Mohamed A Aboulhassan
email: mohamed.abdelkarim@pua.edu.eg

Mohamad Yassin
email: mohamad.yassin@irisa.fr

Samer Lahoud

Marc Ibrahim

Dany Mezher

Bernard Cousin

Essam A Sourour

Classification and Comparative Analysis of Inter-Cell Interference Coordination Techniques in LTE Networks

Keywords: Inter-Cell Interference Coordination, 3GPP LTE, reuse-3 model, FFR, SFR, spectral efficiency

Frequency reuse-1 model is required to satisfy the exponential increase of data demands in mobile networks, such as the Long Term Evolution (LTE) of Universal Mobile Terrestrial radio access System (UMTS). However, the simultaneous usage of the same frequency resources in adjacent LTE cells creates inter-cell interference problems, that mainly affect cell-edge users. Inter-Cell Interference Coordination (ICIC) techniques are proposed to avoid the negative impact of interference on system performance. They establish restrictions on resource usage, such as Fractional Frequency Reuse (FFR), and on power allocation such as Soft Frequency Reuse (SFR). In this paper, we classify the existing ICIC techniques, and investigate the performance of reuse-1, reuse-3, FFR, and SFR schemes under various user distributions, and for various network loads. Performance of cell-center and cell-edge users are inspected, as well as the overall spectral efficiency. System level simulations show the advantages and limitations of each of the examined techniques compared to frequency reuse-1 model under different network loads and user distributions, which helps us to determine the most suitable ICIC technique to be used.

I. INTRODUCTION

Orthogonal Frequency Division Multiple Access (OFDMA) is the multiple access technique chosen by the Third Generation Partnership Project (3GPP) for the downlink of the radio interface in Long Term Evolution (LTE) networks [START_REF] Gpp | Physical Layer Aspects for Evolved Universal Terrestrial Radio Access (UTRA) (Release 7)[END_REF]. Data is transmitted on a large number of parallel, narrow-band subcarriers, and the smallest resource unit that could be allocated for a User Equipment (UE) is called Resource Block (RB). Since subcarriers are orthogonal, intra-cell interference is eliminated; however, Inter-Cell Interference (ICI) remains the major problem for multiuser OFDMA networks such as LTE. It limits system performance and reduces the achievable throughput, especially for UEs located at the edge of the cell. ICI is caused by collisions [START_REF] Bosisio | Interference Coordination Versus Interference Randomization in Multicell 3GPP LTE System[END_REF] between RBs that are simultaneously used in adjacent cells, according to frequency reuse-1 model.

Dense frequency reuse scheme aims at improving system capacity by increasing the number of available RBs in each cell. It is a necessity for mobile network operators seeking to fulfill the huge data demands, due to the proliferation of mobile applications and the exponential increase in the number of connected devices. Therefore, Inter-Cell Interference Coordination (ICIC) techniques are required to avoid the negative impact of ICI on system performance, without largely sacrificing spectral efficiency. ICIC aims at mitigating Signal-to-Interference and Noise Ratio (SINR) degradation by applying cell specific preferences for different RB subsets, or by employing reduced power for colliding RBs [START_REF] Fodor | Intercell Interference Coordination in OFDMA Networks and in the 3GPP Long Term Evolution System[END_REF].

Operators of the Global System for Mobile communications (GSM) exploit the cellular concept along with frequency reuse-N [START_REF] Denes | Frequency Allocation for a Mobile Radio Telephone System[END_REF] model, in order to use the same frequency resources in several distant cells without having restrictive interference problems. Although it largely reduces ICI, frequency reuse-N model has a negative impact on spectral efficiency, and consequently on system capacity. Only 1 N of the available spectrum is used in each cell. Multiuser OFDMA networks require the usage of ICIC techniques that restrict the usage of parts of the spectrum through a frequency reuse factor larger than one for cell-edge UEs [START_REF] Astely | LTE: The Evolution of Mobile Broadband[END_REF]. Such schemes succeed in improving SINR, but they reduce spectral efficiency since frequency bands available in each cell become smaller. Therefore, ICIC consists in managing the trade-off between SINR and spectral efficiency, through restrictions on RB scheduling, power allocation, or both.

Fractional Frequency Reuse (FFR) is a compromise between reuse-1 and reuse-N models. Each cell is divided into cell-center and cell-edge zones, where frequency reuse-1 model is used in the cell-center zone, while a higher frequency reuse factor is used in cell-edge zone. The available spectrum is divided into two sub-bands: the first one is permanently used in cell-center zones, while the second sub-band is used according to frequency reuse-N model in the cell-edge zones. Consequently, SINR for celledge UEs is improved [START_REF] Boudreau | Interference Coordination and Cancellation for 4G Networks[END_REF], since they operate on disjoint spectrum. One disadvantage of FFR is that a portion of the available spectrum is permanently unused in each cell. Soft Frequency Reuse (SFR) performs radio resource management along with power allocation in order to mitigate ICI. In each cell, the available spectrum is divided into a cell-edge band, where RBs are allocated the maximum transmission power, and a cell-center band, where RBs are allocated a lower transmission power, in order to reduce the resulting interference at the neighboring cells.

In [START_REF] Yassin | Non-Cooperative Inter-Cell Interference Coordination Technique for Increasing Throughput Fairness in LTE Networks[END_REF], we introduced a non-cooperative ICIC algorithm, and we investigated its performance compared to other techniques. However, state-of-the-art techniques are not evaluated for different network loads and for other performance parameters such as energy efficiency and UE satisfaction. In this paper, we describe and classify the existing ICIC techniques for multiuser OFDMA networks such as LTE. More specifically, we study the performance of reuse-1, reuse-3, FFR, and SFR schemes under uniform and non-uniform UE distributions. We focus particularly on spectral efficiency and fairness index for each of the compared techniques. We also investigate their impact on the achievable throughput under various UE distributions and network loads. A MATLAB-based LTE downlink system level simulator [START_REF]LTE Downlink System Level Simulator[END_REF] is chosen as the simulation platform for our comparisons. An efficient ICIC technique succeeds in improving UE throughput without reducing spectral efficiency.

The remainder of this paper is organized as follows: in section II we describe FFR and SFR schemes, then we classify the existing ICIC approaches. System model of the LTE network is given in section III, while simulation environment and simulator parameters are explained in section IV. In section V, simulation results for the compared ICIC approaches under various conditions are presented and discussed. Concluding remarks are given in section VI.

II. RELATED WORK

Frequency reuse-N model in GSM networks allows the usage of the same spectrum, several times within the network. For instance, in a cluster of three adjacent GSM cells, the available spectrum is divided into three sub-bands, and each cell operates on a disjoint part of spectrum. Moreover, the same radio channels are used on the same carrier frequency to cover different areas that are separated from one another by sufficient distances so that co-channel interference is almost eliminated [START_REF] Donald | Advanced Mobile Phone Service: The Cellular Concept[END_REF]. Nevertheless, this scheme reduces spectral efficiency, since only one sub-band is used in each cell.

FFR and SFR are introduced to mitigate ICI in multiuser OFDMA [START_REF] Hanzo | MIMO-OFDM for LTE, WiFi and WIMAX: Coherent Versus non-Coherent and Cooperative Turbo-Transceivers[END_REF] networks such as LTE. The former applies restrictions on RB usage, while the latter adjusts transmission power allocated for each frequency sub-band. These two schemes divide each LTE cell into two zones: cell-center zone and cell-edge zone. UEs classification between the different zones is either made according to the distance that separates them from the serving base station, or according to their wideband SINR. When distance is chosen, the optimal cell-center region radius is approximately equal to 2 3 of the overall cell radius [START_REF] Assaad | Optimal Fractional Frequency Reuse (FFR) in Multicellular OFDMA System[END_REF]. However, distance-based classification is not accurate, since we might find cell-center UEs characterized by low SINR values, due to ICI and shadow fading problems. These UEs should be protected from ICI problems, as well as UEs located at cell border, and having low SINR (caused by propagation loss and interference). For these reasons, we divide each cell into two zones: one containing UEs characterized by Good Radio conditions (GR UEs); this zone is commonly known as cell-center zone. The second contains UEs experiencing high ICI, thus characterized by Bad Radio conditions (BR UEs); this zone is commonly known as cell-edge zone. A GR UE is characterized by wideband SINR higher than a predefined SINR threshold , while BR UEs have their wideband SINR lower than this threshold. Another advantage for SINR-based classification is that it does not require any information about geographical positions of UEs.

FFR and SFR techniques are illustrated in Fig. 1(a) and Fig. 1(b) respectively. FFR divides the available spectrum into a few non-overlapping frequency sub-bands [START_REF] Yang | Interference Management in LTE Wireless Networks [Industry Perspectives[END_REF]. GR and BR UEs of the same cell operate on different sub-bands, and BR UEs of the neighboring cells within the same cluster also operate on non-overlapping subbands. When SFR is applied, a portion of the available spectrum is permanently allocated the maximum downlink transmission power, while the remaining RBs are used with a lower transmission power. GR UEs have access to the low power frequency sub-band, while BR UEs use RBs from the high power frequency subset. SFR adopts reuse-1 model, and it protects BR UEs by reducing the transmission power allocated to the interfering RBs in the GR zones of the neighboring cells.

FFR and SFR techniques are compared with reuse-1 model in [START_REF] Novlan | Comparison of Fractional Frequency Reuse Approaches in the OFDMA Cellular Downlink[END_REF], where network throughput, spectral efficiency, and cell-edge UEs SINR are discussed. It has also been proved that SFR balances the requirements of interference reduction and resource efficiency. However, only uniform UE distributions are considered in typical OFDMA deployment. In [START_REF] Qian | Adaptive Soft Frequency Reuse Scheme for Wireless Cellular Networks[END_REF], authors introduced an adaptive SFR technique that dynamically adjusts RB and power allocation in order to improve system capacity. It is a distributed technique that requires an exhaustive search until a stable SFR pattern is found. The proposed technique is compared to traditional frequency reuse schemes under different traffic load scenarios to emphasize the dynamic aspect of the proposed technique. However, authors did not consider non-uniform UE distributions within the network.

We should also mention several contributions that tried to improve FFR and SFR performance, such as [START_REF] Gonzalez | Optimization of Soft Frequency Reuse for Irregular LTE Macrocellular Networks[END_REF][START_REF] Lee | Inter-Cell Interference Coordination for LTE Systems[END_REF], in a distributed or cooperative manner. Authors in [START_REF] Yassin | A Downlink Power Control Heuristic Algorithm for LTE Networks[END_REF] introduce a heuristic power control algorithm to reduce ICI; another technique proposed in [START_REF] Aboul Hassan | Novel Resource Allocation Algorithm for Improving Reuse One Scheme Performance in LTE Networks[END_REF] performs power allocation according to SINR level for each RB. ICIC techniques are classified into frequency reuse-based, such as reuse-3, FFR, and SFR, autonomous techniques, where each cell makes its own interference mitigation decisions, independently of the other cells. Cooperative techniques make use of the communications between adjacent LTE cells over X2 interface, in order to adjust RB allocation, power allocation, or both in a collaborative manner. Several works surveyed the existing ICIC techniques and classified them according to cell cooperation and frequency reuse such as [START_REF] Fodor | Intercell Interference Coordination in OFDMA Networks and in the 3GPP Long Term Evolution System[END_REF][START_REF] Hamza | A survey on inter-cell interference coordination techniques in ofdma-based cellular networks[END_REF]. However, some of them only report qualitative comparisons of the existing ICIC techniques. Others perform simulations under uniform UE distributions and ordinary network scenarios. In our work, we investigate several interference mitigation techniques under various UE distributions, and we show the impact of each technique on throughput distribution and throughput fairness among all the active UEs. These evaluations allow us to draw conclusions on the efficiency of each technique for each of the simulated scenarios.

III. SYSTEM MODEL A. Deployment Model

Our system model consists of seven adjacent Macro Base Stations (MBS). MBS coverage is modeled as a sectorized hexagonal layout, where each site consists of three adjacent sectors. Each sector is served by an eNodeB that has its own scheduler, bandwidth, and power allocation policy. When reuse-1 model is used, the entire bandwidth is available in all the cells, while reuse-3 allows the usage of one third of the available spectrum in each cell. FFR applies restrictions on RB usage in each zone, and SFR adjusts transmission power allocation over RBs used by GR and BR UEs.

B. SINR-Data rate mapping

The value of achievable data rate that corresponds to the SINR value can be obtained from Table I [START_REF] Ramli | Performance of well known packet scheduling algorithms in the downlink 3GPP LTE system[END_REF].

C. UE distribution

We consider UE distribution between cell zones as an essential parameter in our simulations, since it has an important impact on UE throughput and on system performance. UEs are classified into GR and BR UEs according to their mean SINR over the available RBs. GR UEs are the ones characterized by relatively high SINR values in comparison with other UEs. SINR degradation is mainly due to signal path loss, as well as interfering signals received from the neighboring base stations. UEs geographical positions and UE distribution between cell zones have a great impact on ICI, and consequently on system throughput. We simulate scenarios where UEs are uniformly distributed between GR and BR zones, as well as other scenarios characterized by non-homogeneous UE distributions. For instance, the majority of active UEs are either in GR zone or in BR zone.

IV. SIMULATION ENVIRONMENT

System level simulator used in our work is a MATLAB based simulator [START_REF] Ikuno | System Level Simulation of LTE Networks[END_REF] developed by Vienna University of Technology. Reuse-1 model along with homogeneous power allocation are included in the original version of the simulator. FFR scheme is handled as a scheduling policy, where the scheduler of each eNodeB has restrictions on RB allocation for UEs in each zone. We adjusted the original code of the simulator so that non-homogeneous power allocation would be supported. We also made the necessary modifications to implement reuse-3 model and SFR technique. Simulation parameters for the LTE network are given in Table II.

As mentioned earlier in this paper, cell geometry is hexagonal, and each site consists of three adjacent sectors, where each sector is served by an eNodeB. Inter-eNodeB distance equals 500 m, which corresponds to an LTE network deployed in an urban area. In each cell, 25 RBs are available, since the operating bandwidth equals 5 MHz. However, traffic model is full buffer i.e., all the available RBs are permanently allocated for the active UEs in the network. UE scheduling is performed every one millisecond. Path loss model is the one defined by 3GPP in TS 25.814, and feedback reception at eNodeBs is delayed by three milliseconds. When homogeneous power allocation is used, the maximum downlink transmission power is allocated for each RB. However, SFR reduces the transmission power allocated for RBs used by GR UEs. SINR threshold is a predefined parameter, used to classify active UEs into GR and BR UEs. It can be adjusted by mobile network operators according to network load and UE satisfaction. Uniform and non-uniform UE distributions are considered in our simulations, and reuse-1 model is compared to reuse-3, FFR, and SFR schemes.

V. SIMULATION RESULTS AND ANALYSIS A. Performance Metrics 1) Spectral Efficiency: This metric reflects the efficiency of spectrum usage in terms of the achievable throughput using the available bandwidth. It is calculated as follows:

Spectral efficiency = K k=1 R k [bit/s] Total spectrum [Hz] , (1)
where K denotes the set of active UEs in the network, and R k is the mean throughput achieved by UE k.

2) UE throughput: We aim to find how much throughput for each zone is modified; thus, we study the impact of addressed ICIC techniques on UE throughput in each GR and BR zones, as well as mean throughput per UE.

3) Fairness index: Fairness indicates how much resources are distributes among users. Jain's fairness index [START_REF] Jain | A Quantitative Measure of Fairness and Discrimination for Resource Allocation and Shared Computer System[END_REF] is a good measure for fairness and can be given as:

J(R 1 , R 2 , ..., R K ) = ( K k=1 R k ) 2 K. K k=1 R 2 k , ( 2
)
where J rates the fairness of a set of throughput values; K is the number of UEs, and R k is the mean throughput of UE k. Jain's fairness index ranges from 1 N (worst case) to 1 (best case). It reaches its maximum value when all UEs receive the same throughput. An efficient ICIC technique reduces the gap between GR and BR UEs throughputs, and increases Jain's fairness index.

4) UE satisfaction: It is the minimum throughput value required to guarantee an acceptable quality of service. A UE is considered satisfied if his average throughput is higher than satisfaction threshold.

The percentage of unsatisfied UEs among all the active UEs in the network is another parameter for performance comparison. An ICIC technique is better than other techniques when it shows the lowest percentage of unsatisfied UEs.

5) Throughput Cumulative Distribution Function (CDF): Empirical CDF for different values of throughput for each ICIC technique are calculated.

B. Results

1) Mean Throughput per Zone:

The simulated network consists of seven adjacent LTE cells with 10 UEs randomly placed in each cell. Mean throughput for GR and BR zones as well as mean throughput per UE are calculated for 100 simulation runs, and mean results are shown in Fig. 2. It is noticed that FFR technique improves BR UEs throughput, in comparison to reuse-1, reuse-3 and SFR techniques. It prohibits the usage of the same sub-band not only in adjacent BR zones, but also in any other GR zone of the considered cluster. Although ICI is minimized for BR UEs, available spectrum for GR zones become smaller, thus FFR reduces the average throughput per UE when compared to reuse-1 model. Reuse-3 aggravates the disadvantage of FFR, where each cell is assigned only 1/3 of available bandwidth. Thus, mean throughput per UE reaches its lowest value with reuse-3 model. No plot is available for GR UEs with reuse-3 scheme, since all the active UEs are considered as BR UEs when reuse-3 model is applied. SFR technique improves BR UEs throughput without reducing mean throughput per UE for the entire network. Indeed, the power allocation strategy applied by SFR mitigates ICI for BR UEs. Thus, it maximizes the usage of the available spectrum in all network cells, and reduces ICI simultaneously.

2) Throughput Cumulative Distribution Function: Empirical CDF for the compared techniques is calculated for the same simulation scenario. These values allow us to study throughput distribution among active UEs in the network as illustrated in Fig. 3. It is shown that throughput CDF of reuse-3 model is the first technique to reach the maximum. Although ICI is mitigated, the amount of available RBs in each cell is not enough to guarantee high data rates for UEs. FFR CDF throughput function is improved in comparison with reuse-3; however, it is also faster than reuse-1 CDF in reaching the maximum. For SFR, the number of UEs suffering bad quality of service is reduced. For relatively low throughput values (less than 1 Mbit/s) throughput CDF for SFR is the lowest curve; thus, it shows the lowest percentage of UEs served with low throughputs. Moreover, SFR curve is the last one to reach its maximum (at 3 Mbit/s approximately). When SFR is applied, we make use of all the available spectrum in each cell, and BR UEs have access to the portion of bandwidth with less ICI. Consequently, the achievable throughput increases, and BR UEs throughput is improved.

3) UE Satisfaction Versus Network Load: The percentage of unsatisfied UEs for each technique is compared and presented in Fig. 4 for different number of UEs per cell. For each scenario, simulations are repeated 100 times. Satisfaction throughput threshold is set to 512 kbit/s. If the average throughput of a UE is higher than this threshold, it is considered as satisfied; otherwise, this UE is considered as an unsatisfied UE. We notice that reuse-3 technique shows the lowest percentage of unsatisfied UEs for low network loads. When each cell is using a disjoint part of spectrum, ICI problems are eliminated. However, the percentage of unsatisfied UEs becomes the highest among all the compared techniques when the network load increases. Only one third of the available spectrum is used in each cell; thus, network capacity and UE satisfaction are reduced when network load increases.

Despite of the power reduction over RBs allocated for GR UEs, SFR shows approximately the same percentage of unsatisfied UEs as for reuse-1 model. The power allocation strategy reduces ICI, especially for BR users, and GR throughput loss is compensated. Compared to reuse-1 model, FFR increases the percentage of unsatisfied UEs, due to restrictions on RB usage between network cells. A portion of the available spectrum is not allowed to be used in each cell. When network load increases, FFR performance becomes better than reuse-3 model. It is a compromise between reuse-1 model (in cell-center zone) and reuse-3 model (in cell-edge zone).

4) Fairness Index Versus UE Distribution: We then study UEs throughput fairness index when the percentage of GR UEs in the network changes. For each UE distribution, simulations are repeated 100 times, and the obtained results are illustrated in Fig. 5. Reuse-3 model shows the highest throughput fairness index among all the studied techniques. In fact, ICI is eliminated, and the achievable throughput is approximately the same for all UEs. For reuse-1 model, BR UEs suffer from ICI, which has a negative impact on their throughput, while GR UEs achieve higher throughputs. Thus, it shows a lower fairness index. The static RB and power distributions between BR and GR zones, applied in FFR and SFR, are not adequate for all UE distributions, especially when the majority of active UEs are not homogeneously distributed between cell zones. Although they succeed in reducing ICI, FFR and SFR do not improve throughput fairness among all UEs for these particular scenarios, because restrictions made on RB usage between cell zones are not adjusted according to UE demands. Nevertheless, FFR improves Jain's fairness index in comparison with reuse-1 model when 55% to 65% of UEs are GR UEs. Thus, FFR tuning parameters should be adjusted according to network load and UE distribution between the different zones.

5) Spectral efficiency versus UE distribution: The impact of UE distribution on spectral efficiency is then studied for the compared ICIC techniques, and simulation results are reported in Fig. 6. It can be noticed that SFR shows the highest spectral efficiency, as it utilizes entire available spectrum in every cell, while imposing constrains on power allocation for RBs available in each zone. Therefore, it succeeds in reducing ICI while increasing spectral efficiency for all UE distributions, except the case where the majority of UEs are GR UEs: in this case, reuse-1 model is better since it achieves higher throughputs without the need to reduce downlink transmission power. Restrictions on RB usage make reuse-3 technique the one with the lowest spectral efficiency: in a cluster of three adjacent cells, only one third of the available spectrum is used in each cell. FFR is a compromise between reuse-1 and reuse-3 models, since reuse-1 model is used in GR zones, while reuse-3 model is used for BR zones. Thus, spectral efficiency curve for FFR is located between the curves of reuse-1 and reuse-3 schemes.

VI. CONCLUSION

In this paper, the performance of several ICIC techniques, such as reuse-3 model, FFR, and SFR, is analyzed. System-level simulations are performed under uniform and non-uniform UE distributions to compare their performance with that of reuse-1 model. Mean throughput per zone, throughput fairness index, UE satisfaction, and spectral efficiency are investigated under uniform and non-uniform UE distributions. SFR shows the highest spectral efficiency for approximately all UE distributions, unless the majority of UEs have good radio conditions. In this case, the usage of reuse-1 model is better. Reuse-3 outperforms all the other techniques in terms of UE satisfaction and throughput fairness, only when network load is relatively low. However, it permanently shows the lowest spectral efficiency. FFR technique is a compromise between reuse-1 and reuse-3 models. Moreover, FFR and SFR require interventions from mobile network operator to adjust RB and power distribution between cell zones according to UE distribution and throughput demands.

  Fig. 1: FFR and SFR schemes

Fig. 2 :

 2 Fig. 2: Mean Throughput per GR, BR, and all UEs

Fig. 3 :

 3 Fig. 3: Throughput cumulative distribution function

Fig. 4 :

 4 Fig.4: UE satisfaction versus network load

Fig. 5 :

 5 Fig. 5: Fairness index versus percentage of GR UEs

Fig. 6 :

 6 Fig. 6: Spectral efficiency versus percentage of GR UEs

TABLE I :

 I SINR-MCS-Data Rate Mapping Table

	Minimum SINR	Modulation and Coding Scheme (MCS)	Data Rate (kbit/s)
	1.7	QPSK(1/2)	168
	3.7	QPSK(2/3)	224
	4.5	QPSK(3/4)	252
	7.2	16QAM(1/2)	336
	9.5	16QAM(2/3)	448
	10.7	16QAM(3/4)	504
	14.8	64QAM(2/3)	672
	16.1	64QAM(3/4)	756

TABLE II :

 II Simulation Parameters

	Parameter	Value	Description
	Cell geometry	Hexagonal	Sectorized
	Inter-eNodeB distance	500 m	Urban area
	Operating bandwidth	5 MHz	--
	Number of RBs (N )	25	5 MHz bandwidth
	Carrier freq.	2 GHz	--
	Subcarrier frequency	15 kHz	1 RB = 12 sub-carriers
	RB bandwidth	180 kHz	12 × 15 kHz
	TTI	1 ms	--
	Pathloss model	15.3 + 127.6 log 10 (D)	in TS 25.814 D in km
	Thermal noise density	-174 dBm/Hz	--
	Feedback delay	3 ms	3 TTIs
	Scheduler	Round Robin	--
	Traffic model	Full buffer	--
	eNodeB max. power (P )	20 W	43 dBm
	Max. RB power (Pmax)	0.8 W	P N
	SINR threshold	3	UE classification
	SFR power ratio (α)	0.25	P GR = Pmax 4
	Satisfaction threshold	512 kbit/s	UE satisfaction
	Simulation time	100 TTIs	--


A R Bosco De Magalhães

J G Peixoto De Faria

R Rossi

Quantum erasure in the presence of a thermal bath: the effects of system-environment microscopic correlations

Keywords: numbers: 03.65.Ta, 42.50.Pq, 03.65, Yz, 03.75.Dg quantum erasure, cross decay rates, open quantum systems, cavity quantum electrodynamics

We investigate the role of the environment in a quantum erasure setup in the cavity quantum electrodynamics domain. Two slightly different schemes are analyzed. We show that the effects of the environment vary when a scheme is exchanged for another. This can be used to estimate the macroscopic parameters related to the system-environment microscopic correlations.

I. INTRODUCTION

"In reality, it contains the only mystery, the basic peculiarities of all of quantum mechanics." [START_REF] Feynman | Quantum Mechanics and Path Integrals McGraw-Hill[END_REF]. The famous statement by Richard Feynman about the wave-particle duality, gives a glance on the relevance of the subject to quantum theory. The double slit experiment illustrates very well the wave-particle duality. In such an experiment, if the information about which slit the quanton (in the sense of [START_REF] Englert | [END_REF]3]) has crossed (which-way information) is available, the interference fringes are not visible on the screen (particle behavior); however, if the which-way information is not available, there is an interference pattern (wave behavior).

At the Solvay conference (1927), A. Einstein presented a gedanken experiment (the Recoiling Slit Experiment) which consisted in a double slit experiment with a movable slit placed before the double slit. The goal was to detect which-way information of the quanton (recorded by the movable slit) and still see an interference pattern [START_REF] Bohr | Albert Einstein: Philosopher-Scientist, Library of Living Philosophers[END_REF]. The apparent difficulty imposed by such gedanken experiment was solved by N. Bohr, who pointed out that a careful analysis of the movable slit would require the inclusion of uncertainty relations of its position and momentum; this would add random phases in the quanton path and consequently it would make the interference pattern vanish. Therefore, in this argumentation, N. Bohr used the uncertainty principle to sustain the wave particle duality.

In the eighties, another gedanken experiment, the quantum eraser, proposed by M. Scully and collaborators [START_REF] Scully | [END_REF][6][7], brought back to light the debate about wave-particle duality. In the quantum eraser experiments, the quanton interacts with a probe system, and they become entangled. This interaction makes the which-way information available and destroys the interference pattern, even when there is no relevant modifications on the quanton position and momentum degrees of freedom. According to the authors, the entanglement is the essential key behind this phenomenon, and it is not necessary to call upon Heisenberg's uncertainty principle, as it was done in the early discussions between A. Einstein and N.

Bohr. As a result, a debate on the role of the entanglement and uncertainty relations began [8][9][10][11][12]. In a quantum eraser experiment, the which-way information available in the entangled state can be erased, and consequently the interference pattern recovered, by correlating the measurement results of the probe and the interferometric system.

Several experimental observations of the quantum eraser have been reported [13][14][15][16][17][18]. The quantum eraser is an important tool for debating on fundamental questions, but it is also used in practical applications. To quote a few examples: In Ref. [19], it was used as a tool for channel corrections; in Ref. [20], to improve the cavity spin squeezing; in Ref. [21], for imaging applications; in Ref. [22] for experimental entanglement verification.

In the present work, we propose an experimental setup in the context of cavity quantum electrodynamics (CQED) where quantum erasure can be accomplished. We propose an implementation of Ramsey interferometry, where the interferometric paths are represented by two internal states of Rydberg atoms. The which-way information is held by a bipartite system composed of two microwave modes that interact with a common environment modeled by a thermal reservoir [23,24].

The correlation resulting from the interaction between two or more systems and a common bath is responsible for the appearance of a set of states that are robust against decoherence [25,26]-the decoherence-free subspaces. In our model, the coupling of the bipartite system to a common bath leads to a dramatic difference in the process of erasure depending on the class of states chosen to perform it. This fact allows us to propose a measurement scheme of the parameter related to the cross correlations resulting from the interaction between the bipartite which-way register and the environment. This work is organized as follows. In the next section, we describe the basic setup and compute the action of the bath. In Section 3, we describe how a slightly modified scheme can be used to highlight cross decay rates related to the cavity modes. The Conclusion is found in Section 4.

II. SETUP FOR QUANTUM ERASURE

In the Ramsey interferometry experiments, the interference is observed by dealing with the states of the internal degrees of freedom of atoms or molecules: the role of different paths in this type of interferometry is played by them. Accordingly, it is necessary to create coherent superpositions of these states with the ability to manipulate the relative phase. When counting the number of atoms or molecules in a given state as a function of the relative phase, the interference can be observed. Clearly, the which-way information destroys the interference, and the erasure of this information may be used to restore it. The proposed experiment is based on Ramsey interferometry. We examine a setup where there is which-way information related to the states of two microwave modes. The scheme involving two modes is more complex than that possibly designed with only one mode; nevertheless, as will be clear, two cavity modes are necessary to investigate system-environment microscopic correlations.

Consider two superconducting cavities A and B that support the resonant modes M A and M B with frequency ω. Atoms with levels i, e, f , and g relevant to the experiment will go through these cavities and two Ramsey zones (see Fig. 1). The frequencies related to the atomic transitions are illustrated in Fig. 2. The tuning of the atomic transitions with the field modes can be performed by means of the Stark effect. The transition e → g is assumed to be resonant with the modes M A and M B when there is no Stark effect. We assume, in what follows, that the relations between couplings and detunings are such that non resonant transitions can be ignored at every step.

We consider the initial state of the cavities as the vacuum state. An atom prepared in the state i is sent, first passing through the cavity A and then the cavity B. When the atom enters the cavity A, the i → e transition is brought into resonance with the mode M A by the Stark effect during one π/2 Rabi pulse. Next, it goes to the cavity B, where the transition i → f is put into resonance with the mode M B for one π Rabi pulse. Then, the atom flies to a Ramsey zone tuned with the transition f → g. After this transition is performed, the atom travels to another Ramsey zone, tuned with the transition e → g. With a suitable i f e 5 choice of the atomic dipole, the state of the system just before this Ramsey zone will be

|ψ 1 = |g 1 |0 A |1 B + e iφ 1 |e 1 |1 A |0 B √ 2 , (1)
where φ 1 depends on the energies of the modes and atomic states, as well as the distances in the experimental apparatus and the velocity of the atom. When the atom passes through the second Ramsey zone, the system evolves to

|ψ 2 = 1 2 |e 1 |0 A |1 B + e iφ 1 |1 A |0 B -|g 1 |0 A |1 B -e iφ 1 |1 A |0 B . (2)
The probability of finding the atom in the state e (g) is given by P e = 1/2 (P g = 1/2),

showing no interference.

In ordinary Ramsey interferometers, the aim in the first step is to create a state of superposition between the atomic levels e and g with a relative phase that can be varied.

The preparation of the state |ψ 1 is analogous to this first step. However, this state clearly exhibits a perfect path discrimination due to the entanglement between the atom and the cavity modes. This prevents the direct observation of the interference between the paths related to e and g as in usual Ramsey interferometers. To observe the interference, it is necessary to perform the erasure of the which-way information. In the original proposal for quantum erasure, this was performed by a detector that interacted only with the symmetric mode of the field. Here, the erasure is achieved by sending a second atom that absorbs only the energy of the symmetric mode or of the antisymmetric mode of the field. As we will see, the action of the environment can vary according to this choice.

In order to investigate how the bath can disturb the erasure process, we permit a time interval τ between the end of the interaction of the field with the first atom and the beginning of the interaction with the second atom. The environment, at zero temperature, will be considered only during this interval, which should be large compared to the other times involved in the experiment. The decay of the fields and atomic states should be slow enough so that they can be neglected outside this interval. Experiments with atoms with slow decay and cavities with very high quality factors have been reported [27].

For the bath at zero temperature, the action of the environment can be computed using the master equation

d dt ρ = Lρ. L • = -i ωa † a + ωb † b, • (3) +k 2a • a † -•a † a -a † a• + k 2b • b † -•b † b -b † b• +k c 2a • b † + 2b • a -•b † a -•a † b -a † b • -b † a• ,
where a † (b † ) and a (b) are the creation and annihilation operators related to the modes M A (M B ), respectively, k is the decay rate of both the cavity modes, and k c is the cross decay rate. We use the usual notation for superoperators, where the symbol • indicates where the density operator must be placed. According to the discussion presented in [28], we consider

k c ≥ 0.
Let us return to the experimental sequence. Suppose that the state of the first atom is measured immediately after it passes through the second Ramsey zone. If the result of this measurement is e, the state of the field modes after the interval τ will be

ρ M A ,M B ,e = (ζ + |1 A |0 B + η + |0 A |1 B ) (H.c.) + 1 -|ζ + | 2 -|η + | 2 |0 A |0 B 0 A | 0 B | ,
where H.c. stands for Hermitian conjugate and

ζ + = e iφ 1 f (τ ) + l (τ ) √ 2 , η + = f (τ ) + e iφ 1 l (τ ) √ 2 , f (t) = e -iωt 2 e -(k+kc)t + e -(k-kc)t , l (t) = e -iωt 2 e -(k+kc)t -e -(k-kc)t .
Then, a second atom, initially in the state g, absorbs the antisymmetric field mode by interacting with the mode M A during three π Rabi pulses and with the mode M B for one π/2 Rabi pulse. With respect to M A , the interaction time can be adjusted by removing the atomic transition from the resonance with this mode, by means of the Stark effect, in the beginning of the path of the atom inside the cavity A, and then by letting in the resonance for the time necessary for three π Rabi pulses. As regards M B , one can allow the atom to interact with this mode during the time required for the π/2 Rabi pulse, in the beginning of its path inside the cavity B, after which the interaction is interrupted using the Stark effect. When the atom leaves the cavity B, M A is in the vacuum state and the second atom plus M B state can be written as

ρ M B ,A 2 ,e = 1 2 e -iφ 2 (ζ + -η + ) |e 2 |0 B + (ζ + + η + ) |g 2 |1 B [H.c.] +2 1 -|ζ + | 2 -|η + | 2 |g 2 |0 B g 2 | 0 B | ,
where e -iφ 2 corresponds to the phase accumulation during the Stark effect in cavity B. If we consider that the first atom was measured in the state g, we reach a final state concerning the second atom and the mode M B given by

ρ M B ,A 2 ,g = 1 2 e -iφ 2 (ζ --η -) |e 2 |0 B + (ζ -+ η -) |g 2 |1 B [H.c.] +2 1 -|ζ -| 2 -|η -| 2 |g 2 |0 B g 2 | 0 B | ,
where

ζ -= -e iφ 1 f (τ ) + l (τ ) √ 2 , η -= f (τ ) -e iφ 1 l (τ ) √ 2 .
Therefore, the probabilities of measuring the first atom in the state x and the second one in the state y (where x and y stand for e or g) are

P ee = 1 4 (1 -cos φ 1 ) e -2(k-kc)τ , (4)
P eg = 1 2 -P ee , P ge = 1 4 (1 + cos φ 1 ) e -2(k-kc)τ , P gg = 1 2 -P ge .
In equations [START_REF] Bohr | Albert Einstein: Philosopher-Scientist, Library of Living Philosophers[END_REF] we see that the interference is completely recovered in two cases. One of them corresponds to τ = 0, i.e., there is no interaction with the environment. The other is the limiting case k c = k, where the environment does not disturb the erasure process, since it interacts only with the symmetric mode [28] and the erasure is based on the absorption of the antisymmetric mode.

III. INVESTIGATING THE CROSS DECAY RATES

In the sequence described above, the second atom absorbs energy only from the antisymmetric mode. If we change this scheme so that the second atom interacts with the mode M A during the time of one π Rabi pulse, and maintain the remainder unmodified, it will absorb energy from the symmetric mode only. In this case, the limiting case k c = k leads to the maximum attenuation of the interference fringes. Indeed, the probabilities of measuring the state of both atoms, defined analogously as in equation ( 4), are

P ee = 1 4 (1 + cos φ 1 ) e -2(k+kc)τ , (5)
P eg = 1 2 -P ee , P ge = 1 4 (1 -cos φ 1 ) e -2(k+kc)τ , P gg = 1 2 -P ge .
The interference decreases according to e -2(k-kc)τ in equations ( 4), and according to e -2(k+kc)τ in equations ( 5); this result is related to the fact that the environment acts more strongly on the symmetric mode than on the antisymmetric mode [28]. This can be used to measure the cross-decay rate k c . Once the two experimental schemes have been completed, the measures of the frequency of the atomic states can be aggregated by computing the quantity ξ = P ge -P gg -P ee + P eg + P ge -P gg -P ee + P eg = e -2(k-kc)τ 1 -e -4kcτ cos φ 1 , which will be non-zero if, and only if, k c is not null.

IV. CONCLUSION

We explored the effects of the environment on quantum erasure in the cavity quantum electrodynamics domain. We showed that the bath disturbs the erasure process in a way that may depend on the details of the experimental setup. In fact, the attenuation of the interference fringes due to the environment depends, for non zero cross decay rates, on the mode (symmetric or antisymmetric) absorbed by the eraser, namely, the second atom that crosses the apparatus. This can be used to estimate the cross decay rates, which are related to microscopic correlations in the system-environment interaction. These rates, whose conditions for existence may be associated with the construction of modes spatially close in the scale of their wavelengths, are responsible for superradiance and subradiance and, in the limit, for decoherence-free subspaces. In this limit, the antisymmetric mode decouples from the bath, and an erasure scheme not affected by the environment can be envisaged.

Figure 1 :

 1 Figure 1: CQED experimental setup devised to detect the quantum erasure assisted by the environment. A Ramsey interferometer is implemented using two microwave cavities, R 1 and R 2 , where classical fields are stored, resonant or quasi-resonant with transitions f ↔ g and e ↔ g, respectively. A and B are high-Q microwave cavities that work out as path identifiers. The atomic levels e and g play the role of interferometric paths of the atom A 1 . The second atom A 2 is used to probe the state of the high-Q cavities A and B, and the erasing of the which-path information is yielded by coincidence measurements of the states of the two atoms.

, 305380/2012-5 and 306871/2012-2) and CEFET/MG (PROPESQ program-Grant 10122 2012). RRJr acknowledges the Brazilian agency FAPEMIG (Grant APQ-00597-12) for partial financial support.


Anne Clerval
email: anne.clerval@univ-paris-est.fr

Antoine Fleury
email: afleury@parisgeo.cnrs.fr

Urban Policy and Gentrification. A critical analysis using the case of Paris

Keywords: gentrification, public policies, urbanism, housing, public spaces, cultural policy, Paris

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

Introduction

In recent years the phenomenon of gentrification has aroused interest among French researchers, who have use this concept to analyse trends in inner cities (Bidou-Zachariasen, 2003;[START_REF] Simon | Gentrification of old neighborhoods and social integration in Europe[END_REF]Fijalkow & Préteceille, 2006;Authier & Bidou-Zachariesen, 2008). The concept, which has been theorised and studied since the 1970s mainly by researchers in the UK and North America, refers to a particular form of what the French would refer to generally as embourgeoisement, which alters the social composition, housing patterns and public spaces in working-class quarters. The concept of gentrification has the advantage of focusing on the interactions between changes in social structures and changes in the urban space. The term, coined by a Marxist sociologist in the 1960s in relation to London, originally contained a dimension of criticism aiming to denounce the eviction of the working classes from the city centre [START_REF] Glass | Introduction" in Centre for Urban Studies[END_REF]. Even so, despite some early work pointing to the role of public and private agents in this process [START_REF] Smith | Gentrification and capital: practice and ideology in Society Hill[END_REF] the notion of gentrification has since been used to describe a process that is at once inexorable and positive in the "revitalisation" of inner cities [START_REF] Slater | The Eviction of critical perspectives from gentrification research[END_REF]. This removal of the critical perspective in the research on gentrification has been interpreted by the sociologist Wacquant (2008) as being the result of the general marginalisation of the working classes in political, media and scientific discourse, and also of the underestimation of the part played by public authorities in the dynamics of gentrification. Several recent studies have pinpointed the importance of this influence, showing how gentrification can even, in some instances, become a fullscale local political strategy (Smith, 2003;Atkinson & Bridge, 2005). In their introduction to a special issue of the journal Urban Studies devoted to the role of public policies in the phenomenon of gentrification, [START_REF] Lees | Introduction to Special Issue on Gentrification and Public Policy[END_REF] reposition these local policies within the more general context of neo-liberalism and international competition between cities. The "third wave" of gentrification, symbolised by the conversion of the London docks into a new world-class business quarter, appears to federate researchers, who hitherto had opposed one another over the causes of gentrification: [START_REF] Lees | Introduction to Special Issue on Gentrification and Public Policy[END_REF] align themselves on the analysis by [START_REF] Smith | Gentrification and uneven development[END_REF], who argued the key position of gentrification in world capitalism at the time of the neo-liberal turning point. They also recall how the consensus-making objective of a social mix can be used today to favour the gentrification of working-class quarters.

The present paper aims to contribute to this reappraisal of the issues of gentrification as encountered in the literature, and proposes an analysis based on the case of Paris, which has, paradoxically, not been widely studied in this respect. Using material from two theses that have recently been presented [START_REF] Fleury | Les Espaces publics dans les politiques métropolitaines. Réflexions au croisement de trois expériences : de Paris aux quartiers centraux de Berlin et Istanbul[END_REF][START_REF] Clerval | La gentrification à Paris intra-muros : dynamiques spatiales, rapports sociaux et politiques publiques[END_REF], the paper studies the roles of public policies in the process of gentrification in Paris. The approach chosen sets out to avoid being solely technical, and aims to propose a genuinely critical analysis of the issues, and highlight the meaning of public policies implemented today in the Paris inner city. We will first show that gentrification in Paris appears as being mainly the result of private initiative; public authorities played the part of accompanying and accelerating the process, even if this is less easy to perceive than elsewhere. In a second section we will concentrate on the public policies implemented in Paris since the Municipality shifted to the Socialist left in 2001: in the area of housing, public spaces and culture we will endeavour to clarify the issues involved and the effects of these policies on the gentrification process. Finally, we will propose an interpretation of these different policies, showing that they are consistent with an accompaniment of the process of gentrification.

Characteristics of the gentrification process in Paris

The specific features of Paris compared to other metropolises

In Paris, gentrification appears as a process occurring late in comparison to London or New York. There are several reasons to explain this. Some emphasise the fact that the French capital has been the place of residence for the ruling elite for a long time, and this is indeed still true [START_REF] Préteceille | Is gentrification a useful paradigm to analyse social changes in the Paris metropolis?[END_REF]. The heart of the Paris agglomeration did not experience the white flight noted in American cities where the middle and upper class whites left the city centres for the suburbs in the 1950s and 1960s. Only the Marais quarter in Paris has seen a desertion and a later return of the wealthier classes in the course of its history, the first type of trend to be described as gentrification, in reference to numerous Victorian areas of London [START_REF] Glass | Introduction" in Centre for Urban Studies[END_REF]. Nevertheless, the extent of areas of older buildings in working-class Paris represents a considerable potential for gentrification. From the 1960s, the central quarters on the left bank of the Seine (Rive Gauche) were gradually transformed, extending the process of removal of working-class quarters that had begun with the work of Hausmann. Indeed the quality of the old buildings in the Marais led to rehabilitation or upgrading policies that fostered the return of the wealthier classes to this area.

As is the case with cities in the south of Europe, the French capital stands out on account of the existence a considerable mass of old buildings, and the heritage that they represent. Unlike the case of London, these buildings were largely spared the destructions of the Second World War, and the post-war period did not see city planning alterations on the scale on which they occurred in the City in London, nor indeed on the scale of what occurred in a different context in American cities such as Philadelphia [START_REF] Smith | Gentrification and capital: practice and ideology in Society Hill[END_REF]. In addition, in Paris, rents were tightly controlled for a long period, which slowed speculation in real estate, or at least restricted it to certain market sectors. Thus the takeover of working-class quarters by better-off classes occurred in a gradual manner in Paris. It is only in the 1990s that gentrification became really visible, and that the demolition of old, run-down housing was officially abandoned in favour of upgrading.

Actors and factors in gentrification

Gentrification

in Paris thus occurred mainly on private initiative, via the rehabilitation and upgrading of working-class housing [START_REF] Clerval | La gentrification à Paris intra-muros : dynamiques spatiales, rapports sociaux et politiques publiques[END_REF]. The people subsequently moving in were often artists and architects looking for professional premises, who took over former craft and industrial workshops in the eastern part of Paris towards the end of the 1970s, sometimes in the wake of the squats that characterised that decade (Vivant & Charmes, 2008). But more widely, at the same time, middle-class households (among whom cultural professions were over-represented) acquired and rehabilitated homes in the working-class quarters [START_REF] Bidou | Les Aventuriers du quotidien : essai sur les nouvelles classes moyennes[END_REF]. This trend was backed up by estate agents, who increased in number with the rise in prices, by the banks, and by the fall in the cost of mortgages which partially compensated for the rise in prices (APUR, 2007b). The trend was also fostered by promoters who undertook rehabilitation and upgrading for split sales. These promoters move in when prices are sufficiently high in a given area, this being particularly characteristic of the large artisan or industrial cours (enclosed yards housing workshops and living premises) in Eastern Paris [START_REF] Clerval | La Cour de Bretagne. Un cas de gentrification dans un quartier populaire[END_REF], in some cases involving former factories (Clerval, 2008a).

As elsewhere, the gentrification of the working-class areas of Paris can be explained by several structural factors in the spheres of housing and employment. The most obvious of these is the continuous decrease in the number of working-class jobs in Île-de-France (the Paris region) and Paris from the 1960s. This was compounded by a decrease in the number of unskilled jobs in the services sector in Paris from the 1980s, while in the same period the number of managerial, professional and intellectual jobs increased [START_REF] Rhein | Changements sociaux et transformation de l"espace[END_REF][START_REF] Clerval | La gentrification à Paris intra-muros : dynamiques spatiales, rapports sociaux et politiques publiques[END_REF]. This change in the structure of the job market in Ile-de-France was the result of the redistribution and reorganisation of production worldwide, a phenomenon that was accelerated by neo-liberal macro-economic policies from the 1980s. Deregulation and the international integration of the economy increased competition among unskilled workforces on a global scale, and facilitated the globalisation of industrial production, while the metropolisation that resulted led to a concentration of qualified jobs in Ile-de-France. Nevertheless, the changes in the structure of the job market are not sufficient to explain the increasing social selection occurring in the residential space in the inner city of Paris [START_REF] Clerval | La gentrification à Paris intra-muros : dynamiques spatiales, rapports sociaux et politiques publiques[END_REF]. Here the patterns of housing and the structure of the housing market accentuated the contrasting dynamics between professional and intellectual categories and working-class categories. The earlier working-class housing was deteriorating under the effects of strategies aiming at short-term financial returns on the part of landlords, or else as a result of their intention to sell their properties (this being particularly true for complete buildings). The policy of mass construction of "social" housing in city outskirts in the 1960s and 1970s led to the departure of large numbers of working-class people from Paris to its suburbs, generating a phenomenon of vacancy in the workingclass quarters. These were then partly taken over by more precarious populations, often immigrants, and alongside them by gentrifying households. The partial vacancy and the poor state of repair of the housing created a differential in terms of economic viability [START_REF] Smith | Gentrification and uneven development[END_REF] in these quarters close to the centre: after the first gentrifiers moved in and the appearance of trendy retail stores (Van Crieckingen & Fleury, 2006), numerous quarters gained lucrative potential for investors.

Earlier public policies had a considerable part in this process. From the 1960s to the 1990s, the operations consisting in the "renovation" of areas by way of demolition and reconstruction contributed to de-structuring working-class quarters, but at the same time led to the building of a large amount of social housing. In addition, the first gentrifiers were opposed to the demolition of old housing [START_REF] Simon | La Société partagée. Relations interethniques et interclasses dans un quartier en rénovation[END_REF], which demonstrates the lasting hiatus between the public authorities and the gentrification process. However the de-regulation of rents in 1986 encouraged real estate speculation, while at the same time several prestige operations gave impetus to the upgrading process in working-class areas for the benefit of the middle and upper classes, for instance the demolition and development of Les Halles (originally a large covered market) and the construction of the Georges Pompidou Modern Art Centre (Beaubourg) in the 1970s, the rehabilitation of the Marais area in the same period, and the creation of certain large facilities to the east and north of Paris i .

Spatial dynamics: diffusion, outposts and bypass

Figure 1. Paris: study area

Figures 2 and 3 make it possible to apprehend the spatial dynamics of gentrification using a typology of the Paris IRIS (Ilôts regroupés pour l'information statistique -a methods of grouping housing blocks in units of around 2,000 inhabitants for statistical purposes) according to the characteristics of the housing and the socio-professional categories of the inhabitants, for 1982 and 1999. What can mainly be seen is a diffusion from the wealthy quarters (les Beaux quartiers) in the west. On the left bank of the Seine, gentrification is long-standing and based at once on demolition and reconstruction operations such as the Beaugrenelle development (15 th arrondissement ii ) on the Front de Seine, on new private constructions in peripheral quarters, and on the rehabilitation of older housing in the quarters that had become fashionable, such as La Butte-aux-Cailles (13 th ) in the 1990s. In 1999 the left bank displayed an almost exclusively upper class profile, the only distinction being between old and new buildings. On the right bank, the large crescent of former working-class areas extending from the Batignolles (17 th ) to the Faubourg St Antoine (11 th , 12 th ) was progressively taken over and rehabilitated by middle and upper class occupants, moving outwards from the centre to the periphery. This was a continuous process of turnover of occupancy, the gentrifiers becoming increasing wealthy. This explains why in 1999 the central and immediately peri-central arrondissements have a wealthier social profile than those that are more peripheral, although these were also in the process of becoming gentrified. In space, this diffusion process is not completely continuous. Gentrification possesses a sort of a pioneer fringe, with outposts, particular spaces such as the Butte Montmartre, the canals, or the green areas, and by-pass effects, in particular around spaces with marked proportions of immigrant populations, such as the centre of the right bank (around Le Sentier in the 2 nd ) or Belleville (20 th ) and the Faubourg du Temple (10 th , 11 th ). These quarters appear as pockets of resistance bypassed by gentrification, although not completely exempt from the process.

Figures 2 and 3. IRIS typology in Paris according to main residence characteristics and social category of households in 1982 and in 1999

The public policies implemented in Paris since 2001 and their role in the process of gentrification

In 2001, the municipal elections for the first time brought a Socialist majority to power in the Paris Mairie. The new Mayor, Bertrand Delanoë, re-elected in 2008, communicated widely on the theme of a breakaway from the previous team, in particular on the notion of a "shared" city, both among the different social classes, and among the different users of the streets and traffic routes. This discourse made reference to new approaches to public action, both in terms of financial investment, and in terms of the mobilisation of various means to eradicate insalubrious housing, to re-launch urban policies, and to give impetus to cultural action. We shall explore how these different policies have fitted into the Paris space on different scales, and what have been their effects on the social composition of Paris.

We will show that the real turning-point in local public policies in Paris occurred in 1995, when Jacques Chirac was replaced by Jean Tiberi, rather than in 2001. As early as 1996, six arrondissements in the north of the city had gone over to the political left, showing in particular the increasing opposition towards demolition and reconstruction work conducted by the previous municipal Council. The abandoning of this policy went hand in hand with the rallying of public authorities to the process of gentrification. This was occurring mainly by way of rehabilitation of older buildings, pursuing objectives of social upgrading, economic revitalisation and rehabilitation of the local heritage. The terms of "revalorisation" and "revitalisation" were commonly used by public authorities. As has been pointed out by Smith (1996) the fact that gentrifiers were referred to as "pioneers" implies that working-class quarters were undergoing a crisis and that gentrification was a way of "saving" them, thus masking the eviction of the working classes from the inner city.

From this point of view, we will show that the Socialist Council elected in 2001 in fact pursued and amplified the policy initiated by Tiberi, without really calling the issue of gentrification into question.

Urbanism and housing: preserving older buildings and re-launching the production of social housing

At the end of the 1990s, the decline in demolition and construction operations was accompanied by a marked decline in the production of social housing. This can be explained as much by decision on the part of the Council as by an increasing withdrawal on the part of the State. The re-launching of production implemented by Delanoë preceded the return of State investment in social housing from 2004. If this is in marked contrast with the preceding municipality, we will see that the policy cannot claim to have slowed the gentrification process, and that on the contrary in certain areas it has supported it.

The re-launching of the production of social housing in Paris has been particularly highlighted in Council communication. These operations did not involve a return to the demolition and reconstruction process, but were based on more stringent requirements for the rehabilitation of old buildings and the eradication of insalubrious housing. Within these measures, work on housing improvement could be declared to be in the public interest and made compulsory for property owners. The Council could use its right to pre-empt (legally reinforced in certain quarters) to acquire run-down buildings and convert them into social housing after rehabilitation. It instated targeted operations for buildings listed as insalubrious (a task entrusted to SIEMP -Société immobilière d'économie mixte de la Ville de Paris) or for certain quarters such as Château Rouge (18 th arrondissement).

However, the planned housing improvement operations scheme (OPAH -Opérations programmées d'amélioration de l'habitat), an incentive-generating tool providing public grants for the improvement of old buildings, widely used by the previous municipality, continued alongside, without any specific requirements in return from the private property owners, thus fostering the process of gentrification.

The increased production of social housing was accompanied by a diversification in the modes of production and in modes of financing. New constructions were implemented alongside the rehabilitation of old buildings (acquired by the municipality) and in some cases purchase and commissioning of buildings without any work being performed. The two latter novel forms of production of social housing enabled the target of 3,000 approved homes per year to be reached as early as 2001. Production reached 4,000 homes a year in 2004 and 2005, 5,000 in 2006 and 6,000 in 2007 via the resumption of new construction

iii . The objective of 20% of main residence homes being social housing (provided for in the SRU -Solidarité et renouvellement urbain -law of 2000) could be reached by the end of Delanoë"s second term of office. In 2007, the proportion of SRU social housing had reached 15.4% in Paris [START_REF] Apur | L'Accès au logement social à Paris. Analyse de la demande de logement social et bilan des attributions de logements sociaux à Paris en[END_REF]. This quantitative achievement reflects a budgetary investment by the Council, housing being their second-ranking budget item at 437M euros in 2008, or an amount equivalent to the State budget devoted to rented social housing in the country as a whole.

However these figures mask the structural limitations of this endeavour. Indeed, available building plots are increasingly scarce in Paris, and new constructions are most often replacements of old buildings too run-down to be rehabilitated. Thus "official" social housing has in fact replaced de facto social housing, without however enabling the housing of as many individuals, both because the homes are larger, and because the ratio of the number of occupants to the number of rooms is much lower in the first instance than in the second. Thus the production of social housing needs to be positioned within the wider context of the housing pool in Paris overall. This housing, as we have seen, is markedly affected by gentrification. The process contributes to restricting the private housing rental offer, in particular the offer affordable by working-class households, i.e. de facto workingclass accommodation. The table below shows that the production of social housing is insufficient to replace the decrease in de facto social housing which is estimated here via housing lacking basic conveniences iv : between 1982 and 1999 this pool lost around 300,000 units, while "official" social housing gained 60,000, the main residence pool overall remaining more or less stable. Thus to reach the objective of 20% social housing if the rest of the housing pool -which is private -is occupied by the middle and upper classes, equates to maintaining only 20% working class households in Paris, in other words half as many as the proportion they actually represent in the structure of the Île-de-France workforce.

In addition, not all the SRU social housing is allocated to working-class households. In particular a social housing funding system, the Prêt locatif social (PLS), was instated in 2001, producing homes intended for households that were 30% above the income level for obtaining PLUS accommodation (Prêt locatif à usage social), the classic social housing provision. The PLUS also leads to a degree of social mix, since it allows for 10% of the housing to be allocated to households with resources above the maximum level. The re-launching of the production of social housing is nevertheless grounded in this diversification of types of funding: thus PLS housing merely requires State approval without funding, which enables the State to withdraw from the production of social housing. Nearly 30% of the social housing produced in Paris since 2001 comes under the PLS system, while this does not correspond to the social structure of the demand: among the 100,000 applicants for social housing in Paris, 75% have incomes that are below the PLAI housing limit (Prêt locatif aidé d'intégration), intended for the lowest income category, while only 4% could apply for PLS housing (APUR, 2007a). In fact this 30% PLS housing reflects the proportion of Parisian households with incomes allowing them to apply for this type of accommodation. This can be read as a policy aiming to reorient part of the social housing pool towards middle-class households, while at the same time working-class households are experiencing ever-increasing difficulties in finding housing in Paris, whether in the social sector or in the private offer. This diversification of the types of financing, and hence of the social status of households qualifying to apply for social housing, also serves the strategy aiming to rebalance the housing pool geographically and to enhance the social mix promoted by the Socialist municipality. The idea is to create social housing in arrondissements where it is lacking, in particular "very" social housing (for the lowest incomes), and to promote PLS housing in areas where there is already 20% social housing, or more generally in working-class quarters. Thus the Municipality has acquired buildings in the Avenue Mozart (16 th arrondissement -a traditionally wealthy area) to turn them into PLAI (low-rent) housing, and has created PLS housing in the working class areas of Château Rouge (18 th ) or lower Belleville (10 th , 11 th ). While the process of gentrification is already pronounced in Paris, after accelerating in the period of rising property prices from the end of the 1990s up to 2008, these immigrant quarters, as seen earlier, are those in which the process is the least advanced today. Thus in areas where this process could still be controlled -for instance by increasing the offer of genuinely social housing targeting the working class -the municipality contributes, on the contrary, to accompanying gentrification, replacing only a minority of de facto social housing, via demolition or upgrading, and via its conversion into social housing with particular emphasis on the PLS type. In Château Rouge this is accompanied by an attempt to restrict or remove so called "exotic" retail trade, frequented by African populations, and by the overall upgrading of the public space, vigorously called for by the new house-owners in the quarter (Bacqué, 2005;[START_REF] Clerval | La gentrification à Paris intra-muros : dynamiques spatiales, rapports sociaux et politiques publiques[END_REF]).

Public space: streets and green areas. The general embellishment of the city

The first major upgrading operations applied to the public space concerned street areas and were launched in the 1980s. In this period numerous streets and squares (small public gardens) were upgraded in the most emblematic quarters, mainly the historic quarters, where pedestrianised areas were created. During the mandate of Tiberi (1995-2001) a full-scale policy was established, including more local public spaces. This policy developed and became systematic in the first term of office of Delanoë (2001)(2002)(2003)(2004)(2005)(2006)(2007).

Within the "green quarter" policy, in particular, developments diffused across a large part of the Paris area. For the most part, this consisted in setting aside all or part of the street areas from vehicle traffic, and this involved rehabilitation of the street surfaces, widening of pavements, and also the creation of parking areas. In ever increasing number, these developments were aimed at the general embellishment of public spaces, implementing a dual process of "greening" (lining with trees, flower beds etc) and "patrimonialisation", referring to the use of heritage-sensitive quality materials and traditional urban objects. The so-called "espaces civilises" (or "civilianised spaces" where traffic is reduced) are one manifestation of this policy applied to the former "red" traffic routes (axes rouges), comprising specific facilities for buses and bicycles, and the reduction of the number of traffic lanes.

Numerous green areas were created in the 1980s and 1990s, mainly in the form of large parks within vast urban developments (Bercy, André Citroën, La Villette, etc). After 2000 this policy was pursued. It was necessarily restricted, for lack of space, but the 2.4 hectares developed in 2005 followed by 9 hectares in 2006 are far from negligible. This mainly concerns small local area gardens, but also involves larger parks, such as the recent Jardin d'Eole (18 th arrondissement) or the Clichy-Batignolles park (17 th arrondissement).

The Municipality also undertook the rehabilitation of numerous "squares" (which in French refers to rather small squares often with trees, flowerbeds, swings etc). In all instances, the aim has been to make gardens into socialising spaces where many different occupations are possible. This also required new rules allowing public access to lawns (traditionally out-of-bounds in French parks) so that inhabitants of the quarter can picnic there, and teenagers gather. The systematic creation of playgrounds for children was also intended to develop "proximity" activities. And finally the Municipality worked for the creation of "shared gardens": even more than the "squares", these are intended for the resident population and aimed to create areas for socialising among neighbours.

A large proportion of these developments occurred in the arrondissements in the east and north of the city. This is true for the parks and gardens, but also for the streets and squares. The analysis of the logics of the positioning of green areas in relation to socio-demographic characteristics v shows that the public authorities have invested considerable means to enhance and upgrade streets in the most heterogeneous quarters, those undergoing the greatest changes from the socio-demographic point of view. In all, it can thus be said that the developments of public spaces in recent years have mainly involved quarters affected by a more or less advanced process of residential gentrification (table 2). It is certainly true that the effects of a development involving a public space on residential gentrification are as yet not well known. In Paris, most of the elected representatives say they remain "watchful" on the subject. As for associations for the protection or improvement of the living environment, they agree to being aware of residential dynamics, but refuse to link this to their action [START_REF] Fleury | Les Espaces publics dans les politiques métropolitaines. Réflexions au croisement de trois expériences : de Paris aux quartiers centraux de Berlin et Istanbul[END_REF]). Yet the development of "proximity" or neighbourhood public spaces, which play a full part in the embellishment of the city, necessarily contributes to the increase in value of property in quarters where it is implemented. To cite just a few examples, the rehabilitation of the streets in the Marais quarter in the 1980s, the rehabilitation of certain quarters such as the Butte-aux-Cailles or Mouffetard in the 1990s, or the canal banks in the 1990s and beyond (Canal St Martin, Bassin de la Villette), contributed to upgrading certain portions of the urban space and hence to an increase in the value of property. The property agents made no mistake, regularly using the description "green neighbourhood" in their adverts, on a part with proximity to shops and public transport. The presence of a "square", or the aesthetic quality of the street are indeed amenities, and hence an additional selling argument.

Socio-demographic profiles

The cultural policy: facilities and cultural events

The embellishment of the city has been accompanied by efforts in the cultural sphere. Cultural policies, in classic manner, first involved the creation of new facilities. Numerous investments were made by the State, and then by the Paris Municipality, in the north-east of the city. In particular, the developments within the Parc de la Villette complex deserve mention: the Zénith, the Cité des sciences et le l'industrie, the Cité de la musique, built by the State in the 1980s and 1990s. More recently, the Paris Municipality has pursued in the same direction. Several abandoned industrial sites have been converted into venues devoted to cultural production and consumption. Along the Canal Saint-Martin, a former building material warehouse was taken over by the association Usines Ephémères (literally: ephemeral factories) with the support of the Muncipality. It subsequently became "Le Point Éphémère", and the building houses an exhibition area, a concert hall and a café, as well as artist "residences". In the Faubourg du Temple area, the Maison des Métallos, which was a famous trade union venue from the Front Populaire in the 1930s, was rehabilitated and converted into a cultural facility used for live shows, drastically reducing the space devoted to the associations committee that ran the place previously vi . Finally, close to the railway tracks of the Gare de l'Est, 35,000m 2 have been set aside for contemporary artistic creation, also integrating a restaurant, a café, and shops, and this was inaugurated in 2008 in n o 104 rue d'Aubervilliers in the building that previously housed the city funeral enterprise. With an investment of 102M euros, equivalent to almost a quarter of the budget devoted to social housing in 2008, this project, the "Cent Quatre" (104) constitutes the largest project in Delanoë"s first term of office.

Alongside the creation of new facilities, the public authorities are also investing more and more in the organisation of cultural manifestations taking the form of events in public spaces. In particular certain major events gaining wide media coverage can be mentioned, such as Paris Plage or the Nuit Blanche (literally a "sleepless night"), and also festivals like Paris quartier d'été or Cinéma au Clair de Lune. These events take on very diverse forms, very often integrating music and dancing, but also increasingly contemporary art. Most are designed as events based on some form of scenography, and some entail non-permanent fixtures. These events are, all in all, the tools of a cultural policy that has been seeking to reach a wider public since the 1960s.

It is when their geography is observed that one becomes aware of the role that these events can have in the process of gentrification. Indeed new spaces for holding events and celebrations have emerged in the city (figure 4). The parks and the banks of the Seine have gained in frequentation, and even if the historic city centre still has an important role, the diffusion of recreational and cultural events towards the outskirts of the city is an increasing concern among protagonists in these matters [START_REF] Pouessel | Réinventer les espaces publics par la fête : le cas de Paris, mémoire de maîtrise en géographie[END_REF][START_REF] Fleury | Les Espaces publics dans les politiques métropolitaines. Réflexions au croisement de trois expériences : de Paris aux quartiers centraux de Berlin et Istanbul[END_REF]). Among the events and festivals organised completely or partly out of doors, this trend has strengthened from one season to the next. Paris Plage (where stretches of the banks of the Seine are converted into a "beach" with sand and deckchairs) clearly appears as the most emblematic operation in the diffusion of festive events towards the city outskirts: it was created in 2001 on the banks of the Seine in the historical area of the city and then, under the altered name of Paris Plages, was extended in 2006 to the 13 th arrondissement, along the Paris Rive gauche ZAC, before further extension to la Villette in the 19 th arrondissement.

Behind the laudable objectives of upgrading public space, familiarising people with contemporary art, or providing a beach for those who could not go away on holiday, these operations have contributed to turning the city into a sort of stage set or backdrop. They also draw a large number of citizens to quarters hitherto little frequented except by those living there [START_REF] Pouessel | Réinventer les espaces publics par la fête : le cas de Paris, mémoire de maîtrise en géographie[END_REF][START_REF] Fleury | Les Espaces publics dans les politiques métropolitaines. Réflexions au croisement de trois expériences : de Paris aux quartiers centraux de Berlin et Istanbul[END_REF]). The public authorities, by the same token, have contributed to a change in image and in frequentation of these former working-class quarters. The effects of this are all the more marked because they reinforce the impact of the developments in public spaces completed in the 1980s and 1990s. In addition, despite the fact that the centralising phenomenon is short-lived, these events make use of the large facilities already achieved or under construction in the former working-class quarters, creating new cultural centralities in a durable manner.

The political meaning of urban policies

Accompanying gentrification

Thus the regeneration of municipal public action in Paris since 2001, in fields as varied as housing, public space and culture, presents a degree of consistency, despite the ambiguity of the objectives set out to justify it. To interpret public action, it is not sufficient to stop at actual investment in housing and public facilities, nor at the objective, which is at once vague and consensus-prone, of maintaining a social mix in Paris. These different policies form a whole that is fully consistent with the wishes of the gentrifiers. These protagonists are often the first to speak up in the local councils (conseils de quartier), established by the Socialist Municipality in the different quarters to foster participation in urban policy-making by the citizens [START_REF] Blondiaux | La démocratie par le bas. Prise de parole et délibération dans les conseils de quartier du 20 e arrondissement de Paris[END_REF]. Alongside this, in quarters undergoing a process of gentrification, these local councils appear as a tool enabling justification of public policies a posteriori by the very people who put energy into campaigning for them, and who benefit from them.

The geography of these policies is revealing of the way in which they accompany gentrification: the working-class quarters taken over by the middle and upper classes are those that have had the benefit of the largest numbers of creations of green areas, "green quarters", and new sporting and cultural facilities. The extension of the itinerary for the Nuit Blanche and the shift of Paris Plages to the north-east constitute explicit public support for the arrival of the gentrifiers in quarters that are still marginal in relation to this process.

Policy serving the middle classes?

More generally, the absence of reference to the process of gentrification that is underway, the manner in which it is progressing, and its spatial dynamics, casts light on the motives for embellishing the city and promoting culture within its boundaries. The establishment of large cultural facilitiesdespite the fact that Paris was not lacking in these -in the north-east of Paris clearly contributes to gentrification by way of culture, as in Los Angeles [START_REF] Davis | City of Quartz: Excavating the Future in Los Angeles[END_REF] or Bilbao (Vicario & Matinez Monje, 2003). Culture is not only instrumentalised in order to upgrade the image of the city and draw wealthier occupants, tourists and private investments; the creation of cultural facilities is also addressed to professionals in the cultural field, and it has been seen that these are the avantgarde of the gentrification process. The intellectual middle class, cultural professionals and highly qualified managerial staff are targeted by the policies implemented by the municipality, which is in turn composed of elected members from these same social groups. In practical terms, they are in a better position to make use of the local councils (conseils de quartier), the new sporting, educational and cultural facilities, or the infrastructures set aside for what is known as "circulation douce" (or "soft" transport, i.e. bicycles, roller skates etc.). Ideologically, it is their vision of the city that wins the day, aesthetics, heritage, and recreation: present public policies go a long way towards creating urban landscapes for "consumption" by the middle and upper classes (Smith, 2003, p.58). The orientation of urban policies in favour of the middle class is often operated under cover of the common interest, the middle class being considered to be in a clear majority vii , along the ideological lines of the averaging-out of society in the post-war boom, despite the fact that this has been completely contradicted by the tendency for the income gap to widen [START_REF] Bihr | Déchiffrer les inégalités[END_REF].

In reality, the Parisian middle class are generally well above the median for incomes in France. Among middle-class voters, the gentrifiers vote widely for the Socialist party or the Greens, so that the municipal policies implemented in Paris are also intended to meet the approval of an electorate. The re-election of Delanoë in 2008 shows the success of this strategy: in particular the takeover of environmental concerns generally promoted by the Greens finally benefited the Socialist party, since it strengthened its hegemony across the political left at the expense of the Greens.

The eviction of the working classes and the "undesirables"

The areas where the clear distinction between the middle classes and common interest is verified are the de-structuring of the working-class quarters resulting from these different policies and the eviction of the "undesirables" [START_REF] Belina | Evicting the undesirables. The idealism of public space and the materialism of the bourgeois State[END_REF]. In housing, rehabilitation by private agents, as well as the eradication of insalubrious housing, have gradually led to a reduction in the housing offer affordable by the working classes. From this point of view, public policies have enabled improvement of housing conditions solely for the minority that can afford to remain in the area. More generally, public policies have tended to eliminate public spaces as they existed previously in the working-class areas,-ordinary public space, sometimes in poor condition but with a symbolic impact; they have also, in some instances, actually challenged the traditional function of spaces where people spent time as constituting public nuisances. The public authorities have thus tended to erase popular memory and belonging from places, thereby accentuating the exclusion of former inhabitants. In the same manner as commercial developments or cultural venues, a good quality "up-market" public space can constitute a symbolic barrier for certain social groups: the streets no longer draw the same public, and certain social groups no longer have reason to be there because they are not sensitive to the atmosphere of the street and its shops, or indeed they may no longer dare to go there because they are not welcomed into spaces for which they do not possess the access codes. At this point the public space is no longer hospitable [START_REF] Joseph | L"hospitalité et l"univers des rencontres" in La ville sans qualités[END_REF].

Exclusion can be even more radical in the case of groups considered as "undesirable". The most obvious instance is the homeless. If in Europe there are not the extreme situations found in the USA, where parks are fenced and boundaries set to avoid the homeless settling in [START_REF] Smith | New city, new frontier: the Lower East Side as wild, wild West[END_REF], new designs for urban objects and amenities are emerging where the intention is the same (Zeneidi-Henry, 2003). For instance, park benches are done away with or redesigned to avoid people lying on them, and watering systems are set up on lawns. In Paris the trend is certainly less marked than in the USA, but is does exist, in particular on the strength of pressure from local inhabitants [START_REF] Froment-Meurice | Le contrôle par le mobilier urbain. Les aménagements dissuasifs dans les espaces publics parisiens[END_REF]. The presence of homeless people, drug dealers or prostitutes frequently constitutes the more or less explicit reason for a particular development, or for the organisation of events. It is the case for the area along the Canal Saint-Martin and the Bassin de la Villette (including Place Stalingrad). It is also the case for numerous "squares" that have been refurbished with a view to enabling re-appropriation by the local inhabitants (in particular mothers and children) of spaces that had been partially taken over by specific groups such as drug traffickers, homeless people or working-class youths. It is via an analysis of the place of the homeless in urban policies that [START_REF] Mitchell | The annihilation of space by law: the roots and implications of anti-homeless laws[END_REF] came to refer to the "public landscape", considering that public spaces were becoming not so much spaces in which political expression and social conflict were played out, but stage sets for the use of well-off citizens and places from which others were excluded.

Thus analysing the policies accompanying gentrification as being aimed at the middle class is insufficient. Indeed it more generally serves the interests of the dominant class by way of an eviction of the working classes both from the city and from public debate -i.e. from the public space in its various dimensions. It contributes to the erosion of the class consciousness of the dominated population, which is an efficient means of avoiding the emergence of social conflict that might question the present class relationships.

Conclusion

It thus appears that the specific features of public policies implemented in Paris in the past, as well as the maintaining of rental controls up to 1986, and the production of social housing up to the 1980s, all partly explain the late occurrence of gentrification in Paris. Nevertheless, since Haussmann, public authorities have sought to upgrade or add value to the inner city, fostering gentrification in various modes. Even today, public policies in Paris show this trend, although they cultivate a certain degree of ambiguity as to their objectives and the populations targeted. The regeneration of public action promoted by the Socialist left in Paris from 2001 has been unable to halt the process of gentrification in housing, and on the contrary has contributed to accompanying the trend by development of the public spaces and by its cultural policies.

These complex issues of public policy render research in social sciences all the more necessary on the role of such policies in the urban and social transformations occurring. This article is an invitation to explore interactions between public policies and structural changes. The action of public authorities in favour of the dominant classes is not always visible, and is tending increasingly to cover up its own footprints via communication techniques. The role of social sciences is to highlight relationships of power underpinning these policies and their effects, avoiding stopping short at the often over-moralising issues of the intentions of public agents in their effective accompaniment of gentrification, even if they are unaware of the fact. Smith, N. (1996) The New urban frontier: gentrification and the revanchist city, New York, Routledge, XX-262 p. i For instance the following can be quoted: the Cité des sciences et de l'industrie opened in 1986 on the site of the former La Villette abattoir, or the Opéra Bastille opened in 1989 on the edge of the Faubourg St Antoine which was in the process of marked gentrification. ii Paris is divided into arrondissements that are numbered from the centre outwards in spiral form. iii A new mode of calculation included extensive rehabilitation in the category of new constructions. iv This refers to housing used as main residence and provided with neither toilets nor bath/shower. v What is referred to here is the definition of socio-demographic profiles for the Paris IRIS developed in a study on the Paris quarters [START_REF] Saint-Julien | Les contextes socio-résidentiels[END_REF]. Eight socio-demographic profiles were defined by considering the distribution of inhabitants over 15 on the basis of age, educational level, social category of households reference person, and employment status. This typology was then set against the map of the "green quarters" [START_REF] Fleury | Les Espaces publics dans les politiques métropolitaines. Réflexions au croisement de trois expériences : de Paris aux quartiers centraux de Berlin et Istanbul[END_REF]. vi The "Comité métallos" groups local associations and inhabitants who mobilised between 1997 and 2001 to save the site, which was at the time up for part sale by the Union, from property speculation. Today this collective denounces the fact that it has been cut off from the new Maison des Métallos which has become a cultural establishment run by the municipality, and also the fact that it is rented out to private enterprises 30 days a year [see www.lesmetallos.org]. vii This is reported in a study for APUR, the urban development agency of the city of Paris, entitled Les classes moyennes et le logement à Paris (Nov 2006), in which the middle classes are defined partly arbitrarily so as to correspond to 60% of the Paris population.

Figure 4 .

 4 Figure 4. Towards a redefinition of leisure centralities by the public authorities

  Smith, N. (2003) "La gentrification généralisée : d"une anomalie locale à la "régénération" urbaine comme stratégie urbaine globale" inBidou-Zachariasen, C. (ed.) Retours en ville : des processus de "gentrification" urbaine aux politiques de "revitalisation" des centres, Paris, Descartes & Cie, pp. 45-72. Van Criekingen, M. et Fleury, A. (2006) "La "ville branchée" : gentrification et dynamiques commerciales à Bruxelles et à Paris", Belgéo, n o 1-2, pp. 113-133. Vicario, L. et Martínez Monje, P. M. (2003) "Another "Guggenheim effect", Urban Studies, vol. 40, n o 12, pp. 2383-2400. Vivant, E. et Charmes, É. (2008) "La gentrification et ses pionniers : le rôle des artistes off en question", Métropoles, 3, Varia, <http://metropoles.revues.org/document1972.html> [mis en ligne le 22 septembre 2008]. Wacquant, L. (2008) "Relocating gentrification: the working-class, science and the state in recent urban research", International Journal of Urban and Regional Research, vol. 32, n o 1, pp. 198-205. Zeneidi-Henry, D. (2003) Les SDF et la ville. Géographie du savoir-survivre, Paris, Bréal, 288 p.

Table 1 . Evolution of housing lacking basic conveniences compared to "official" social housing in Paris since 1982 (source INSEE RGP 1982, 1990 and 1999; Mairie de Paris, 2006)

 1

	Year	Main residences	Homes lacking basic conveniences numbers proportion(%) numbers Social housing proportion (%)
	1982	1,097,452	322,916	29.4	125,180	11.4
	1990	1,095,108	188,709	17.2	134,455	12.3
	1999	1,110,912	40,069	3.6	185,328	16.7
	Yearly evolution	Homes lacking basic conveniences	Social housing	Year	SRU Social housing
	1982-1990	-16,776	+1,159	2001	154,314
	1990-1999	-16,516	+5,653	2005	167,393
	Overall evolution1982-1999	-282,847	+60,148	Yearly evolution	+3,270

Characterization Total number of Parisian IRIS Number of IRIS included in "green quarters" Proportion of IRIS included in "green quarters" in the total number of Parisian IRIS (in %)

		Young low-skilled and
	Working-class quarters with ongoing gentrification	unskilled non-manual workers, intermediate	182	68	37
		professions
		Low-skilled and
	"Precarious" quarters	unskilled workers,	84	27	32
		unemployed
	Gentrified quarters	Young highly qualified workers	174	51	29
	Prosperous quarters on the fringes of the wealthy quarters	Middle-aged people, professions managerial	143	32	22
	Residual working-class quarters	Elderly and retired skilled people, medium-	92	18	20
	Working-class quarters made up of social housing	Low-skilled and unskilled workers	61	11	18
	Working-class quarters with ageing population	Unskilled, populations unemployed, low paid	34	5	15
		Highly qualified
	Traditionally wealthy quarters	professionals, upper class, middle aged	149	10	7
		and elderly people

Table 2 . The "green quarters" in their socio-demographic context (source INSEE RGP 1999, Mairie de Paris/Direction de la voirie ; based on Fleury, 2007, pp. 305-306)

 2