A)I β N (θ, B) sup U ∈E x N,δ I β N (θ, A + U BU * ) I β N (θ ′ , A + U BU * )
We now have to estimate sup U ∈E x N,δ I β N (θ, A + U BU * ): we will use the continuity of spherical integrals derived in [START_REF] Maïda | Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles[END_REF] that states as follows. Let (G N ) N ≥1 a sequence of deterministic matrices such that sup N ≥1 G N < ∞ and for any N ≥ 1, [START_REF] Maïda | Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles[END_REF], for any θ ≥ 0, there exists a continuous function g θ such that g θ (0) = 0 and for any

λ (G N ) 1 = x and d(μ G N , ν β N ) ≤ N -1/4 . According to Proposition 2.1 in
U ∈ E x N,δ , 1 N log I β N (θ, A + U BU * ) - 1 N log I β N (θ, G N ) ≤ g θ (δ). Therefore, lim sup N →∞ 1 N log m β,θ N (E x N,δ ) ≤ lim N →∞ (J β N (θ ′ , A) + J β N (θ ′ , B) -J β N (θ, A) -J β N (θ, B)) + lim N →∞ (J β N (θ, G N ) -J β N (θ ′ , G N )) + g θ (δ) + g θ ′ (δ), ≤ -(I β (θ ′ , x) -I β (θ, x)) + g θ (δ) + g θ ′ (δ),
where at the last line, we have used Theorem 6 in [START_REF] Guionnet | A Fourier view on the R-transform and related asymptotics of spherical integrals[END_REF].

Letting δ going to zero and then optimizing over θ ′ ≥ 0, we get the required upper bound.

Properties of the rate function I β

We now check the properties of the rate function I β defined in (2.1).

Proof of Lemma 1. An ingredient for the proof if the following: for any compactly supported µ, for any θ ≥ 0 and ρ ≥ r(µ) such that θ ≤ G µ (ρ), we have

ρ - 1 θ ≤ R µ (θ) ≤ ρ - 1 G µ (ρ) . (4.1) Indeed, as K µ is a decreasing function, we have R µ (θ) = K µ (θ) -1 θ ≥ ρ -1 θ . On the other hand, the limit of R µ (θ) as θ grows to G µ (ρ) is ρ -1 Gµ(ρ) .
As R µ is nondecreasing, we get the upper bound. Moreover, it is easy to check that, for any x ≥ 0, there exists C, C ′ ∈ R (depending on µ and x but not on θ) such that, for θ large enough, we have

θx - β 2 log θ + C ≤ J β µ (θ, x) ≤ θx + C ′ ,
so that, for any x ≥ 0, there exists c, c ′ ∈ R such that, for θ large enough,

θ(x -ρ a -ρ b ) - β 2 log θ + c ≤ I β (θ, x) ≤ θ(x -ρ a -ρ b ) + β log θ + c ′ .
If x > ρ a + ρ b , letting θ grow to infinity, we obtain that I β (x) = +∞.

If θ ≥ 0 is small enough, To perform the tilt leading to the lower bound, we will need to further study the properties of the function

I β (θ, x) = β 2 2θ β 0 (R µa⊞µ b (u) -R µa (u) -R µ b (u))du =
I β . Lemma 9. Under Assumption 1, for any r(µ a ⊞ µ b ) ≤ x < ρ a + ρ b such that G µa⊞µ b (x) ≤ min(G µa (ρ a ), G µ b (ρ b )),
then, for β = 1 or 2, there exists a unique θ ≥ 0 such that

I β (θ, x) = sup θ ′ ≥0 I β (θ ′ , x).

We denote by θ

β x := argmax θ≥0 I β (θ, x). For any r(µ a ⊞ µ b ) ≤ x < ρ a + ρ b and r(µ a ⊞ µ b ) ≤ y ≤ ρ a + ρ b such that x = y, sup θ≥0 I β (θ, y) > I β (θ β x , y). Proof of Lemma 9. Let r(µ a ⊞ µ b ) ≤ x < ρ a + ρ b such that G µa⊞µ b (x) ≤ min(G µa (ρ a ), G µ b (ρ b )).
The first remark is that if With the function I β defined in (2.1), if we denote by I β x the function θ → I β (θ, x), then there exist some constants C 1 , C 2 and C 3 (that may depend on µ a , ρ a , µ b , ρ b and x but not on θ) such that

G µa (ρ a )
I β x (θ) =                0, if 0 ≤ 2θ β ≤ G µa⊞µ b (x), θx -β 2 log θ -β 2 2θ β 0 (R µa + R µ b )(u)du + C 1 , if G µa⊞µ b (x) ≤ 2θ β ≤ G µa (ρ a ), θ(x -ρ a ) -β 2 2θ β 0 R µ b (u)du + C 2 , if G µa (ρ a ) ≤ 2θ β ≤ G µ b (ρ b ), θ(x -ρ a -ρ b ) + β 2 log θ + C 3 , if 2θ β ≥ G µ b (ρ b )
, where the last line does not occur if G µ b (ρ b ) = ∞. In the computation, we have used the well known fact that R µa⊞µ b = R µa + R µ b when the three functions are well defined. Therefore, one can check that the function I β

x is continuously differentiable and its derivative is given by:

(I β x ) ′ (θ) =              0, if 0 ≤ 2θ β ≤ G µa⊞µ b (x), x -K µa⊞µ b 2θ β , if G µa⊞µ b (x) ≤ 2θ β ≤ G µa (ρ a ), x -ρ a -R µ b 2θ β , if G µa (ρ a ) ≤ 2θ β ≤ G µ b (ρ b ), x -ρ a -ρ b + β 2θ , if 2θ β ≥ G µ b (ρ b ).
We now set α x :=

1 ρa+ρ b -x . We claim that α x ≥ G µa (ρ a ). Indeed, K µ b is well defined on the interval (0, G µ b (ρ b )), so that K µ b (G µa (ρ a )
) and therefore K µa⊞µ b (G µa (ρ a )) are well defined. As K µa⊞µ b is a decreasing function, we have:

G µa⊞µ b (x) ≤ G µa (ρ a )
and this implies:

x ≤ K µa⊞µ b (G µa (ρ a )) = K µa (G µa (ρ a )) + K µ b (G µa (ρ a )) - 1 G µa (ρ a )
As K µ b is also a decreasing function, this yields:

x ≤ K µa (G µa (ρ a )) + K µ b (G µ b (ρ b )) - 1 G µa (ρ a ) = ρ a + ρ b - 1 G µa (ρ a ) ,
which is equivalent to α x ≥ G µa (ρ a ). There are therefore two cases to consider and we claim that:

Case 1: If G µa (ρ a ) ≤ α x < G µ b (ρ b ), then I β
x reaches its maximum at

θ β x := β 2 R (-1) µ b (x -ρ a ); Case 2: if α x ≥ G µ b (ρ b ), then I β
x reaches its maximum at θ β x := β 2 α x . Let us now prove this claim. On the interval 0, β 2 G µa (ρ a ) , the function (I β x ) ′ is nondecreasing and it vanishes at zero, it is therefore nonnegative so that I β

x is nondecreasing on this interval. We have

(I β x ) ′ β 2 G µa (ρ a ) ≥ 0 and (I β x ) ′ β 2 G µ b (ρ b ) = - 1 α x + 1 G µ b (ρ b )
.

Moreover, as R µ b is an increasing function, (I β x ) ′ is decreasing on the interval

β 2 G µa (ρ a ), β 2 G µ b (ρ b )
. We now distinguish the two cases. In Case 1,(I β x ) ′ β 2 G µ b (ρ b ) < 0, and therefore there exists

θ x ∈ β 2 G µa (ρ a ), β 2 G µ b (ρ b )
such that I β x is increasing on β 2 G µa (ρ a ), θ x and then decreasing. One can check that the point where (I β x ) ′ cancels is given by β 2 R

(-1)

µ b (x -ρ a ). Moreover, (I β x ) ′ is decreasing on β 2 G µ b (ρ b ), ∞ and negative at β 2 G µ b (ρ b
) so it remains negative and I β

x is decreasing on this interval. The first claim holds true.

In Case 2,

(I β x ) ′ β 2 G µ b (ρ b ) ≥ 0, and therefore I β x is increasing on the inter- val β 2 G µa (ρ a ), β 2 G µ b (ρ b ) . But (I β x ) ′ is nonnegative at β 2 G µ b (ρ b ), decreasing on β 2 G µ b (ρ b
), ∞ and converges to x -ρ a -ρ b < 0 as θ grows to ∞. Therefore, there exists

θ x ∈ β 2 G µ b (ρ b ), ∞ such that I β x is increasing on β 2 G µ b (ρ b )
, θ x and then decreasing. One can check that the point where (I β x ) ′ cancels is given by β 2 α x and the second claim holds true. This concludes the proof of the uniqueness of θ.

Moreover, looking carefully at the definition of θ β x in Case 1 and Case 2, one can see that it is an increasing function of x. In particular, for x = y such that r(µ a ⊞ µ b ) ≤ x, y < ρ a + ρ b , θ β x = θ β y and therefore sup θ≥0 I β (θ, y) > I β (θ β x , y). We now have to deal with the case when y = ρ a + ρ b , that is to show that:

sup θ≥0 I β (θ, ρ a + ρ b ) > I β (θ β x , ρ a + ρ b ). (4.2) If G µ b (ρ b ) is finite, for θ > β 2 G µ b (ρ b ), I β (θ, ρ a + ρ b ) = β 2 log θ + C 3
and therefore the supremum is infinite and (4.2) holds. Otherwise let us first consider the case where µ b = δ ρ b . We claim that in this case, the condition r(

µ a ⊞ µ b ) ≤ x < ρ a + ρ b and G µa⊞µ b (x) ≤ min(G µa (ρ a ), G µ b (ρ b )
) are never simultaneously satisfied. Indeed, in this case, µ a ⊞ µ b is just a shift of µ a by ρ b , so that, for any

x < ρ a + ρ b , G µa⊞µ b (x) = G µa (x -ρ b ) > G µa (ρ a ), as G µa is decreasing. If µ b = δ ρ b , then, there exists α ∈ (0, 1] and M finite such that, for any x ≥ ρ b , G µ b (x) ≤ 1 -α x -ρ b + M.
From there, we get that, for any u > G µa (ρ a )

∨ 2M α , u ≤ 1 -α K µ b (u) -ρ b + M so that R µ b (u) ≤ ρ b - α 2u .
Therefore, there exist c, c ′ ∈ R, such that for any θ ≥ G µa (ρ a ) ∨ 2M α ,

I β (θ, ρ a + ρ b ) ≥ θρ b - β 2 2θ β 2M α ρ b - α 2u du + c = βα 4 log θ + c ′
so that, letting θ grow to infinity, we get again that I β (ρ a + ρ b ) = ∞ and (4.2) holds. This concludes the proof of Lemma 9.

Large deviation lower bound

The goal of this section is to show Proposition 3. A classical strategy to get a large deviation lower bound is to tilt the measure in such a way that the rare event {λ N max ∈ [x -δ, x + δ]} becomes typical under the tilted measure. We now check that it is possible to make such a tilt:

Lemma 10. Under Assumption 1, for any x ∈ [r(µ a ⊞ µ b ), ρ a + ρ b ) such that G µa⊞µ b (x) ≤ min(G µa (ρ a ), G µ b (ρ b )), for β = 1 or 2, we have lim δ↓0 lim inf N →∞ 1 N log m β,θ β x N E x N,δ ≥ 0,
where E x N,δ was defined in (3.5) and θ β x in Lemma 9. Proof of Lemma 10. Let β = 1 or 2 and r(µ a ⊞ µ b ) ≤ x < ρ a + ρ b be fixed. Let y = x be such that y < r(µ a ⊞ µ b ) or y > ρ a + ρ b . By Lemma 1, we know that I β (y) = ∞, so that, by Proposition 6, we have lim δ↓0 lim sup

N →∞ 1 N log m β,θ β x N λ N max ∈ [y -δ, y + δ] = -∞.
Let now y = x be such that r(µ a ⊞ µ b ) ≤ y ≤ ρ a + ρ b . Then, by Proposition 6 , we have

lim δ↓0 lim sup N →∞ 1 N log m β,θ β x N λ N max ∈ [y -δ, y + δ] ≤ -(sup θ≥0 I β (θ, y) -I β (θ β x , y))
As a consequence, if we denote by

L β x (y) := sup θ≥0 I β (θ, y) -I β (θ β x , y), if r(µ a ⊞ µ b ) ≤ x ≤ ρ a + ρ b , ∞, otherwise,
we know that the law of λ N max under m β,θ β x N satisfies a weak large deviation upper bound with good rate function L β x . Moreover, for N large enough, λ N max lies with probability one in the compact set [r(µ a ⊞ µ b ) -1, ρ a + ρ b + 1], so that it is in fact a large deviation upper bound. By Lemma 9, we know that L β

x is nonnegative and vanishes only at x. Therefore, we deduce that, for any δ > 0, for N large enough,

m β,θ β x N λ N max ∈ [x -δ, x + δ] ≥ 3 4 .
But, in virtue of Lemma 8, for N large enough, we also have

m β,θ β x N d(μ N , ν β N ) ≤ N -1/4 ≥ 3 4 so that m β,θ β x N E x N,δ ≥ 1 2 ,
and Lemma 10 follows.

From there, one can easily get the large deviation lower bound.

Proof of Proposition 3. Let β = 1 or 2 and x ≥ r(µ a ⊞ µ b ) be fixed. If x > ρ a + ρ b or x < r(µ a ⊞ µ b ), Lemma 1 gives that I β (x) = ∞, so that the lower bound obviously holds. Moreover, as we have seen at the end of the proof of Lemma 9, as µ b is not a Dirac mass at ρ b , then I β (ρ a + ρ b ) = ∞ and the lower bound also holds for

x = ρ a + ρ b .
Let us now assume that r(µ a ⊞ µ b ) ≤ x < ρ a + ρ b and let θ β x be the corresponding shift defined in Lemma 9. Then, with E x N,δ defined in (3.5), we have:

m β N (λ N max ∈ [x -δ, x + δ]) ≥ m β N (E x N,δ ) = E m β N 1 E x N,δ I β N (θ β x , H) I β N (θ β x , H) ≥ inf U ∈E x N,δ 1 I β N (θ β x , A + U BU * ) ×I β N (θ β x , A)I β N (θ β x , B)m β,θ β x N (E x N,δ )
so that, using again Proposition 2.1 in [START_REF] Maïda | Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles[END_REF], we get:

lim inf N →∞ 1 N log m β N λ N max ∈ [x -δ, x + δ] ≥ -I β (θ β x , x) -g θ β x (δ) + lim inf N →∞ 1 N log m β,θ β x N E x N,δ .
Letting δ going to zero and using Lemma 10, we get that lim δ↓0 lim inf

N →∞ 1 N m β N (λ N max ∈ [x -δ, x + δ]) ≥ -I β (θ β x , x) ≥ -I β (x).
This concludes the proof.

Proof of the main theorem and its corollary

Proof of Theorem 4. Assume that Assumption 1 and the condition (NoOut) are satisfied. If we denote by K := sup n≥1 ( A N + B N ), which is assumed to be finite, we have that for any N ≥ 1,

m β N (λ N max > 2K) = 0
, so that the exponential tightness is obviously satisfied. By [Dembo and Zeitouni, 1998, Lemma 4.1.23], it is therefore enough to show a weak large deviation principle. The upper bound is given by Proposition 2 for θ = 0.

As for the lower bound, we distinguish three cases, if G µa (ρ a ) = G µ b (ρ b ) = ∞, as we have seen if the proof of Lemma 9, we have that r(µ a ⊞ µ b ) = ρ a + ρ b . In particular, λ N max converges almost surely to ρ a + ρ b , so that the lower bound holds.

If µ b = δ ρ b , then µ a ⊞ µ b is just a shift of µ a by ρ b , so that r(µ a ⊞ µ b ) = r(µ a ) + ρ b and G µa⊞µ b (r(µ a ⊞ µ b )) = G µa (r(µ a )). Assume that G µa (ρ a ) < ∞. If r(µ a ) < ρ a ,
then the condition (NoOut) is not satisfied, because G µa is a decreasing function. If r(µ a ) = ρ a , then we have a similar situation as in the previous case, λ N max converges almost surely to ρ a + ρ b , so that the lower bound holds. By symmetry, the same holds true if µ a = δ ρa . Otherwise and if the condition (NoOut) holds, as G µa⊞µ b is decreasing, then for any x ≥ r(µ a ⊞ µ b ), we have

G µa⊞µ b (x) ≤ min (G µa (ρ a ), G µ b (ρ b )) .
The lower bound is given by Proposition 3. We now prove Corollary 5. Our goal is to show that if A N and B N have no outliers, then the condition (NoOut) is automatically satisfied. Indeed, if A N and B N have no outliers, it means that their respective largest eigenvalues converge to the edge of the support of the limiting measure, that is to say ρ a = r(µ a ) and ρ b = r(µ b ). Therefore, Corollary 5 is a direct consequence of the following lemma: Lemma 11. For any probability measures µ and ν compactly supported on R, we have

G µ⊞ν (r(µ ⊞ ν)) ≤ min(G µ (r(µ)), G ν (r(ν))).
Proof. If one of the measures µ or ν is a single point mass, the additive free convolution is just a translation and we have equality. We now assume that none of them is a single point mass. In general, we know (see e.g. [START_REF] Belinschi | The Lebesgue decomposition of the free additive convolution of two probability distributions[END_REF]) that there exists a function ω, called the subordination function, which is analytic on

C + := {z ∈ C, Im z > 0} such that, for all z ∈ C + , G µ⊞ν (z) = G µ (ω(z)) (6.1)
By [Belinschi, 2006, Theorem 2.3], as µ or ν are not a single point mass, G µ⊞ν can be continuously extended to C + ∪ R with values in C := C ∪ ∞. Moreover, as µ and ν are compactly supported, by [Belinschi, 2008, Theorem 3.3(3)], ω can also be continuously extended to C + ∪ R. From (6.1), we have that, for any z ∈ C + ∪ R,

Im G µ⊞ν (z) = -Im ω(z). dµ(t) |t -ω(z)| 2 .
Let z be a real number in the interval (r(µ ⊞ ν), ∞). Then

dµ(t) |t-ω(z)| 2 > 0 and Im G µ⊞ν (z) = 0, so that Im ω(z) = 0. Therefore, ω restricted to the interval (r(µ ⊞ ν), ∞) takes values in R ∪ ∞. Moreover ω(z) goes to ∞ as z goes to ∞, so that ω((r(µ ⊞ ν), ∞)) is an interval I ω containing a neighborhood of ∞.
Let a < r(µ) such that (a, ∞) ⊂ I ω . For any y > 0, we have

- r(µ) a Im G µ (x + iy) = r(µ) a dµ(t) arctan r(µ) -t y -arctan a -t y .
As y decreases to zero, the right hand-side converges to πµ((a, r(µ))) > 0. On the other hand, for any x ∈ (a, r(µ)) ⊂ ω((r(µ ⊞ ν), ∞)), there exists x ′ > r(µ ⊞ ν), such that x = ω(x ′ ) and

Im G µ (x) = Im G µ (ω(x ′ )) = Im G µ⊞ν (x ′ ) = 0.
As G µ is continuous on C + ∪ R, by dominated convergence, we get that the left hand-side goes to zero, as y decreases to zero. This leads to a contradiction and we deduce that

I ω ⊂ [r(µ), ∞), which means, by continuity of ω, that ω(r(µ ⊞ ν)) ≥ r(µ).
As G µ is decreasing on (r(µ), ∞), this gives

G µ⊞ν (r(µ ⊞ ν)) = G µ (ω(r(µ ⊞ ν))) ≤ G µ (r(µ)) .
As µ and ν play symmetric roles, this concludes the proof of Lemma 11.

Appendix A. Study of the deformed model (2.3)

In order to study the deviations of the largest eigenvalue of the deformed model below its expected value, we will need a counterpart of Theorem 4 for the smallest eigenvalue of H N . We first state the counterpart of the condition (NoOut).

G µa⊞µ b (l(µ a ⊞ µ b )) ≥ max (G µa (ℓ a ), G µ b (ℓ b )) .
As in Lemma 11, one can check that this condition is satisfied if A N and B N have no outliers. We now extend the definition of the rate function I β introduced in (2.1). For any compactly supported probability measure µ, we denote by l(µ) the left edge of the support of µ. For β = 1 or 2, θ ≤ 0, µ a compactly supported probability measure and ℓ ≤ l(µ):

J β µ (θ, ℓ) :=    β 2 2θ β 0 R µ (u)du, if G µ (ℓ) ≤ 2θ β ≤ 0, θℓ -β 2 log(-θ) -β 2 log(y -ℓ)µ(dy) + β 2 log β 2 -1 , if 2θ β < G µ (ℓ).
For any θ ≤ 0 and x ≤ l(µ a ⊞ µ b ), we denote by

I β (θ, x) := J β µa⊞µ b (θ, x) -J β µa (θ, ℓ a ) -J β µ b (θ, ℓ b ),
and For the sake of simplicity, when treating the deformed model, we will stick to the case β = 1. For any x > r(µ a ⊞ µ b ), we denote by µ x the measure defined as follows: for any bounded measurable function f,

I β min (x) := sup θ≤0 I β (θ, x), if x ≤ l(µ a ⊞ µ b ), ∞,
f (λ)µ x (dλ) = f 1 x -λ µ a ⊞ µ b (dλ). If x = r(µ a ⊞ µ b ), we set f (λ)µ x (dλ) = lim y↓x f 1 y -λ µ a ⊞ µ b (dλ),
whenever it exists. In particular, for any

x ≥ r(µ a ⊞ µ b ), λµ x (dλ) = G µa⊞µ b (x). For any x ≥ ρ ≥ r(µ a ⊞ µ b ) and ℓ ≤ l(µ a ⊞ µ b ) we define α + (ρ) := G µa⊞µ b (ρ) 1 + (x -ρ)G µa⊞µ b (ρ) and α -(ℓ) := G µa⊞µ b (ℓ) 1 + (x -ℓ)G µa⊞µ b (ℓ)
.

For α ∈ 1 x-ℓ , 1

x-ρ and κ / ∈ 1 x-ℓ , 1

x-ρ , we set h α,x (κ) := log κ -λ κ -α µ x (dλ).

We finally set

T + x,ρ (α) :=      h α,x (K µx (Q µx (α))), if α ∈ [G µa⊞µ b (x), α + (ρ)], h α,x 1 x-ρ , if α ∈ α + (ρ), 1 x-ρ , ∞, if α > 1 x-ρ , (A.2)
and

T - x,ℓ (α) :=      h α,x (K µx (Q µx (α))), if α ∈ [α -(ℓ), G µa⊞µ b (x)], h α,x 1 x-ℓ , if α ∈ 1 x-ℓ , α -(ℓ) ∞, if α < 1 x-ℓ . (A.3)
Before proving Theorem 14, we need to state a variant of Proposition 16 in [START_REF] Guionnet | A Fourier view on the R-transform and related asymptotics of spherical integrals[END_REF]. Let (λ i ) i∈N * be a sequence of real numbers such that 1 N N i=1 δ λ i converges to µ a ⊞ µ b . We denote by P the standard Gaussian measure on R and we assume that (g 1 , . . . , g N ) follows the law P ⊗N . For any x / ∈ {λ i , i ∈ N * },

we denote by v N (x) = N i=1 1 x-λ i g 2 i N i=1 g 2 i .
Proposition 13. Assume that max N i=1 λ i converges, as N grows to ∞, to ρ ≥ r(µ a ⊞ µ b ). Then, for any x ≥ ρ and α ∈ R such that α ≥ G µa⊞µ b (x), we have

lim δ↓0 lim N →∞ 1 N log P ⊗N (v N (x) ∈ [α -δ, α + δ]) = -T + x,ρ (α) .
Assume that min N i=1 λ i converges, as N grows to ∞, to ℓ ≤ l(µ a ⊞ µ b ). Then, for any x ≥ r(µ a ⊞ µ b ) and α ∈ R such that α ≤ G µa⊞µ b (x), we have

lim δ↓0 lim N →∞ 1 N log P ⊗N (v N (x) ∈ [α -δ, α + δ]) = -T - x,ℓ (α) .
We will not give a full proof of Proposition 13. This follows from an adaptation of Lemma 18 and Proposition 16 in [START_REF] Guionnet | A Fourier view on the R-transform and related asymptotics of spherical integrals[END_REF]. In Lemma 18 in particular, one can check that the deviations above the mean may involve not only the limiting empirical distribution but also the limiting largest particle, whereas the deviations below the mean may depend on the limiting smallest particle.

Ror γ := (γ 1 , . . . , γ p ), we now define by recursion, for any 1 ≤ i ≤ p,

L (i) γ (x) :=          inf y≤l(µa⊞µ b ) T - x,y 1
γ i + I 1 min (y) , if r(µ a ⊞ µ b ) ≤ x ≤ K µa⊞µ b 1 γ i , inf r(µa⊞µ b )≤y≤x T + x,y 1
γ i + L (i-1) γ (y) , if x ≥ K µa⊞µ b 1 γ i , ∞, if x < r(µ a ⊞ µ b ),
with the convention that

L (0) γ (y) := I 1 (y), if y ≥ r(µ a ⊞ µ b ) and K µa⊞µ b 1 γ i = r(µ a ⊞ µ b ) if G µa⊞µ b (r(µ a ⊞ µ b )) ≤ 1 γ i
We can now state our main result Theorem 14. Under the assumptions (H bulk ), (NoOut) and (NoDown), for any p ∈ N * and any γ ∈ (R + ) p , the law of the largest eigenvalue λ N max of the matrix X N defined in (2.3) under (m 1 N ) ⊗(p+1) satisfies a large deviation principle in the scale N with good rate function

L (p) γ .
The rest of this section is devoted to the proof of Theorem 14 in the case p = 1. For p > 1, the proof is very similar, except that instead of conditioning by the deviations of the extreme eigenvalues of H N , we will condition of the deviations of extreme eigenvalues of the model at step p -1.

Proof of Theorem 14 in the case p = 1. As in the proof of Theorem 4, the exponential tightness is straightforward : for any N ≥ 1, (m 1 N ) ⊗2 ( λ N max ≥ 2K + γ 1 + 1) = 0. We now prove a weak large deviation principle. For γ 1 > 0, for any z which does not belong to the spectrum of H N , one can write det

(zI N -X N ) = det(zI N -H N )γ 1 1 γ 1 -(U (1) 1 ) * (zI N -H N ) -1 U (1) 1 .
Therefore, z is an eigenvalue of X N which is not an eigenvalue of H N if and only if (U

) * (zI N -H N ) -1 U (1) 1 = 1 γ 1 . (1) 1
By invariance by unitary conjugation, one can always assume that H N is diagonal, so that the latter reads

N i=1 1 z -λ (H N ) i v 2 i = 1 γ 1 , where v 2 i = g 2 i 1 N N i=1 g 2 i
, with (g 1 , . . . , g N ) having distribution P ⊗N .

For any (λ 1 , . . . , λ N ) fixed, the function

f λ : z → 1 N N i=1 1 z -λ i v 2 i is decreasing and continuous, on (max N i=1 λ i , ∞), uniformly on (v 1 , . . . , v N ) such that N i=1 v 2 i = 1. Therefore, f λ ( λ N max ) = 1 γ 1 ,
if and only if there exists a function ε λ going to zero at zero, such that for any δ > 0 small enough, for any

x ∈ [ λ N max -δ, λ N max +δ], f λ (x) ∈ 1 γ 1 -ε λ (δ), 1 γ 1 + ε λ (δ) . If we assume that η, δ < |x-y| 4
and for all i ∈ N * , λ i ≤ y + η, one can choose ε λ uniformly in (λ 1 , . . . , λ N ). Moreover, if we denote by

v N (x) := 1 x-y v 2 1 + N i=2 1 x-λ (H N ) i v 2
i , we have the following: for any r(µ a ⊞µ b ) ≤ y < x, there exists a function ε going to zero at zero such that, for η < |x-y| 4 and δ small enough,

(m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ]) ≥ (m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ] ∩ E y N,η ) ≥ (m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ]|E y N,δ )m 1 N (E y N,η ) ≥ (m 1 N ) ⊗2 v N (x) ∈ 1 γ 1 -ε(δ), 1 γ 1 + ε(δ) |E y N,δ × m 1 N (E y N,η ),
where E y N,η was defined in (3.5). Assume that G µa⊞µ b (x) ≤ 1 γ 1 . By Proposition 13,

lim δ↓0 lim N →∞ 1 N log P ⊗N v N (x) ∈ 1 γ 1 -ε(δ), 1 γ 1 + ε(δ) |E y N,η = -T + x,y 1 γ i , so that lim δ↓0 lim N →∞ 1 N log(m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ]) ≥ -T + x,y 1 γ 1 + lim N →∞ 1 N log m 1 N (E y N,η ).
Taking the limit of the right hand-side as η goes to zero, we get using Theorem 4 that lim

δ↓0 lim N →∞ 1 N log(m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ]) ≥ -T + x,y 1 γ 1 -I 1 (y) ≥ -L (1) γ (x),
where the last inequality was obtained by optimizing on y. x-λ

(H N ) i v 2 i + 1
x-y v 2 N . We can then write

(m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ]) ≥ (m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ] ∩ E y,- N,η ) ≥ (m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ]|E y,- N,η )m 1 N (E y,- N,η ) ≥ (m 1 N ) ⊗2 v N (x) ∈ 1 γ 1 -ε(δ), 1 γ 1 + ε(δ) |E y,z,- N,η
× m 1 N (E y,- N,η ). In this case, by Proposition Then, taking the limit as η goes to zero in (A.4) and optimizing in y gives the required lower bound.

We now prove (A.5). Similarly to Lemma 9 and 10 (by symmetry between the smallest and largest eigenvalue), one can show that there exists a unique θ y ≤ 0 such that, for any η > 0 and N large enough, Indeed, (A.6) comes from the following remark: if we set ϕ(θ) := m 1,θ N (λ N max ≥ r + η), the function ϕ is convex so that its derivative is increasing. At θ = 0, ϕ and its derivative go exponentially fast to zero by the previous large deviation upper bound. Hence, for θ ≤ 0 ϕ goes exponentially fast to zero.

With this ingredient, the proof of (A.5) goes as in the proof of Proposition 3: The strategy to get the upper bound is similar : we know that, for N large enough, λ

m 1 N (E y,- N,η ) = E m 1 N 1 E y,- N,η
(H N ) 1
∈ [r(µ a ⊞ µ b ), r(µ a ) + r(µ b ) + 1] almost surely, so for any δ > 0, there exists p ∈ N * and ρ 1 , . . . , ρ p such that

m 1 N λ N max ∈ ∪ p i=1 [ρ i -δ, ρ i + δ] = 1.
Similarly, for any δ > 0, there exists ℓ 1 , . . . , ℓ p such that

m 1 N λ N min ∈ ∪ p i=1 [ℓ i -δ, ℓ i + δ] = 1.
Assume that G µa⊞µ b (x) ≤ 1 γ 1 .

(m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ]) ≤ (m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ] ∩ {d(μ N , ν 1 N ) ≤ N -1/4 }) + m 1 N (d(μ N , ν 1 N ) > N -1/4 ) ≤ p i=1 (m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ] ∩ E ρ i N,δ ) + m 1 N (d(μ N , ν 1 N ) > N -1/4 ) ≤ p i=1 (m 1 N ) ⊗2 ( λ N max ∈ [x -δ, x + δ]|E ρ i N,δ )m 1 N (E ρ i N,δ ) + m 1 N (d(μ N , ν 1 N ) > N -1/4 )
We then use Lemma 8 to get rid of the last term and then let δ go to zero.

Assume now that G µa⊞µ b (x) ≤ 1 γ 1 . We apply the very same strategy with E ℓ i ,- N,δ

instead of E ρ i N,δ and use the same ingredient together with the bound:

E ℓ i ,- N,δ ⊂ {λ N min ∈ ∪ p i=1 [ℓ i -δ, ℓ i + δ]} ∩ {d(μ N , ν 1 N ) ≤ N -1/4 }.

  .4) Using Corollary 5.4.11 for β = 2 and Exercise 5.4.18 for β = 1 in [Anderson et al., 2010], we know that ν β N converges weakly to µ a ⊞ µ b as N goes to infinity. As the distance d metrizes the weak convergence, for N large enough,

  0, by the properties of the R-transform. The function I β is therefore nonnegative. If we denote by g the lower semi-continuous function which is equal to -∞ on [r(µ a ⊞ µ b ), +∞) and +∞ outside, then I β = sup(g, sup θ I β (θ, •)) is lower semicontinuous as a supremum of lower semi-continuous functions. As it is infinite outside the interval [r(µ a ⊞ µ b ), ρ a + ρ b ], it is a good rate function.

  and G µ b (ρ b ) are infinite, then r(µ a ⊞µ b ) ≥ ρ a +ρ b and there is nothing to check. Indeed, if G µa (ρ a ) = G µ b (ρ b ) = ∞, we see by the inequalities (4.1), that lim x→∞ R µa (x) = ρ a and lim x→∞ R µ b (x) = ρ b , so that lim x→∞ K µa⊞µ b (x) = ρ a + ρ b and lim x→ρa+ρ b G µa⊞µ b (x) = ∞, leading to r(µ a ⊞ µ b ) ≥ ρ a + ρ b . By symmetry of the problem, without loss of generality, one can now assume that G µa (ρ a ) ≤ G µ b (ρ b ) and G µa (ρ a ) < ∞.

(

  NoDown) The smallest eigenvalues λ (A N ) N and λ (B N ) N converge as N grows to infinity to ℓ a and ℓ b respectively and

1 γ 1 .

 11 Assume now that r(µ a ⊞ µ b ) < x < K µa⊞µ b We denote by r := r(µ a ⊞ µ b ), we define, similarly to (3.5), for y ≤ l(µ a ⊞ µ b )E y,- N,η := λ N min ∈ [y -η, y + η], λ N max ∈ [r -η, r + η], d(μ N , ν 1 N ) ≤ N -1/4 ,and we change the definition of v N (x) := N -1 i=1 1

1 Nlog m 1 N→∞ 1 N log m 1 N

 1111 ) ⊗2 ( λ N max ∈ [x-δ, x+δ]) ≥ -T - (E y,- N,η ). (A.4) The last step to prove the lower bound in this case is to check lim η↓0 lim N (E y,- N,η ) ≥ -I min (y). (A.5)

m 1 , 3 .

 13 θy N λ N min ∈ [y -η, y + η], d(μ N , ν 1 N ) ≤ N -1/4 ≥2One can also check that, for any θ y ≤ 0 and for any η > 0 and N large enough,m 1,θy N (λ N max ∈ [r -η, r + η]) that, for any η > 0 and N large enough,

I 1 Nlog m 1 N

 11 (θ y , H) I 1 N (θ y , H) ≥ inf U ∈E y,- N,η 1 I 1 N (θ y , A + U BU * ) I 1 N (θ y , A)I 1 N (θ y , B)m 1,θyN (E y,- N,η ), so that, using again Proposition 2.1 in[START_REF] Maïda | Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles[END_REF], we get: E y,- N,η ≥ -I min (θ y , y) -lim η↓0 g θy (η) = -I min (y).

This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) and by the Labex CEMPI (ANR-11-LABX-0007-01) of Université de Lille operated by the French National Research Agency (ANR).


Olivier Nguyen

Natacha Krins

Gianguido Baldinozzi

Analysing Mesoporous Thin Films Electrodes by Glancing-Incidence XRD-Application to Photorechargeable Li-ion Batteries

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Outline

Photo-rechargeable lithium-ion batteries, which are able to both harvest and store solar energy within electrodes, are a promising technology for a more efficient use of intermittent solar radiation [1]. However, there is a lack of understanding on how the light-induced lithium extraction reaction occurs within the electrode host lattice. In fact, this technology is based on bifunctional nanostructured electrodes, which undergo simultaneous complex modifications: structural reorganization through lithium-ion insertion and photo-excited charges creation through solar illumination. In order to get more insight on the mechanism governing the photo-recharge of a single device based on a TiO 2 thin film electrode made of mesoporous anatase, we perform ex-situ glancing-incidence XRD experiments. Using laboratory glancing incidence diffraction, and implementing proper corrections for the instrumental broadening of this diffraction geometry in the Rietveld refinement software XND, it was possible to extract quantitative information about the structure (lattice, strain, and atomic positions) and the microstructure (crystallite size and micro-strain), selectively probing the material on a depth of few nanometers.

Analysing Mesoporous Thin Films Electrodes by Glancing-Incidence XRD -Application to Photorechargeable Li-ion Batteries

Olivier Nguyen 1 , Natacha Krins 1 , Gianguido Baldinozzi 2 , Christel Laberty-Robert Thin Film Electrode Synthesis Glancing Incidence XRD [2] Tunable Electrode Architecture

Conclusion

• The characteristics of architectured mesoporous TiO 2 thin films, produced by dip-coating on a FTO/glass substrate, can be studied across their thickness. • Probing the films at multiple glancing angles shows evidence of single-phase nanocrystalline anatase with a same structure and microstructure across the entire film thickness, a result in very good correlation with SEM-FEG and TEM analyses that involve the destruction of the sample.

n cosα ' = cosα a I + a R ( ) nsinα ' = a I -a R ( ) sinα " # $ % $ n = 1-δ + iβ ≡ 1- α c 2 2 + iβ ; α c 2 ≡ 4π k 2 # $ % & ' ( ρ , β= µ 2k 1- α c 2 2 + iβ " # $ % & ' 1- α ' 2 2 " # $ % & ' = 1- a 2 2 " # $ % & ' → α ' 2 = α 2 -α c 2 + 2iβ 1+ a R a I " # $ % & ' α ' = 1- a R a I " # $ % & ' α → R ≡ a R a I = α -α ' α +α ' * + , , -

  Place Jussieu, 75005 Paris, France 2 SPMS, LRC Carmen, CNRS CEA CentraleSupélec, Université Paris-Saclay, Gif-Sur-Yvette, France

	Ellipsometric
	porosimetry
	(Isopropanol)
	9nm	40nm
	4nm
	15nm
	Pores dimensions

1 Laboratoire de Chimie de la Matière Condensée de Paris, UPMC / Collège de France, CNRS UMR-7574, Sorbonne Universités, 4

Acknowledgements

Elisa Bindini, Cédric Boissière (LCMP, UPMC) Env. Ellipsometry Porosimetry David Montero (IMP, UPMC) -SEM-FEG


Frédérick Lemarchand

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Encouraging behavioural change

Encouraging behavioural change

Frédérick LEMARCHAND, socio-anthropologist at the University of Caen, Co-director of the Risks-MRSH/CNRS cluster. Researcher at the Centre for risk and vulnerability study and research (CERREV).

Encouraging change, whether it be in a way of life or in behaviour, is nothing new. One might even argue that the advent of the consumer society, the consequences of which today are the destruction of resources and common goods, rests entirely on the possibility of encouraging consumers to adopt the expected behaviour, i.e. to desire the merchandise, both in itself (technical object) and for itself (what another does not have). We might equally state that this enterprise, combining mass production, advertising, the promotion of certain individualist values, resting on a cornucopian conception of the world-that is to say an optimistic vision of the latter offering unlimited resources in order to satisfy equally unlimited needs-has, to a large extent, succeeded… to the detriment of the sustainability of the said resources and the economic system that gave rise to it.

There is then a great temptation to turn to social even behavioural psychology in order to build an empirical and simplistic approach to the problem, which most often takes the form of delivering communication messages of a more or less educational and sometimes even childish nature to set citizens on the path to 'best practices'. But this behaviourist and mechanistic-and therefore reductionist-approach has its limits, as sociology has been able to measure in the field of risk management, for example. If, in the majority of cases, these incentive policies do not work at all or to only a limited degree, it is not so much because the populations concerned haven't understood the challenges and cannot change but that they do not want to. Numerous examples make this clear. They include the difficulty of fighting AIDS in Africa in the face of deeply rooted cultural practices or policies for the prevention of industrial risks at work, from the farmer who refuses to wear protective clothing when spraying pesticides in order to save face in front of his neighbours to the worker who handles dangerous products without gloves in the belief that the skin offers sufficient protection (he will wear gloves once it is damaged). The externalization of many determinisms underlying actions such as the nature of social relations and institutions that transcend the individual, challenges of a symbolic nature or even the existence of social constructs (such as that of productivism) in most cases overwhelm sector policies that aim to produce changes in behaviour, such as the action of using fewer resources (water, energy). Especially in the case of sustainable development, changing practices means that one must not simply question oneself on what is at stake in terms of benefits to be gained but also about what an individual or group of individuals is ready to renounce, or not! This question only makes sense if it is addressed in a story (and elements of this story that could positively or negatively influence the possibility of a given population choosing the change or otherwise) and integrated into its collective dimension (taking into account the forms and social stratifications, the interactions between individuals and groups of individuals...).

The idea of producing changes in behaviour and therefore the effects on the environment through consumption and/or the market, poses the same problem. As Michèle Dobré, a sociologist who studies ways of life, has identified, there are at least three types of environmental mismatches when it comes to consumption. Firstly, products said to be 'environmentally sustainable' are either not available or only available to a limited degree in a market that continues to feature planned obsolescence and waste (this was the case with solar panels in the 80s and 90s) or they are not competitive economically (the products or organic farming); secondly, there are significant personal contradictions associated with a consumerist hedonism, itself linked to income, that drives us to consume well beyond our vital needs and, above all, the environmental balance (if humanity lived as an American then eight Earths would be needed to satisfy its needs and four if it lived as a European); finally, a third type of constraint and one that is more difficult to overcome is the interdependence of practices, which in turn leads on to the social, to the being-whole, where complex mechanisms are at play and where systemic factors come together and mutually reinforce each other. As a result, the unconsidered neglect by public policies of conditions and ways of living collides with what, as the following examples show, appears-perhaps wrongly-to be a passive resistance to change.

Why change? The example of restoring living conditions after Chernobyl.

If Chernobyl is seen as providing a foretaste of the catastrophic situation of tomorrow resulting from the creation of a lasting and irreversible state of environmental crisis, then lessons need to be learned (The Silence of Chernobyl). Following the accident on 26 th April 1986, a policy of large-scale rehousing was ordered by the authorities in order to protect the worst affected population and then later, during the 90s, numerous 'pilot' programmes were launched in order to encourage the inhabitants of the contaminated zones to change their eating habits. Neither the one nor the other led to the expected results and all these measures came, more or less violently, up against a resistance to change to which we will give a heuristic value. In short, a high proportion of the inhabitants of the contaminated zones chose to remain living in their home neighbourhoods, preserving their habits to a greater or lesser degree, and, if possible, their ways of life, whilst a portion of those who left have returned to live where they lived before the accident (as at Fukushima today). Furthermore, the health education policies encouraging the inhabitants to change their food habits came up against the same resistance. What have both the one and the other forgotten? Precisely the fact that the question of rehousing is not just a spatial question (simply moving the population) but of territorial roots ramifying into every dimension of existence. So much so that for the inhabitants of Chernobyl the problem was not the contamination but the rehousing itself. In the same way, the pilot 'educational' programmes encouraging changes in behaviour overlooked the fact that in a subsistence economy food is linked to traditional practices (fishing, hunting, foraging, garden cultivation), which form the group culture in the ethnological sense. Ultimately, what Chernobyl teaches us is that, besides the confidence placed in the institution promoting the change, one cannot contemplate asking a population to give up its habits and practices without putting forward an acceptable substitute. In this case, the post-accident management policy restricted itself to producing a set of negative proscriptions holding out no prospect of a better everyday life. The choice was quickly made by the inhabitants, who maintained their way of life… at the price of denying the health threats. These questions will arise again on a large scale given the prospect of the displacement of populations linked to climate change. The first projections were made by the Pentagon back in the early 90s and there is good reason to think that the question of 'climate refugees' will be one of the major geopolitical questions of the century to come.

Can fear serve as an engine for social change?

The belief that the environment question requires a rethink of territorial development policies is shared by numerous researchers, analysts and development agencies around the world. Nevertheless, one might take the view that for the moment the extensive changes required in order to maintain human life on earth have yet to feed into major public and private sector development policies, as well as into the ways of life in the richest countries. It must be said that despite the catastrophism of our time-we cannot say that we don't know when the mass of indicators (and warnings) circulating around us is such that they forma a sort of ambient environment-we encounter two major difficulties when it comes to the possibility of us actually grasping the situation and transforming it into a collective effort for social structure. The first is tied up with the fact that many of the dangers (and/or the potential risks) in the form, of pollution, the degradation of nature, the exhaustion of energy resources and other raw materials, as well as all the substances that have an unseen effect on human health and on the balance of ecosystems by threatening biodiversity (endocrine disruptors), are not perceptible and operate at 'low volume'. These weak signals, which, Günther Anders terms infraliminary, escape our senses and so limit our ability to react, to make rational choices enlightened by experience. When the individual or collective ill emerges it is already too late. If one adds to this that certain consequences, such as the expansion of deserts or the rise in temperature at the poles, are geographically distant, or delayed in time, such as the end of 'peak oil', after which oil production will decline, and, finally, are indeterminate, such as the systemic effects of human activity on living organisms and biodiversity, it is no longer possible to believe that a change in behaviour could be solely based on the empirical fact of the existence of contradictions that would lead to an immediate awareness, which would result, in its turn, in action. The infraliminary nature of environmental problems, which constitutes the most powerful engine for the denial of reality, forces us to question the hope expressed by Hans Jonas about the role of fear in the foundation of an ethic for techno-scientific societies-a fear that would become a 'good counsellor'. How can one have a fear of what one cannot sense, of something that one knows almost nothing about?

The question becomes still more pertinent, and our hopes take flight, when considers, again with G.

Anders, that the environmental phenomena triggered by man-and above all by modern western man-also evade our awareness because of their supraliminary character, that is to say by the fact that they are 'too large'. Climate change belongs to this kind of event, like the threat of total nuclear war and the emerging threat of mass extinction, in that they literally exceed the capacity of our imagination through the extent and time scale of their consequences. "We are in the process of bringing the interlude of stable climate that we have known since the end of the last glaciation, over 10,000 years ago, to a close without knowing if it will ever return" argue Dominique Bourg and Kerry Witheside, before adding that "the losses of biodiversity that we are causing will take millions of years to make good". It is therefore the hubris, the excessiveness of the processes that we have triggered-rather than the risks that we would have produced and that we would be called on to master-that impairs our ability to understand (verstehen) what we do (herstellen) and consequently awareness of the moral evil to be found in our most ordinary daily activities (G. Anders, The Obsolescence of Humankind).

Catastrophism or democratic renewal?

Two authors have attempted to answer the question of 'what to do', once we have faced up to our difficulty in envisaging the worst. For the philosopher Dominique Bourg, re-establishing our responsibility-that is to say having one day or another to render accounts to future generationsinvolves a renewal of democratic institutions. Our wait and see attitude in the face of catastrophe is then seen as being linked to the fact that our democratic institutions are not suited to the nature of contemporary environment problems. According to him, it is only by regaining our sense of the common good that we will be able to hope for a complete change in politics, in the sense of the government of individual behaviours, by enlarging the existing representative process so as to redefine the rights and duties of each person (with regards to future generations, for example), from that of the national level to that of world governance. For his part, the philosopher Jean-Pierre Dupuy (For an enlightened catastrophism) believes that we can no longer content ourselves with envisaging the worst as a 'simple possibility' if we are effectively to prevent it. The idea of 'risk' as possibility and its corollary, 'precaution' are no longer sufficient in a situation where we are continuing to decide for the present in the face of uncertainty over future consequences. We need to adopt a new temporal perspective in order to envisage the apocalyptic destiny of humanity, that is to say the catastrophe of our self-destruction, as a certainty. By drawing on the example of nuclear deterrence, Jean-Pierre Dupuy proposes a fundamental consideration of the change in attitude towards the future that we would need to adopt if we do not want to be swallowed up by the real catastrophe. Enlightened catastrophism, a proposition for a new metaphysics for our time, aims to remove our blindness towards the possibility of the catastrophe occurring by depicting it as certain … in order the better to avoid it. Ultimately, we might consider whether the conditions required for a complete change of direction in policies and individual practices might be met that if the philosophical proposition of J.-P.

Dupuy found expression in the political one of D. Bourg.