Pauline Moreau

Isabelle Duhamel-Achin

Blandine Gourcerol

Philippe Lach

Catherine Lerouge

Nicolas Maubec

Philippe Négrel

Guillaume Wille

ION4RAW: Improving metal recovery in Cu-Pb-Zn-(Au-Ag) ore deposits through inventory of by-products and critical raw materials

ION4RAW: Improving metal recovery in Cu-Pb-Zn-(Au-Ag) ore deposits through inventory of by-products and critical raw materials Moreau P., Duhamel-Achin I., Gourcerol B., Lach P., Lerouge C., Maubec N., Negrel Ph., Wille G. Long-term management of the mineral resource supply incorporating anthropogenic environmental impacts is crucial for sustaining human society. This is especially true for recovery of by-products and critical raw materials (CRM) whose production is often unable to respond quickly to rapid changes in consumption trends. As part of European H2020 research and innovation, the ION4RAW project aims at obtaining reliable estimates of by-products and CRM, and at developing ionometallurgy processes to improve their extraction from primary resources. Targeted metals are by-products (Te, Se, Re and Mo) and CRM (Bi, Ge, In, Co, Pt, Sb) in 5 selected Cu-Ag-Au ore deposits through the world (Cononish Gold mine, Scotland; Cobre Las Cruces and El Valle Boinas, Spain; El Porvenir and Cerro Lindo, Peru). The final objective of this study is to determine the carrier minerals of CRM and by-products, the variability of their chemistry, their distribution and quantification, in order to improve their recovery during ore treatment processes. We currently inventory byproducts and CRM by characterizing ores and gangue, using a multi-technical approach (bulk chemistry and X-Ray diffraction, optical and scanning electron microscopic observations, µX-ray fluorescence mapping, EPMA spot analyses and laser ablation-ICP-MS). We present here the preliminary results of El Porvenir and El Valle Boinas that are two calcic skarn-related deposits defined by their garnet composition.

El Porvenir (Peru), owned by Nexa Resources, is a Pb-Zn ore deposit associated with andradite-bearing skarn, exploited in an underground mine located in the Western Cordillera of the Andes mountain range in central Peru. El Valle-Boinas (Spain), owned by Orvana Minerals Corp, is a Cu-Au ore deposit associated with a grossular-bearing skarn, exploited in an underground mine located in Cantabrian Mountains, 60 km southwest from Aviles in Spain.

Mineralogical investigations indicate that the major ore consists of chalcopyrite, galena, sphalerite, pyrite with minor pyrrhotite, tennantite-tetrahedrite-series minerals and tellurides. The electron microprobe allows analyzing micron-sized metal-carrier minerals, including electrum, Bi-Pb sulfosalts, hessite [Ag2Te], stannoidite, determining the composition of tennantite-tetrahedrite-series minerals (argentotennantite containing up to 12 wt% Ag and 5 wt% Bi) and detecting traces in major ore at a detection limits of 200-1000 ppm (for example, galena significantly contains Ag, Sb and Te). The laser ablation-ICP-MS was tested at maximum power of the laser and at different beam diameters adapted to the grain sizes (from 85 to 10 µm). Laser ablation-ICP-MS analyses with a beam diameter of 10 µm confirms EPMA data and allows detecting lower metal contents, such as Se, Rh, Pd, In, Te, in main ore minerals at detection limits of the ppm.


Christophe Goupil

Henni Ouerdane

Arnold Khamsing

Yann Apertet

François Bouteau

Stefano Mancuso

Rodrigo Patiño

Philippe Lecoeur

Green thermoelectrics: Observation and analysis of plant thermoelectric response

Plants are sensitive to thermal and electrical effects; yet the coupling of both, known as thermoelectricity, and its quantitative measurement in vegetal systems never were reported. We recorded the thermoelectric response of bean sprouts under various thermal conditions and stress. The obtained experimental data unambiguously demonstrate that a temperature difference between the roots and the leaves of a bean sprout induces a thermoelectric voltage between these two points. Basing our analysis of the data on the force-flux formalism of linear response theory, we found that the strength of the vegetal equivalent to the thermoelectric coupling is one order of magnitude larger than that in the best thermoelectric materials. Experimental data also show the importance of the thermal stress variation rate in the plant's electrophysiological response. Therefore, thermoelectric effects are sufficiently important to partake in the complex and intertwined processes of energy and matter transport within plants.

I. INTRODUCTION

Plants are complex biological systems, which feed, develop and function thanks to a variety of elementary and cooperative processes that ensure transport of energy and matter in the form of mineral elements, carbohydrates, and hormones in watery solutions called sap. Two categories of sap are transported in conductive tissues: xylem sap transported in xylem, a continuum of dead cells forming a pipe that opposes little resistance to the transport of water and mineral elements from roots to leaves; and phloem sap, containing water and sugar, formed in the leaves by photosynthesis, and transported in phloem (sieve tube elements) from the leaves to the roots. Explanations for xylem sap transport includes the cohesion-tension theory 1,2 and multiforce theories; [3][4][5] and the pressure flow hypothesis proposes a mechanism for phloem sap transport. 6 Notwithstanding their relative merits, it is of interest to note that as saps contain ions, a net voltage may be recorded within plants. Furthermore, as a living plant is naturally submitted to a temperature gradient between its roots and its leaves, heat may also flow from its hotter to its colder regions. These two facts lead to the questions of the existence and strength of a coupling between electrical current and heat flux in plants, and whether this coupling plays a significant role in the plants' lives. Answers to these questions are of importance to gain further insight into plant homeostasis, closely related to the Le Chatelier-Braun principle in chemical thermodynamics, and to develop further the field of plant electrophysiology by analysing the im-pact of thermoelectric coupling on photosynthesis, plant respiration, and any other processes involving bioelectrochemical phenomena. [START_REF] Volkov | Plant Electrophysiology: Theory and Methods[END_REF] Biophysical processes, which include energy and matter transport in dedicated structures, are essentially nonequilibrium in nature. Their description involves the notions of thermodynamic forces and their conjugate fluxes which, in a linear response description, are proportional to each other. In a general manner, transport phenomena are nonequilibrium irreversible processes: 8,9 fluxes within a system result from applied external constraints, and they are accompanied by energy dissipation and entropy production. As the forces applied to a system derive from potentials, the description of the system's nonequilibrium properties relies on the definition of these potentials and their degree of coupling. [START_REF] Kedem | [END_REF] Thermoelectricity is sometimes mistakenly viewed as a phenomenon pertaining to solid-state systems, [START_REF] Rowe | CRC Handbook of Thermoelectrics[END_REF][START_REF] Koumoto | Thermoelectric Nanomaterials[END_REF] but this effect has also been reported in liquids [START_REF] Liu | Copper ion liquid-like thermoelectrics[END_REF] and gels. 14 Now, considering more specifically electrical phenomena in plants, one should also account for their dependence on temperature and the local variations of this latter. As a matter of fact, one learns from the equilibrium thermodynamics of charged fluids that heat and electricity are fundamentally related through the Gibbs-Duhem relation; so, since plants, as any other system, are subjected to the laws of thermodynamics, one may anticipate that a proper understanding of the coupled transport of heat and electricity in plants will pave the way towards the potentially rich new field of thermoelectricity-based plant energy conversion, and harvesting metabolic energy from plants. [START_REF]Voltree Power[END_REF] Thermoelectricity as part of non-isothermal processes in living organisms, has already been mentioned; but except for the case of insects like hornets and bees, [START_REF] Ishay | [END_REF][17][18] for which genuine thermoeletric effects have been analysed, the other studies simply reported the record of the temperature rise of Colocasia odora leaves using a thermocouple, [START_REF] Van Beek | Observations thermoélectriques sur l'élévation de température des fleurs de Colocasia odora[END_REF] or analysed pyroelectricity, mistakenly taken as a thermoelectricity, amongst the nonequilibrium processes at the origin of life. [START_REF] Muller | [END_REF] We propose here to consider bean sprouts as an illustrative case of a living plant system whose physiological properties also entail thermoelectric effects, with a particular focus on the thermovoltage response of the plant. While a voltage difference along the stem of a plant has already been measured and analysed, 21 its interpretation does not account for a likely the thermoelectric origin. Nevertheless, it is clear that the source of any voltage difference is due to concentration gradients of charges inside the plant, regardless of the cause of the gradients. The underlying processes may thus originate in various electrochemical processes, including acid-base and redox effects. 21 In the present work we focus on the voltage response due to a temperature difference. The system is presented in Fig. 1. The stem and leaves are kept at constant temperature while the temperature of the roots can be modified; the thermoelectric response is recorded on both the plant (a) and a control wire (b).

II. THEORETICAL CONSIDERATIONS

Thermoelectricity differs from electrochemistry as the latter considers isothermal systems only. Thermoelectric effects, on the contrary, manifest themselves in nonisothermal conditions and their thermodynamic description is best done with the coupled variables (µ, T ), where µ is the electrochemical potential and T the temperature of the considered system. In a force-flux approach, as a temperature bias is applied to a conductor, both heat and charges are transported in coupled flows since each electron carries an electric charge, e, and energy; the thermoelectric coupling parameter or Seebeck coefficient α is thus defined as:

α -→ ∇T = - 1 e -→ ∇µ (1)
where -→ ∇ denotes the spatial gradient. Equation ( 1) may be viewed as a generalization of the Gibbs-Duhem relation to the dynamical response of the coupled intensive variables (µ, T ) by defining the thermodynamic forces acting on the system as the gradients of the thermodynamic potentials. Each force is conjugated to a flux which is proportional to the time derivative of the corresponding extensive parameter, and it follows that forces and fluxes are linearly combined as proposed by Onsager. 23 Note that the coefficient α, which in Eq. ( 1) is defined locally, is also often expressed as α ≈ -e∆V /∆T , where ∆ denotes the global difference between two distant points of the system.

These general considerations apply for any system containing free charge carriers. As a consequence, for a given temperature difference, the measurement of a thermovoltage directly gives an estimation of the average ratio of charge concentrations in separate parts of the system. Therefore, any system containing free mobile carriers may be exhibit a thermoelectric signature. One should note that depending on the specific properties of the considered system, this signature may intertwine or not with those of other temperature-dependent processes; hence the need for specific system-dependent approaches to extract the thermoelectric signal. In the case of liquids and gels, the studies usually focus on the thermoelectrophoresis parameters. [START_REF] Liu | Copper ion liquid-like thermoelectrics[END_REF]14,24 For solid state-systems, the conduction is ensured by mobile electrons and holes, leading to a possible steady-state electrical current. For ionic carriers, in liquids and gels, such a steady-state current may only take place if a redox process occurs at the ends of the system.

III. EXPERIMENT

The experiment aims to identify and characterize the thermoelectric response of a living plant (a bean sprout in the present work). As the response of the plant necessarily entails physiologic effects, the measured data is compared to the purely thermoelectric response of a control wire. As depicted on Fig. 1, the temperature of the plant's roots is imposed by a thermostatic bath; two thermocouples are used to measure the temperatures of the roots and the leaves, and two electrodes are placed respectively in contact with the roots and the leaves of the living plant. The roots are bathed in an aqueous dilute KCl solution. The electrodes connected to the leaves are made of Ag/AgCl wires. All the temperatures and voltages are recorded using a Keithley K2700 scanning multimeter and a K182 nanovoltmeter. The thermoelectric control wire (CW) is made of two chromium-nickel-steel junctions, with a Seebeck coefficient α junction = 4.6 µV/K; it is placed in the same configuration as that of the plant. The recorded voltage at the ends of the control wire describes a pure thermoelectric response, exempt from any influence of the physiologic process. Response

∆T (K) V cw V plant V cw (x 2000) V plant FIG. 2.
Experimental data including the voltage response of the control wire Vcw (upper left panel) and of the plant V plant (upper right panel), both submitted to a time-varying temperature difference ∆T , reported on the lower panel. The magnitude of measured voltage Vcw has been multiplied by a factor 2000 in order to report it on the same scale as that of V plant . The red-dotted lines indicate the average slope of the collected data. In the lower panel, the scale is that of the temperature difference; the voltages are also depicted in this lower panel to see how close their time-dependent behaviour is to that of the measured temperature difference.

The voltage response of the living plant and that of the control wire to the temperature difference ∆T are both shown on Fig. 2, where the time evolution of ∆T is also displayed, illustrating the dynamics of the plant and that of the control wire. These data were recorded over a period of four hours for various temperature differences and variation rates. The two upper panels show separately the parametric plots characterising the thermoelectric response of the control wire and that of the plant. Notice that the response of the control wire is strictly proportional to the temperature difference, as expected for a thermoelectric material, with a passive See-beck coefficient. Conversely, the plant exhibits a complex response, which includes the effects of its physiological processes thus giving rise to an active Seebeck response. The curves represented on the lower panel are displayed together on the same scale to qualitatively show how the responses (voltages V cw and V plant ) follow the applied constraint (temperature difference ∆T ). The figures on the y-axis correspond to ∆T in • C. The values taken by the voltages V cw and V plant may be determined from the curves using a scaling factor that corresponds to the relevant Seebeck coefficient. The control wire shows a faithful homothetic response to the temperature difference, with a linear factor of scaling of 4.6 µV/K as expected for the Seebeck coefficient of the corresponding chromium-nickel-steel thermocouple. The response of the plant gives an average Seebeck coefficient of 2.5 mV/K. Such a large value in a solid-state material would correspond to a system with low electrical conductivity. Here, this results from ion movements through xylem and phloem saps circulating in opposite directions.

To analyze precisely these data, it is convenient to split each panel of Fig. 2 into three parts, which correspond to three distinct time domains. The first and third ones, Figs. 3 and5, respectively, describes the plant's response when its roots are heated; while the second, Fig. 4, reports the response over the cooling sequence.

Response Depending on the precise thermal condition supported by the plant, we can identify different thermoelectric responses. Consider the first warming sequence of Fig. 3. One may identify five events (labelled from a to e). During the first period (Fig. 3a), the temperature difference is roughly constant. The plant presents a large fluctuating thermoelectric response while the control wire is characterized by a constant Seebeck coefficient. During the second period, shown on Fig. 3b, the plant's reaction is essentially sensitive to the temperature variation rate dT /dt. This sensitivity has already been observed in studies devoted to plant sensing. 25,26 The processes taking place within the zones a and b can be also observed in the c and d zones; and when the rate dT /dt is reduced, we observe a linear response for the plant, as shown in the zone e. It thus appears clearly that the passive response, probably mostly due to the xylem properties, can be strongly modified by active physiological processes. This latter appears to be particularly sensitive to the time derivative of the stress, but not so much on the intensity of the stress itself; this observation is perfectly consistent with previous findings. 25,26 We now turn to the cooling sequence, Fig. 4. The cooling rate is approximately constant, except for the two events when it is clearly larger than the average as indicated with the labels a and b, on the upper right panel. These events relate to time delays, also apparent in the measured data for the control wire on the upper left panel: these are signatures of the change of rate dT /dt. Note that although these may serve as a probe, time delays must be considered with care, since in the case of very large time response of a system, they may lead to erroneous interpretations. 14 The shape of the plant response for events taking place in the zones a and b is very similar to that of Figs. 3b and3d, which confirms the acute sensitivity of the plant to the rate dT /dt. This physiological response is the fastest response to a stress that a plant can use, as it has no means to avoid it.

∆T (K) V cw V plant V cw (x 2000) V plant
The last sequence depicted on Fig. 5 concerns the plant's response to warming at a reduced rate. It is essentially linear, thus confirming the presence of a threshold rate process for the plant's response to a given stress. It also shows unambiguously that thermoelectricity naturally partakes in the complex processes that contribute to a plant's life.

IV. DISCUSSION AND CONCLUDING REMARKS

All the effects discussed in the present work are related to the internal modifications of the ionic concentrations inside the plant, including the regulation of cellular ion transporters allowing ion release into the xylem sap toward the shoots. It is clear that while the response is mainly driven by a classical thermoelectric process with a giant Seebeck coefficient, the physiology of the plant interferes in a complex way with it. A complete numerical model, including series and parallel assemblies of living (phloem sieve tube elements, root and endodermis cells) and dead cells (xylem tracheids and vessels), will be considered for further investigation of the interplay between physiologic and thermoelectric response to a thermal stress. Further, since there is no limitation in the number of thermodynamic potentials which may be coupled, 23 we may account for pressure and derive a complete response of a system to the direct and the crosseffects of the three thermodynamic potentials, (P, µ, T ). Then the lowering of a local pressure gradient, by the coupled effect of the temperature and the electrochemical potential becomes possible. This latter process may contribute to the motion of saps in plant, and especially in trees.

The reported measurements are fully scalable from the unique cell size to a complete plant or tree. In addition to the stationary response observed here, the voltage fluctuations are also of great interest since they permit the study of the threshold of the physiological response through its noisy signals. As observed in other systems, the noise response is a very sensitive probe of the emergence of a macroscopic response. 27,28 Taking each cell as a fluctuator, the convolution of the individual responses may lead to different signatures depending on the considered scale of the sample.

FIG. 1 .

 1 FIG. 1. General view of the experimental setup. The voltage records are respectively on the plant (a), and the control wire (b).

FIG. 3 .

 3 FIG.3. First heating sequence. The plant's response is strongly influenced by the temperature variation rate dT /dt.

FIG. 4 .FIG. 5 .

 45 FIG.4. Cooling sequence. The response is roughly linear but visible discrepancies are due to the plant's physiological processes.

Appendix A: Plant material Seeds of bush bean (Phaseolus vulgaris L. cv. contender) were sown into pots containing vermiculite as soil. The bean seedlings were grown in a growth chamber at 25 ± 2 • C with a cycle of 12 hours of light (40 µmol photon m -2 • s -1 ), 12 hours of darkness. The plants were watered at the bottom of the pots every three days. The 21 days old plants were removed from the vermiculite. Roots were carefully rinsed with water to remove vermi-culite particles and then transferred in dilute KCl solution (120 mM) for experiments.

Appendix B: Measurement electrodes

Two electrodes were placed respectively at the surface of the root and the stem of the living plant. The electrodes connected to the stems are made of Ag/AgCl wires moistened with 200 mM KCl and wrapped in cellophane to provide appropriate contact with the plant surface.


A Chambinaud

M Cordier

A Reigue

Y Asselah

G Bonamis

C Hoenninger

B Debord

F Gérôme

F Benabid
email: f.benabid@xlim.fr

High brightness single photon source at the telecom wavelength based on inhibited-coupling hollow-core fiber

We report on a Raman-free heralded single-photon source based on xenon-filled hollowcore fiber pumped with GHz-repetition rate femtosecond laser. The measured brightness of 720 kHz is one order of magnitude improvement over our previous demonstration.

Introduction

The emerging second quantum revolution entails the development of several key quantum and photonic components and devices with stringent specification requirements [START_REF] Flamini | Photonic quantum information processing: a review[END_REF]. For example, single photon sources, which are required in quantum information, must combine high signal-to-noise ratio, high brightness, high purity and integrability in the telecommunications network. A combination that is yet to be achieved. Recently Cordier et al. [START_REF] Cordier | Raman-free fibered photon-pair source[END_REF] demonstrated that the generation of photon-pairs by spontaneous four-wave mixing (SFWM) in inhibited-coupling hollow-core photonic crystal fiber (IC-HCPCF) filled with an inert gas is a very promising platform for the generation of both noise-free pure single photon and entangled photon-pairs. This configuration stands out with three enabling features: i) the IC-HCPCF can be designed and fabricated in such a way that the dispersion meets very strict phase matching conditions to generate the photon-pairs with tailored spectral correlations, ii) the fiber can be designed to operate at wide ranges of wavelengths, as illustrated in [START_REF] Cordier | Raman-free fibered photon-pair source[END_REF][START_REF] Cordier | Active engineering of four-wave mixing spectral entanglement in hollow-core fibers[END_REF] whereby the idler wavelength lies in the telecom wavelength range (~ 1550 nm), and the signal wavelength is in the range of atomic transitions and Silicon single photon detectors (~ 780 nm), and iii) the gas-filled HCF does not suffer from the Raman scattering effect, which has hitherto been an obstacle in the development of a noise-free fiber-based single-photon sources. However, despite all these promising results, the maximum brightness of the source was found to be 50 kHz, mainly limited by the time regime of the pump beam used. Here, we report on a SFWM based heralded single photon at 1550 nm pumped with GHz-repetition rate femtosecond laser, with the brightness corresponding to a photon-pair rate of 720 kHz, and a maximum signal-to-noise ratio given by the coincidence-to-accidentals ratio (CAR) of 7400, resulting in a new state-of-the-art of fiber photon-pairs sources.

Experimental set-up and quantum state preparation

Figure 1 summarizes the set-up used in the present work and the photon-pair correlation properties obtained [START_REF] Cordier | Active engineering of four-wave mixing spectral entanglement in hollow-core fibers[END_REF]. Here, we use a femtosecond laser source delivering light pulses of 16 W maximum average power at 1033 nm with a duration of 280 fs and a repetition rate of 1 GHz. The generated light pulses are then coupled to a 93 cm long gas-filled IC-HCF. The fiber used is a IC-HCPCF with tubular cladding lattice, a core diameter of 44 µm and a silica thickness of 628 nm. The fiber was designed so the SFWM induced photons are emitted near 1550 nm and 780 nm (Fig. 1b). In order to set the SFWM photon-pair in the desired separable quantum state, the gas pressure is varied for fine dispersion control. This was demonstrated by the reconstructed JSI using stimulated emission tomography technique as in [START_REF] Grice | Eliminating frequency and space-time correlations in multiphoton states[END_REF], with the seed laser being a continuous-wave laser tunable in the telecom range. The average power of the pump and the seed is then set respectively to 5.8 W and 10 mW. The results of this quantum state preparation show a JSI with a Schmidt number K of 1.049, corresponding to a high purity factorable pair states of 95% (Fig. 1c). This state was achieved with a gas pressure of 4.3 bar, a signal wavelength of 769 nm and an idler wavelength of 1570 nm.

Heralded single photon source

The resulted separable photon-pairs are then implemented to demonstrate a high brightness heralded single photon source. Here, the set-up described in Fig. 1a is modified by blocking the seed laser (spontaneous scheme) and adding Si and InGaAs single-photon detectors at the fiber output. The signal and idler photons are filtered and detected individually using standard coincidence correlation scheme (see the inset of Fig. 2a). Figure 2 shows the occurrences obtained for both photons and the associated CAR as a function of the average pump power ranging between 0 to 10 W. The evolution with a quadratic polynomial (∝ P 2 ) demonstrates photon-pair generation with negligible Raman Scattering as expected. This is further confirmed by the measured CAR reaching a record for any fiber architecture of 7400 at 1 W over 250 mW. Finally, the photon-pair rate was found to a maximum of 720 kHz, value corresponding to an improvement of more than one order of magnitude compared to our previous work based on a 2 MHz pump laser. This figure is in good agreement with the theoretical calculations which predicts an enhancement factor of 16 for the laser powers considered (100 mW at 2 MHz and 10 W at 1 GHz). This improvement of the brightness will be a strong advantage for future experiments, especially for correlations measurements which will experience a time reduction factor of two orders of magnitude.

Conclusion

To conclude, by pumping a Xe-filled IC-HCF with a GHz-repetition rate femtosecond laser, we improved the current state-of-the-art in signal-to-noise and brightness of fiber photon-pairs sources. This result is a new step to configure our platform as an entangled pair generator to demonstrate quantum key distribution.

Fig. 1 .

 1 Fig. 1. (a) Experimental set-up (b) Loss and phase matching curves of the Xe-filled IC-HCF. (c) Measured and simulated JSI of the source associated with the Schmidt decomposition (assuming a flat phase).

Fig. 2 .

 2 Fig. 2. (a) CAR and brightness of the heralded single photon source. In inset, the detection protocol is added. (b) Signal and idler counts as a function of the average pump power. The measured coincidences and accidentals are also plotted.

The authors acknowledge support from la Région Nouvelle Aquitaine.


HAFEDH Maxime Clenet

E L Ferchichi

Jamal Najim

EQUILIBRIUM IN A LARGE LOTKA-VOLTERRA SYSTEM WITH PAIRWISE CORRELATED INTERACTIONS

Keywords: May 30, 2022. 2010 Mathematics Subject Classification. Primary 15B52, 60G70, Secondary 60B20, 92D40 Linear systems, large random matrices, Gaussian concentration, Lotka-Volterra equations

Consider a Lotka-Volterra (LV) system of coupled differential equations: 9

x k " x k pr k ´xk `pBxq k q , x " px k q , 1 ď k ď n , where r " pr k q is a n ˆ1 vector and B a n ˆn matrix. Assume that the interaction matrix B is random and follows the elliptic model:

where A " pA ij q is a n ˆn matrix with N p0, 1q entries satisfying the following dependence structure piq the entries A ij on and above the diagonal are i.i.d., piiq for i ă j each vector pA ij , A ji q is standard gaussian with covariance ρ, and independent from the other entries; vector 1n stands for the n ˆ1 vector of ones. Parameters α, µ are deterministic and may depend on n.

Leveraging on Random Matrix Theory, we analyse this LV system as n Ñ 8 and study the existence of a positive equilibrium. This question boils down to study the existence of a (componentwise) positive solution to the linear equation:

xn " rn `Bnxn , depending on B's parameters pα, µ, ρq, a problem of independent interest in linear algebra.

In the case where no positive equilibrium exists, we provide sufficient conditions for the existence of a unique stable equilibrium (with vanishing components), and following Bunin [9], present a heuristics estimating the number of positive components of the equilibrium and their distribution.

.

Such LV systems are widely used in mathematical biology to model populations with interactions, and the existence of a positive equilibrium known as a feasible equilibrium corresponds to the survival of all the species x k within the system.

Introduction

Lotka-Volterra system of coupled differential equations. Lotka-Volterra (LV) systems are widely used in mathematical biology, ecology, chemistry to model populations with interactions or chemical reactions [START_REF] Gopalsamy | Global asymptotic stability in volterra's population systems[END_REF][START_REF] Hofbauer | Evolutionary games and population dynamics[END_REF][START_REF] Kiss | Qualitative behavior of n-dimensional ratio-dependent predator-prey systems[END_REF][START_REF] Hering | Oscillations in lotka-volterra systems of chemical reactions[END_REF]. In the context of theoretical ecology (that we shall adopt hereafter without loss of generality), consider a given foodweb and denote by x t n " px k ptqq 1ďkďn the vector of abundances 1 of the various species at time t ě 0. In a LV system, the abundances are connected via the following coupled equations:

dx k ptq dt " x k ptq ˜rk ´xk ptq `n ÿ "1
B k x ptq ¸for k P rns :" t1, ¨¨¨, nu , where B n " pB k q stands for the interaction matrix, and r k stands for the intrinsic growth of species k. Notice that standard results yield that if the initial condition x 0 n " x n | t"0 is componentwise positive, then x t n remains positive for every t ą 0. At the equilibrium dxn dt " 0, the abundance vector x n " px k q kPrns is solution of the system:

x k ¨rk ´xk `ÿ Prns B k x '" 0 for x k ě 0 and k P rns .

An important question, which motivated recent developments [START_REF] Dougoud | The feasibility of equilibria in large ecosystems: A primary but neglected concept in the complexity-stability debate[END_REF][START_REF] Bizeul | Positive solutions for large random linear systems[END_REF], is the existence of a feasible solution x n to [START_REF] Akjouj | Feasibility of sparse large lotka-volterra ecosystems[END_REF], that is a solution where all the x k 's are positive, corresponding to a scenario where no species disappears. Notice that in this latter case, the system (1) takes the much simpler form:

x n " r n `Bn x n , r n " pr k q kPrns .

In this article, we will investigate the existence of an equilibrium, potentially feasible, for a large foodweb (n Ñ 8) whenever the interaction matrix B n is random.

In various models of interest for B n , Random Matrix Theory (RMT) provides an accurate description of the asymptotic properties of a large random matrix (its spectrum, spectral norm, etc.). We will leverage on RMT to infer the existence of an equilibrium in the case where B n follows a random elliptic model, to be described hereafter.

To simplify the analysis, we will consider the case where r n " 1 n .

Random elliptic model for the interaction matrix. In the spirit of May 2 , we model the interaction matrix B n as a non-centered random matrix with pairwise correlated entries:

B n " A n α n ? n `µ n 1 n 1 T n , (2)
where A n " pA ij q i,jPrns is a random matrix satisfying the two conditions piq pA ij , i ď jq are standard Gaussian N p0, 1q independent and identically distributed (i.i.d.) random variables piiq for i ă j the vector pA ij , A ji q is a standard bivariate Gaussian vector, independent from the remaining random variables, with covariance covpA ij , A ji q " ρ with |ρ| ď 1. The sequence of positive numbers pα n q is either fixed or goes to infinity. Parameter µ is a fixed real number. As a consequence, the Gaussian entries of the interaction matrix B n admit the following moments:

EpB ij q " µ n , varpB ij q " 1 α 2 n , covpB ij , B ji q " ρ α 2 n pi ‰ jq .
1 A species abundance is a quantity proportional to the number of individuals for this species.

2 Beware that May did not consider LV systems but rather used a random matrix model for the Jacobian at equilibrium of a generic system of coupled differential equations.

Such a matrix model is often called a random elliptic model for |ρ| ă 1 since the spectrum of matrix A n { ? n is asymptotically an ellipse, see Fig. 1, in the sense that the empirical distribution of the eigenvalues of A n { ? n converges towards the uniform distribution on the ellipsoid

E ρ " " z P C, Re 2 pzq p1 `ρq 2 `Im 2 pzq p1 ´ρq 2 ď 1 * .
Originally introduced by Girko [START_REF] Girko | Elliptic law[END_REF], this model has since been widely studied [START_REF] Girko | The elliptic law: ten years later i[END_REF][START_REF] Naumov | Elliptic law for real random matrices[END_REF][START_REF] Nguyen | The elliptic law[END_REF][START_REF] O'rourke | Low rank perturbations of large elliptic random matrices[END_REF]. The spectral norms of A n and

1 n 1 T n satisfy › › › › A n ? n › › › › " O p1q and › › › › 1 n 1 n 1 T n › › ›
› " 1 hence both the random and deterministic parts of the interaction matrix B n may have an impact as n Ñ 8.

Presentation of the main results. In this article, we address the following issues.

Feasibility. We first describe the conditions over parameters pρ, α n , µq for which system (1) admits a unique feasible equilibrium. We prove that feasibility is reached whenever α n " a 2 logpnq and µ ă 1, and that there is no feasibility otherwise, see Theorem 2.1. Notice that the correlation parameter ρ has no influence since the phase transition threshold is the same as in the i.i.d. case [START_REF] Bizeul | Positive solutions for large random linear systems[END_REF]: the induced correlations between components x k 's of solution x n are too weak. Pushing this remark further, we prove that the same phase transition holds if we consider a covariance profile pρ ij , i ă jq where ρ ij " covpA ij , A ji q instead of a fixed covariance parameter ρ.

In [START_REF] Bizeul | Positive solutions for large random linear systems[END_REF], Bizeul and Najim established the conditions for feasibility in the centered (µ " 0) model with i.i.d interactions pA ij q. In [START_REF] Akjouj | Feasibility of sparse large lotka-volterra ecosystems[END_REF], Akjouj and Najim studied a sparse model of interactions where only d n ě logpnq interactions are non-null in each row and column of A n . The study of the feasibility for an elliptic model completes this picture.

Stability without feasibility. If α is fixed, Dougoud et al. [START_REF] Dougoud | The feasibility of equilibria in large ecosystems: A primary but neglected concept in the complexity-stability debate[END_REF] showed that no feasible solution can arise. Under this assumption, we establish in Proposition 2.3 sufficient conditions for the existence of a unique stable equilibrium to system [START_REF] Akjouj | Feasibility of sparse large lotka-volterra ecosystems[END_REF]. In this case, some species will vanish (some of the components x k 's of solution x n are equal to zero). In order to proceed we combine results by Takeuchi [START_REF] Takeuchi | Global dynamical properties of Lotka-Volterra systems[END_REF] on stability of LV systems with Random Matrix Theory (RMT) results.

Estimating the number of surviving species. We finally conclude with an important question: given a set of parameters pρ, α, µq which yields to a unique stable equilibrium, is it possible to estimate the proportion of surviving species? From a mathematical point of view, this is an open question. At a physical level of rigor, Bunin [START_REF] Bunin | Ecological communities with lotka-volterra dynamics[END_REF] (relying on the cavity method) and Galla [START_REF] Galla | Dynamically evolved community size and stability of random Lotka-Volterra ecosystems[END_REF] (relying on generating functionals techniques) provide a closed-form system of equations to compute the proportion of surviving species. We state the open problem, recall Bunin's and Galla's equations and provide simulations.

In [START_REF] Clenet | Equilibrium and surviving species in a large lotka-volterra system of differential equations[END_REF], equations and simulations are provided in the simpler case where ρ " 0, together with heuristics supporting these equations.

Organisation of the article. Feasibility and stability results together with the open question on the estimation of the number of surviving species are presented in Section 2. Section 3 is devoted to the proof of the feasibility result, Theorem 2.1. Proof of the stability result, Proposition 2.3, is provided in Section 4. Simulations were performed in Python. All the figures and the code are available on Github [START_REF] Clenet | Feasibility in a large lotka-volterra system with pairwise correlated interactions[END_REF].

Notations. If A is a matrix A T stands for its transpose. We denote by logpxq the natural logarithm. If x " px i q iPrns is a vector, we denote by x ą 0 (resp. x ě 0) the componentwise positivity (resp. non-negativity), that is the fact that x i ą 0 (resp. x i ě 0) for every i P rns.

2.

Main results: Feasibility, stability and surviving species 2.1. Feasibility. To simplify the analysis, we consider the case where r k " 1 pk P rnsq. Hence, the LV system takes the following form in the sequel:

dx k ptq dt " x k ptq ¨1 ´xk ptq `ÿ Prns B k x ptq ' for k P rns . (3)
In the next theorem, we describe the conditions to reach a feasible equilibrium. We either assume that matrix B is given by the elliptic model or has a more general covariance profile.

Theorem 2.1 (Feasibility for the elliptic model). Assume that matrix B n is given by the elliptic model (2), or that B n has a covariance profile, i.e.

B n "

Ãn

α n ? n `µ n 1 n 1 T n , (4)
where Ãn is a n ˆn matrix with entries p Ãij , i ď jq i.i.d. N p0, 1q and where p Ãij , Ãji q is a standard bivariate gaussian vector for i ă j, independent from the remaining random variables, with covariance covp Ãij , Ãji q " ρ pnq ij , where pρ pnq ij ; i ă j; n ě 1q is a collection of deterministic real numbers in r´1, 1s.

Let α n ÝÝÝÑ nÑ8 8 and denote by α n " ? 2 log n. If µ ‰ 1 then the following equation

x n " 1 n `Bn x n almost surely admits a unique solution x n " px k q kPrns .

(1) (feasibility) If µ ă 1 and there exists ε ą 0 such that, for n large enough, α n ě p1 `εqα n then

P " min kPrns x k ą 0 * ÝÝÝÑ nÑ8 1 .
( Proof of Theorem 2.1 is established in Section 3 under the assumption that B n follows the elliptic model. The adaptations needed to cover the covariance profile case are provided in Appendix A.

2.2.

No feasibility but a unique stable equilibrium. Aside from the question of feasibility arises the question of stability: for a complex system, how likely a perturbation of the solution x n at equilibrium will return to the equilibrium? Gardner and Ashby [START_REF] Gardner | Connectance of large dynamic (cybernetic) systems: critical values for stability[END_REF] considered stability issues of complex systems connected at random. Based on the circular law for large random matrices with i.i.d. entries, May [START_REF] May | Will a large complex system be stable?[END_REF] provided a complexity/stability criterion and motivated the systematic use of large random matrix theory in the study of foodwebs, see for instance Allesina et al. [START_REF] Allesina | The stability-complexity relationship at age 40: a random matrix perspective[END_REF]. Recently, Stone [START_REF] Stone | The feasibility and stability of large complex biological networks: a random matrix approach[END_REF] and Gibbs et al. [START_REF] Gibbs | Effect of population abundances on the stability of large random ecosystems[END_REF] revisited the relation between feasibility and stability.

For a generic LV system

d y k ptq dt " y k pr k `pCyq k q , k P rns , (5)
Takeuchi and Adachi provide a criterion for the existence of a unique equilibrium y ˚and the global stability of LV systems, see Theorem 3.2.1 in [START_REF] Takeuchi | Global dynamical properties of Lotka-Volterra systems[END_REF].

Theorem 2.2 (Takeuchi and Adachi 1980). If there exists a positive diagonal matrix ∆ such that ∆C `CT ∆ is negative definite, there is a unique non-negative equilibrium y ˚to [START_REF] Bai | Spectral analysis of large dimensional random matrices[END_REF], which is globally stable:

@y 0 ą 0 , # yp0q " y 0 yptq satisfies (5) , yptq Ý ÝÝ Ñ tÑ8 y ˚.
Combining this result (setting I ´B " ´C) with results from Random Matrix Theory, we can guarantee the existence of a globally stable equilibrium x ˚of (3) for a wide range of parameters pρ, α, µq. Denote by

A " " pρ, α, µq P p´1, 1q ˆp0, 8q ˆR , α ą a 2p1 `ρq, µ ă 1 2 `1 2 c 1 ´2p1 `ρq α 2 * ( 6
)
the set of admissible parameters. is eventually positive definite: with probability one, for a given realization ω, there exists N pωq such that for n ě N pωq, pI ´Bω q `pI ´Bω q T is positive definite. In particular, there exists a unique globally stable non-negative equilibrium x ˚.

Proof of Proposition 2.3 is provided in Section 4. For a fixed matrix of interactions B 10 with parameters pρ " 0, α " 2, µ " 0q P A, we consider two distinct initial conditions. Simulations show that the abundances converge in both cases toward the unique globally stable equilibrium x ˚predicted by Proposition 2.3. Notice that since α ă a 2 logp10q » 2.14, we witness vanishing species.

2.3.

Estimating the number of surviving species: Towards Bunin and Galla's equations. After giving conditions for the realization of a feasible equilibrium and investigating the existence and uniqueness of a stable sub-population (i.e some species vanish), we address the question of estimating the proportion of surviving species as a function of the model paramaters pρ, α, µq.

To our knowledge, this question has not received yet an answer at a mathematical level of rigor and remains open. However theoretical physicists/ecologists provided a solution to this problem supported by simulations. Tools from physics to study population dynamics in the context of Lotka-Volterra equations were first introduced by Opper et al. [START_REF] Diederich | Replicators with random interactions: A solvable model[END_REF][START_REF] Opper | Phase transition and 1/f noise in a game dynamical model[END_REF]. In 2017, Bunin [START_REF] Bunin | Ecological communities with lotka-volterra dynamics[END_REF] precisely answers the question of estimating the proportion of surviving species for the model under investigation (non-centered elliptic model B). He uses the dynamical cavity method (a review of which can be found in [START_REF] Barbier | The cavity method for community ecology[END_REF]). The key concept consists of assuming that a unique fixed point exists and introducing a new species with new interactions in the existing system. Provided the coherence of the assumption, an analogy between the properties of the solutions with n and n `1 species yields closed-form equations that we present hereafter.

Notice that recently, similar equations were obtained by Galla [START_REF] Galla | Dynamically evolved community size and stability of random Lotka-Volterra ecosystems[END_REF] using generating functional techniques.

The system of equations presented hereafter is a version of Bunin's equations without the carrying capacity. It is similar to the equations obtained by the replicator equations [START_REF] Diederich | Replicators with random interactions: A solvable model[END_REF][START_REF] Opper | Phase transition and 1/f noise in a game dynamical model[END_REF]. Notice that we mention but do not discuss the many implicit assumptions yielding the system of equations.

Let pρ, α, µq P A and x ˚given by Proposition 2.3. We first introduce the following quantities:

φ " Cardtx i ą 0, i P rnsu n , xxy " 1 n n ÿ j"1 x j , @ x 2 D " 1 n n ÿ j"1 px j q 2 . ( 7
)
Denote by Z " N p0, 1q and set ∆ " p1 `xxy µq α a xx 2 y .

The following system of 4 equations has 4 unknowns, among which the (supposedly existing) asymptotic limits of φ, xxy , @ x 2 D , denoted (by abuse of notations) by the same notations. The fourth unknown v is a parameter essentially related to the dynamical cavity method. This system is supposed to admit a unique solution:

φ " 1 ? 2π ż `8 ´∆ e ´z2 2 dz (8)
xxy "

φ 1 ´ρv α ˜p1 `xxy µq `axx 2 y α EpZ|Z ą ´∆q ¸(9) @ x 2 D " ˆ?φ 1 ´ρv α ˙2 ˆp1 `xxy µq 2 `2p1 `xxy µq a xx 2 y α EpZ|Z ą ´∆q `@x 2 D α 2 EpZ 2 |Z ą ´∆q ˙(10) v " φ ˆ1 α ´ρv ˙(11)
The theoretical solutions of system ( 8)-( 11) are compared with the empirical values obtained by Monte-Carlo experiments. As shown in Fig. 5, the matching is remarkable. The impact of the correlation ρ on the proportion of the surviving species is shown in Figure 6.

Remark 2.4. From a theoretical ecology point of view, notice that a negative correlation (prey-predator) seems to slow down the decline of the surviving species, whereas a positive correlation (mutualism and competition) reverses the trend. These types of results are similar to Allesina and Tang [START_REF] Allesina | Stability criteria for complex ecosystems[END_REF] where they notice that preypredator interactions seem to stabilize the system. Figure 6. Effect of the correlation ρ and the interaction strength α on the proportion of surviving species φ. Each curve is plotted by resolving the system ( 8)-( 11) in the centered case µ " 0.

3. Feasibility: Proof of Theorem 2.1 assume that matrix B n is given by (2) (elliptic model). The case where matrix B n is given by (4) (covariance profile model) needs extra arguments which are provided in Appendix A.

Preliminary results.

Extreme Value Theory (EVT) and the Normal Comparison Lemma. Let pZ k q kPN be a sequence of i.i.d. N p0, 1q random variables and denote:

# M n " max kPrns Z k | M n " min kPrns Z k , α n " a 2 log n , β n " α n ´1 2α n logp4π log nq . ( 12
)
Let Gpxq " e ´e´x be the Gumbel cumulative distribution function, then classical EVT results (see for instance [26, Theorem 1.5.3]) yield that for every x P R,

P tα npM n ´βn q ď xu ÝÝÝÑ nÑ8 Gpxq , P ! α np | M n `βn q ě ´x) ÝÝÝÑ nÑ8 Gpxq . ( 13
)
We consider the following dependent framework: Let pZ k,n q kPrns be a Gaussian vector whose components are N p0, 1q with covariance cov pZ k,n ; Z ,n q " ρ n , |ρ| ď 1 , k ‰ .

We are interested in the behaviour of M n " max kPrns Z k,n and | M n " min kPrns Z k,n , and shall prove the counterpart of ( 13) with the help of the Normal Comparison Lemma (NCL): Theorem 3.1 (Theorem 4.2.1, [START_REF] Leadbetter | Extremes and related properties of random sequences and processes[END_REF]). Suppose that pξ i , i P rnsq is a gaussian vector where the ξ i 's are standard normal variables, with covariance matrix Λ 1 " `Λ1 ij ˘. Similarly, let pη i , i P rnsq be a gaussian vector where the η i 's are standard normal, with covariance matrix

Λ 0 " `Λ0 ij ˘. Denote by ρ ij " max |Λ 0 ij |, |Λ 1 ij |
( and let pu i , i P rnsq be real numbers. Then: |P tξ j ď u j , j P rnsu ´P tη j ď u j , j P rnsu| ď 1 2π

ÿ 1ďiăjďn ˇˇΛ 1 ij ´Λ0 ij ˇˇ`1 ´ρ2 ij ˘´1{2 exp ˜´1 2 pu 2 i `u2 j q 1 `ρij ¸. ( 14
)
Corollary 3.2. Recall the definition of pZ k, q kPrns , M n and | M n above, then

P tα npM n ´βn q ď xu ÝÝÝÑ nÑ8 Gpxq , P ! α np | M n `βn q ě ´x) ÝÝÝÑ nÑ8 Gpxq . ( 15
)
Proof. We apply the NCL to pZ k q kPrns and pZ k,n q kPrns . Let ρ ij " |ρ| n and u n pxq "

x α n `βn , then |Ptα npM n ´βn q ď xu ´Ptα npM n ´βn q ď xu| " |PtZ j ď u n pxq , j P rnsu ´PtZ j,n ď u n pxq , j P rnsu| ,

ď 1 2π npn ´1q 2 |ρ| n ˆ1 ´ρ2 n 2 ˙´1 2 exp ˜´u 2 n pxq 1 `|ρ| n ¸ď K n exp ˆ´u 2 n pxq 1 `1 n ˙.
Now eventually u n pxq " α n p1 `op1qq ě κα n for any κ ă 1 and eventually n exp

ˆ´u 2 n pxq 1 `1 n ˙ď n exp ˆ´2κ 2 logpnq 1 `1 n ˙" n ´´2κ 2 1`ρ{n ´1¯.
This last term goes to zero as n Ñ 8 for a well-chosen κ sufficiently close to one. This concludes the proof for M n . The proof for | M n can be handled similarly with minor modifications.

Random Matrix Theory. Let B n be given by model [START_REF] Allesina | Stability criteria for complex ecosystems[END_REF]. Lemma 3.3. Let A n a n ˆn matrix with i.i.d. N p0, 1q entries for i ď j and pA ij , A ji q a standard bivariate Gaussian vector with covariance ρ for i ă j, then the following estimate holds true: almost surely,

lim sup nÑ8 › › › › A n ? n › › › › ď ? 2 ´a1 `ρ `a1 ´ρ¯ď 2 ? 2 .
Proof. The proof relies on two arguments: the classical estimate of the asymptotic spectral norm of a Wigner matrix [4, Th. 5.1] and the following decomposition of matrix A n { ? n as linear combination of Hermitian Wigner matrices:

A n ? n " A n `AT n 2 ? n ´i " i `An ´AT n ˘‰ 2 ? n , pi 2 " ´1q . ( 16
)
Notice that both matrices W 1 n "

An`A T n 2 ? n and W 2 n " ripAn´A T n qs 2 ? n
are Wigner matrices, with off-diagonal variances pi ă jq:

var ˜" A n `AT n 2  ij ¸" 1 `ρ 2 and var ¨« i `An ´AT n 2 ff ij '" 1 ´ρ 2 .
Hence,

lim sup n › › › › A n ? n › › › › ď lim sup n }W 1 n } `lim sup n }W 2 n } " 2 ˜c 1 `ρ 2 `c 1 ´ρ 2
An elementary analysis yields ? 2p ? 1 `ρ `?1 ´ρq ď 2 ? 2 for |ρ| ď 1.

3.2. Proof of Theorem 2.1 -the centered case µ " 0.

Some preparation and strategy of proof. We first prove Theorem 2.1 in the case where µ " 0 and focus on the equation

x n " 1 n `An α n ? n x n . (17)
By Lemma 3.3, lim sup n }A n { ? n} is a.s. bounded hence › › › › A n α n ? n › › › › a.s. ÝÝÝÑ nÑ8 0 .
As a consequence, the resolvent Q n " pI n ´An {pα n ? nqq ´1 is a.s. eventually welldefined and the solution x n " px k q kPrns of (17) writes x n " Q n 1 n . Denote by e k the kth canonical vector of R n . The following representation holds true (we shall often drop index n in the following)

x k " e T k x " e k Q1 " 8 ÿ "0 e T k ˆA α ? n ˙ 1 , " 1 `1 α e T k ˆA ? n ˙1 `1 α 2 e T k ˆA ? n ˙2 Q 1 . (18)
Denote by

Z k,n " e T k ˆA ? n ˙1 " 1 ? n ÿ i A ki and R k,n pAq " e T k ˆA ? n ˙2 Q 1 . (19)
Notice that the Z k,n 's are standard N p0, 1q however they are not independent as

covpZ k,n , Z ,n q " 1 n covpA k , A k q " ρ n , k ‰ .
Introducing M n " max kPrns Z k,n and | M n " min kPrns Z k,n , we proved in Corollary 3.2 that

P tα npM n ´βn q ď xu , P ! α np | M n `βn q ě ´x) ÝÝÝÑ nÑ8 Gpxq . (20)
In the sequel, we often drop n and simply write R k pAq instead of R k,n pAq. Following the same strategy as in [START_REF] Bizeul | Positive solutions for large random linear systems[END_REF], we notice that (18) yields

# min kPrns x k ě 1 `1 α | M `1 α 2 min kPrns R k pAq min kPrns x k ď 1 `1 α | M `1 α 2 max kPrns R k pAq , which we can rewrite min kPrns x k ě 1 `αn αn ´| M `βn α n ´βn α n `min kPrns R k pAq α n αn " 1 `αn αn ´´1 `oP p1q `min kPrns R k pAq α n αn ¯,
where we have use the fact that M `βn α n " op1q, cf. [START_REF] Gopalsamy | Global asymptotic stability in volterra's population systems[END_REF]. Similarly, we have:

min kPrns x k ď 1 `αn α n ˆ´1 `oP p1q `max kPrns R k pAq α nα n ˙(21)
The proof in the centered case follows then from the following lemma: The remaining of the section is devoted to the proof of Lemma 3.4.

Lemma 3.4. Let R k,
Lipschitziannity and Gaussian concentration. We first introduce a truncated version of R k,n pAq. Let η P p0, 1q and ϕ : R `Ñ r0, 1s a smooth function satisfying:

ϕpxq "

# 1 if x P r0, 2 ? 2 `ηs 0 if x ě 4 , (22)
decreasing from 1 to 0 gradually as x goes from 2 ? 2 `η to 4. Let r R k,n pAq " ϕ n R k,n pAq where

ϕ n " ϕ ˆ› › › › A n ? n › › › › ˙. (23)
Notice that r R k pAq differs from R k pAq if ϕ n ă 1 which happens with vanishing probability as P tϕ n ă 1u " P s n ą 2 ? 2 `η( ÝÝÝÑ nÑ8 0 by Lemma 3.3. The following lemma is a first step towards Gaussian concentration. Lemma 3.5. Let r R k defined by [START_REF] Janson | Gaussian hilbert spaces[END_REF] and M an n ˆn matrix. Then the function

M Þ Ñ r R k pM q " e T k ˆM ? n ˙2 ˆI ´M α ? n ˙´1 1 is K-Lipschitz, i.e. ˇˇr R k pM q ´r R k pN q ˇˇď K }M ´N } F ( 24
)
where M, N are n ˆn matrices,

}M } F " b ř ij |M ij | 2
is the Frobenius norm and K a constant independent from k and n.

The second step is to notice that r R k pAq (where A has Gaussian entries but with off-diagonal pairwise correlations) can be in fact expressed as a Lipschitz function of i.i.d. N p0, 1q entries. Lemma 3.6. Consider the linear function Γ : R nˆn Ñ R nˆn defined by

Γ ii pXq " X ii and $ & % Γ ij pXq " b 1`ρ 2 X ij `b 1´ρ 2 X ji pi ă jq , Γ ji pXq " b 1`ρ 2 X ij ´b 1´ρ 2 X ji pi ă jq . Then (1) We have }ΓpXq} F ď K ρ }X} F where K ρ " 2 a 1 `|ρ| hence Γ is K ρ - Lipschitz.
(2) If matrix X n " pX ij q has i.i.d. N p0, 1q entries, then A n " ΓpX n q has i.i.d.

N p0, 1q entries on and above the diagonal (i ď j) and each vector pA ij , A ji q is a standard bivariate Gaussian vector with covariance ρ for i ă j.

The proof is straightforward and is thus omitted. A consequence of this lemma is that r R k pAq " r R k pΓpXqq is K ˆKρ -Lipschitz. Applying Tsirelson-Ibragimov-Sudakov inequality [8, Theorem 5.5] finally yields: Proposition 3.7. Let K the Lipschitz constant of Lemma 3.5 and

K ρ " 2 a 1 `|ρ|. Then E max kPrns ´r R k pAq ´E r R k pAq ¯ď 2 K ρ K a log n .
Details of the proof are similar to those in [START_REF] Bizeul | Positive solutions for large random linear systems[END_REF] and are thus omitted.

Remark 3.8. Notice that ϕ n ď 1 and that ϕ n " 0 if }A{ ? n} ě 4. In particular,

ϕ n › › › › A ? n › › › › ď 4 and ϕ n }Q} ď 1 1 ´4α ´1 ď 2
for n large enough. For the latter estimate, write

Q " ´I ´A α ? n ¯´1 , Q ´1Q " I and Q " I `A α ?
n Q, then apply the triangular inequality.

Proposition 3.9. The following estimate E r R k pA n q " Op1q holds true, uniformly for k P rns.

Proof. We shall prove that the variables r R k have a common distribution for k P rns, which in particular implies that

E r R k " E r R i , @k, i P rns and E r R k " 1 n ÿ iPrns E r R i . (25)
Once this fact is established, the proof is straightforward:

ˇˇE r R k ˇˇ" ˇˇˇˇˇ1 n ÿ iPrns E r R i ˇˇˇˇˇ" ˇˇˇˇ1 n Eϕ n 1 T ˆA ? n ˙2 Q1 ˇˇˇˇď › › › › 1 ? n › › › › 2 Eϕ n › › › › A ? n › › › › 2 }Q} " Op1q ,
where the last equality follows from the arguments developed in Remark 3.8.

Let us now establish [START_REF] Lata La | Some estimates of norms of random matrices[END_REF]. Denote by ∆ σ the matrix associated to the permutation σ : rns Þ Ñ rns and defined by

r∆ σ s ij " # 1 if i " σpjq 0 else .
Notice in particular that ∆ σ e i " e σpiq , ∆ σ ∆ τ " ∆ στ for σ, τ two permutations and ∆ σ ´1 " ∆ T σ . Denote by pijq the transposition swapping i and j, i.e. pijqi " j, pijqj " i and pijq " otherwise. We consider q A " ∆ pijq A∆ pijq , that is q A is obtained by swapping A's ith and jth column, then the ith and jth row. Observe that A and q A have the same distribution and so is the case for R k pAq and R k p q Aq.

We have ∆ 2 pijq " I n , implying that q A k " ∆ pijq A k ∆ pijq and then

R i p q Aq " e T i ÿ kě2 ˜q A α ? n ¸k 1 " e T i ∆ pijq ÿ kě2 ˆA α ? n ˙k ∆ pijq 1 " e T j ÿ kě2 ˆA α ? n ˙k 1 " R j pAq .
This proves that R i pAq, R i p q Aq, R j pAq have the same law, hence the same expectation. Eq.( 25) is established, which concludes the proof.

We are now in position to prove Lemma 3.4.

Proof of lemma 3.4. Recall that E r R k pAq " E r R 1 . Since max kPrns r R k pAq ´r R 1 pAq ě 0, Markov inequality yields: nq} Ñ 0 a.s. We shall also rely on the fact that }Q ´I} ÝÝÝÑ Denote by x and x the vectors solutions of the equations:

P # max kPrns r R k pAq ´r R 1 pAq α ? 2 log n ě ε + ď E ´max kPrns r R k pAq ´r R 1 pAq εα ? 2 log n , " E ´max kPrns r R k pAq ´E r R k pAq εα ? 2 log n , " E ´max kPrns ´r R k pAq ´E r R k pAq ¯εα ? 2
x " 1 `B x " 1 `ˆA α ? n `µuu T ˙x and x " 1 `A α ? n x .

The following representations hold:

x " pI ´Bq ´1 1 and x " ˆI ´A α ? n ˙´1 1 .

Recall that Q " pI ´A{pα ? nqq ´1. By rank one perturbation identity (Woodbury), we have:

pI ´Bq ´1 " Q `Quu T Q 1 ´µu T Qu and
x "

Q1p1 ´µu T Quq `µQuu T Q1 1 ´µuTQu " x 1 ´µu T Qu .

If µ ă 1 and α ě p1 `εqα ˚then eventually, x has positive components. This is no longer the case if µ ą 1 or α ď p1 ´εqα ˚. This concludes the proof of Theorem 2.1. ˙.

We will rely on the following condition:

2I ´pB `BT q is positive definite ô λ max pB `BT q ă 2 .

Notice that pA `AT q{α is a symmetric matrix with independent N p0, 2p1 `ρq{α 2 q entries above the diagonal (the diagonal entries have a different distribution from the off-diagonal entries, with no asymptotic effect). In this case, it is well known that the largest eigenvalue of the normalized matrix (or equivalently its spectral norm since the matrix is symmetric) almost surely converges to the right edge of the support of the semi-circle law (see [ ˙.

In the centered case (i), condition [START_REF] Leadbetter | Extremes and related properties of random sequences and processes[END_REF] asymptotically occurs whenever α ą a 2p1 `ρq. Before studying subcases (ii) and (iii), we recall a result on small rank perturbations of large random matrices.

Notice that the rank-one perturbation matrix P " 2µ n 11 T admits a unique non zero eigenvalue 2µ. Denote by Ǎ " A`A T α ? n . We are concerned with the top eigenvalue

5 Figure 1 . 2 p1`ρq 2 ỳ2p1´ρq 2 "

 51222 Figure 1. Spectrum of non-Hermitian matrix B n (n " 500) in the centered case (µ = 0) with distinct parameter ρ P t´0.5, 0, 0.5u. The solid line represents the ellipse tz " x `iy P C, x 2 p1`ρq 2 ỳ2 p1´ρq 2 " 1u which is the boundary of the support of the limiting spectral distribution for an elliptic model.

  ) If µ ą 1 or there exists ε ą 0 such that, for n large enough, α n ď p1 ´εqα

Figure 2 .

 2 Figure 2. Transition towards feasibility for the elliptic model (2). For each κ on the x-axis, we simulate 1000 matrices B n of size n " 1000 and compute the solution x n of Theorem 2.1 at the scaling α n pκq " κ a logpnq. Each curve represents the proportion of feasible solutions x n obtained for the 1000 simulations. Three distinct values ρ P t´0.5, 0, 0.5u are used. The dotdashed vertical line corresponds to κ " ? 2 i.e. the critical scaling α n " ? 2 a logpnq.

Figure 3 .

 3 Figure 3. Representation of the set of admissible parameters A by a heat map. The set A given by (6) yields the existence of a unique (random) globally stable equilibrium x ˚. The x-axis corresponds to ρ, the y-axis to σ and the intensity of the color µ.

  (a) Initial conditions drawn in p0, 2q, (b) Initial conditions equal to 1.

Figure 4 .

 4 Figure 4. Representation of the dynamics of a ten-species system. For a fixed matrix of interactions B 10 with parameters pρ " 0, α " 2, µ " 0q P A, we consider two distinct initial conditions. Simulations show that the abundances converge in both cases toward the unique globally stable equilibrium x ˚predicted by Proposition 2.3. Notice that since α ă a 2 logp10q » 2.14, we witness vanishing species.

Figure 5 .

 5 Figure 5. Theoretical values of φ, xxy and xx 2 y (solid line) obtained by solving the system (8)-[START_REF] Clenet | Feasibility in a large lotka-volterra system with pairwise correlated interactions[END_REF] given the parameters (µ " 0.2, ρ " 0.5), compared to the empirical values (dots) obtained by Monte-Carlo simulations (size of matrix n " 500, number of random samples P " 200). The x-axis corresponds to the interaction strength α.

ÝÝÝÑ

  n pAq be defined as in[START_REF] Girko | The elliptic law: ten years later i[END_REF] and recall that α n Ý ÝÝÝÝ Ñ

					nÑ`8	`8,
	then:
	max kPrns R k,n pAq α n ? 2 log n	P ÝÝÝÑ nÑ8	0	and	min kPrns R k,n pAq α n ? 2 log n

P ÝÝÝÑ nÑ8 0 .

  The arguments are similar to those in[START_REF] Bizeul | Positive solutions for large random linear systems[END_REF] Section 2.3]. Proof of the second assertion of Lemma 3.4 can be done similarly. This concludes the proof.3.3.Proof of Theorem 2.1 -the non centered case. Recall that α Ñ 8 as n Ñ 8. Denote by u n " 1 ? n 1 n and notice that the spectrum of I n ´µu n u T n is t1´µ, 1u, the eigenvalue 1 with multiplicity n´1. Notice in particular that if µ ‰ 1, then I ´µuu T is invertible. So is (eventually) I ´A α

					ď	?	2K ˆKρ εα	,	log n	,
	where the last inequality follows from Proposition 3.7.
	This implies that
			max kPrns r R k pAq ´r R 1 pAq α ? 2 log n	P ÝÝÝÑ nÑ8	0 .
	It remains to prove that
	α	r R 1 pAq ? 2 log n	P ÝÝÝÑ nÑ8	0	and	max kPrns R k pAq α ? 2 log n
							?	n ´µuu T as }A{pα	?

P ÝÝÝÑ nÑ8 0 .

4 .

 4 Stability: Proof of Proposition 2.3

	Proof. We have
	I ´B `I ´BT " 2I ´pB `BT q " 2I	´ˆA `AT α ? n	`2µ n	11 T

  5, Theorem 5.2]):

	λ max	ˆA `AT α ? n	˙a.s. ÝÝÝÑ nÑ8	2	a 2p1 `ρq α	.	(27)
	Suppose that pρ, α, µq P A. Notice that in this case,
	? α 1 ? `ρ 2		ă	1 2	ă	1 2 `1 2	c 1	´2p1 `ρq α 2	.
	We consider three subcases
	(i) µ " 0, (ii) µ ď ? 1`ρ α ? 2 ,
	(iii) µ P ˆ?1`ρ α ? 2 , 1 2 `1 2	b	1 ´2p1`ρq α 2

of the symmetric matrix Ǎ `P . Based on a result by Capitaine et al. [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations[END_REF]Theorem 2.1], we have:

2p1`ρq α else.

Consider now subcase (ii), then λ max p Ǎ `P q a.s. , which is strictly lower than 2 since pρ, α, µq P A. Hence λ max p Ǎ `P q is eventually strictly lower than 2 in this case. We finally consider subcase (iii). In this case,

We shall prove that 2µ `1`ρ

An elementary study of the polynomial ΩpXq " 2α 2 X 2 ´2α 2 X `1 `ρ yields that Ω's discriminant is positive if α ą a 2p1 `ρq and Ω's roots are given by

Also remark that Ω ´?1`ρ α ?

2 ¯ă 0, so that ? 1`ρ α ?

2 P pµ ´, µ `q. In particular condition (28) is fulfilled for µ P ´?1`ρ α ?

2 , µ `¯, which is precisely subcase (iii). Hence a.s. lim sup nÑ8 λ max p Ǎ `P q ă 2. We can then rely on Theorem 2.2 to conclude.

Champs sur Marne, 77454 Marne-la-Vallée Cedex 2, France e-mail: najim@univ-mlv.fr Appendix A. Proof of Theorem 2.1: adaptations to the case of a covariance profile

In this section, we provide the arguments to prove Theorem 2.1 in the case where matrix B follows the model (4), i.e.

B "

Ãn

where Ãn 's entries are i.i.d. N p0, 1q on and above the diagonal (i ď j), and p Ãij , Ãji q is a standard bivariate Gaussian vector pi ă j) with covariance covp Ãij , Ãji q " ρ ij , and independent from the remaining random variables.

There are essentially 3 issues to resolve, to fully adapt the proof developed in Section 3 to the covariance profile case:

(1) The decomposition [START_REF] Gardner | Connectance of large dynamic (cybernetic) systems: critical values for stability[END_REF] (2) The Lipschitz property for r R k,n p Ãn q. Essentially, we need the counterpart of Lemma 3.6 to the context of a covariance profile.

(3) The control of the term E r R k,n p Ãq.

A.1. Proof of issue 1: Control of the spectral norm of a Hermitian matrix with a variance profile. Applying Lata la's theorem [START_REF] Lata La | Some estimates of norms of random matrices[END_REF], we easily show that

matrix X n " pX ij q is a Wigner matrix with i.i.d. N p0, 1q entries on and above the diagonal, and ˝stands for the Hadamard product, i.e. Υ n ˝Xn " pΥ ij X ij q. Notice that ? nW 1 n is 1-Lipschitz with respect to the Frobenius norm

Hence by Gaussian concentration, we have

Taking δ " ε ? n, we obtain

The same holds for W

2 X ji pi ă jq . and to modify accordingly the Lipschitz constant by r

To address this issue, we provide a quick argument which relies on Isserlis' theorem also called Wick's formula (see [START_REF] Janson | Gaussian hilbert spaces[END_REF]Th. 1.28]), highly dependent on the Gaussiannity of the entries.

where the sum is over all the partitions Π of rns into pairs ti, ju, and the product over all the pairs contained in Π.

Recall that:

Consider a matrix A n where the pairwise covariance covpA ij , A ji q " 1. Denote by

From this, we deduce that |C | ď C , hence |E r R k p Ãq| ď E r R k pAq. This gives the desired bound since E r R k pAq " Op1q.