Alice Chauvin

Mélodie Monteil

Sergio Bellizzi

Renaud Côte
email: cote@lma.cnrs-mrs.fr

Philippe Herzog

M Pachebat

Acoustic characterization of a nonlinear vibroacoustic absorber at low frequencies and high sound levels

Keywords: Passive Absorber, Acoustic Impedance Measurement, Impedance Tube, High Level, Low Frequency

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Introduction

Nonlinear vibroacoustic absorbers are passive devices dedicated to noise reduction at low frequencies. Such devices consist in a thin structure submitted to large deformations, which exhibits non linear resonances used to absorb sound energy. Such a structure is a thin viscoelastic membrane in [START_REF] Bellet | Experimental study of targeted enrgy transfer from an acoustic system to a nonlinear membrane absorber[END_REF], whereas in [START_REF] Mariani | Toward an adjustable nonlinear low frequency acoustic absorber[END_REF] the diaphram of a loudspeaker without motor assembly is used. This kind of device takes advantage of a phenomenon called "Targeted Energy Transfer" or "Energy Pumping", due to the coupling of the nonlinear resonance of the absorber with the acoustic field that has to be reduced. It is described in detail in [START_REF] Vakakis | Nonlinear targeted energy transfer in mechanical and structural systems[END_REF] in terms of resonance capture and nonlinear modes showing an irreversible energy transfer from the acoustic medium toward the absorber, the energy being thus dissipated within the absorber. In the literature, such a device is often designated as a "Nonlinear Energy Sink" (NES). Nonlinear vibroacoustic absorbers have shown their ability to reduce sound especially at low frequencies range and high enough sound field levels. They may thus overcome some limitations of classical devices such as porous materials, Helmholtz resonators, perforated plates [START_REF] Bellet | Experimental study of targeted enrgy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Mariani | Toward an adjustable nonlinear low frequency acoustic absorber[END_REF][START_REF] Bellet | Enhancing the dynamic range of the targeted energy transfer in acoustics using several nonlinear membrane absorbers[END_REF][START_REF] Côte | Experimental evidence of simultaneous multi-resonance noise reduction using an absorber with essential nonlinearity under two excitation frequencies[END_REF][START_REF] Shao | Theoretical and numerical study of targeted energy transfer inside an acoustic cavity by a non-linear membrane absorber[END_REF].

In these early papers, the nonlinear behavior of the vibroacoustic absorbers was modelled, using nonlinear plate equations of the Von-Kármán type for the thin structure, coupled with the acoustic medium through an impedance boundary condition. In addition to these models, experiments were based on vibration measurements focused on the mechanical NES behavior. Conversely, the present paper focuses on the NES considered as an acoustic device with desirable absorbing properties. Dealing with low frequencies, it uses an approach common to the characterization of other kinds of absorbers, i.e. a cylindrical tube connecting a sound source to the NES. At the considered frequencies, plane waves may safely be assumed into the tube, so the NES may be considered as a single port acoustic device.

A review about linear impedance measurements can be found in [START_REF] Dalmont | Acoustic impedance measurement, part i: A review[END_REF]. The most employed method is the Two Microphone Method (TMM). It was initially developed by Seybert and Ross [START_REF] Seybert | Experimental determination of acoustic properties using a two-microphone random-excitation technique[END_REF] and expanded by Chung and Blaser using transfer functions [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. i. theory[END_REF][START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. ii. experiment[END_REF]. Abom and Boden [START_REF] Abom | Error analysis of the two-mircophone measurements in ducts with flow[END_REF] later reviewed different implementations of this method, the most usual being standardized as ISO 10354-2. Basically, it involves two microphones placed between the sample and the source, their relative calibration requiring them to be switched. The microphones spacing must be adapted to the targeted frequency band, which becomes problematic when dealing with wide frequency ranges. An improvement of the calibration method, using three loads, was later proposed by Gibiat and Laloë [START_REF] Gibiat | Acosutic impedance measurements bu the two-microphone-three-calibration (tmct) method[END_REF]. Recently, Boonen et al. [START_REF] Boonen | Calibration of the two microphone transfer function method with hard wall impedance measurements at different reference sections[END_REF] proposed a method to deal with high impedance magnitudes over a wide frequency range : the calibration is then again performed by measuring several different acoustic loads instead of switching the microphones.

In our study, the absorber is essentially nonlinear and its absorbing properties have to be dug out at low frequencies, under controlled high excitation levels. No standard equipment is available to deal with this need, so we built an impedance tube (thereafter named "Short Kundt's Tube", i.e. SKT) able to reach very high levels at low frequencies (typically several hundreds of Pascals between 10 and 200 Hz). The measurement of the acoustic characteristics of the absorber device are obtained through an original method inspired from the indirect source characterization methods proposed in [START_REF] Prasad | A four load method for evaluation of acoustic source impedance in a duct[END_REF][START_REF] Desmons | A least squares method for evaluation of characteristics of acoustical sources[END_REF][START_REF] Bodèn | On multi-load methods for determination of the source data of acoustic one-port sources[END_REF]. A single microphone is needed, as the source is first characterized using different known acoustic loads. An "equivalent impedance" and the apparent reflection coefficient of the NES are then deduced from this single pressure measurement.

The SKT is named "short" because it is shorter than should be a Kundt's tube according to ISO 10354-2. A single microphone method can use a tube as short as possible with respect to the plane waves approximation. Such a short device is easy to move and occupies a small area. With this setup we expect to reduce errors due to thermal gradient and instabilities along the tube, and we expect to reject the source resonances above the measurement range, as shown in this article.

The structure of this paper is as follows. Section 2 presents the experimental set-up and the identification method allowing to estimate the reflection coefficient of a device under study (DUT). Section 3 applies this method to the estimation of the reflection coefficient of a sample nonlinear absorber, and its measurements are then compared to an analytic model previously published (Section 3.3) and to a set of equivalent resonators 2 (Section 3.4).

In this paper, all quantities, unless otherwise stated, will be considered in the frequency domain, assuming a e +jωt temporal dependence.

Measurement method

Basic principle

Fig. 1(a) presents the acoustical scheme of the set-up. It is made of a sound source (controlled by a voltage U ) and a sample (or DUT), connected through a tubular section of minimal length (hence the name "Short" Kundt's Tube). The source must be linear over the full dynamic range, and therefore features several driver units (only two are shown in Fig. 1(a)). The left hand side of the tube is closed next to the source, and the DUT is mounted on the right hand side. A high pressure microphone is positioned on the axis of the tube, in the measurement plane separating the source from the DUT which is taken as the origin for the x-axis.

The DUT is characterized by an equivalent impedance Z T , which is defined as the ratio of the fundamental components of the spectra of the pressure and the volume velocity, both considered over the tube section in the measurement plane. Plane waves are assumed along the cylindrical part of the system, at sufficient distances from the source and the DUT: the measurement plane is distant at least of the magnitude of one tube diameter from the loudspeaker connectors, and one membrane diameter from the sample membrane. Tests were made with a 0.8 m long extension of the tube between the DUT and the microphone, and no significant difference was noticed between the results. The local pressure P measured by the microphone is thus considered as a good estimate of the pressure averaged over the measurement section. The local volume velocity is not measured, but is indirectly obtained through the setup calibration. When the DUT is nonlinear, Z T corresponds to a linear approximation of the relationship between the pressure and volume velocity. In this case, Z T is an indicator depending on the DUT, obviously, and depending also of the conditions of the experiment (i.e. U and the source frequency content).

The source is considered as linear and time-invariant, so it can be represented using the Norton equivalence by the electro-acoustic scheme shown in Fig. 1(b). It is characterized by its impedance Z S and its volume velocity Q a . Equivalently, it may be characterized by Z s and the transfer function H ae between the acoustic volume velocity Q a and the control voltage U , defined as Q a = H ae U . The equivalent impedance Z loading the volume velocity source is therefore :

Z = Z T Z s Z T + Z s . (1)
Considering the transfer function H m between the control voltage U of the source and the acoustic pressure P at the microphone, Eq. ( 1) reduces to :

H m = 1 Z s + 1 Z T -1 H ae (2)
and the impedance Z T and the reflection coefficient R T of the DUT can be expressed as :

Z T = H m Z s H ae Z s -H m and R T = H m Z s -Z c (Z s H ae -H m ) H m Z s + Z c (Z s H ae -H m ) (3)
where Z c = ρc/S is the specific acoustic impedance of air, ρ is the density of air and c the sound wave velocity. Thus, the determination of the impedance Z T (and the reflection coefficient R T ) of the DUT only requires transfer function H m measurement and the prior knowledge of the source characteristics H ae and Z s . This will be discussed in the following two sections.

Note that Z T and R T are defined at the microphone axial position but they can be deduced at the location of the load thanks to the impedance translation theorem between two axial positions x 1 and x 2 in the tube, recalled hereafter :

P (x 2 ) Q(x 2 ) = cos(k(x 1 -x 2 )) Z c j sin(k(x 1 -x 2 )) Z -1 c j sin(k(x 1 -x 2 )) cos(k(x 1 -x 2 )) P (x 1 ) Q(x 1 ) (4)
where k = ω/c is the wavenumber.

Combining the source characteristics H ae and Z s and the load impedance Z T , it is therefore possible to estimate the transfer function between control voltage U and the acoustic pressure at a new axial position x n as :

H n m = Z n s Z n T Z n s + Z n T H n ae ( 5
)
where

Z n T = cos(-kx n ) + j sin(-kx n )Z c /Z T j sin(-kx n )/Z c + cos(-kx n )/Z T , Z n s = cos(kx n )Z s + jZ c sin(kx n ) cos(kx n ) + j sin(kx n )Z s /Z c , (6)
H n ae = (cos(kx n ) + jZ c sin(kx n )Z n -1 s )H ae . (7)
Transformation ( 5) will be used for the comparison with mechanical models of the NES in Section 3.

Setup calibration

The setup calibration consists in determining the quantities H ae and Z s , from measurements of known acoustic loads. In such a case, Eq. ( 2) becomes :

H m = Z s Z T Z s + Z T H ae . (8)
The measurements H m1 and H m2 obtained respectively from two different (but known) loads Z T1 and Z T2 are thus theoretically sufficient to determine the source parameters H ae and Z s as :

H ae = H m1 H m2 (Z T1 -Z T2 ) Z T1 Z T2 (H m1 -H m2 ) and Z s = Z T1 Z T2 (H m1 -H m2 ) H m2 Z T1 -H m1 Z T2 . (9)
Since the transfer functions H m1 and H m2 are obtained using the same microphone, Z s (given by Eq. ( 9)) does not depend on the microphone calibration. The same property applies to the determination of Z T (given by Eq. ( 3)). Hence with this approach the microphone calibration is not needed.

A simple sensitivity study of Eqs. ( 9) brings some observations. For frequencies where H ae = 0, Z s = 0 or H m = 0, the load impedance cannot be estimated. Moreover, as pointed out by Bodén et al. [START_REF] Bodèn | Error analysis for the two-load method used to measure the source characteristics of fluid machines[END_REF], if Z T1 and Z T2 are close to each other, the error on Z s prevents a reliable calibration. Conversely, when Z s is of the same order of magnitude as the load impedance Z T , the error in the estimation of H ae is smaller.

In order to avoid errors and improve the results, an over-determination of the calibration data is therefore desirable. It can be performed by combining more than two known loads, a technique which is briefly summarized in Appendix A.

Setup consistency

A very basic assumption behind the proposed method is that the source is assumed linear and time-invariant. Its specific electroacoustic design ensures that it may deliver a high volume velocity, required at low frequencies, while keeping a sufficient margin compared to the maximal displacement specified by the loudspeakers manufacturer. Care has also been taken in order to minimize parasitic nonlinear behaviors related to high local acoustic over-pressures, rapid air flows around edges, leakages, etc. Last, the electric powers required during the NES measurements are significantly smaller than the thermal limits specified for the voice-coil heating of loudspeakers, thus reducing the risk of thermal compression during the measurements. A brief description of the source design is given in Appendix B.

Beyond these basic design steps, the source linearity has been checked thoroughly by increasing the RMS level when measuring linear acoustic loads. No significant discrepancy has been found between measurements at different levels, strenghtening the hypothesis of a linear setup. A set of calibration loads has been designed, and used within a minimization process in order to determine a suitable number of loads for calibration. This led to using six reference loads (see Appendix C) spanning a wide impedance range. The full description of these technical aspects is beyond the scope of the present paper, and only cited here to back up the measurement results presented in this paper.

It should be noted that the above-mentionned calibration process is started before every measurements series, as small discrepancies have been detected between calibrations performed at several days intervals. The loudspeaker behavior impacts the calibration data. Such high-power devices do not have the same stability over time than measurement sensors. Frequent calibrations are therefore mandatory to maintain a sufficient measurement accuracy. This time consuming and as such a limitation of our approach.

Nonlinear vibroacoustic absorber behavior

Description of the vibroacoustic absorber

The nonlinear vibroacoustic absorber considered in this study (hereafter named NES) has been described in detail by Bellet et al. [START_REF] Bellet | Experimental study of targeted enrgy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Bellet | Enhancing the dynamic range of the targeted energy transfer in acoustics using several nonlinear membrane absorbers[END_REF]. It consists in a thin viscoelastic circular membrane (see Fig. 2(a)) clamped by a sliding system, allowing to apply a constant inplane pre-stress to the membrane. When the membrane is subjected to a large acoustic pressure, its flexural displacement can be several orders of magnitude larger than the membrane thickness, resulting in a nonlinear behavior. This NES is mounted at the end of the source as illustrated by Fig. 2(b). The measurements presented below were made using a membrane made of latex, with a diameter equal to 0.06 m and a thickness equal to 1.8 10 -4 m.

Acoustic characterization of the NES

The acoustic characterization of the NES was carried out using [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. ii. experiment[END_REF]200] Hz bandlimited white noise with a control voltage U . A multi-channel analyzer/recorder (OROS OR38) including a signal generator was used with a sampling frequency of 3200 Hz. The estimations of the transfer function H m were obtained with a ∆f = 0.5 Hz frequency resolution corresponding to 2 sec time length segments and 50 averages. Seven excitation levels were considered starting from U RMS = 0.01 V to 11.68 V, resulting in acoustic pressures between 2 • 10 -2 Pa and 350 Pa at the microphone position (see Fig. 3). We chose white noise excitation because it allows fast measurements and because it is representative of some NES application environments (e.g. room noise absorption). The NES is nonlinear, some transient state may be several orders of magnitude longer than the period of the excitation and exhibit hysteresis (e.g. SMR [START_REF] Vakakis | Nonlinear targeted energy transfer in mechanical and structural systems[END_REF]), so a frequency stepped excitation may take a long time, and a chirped excitation may give results difficult to understand. Note that these measurements on a nonlinear DUT are not a priori representative of other excitation schemes. Nevertheless, our hypothesis is that this excitation and an analysis based on transfer functions should give a relevant characterization of a NES. It would be interesting in a future work to further study the temporal dynamics of the NES response. At low excitation levels, U RMS = 0.01 V and 0.02 V, the system shows a linear behavior as observed in Fig. 3(a). The transfer function exhibits a minimum near 30 Hz. This minimum of pressure within the tube corresponds to a maximum of the membrane motion characterizing the linear resonance frequency of the NES (at low excitation levels).

A nonlinear behavior appears when increasing the excitation level, see Fig. 3(b). The resonance frequency is shifted toward higher frequencies with a broadening resonance dip. This is characteristic of a system hardening with level.

For each excitation level, the acoustic impedance Z T , Eq. ( 3), and the corresponding reflection coefficient R T , Eq. ( 3), can be deduced from the measured transfer function H m and the estimated source characteristics H ae and Z s (calibration process). The calibration process was carried out before the measurement series of the NES and using the six reference loads (see Appendix C). The results are reported Fig. 4 showing a smooth and flat behavior of H ae and Z s over the considered frequency range. The source is linear without obvious resonances over the frequency range of the study. As a consequence, H ae and Z s curves should be continuous and smooth. On the actual measurements, each point of these quantities has been estimated independently of the others, and the smoothness of the actual curves is an indication about the quality of the measurements. The flatness of the curves can be viewed as a property of the source. It means that the response of the system has no sharp differences along the frequency range, that resonances of the source do not screen DUT characteristics, or in other words that the sensitivity to measurement errors is similar along the frequency range.

Figure 5(a) shows modulus and phases of Z T for the five excitation levels considered in Fig. 3(b). A similar nonlinear behavior as the transfer function H m is observed for Z T . The reflection coefficient R T (modulus and phases) is shown by Fig. 5(b). It allows to assess the amount of energy removed out of the system by the termination, compared to the case of a total energy reflection |R T | 2 = 1. Here, the energy is extracted by the NES over a frequency range widening with the increase of the excitation level. It is important to note that only a part of extracted energy is dissipated (absorbed) by the NES while another part is radiated by the outward face of the membrane or converted to different frequencies [START_REF] Côte | Experimental evidence of simultaneous multi-resonance noise reduction using an absorber with essential nonlinearity under two excitation frequencies[END_REF].

The maximum of extraction of energy (|R T | ≈ 0.05) is observed for U RMS = 5.84 V at a frequency f 80 Hz. Considering these observations, up to 90% energy can be extracted by the NES near its resonance frequency. Further increase of U RMS however tends to slightly reduce the energy extracted by the NES.

Comparison with a viscoelastic model

The measurements resulting from the behavior of the NES are now compared to results from the viscoelastic model proposed by Bellet et al. [START_REF] Bellet | Experimental study of targeted enrgy transfer from an acoustic system to a nonlinear membrane absorber[END_REF]. The nonlinear equation of motion of the membrane (NES) is thus obtained considering the membrane as a thin elastic structure with geometric nonlinearities. The transverse displacement of the membrane is described using a one degree of freedom Rayleigh-Ritz reduction with a single parabolic deflection shape function. The nonlinear equation of motion takes the following form : where u me denotes the (instantaneaous) displacement of the center of the membrane, p me denotes the resulting acoustic pressure in the vicinity of the membrane inside the tube and the dot represents time-differentiation. The coefficient 1/2 in the right hand side of Eq. ( 10) results from the parabolic form used in the Rayleigh-Ritz reduction [START_REF] Bellet | Experimental study of targeted enrgy transfer from an acoustic system to a nonlinear membrane absorber[END_REF]. The model parameters are given by :

m me üme + k 1 [( f 1me f 0me ) 2 u me + η ume ] + k 3 (u 3 me + 2ηu 2 me ume ) = S me 2 p me (t) (10
k 1 = 2πEh 3 me 3(1 -ν 2 )R 2 me , k 3 = 1 2 8πE me h me 3(1 -ν 2 )R 2 me (11) m me = ρ me h me S me + m a 3 , f 0me = 1 2π 1.015 4 π 4 Eh 2 me 12(1 -ν 2 )ρ me R 4 me ( 12
)
where ρ me is the density, ν the Poisson's coefficient and E the Young's modulus of the membrane made in latex, h me is its thickness, R me its radius and S me its surface. The coefficients k 1 and k 3 respectively stand for the linear and nonlinear stiffnesses. As justified by Bellet et al. term m a has also been added to account for the mass of the air column resulting from the waveguide effect of the sliding system. Here f 0me represents the resonance frequency of the membrane alone (without pre-stress) and f 1me is related to the pre-stress applied to the membrane. When no pre-stress is considered, f 1me = f 0me .

For a given excitation level, the acoustic impedance of the NES (close to the membrane) may be defined as :

Z me = Ŝpmeqme Ŝqmeqme ( 13
)
where Ŝqmeqme (respectively Ŝpmeqme ) denotes the power spectral density (respectively cross spectral density) function of the volume velocity q me (respectively between the acoustic pressure p me and the volume velocity q me ). The volume velocity q me is obtained from the velocity of the membrane ume as q me = Sme 2 ume . This definition is in accordance with the impedance quantity considered in Section 2.1.

The NES behavior being nonlinear, comparing model results to measured data requires to use temporal investigation of the nonlinear model. Provided that we use a time trajectory of the acoustic pressure p me which respects similar characteristics as in ex-perimental configuration, Eq. ( 10) can be numerically solved. The resulting trajectories are sufficient to estimate the NES impedance Eq. ( 13) via classical spectral estimation method.

As a gaussian white noise was fed to the source during the measurements, the modulus of the measured transfer function H m (given Fig. 3) is sufficient to characterize the frequency content of the acoustic pressure in the vicinity of the membrane. H m being defined at the microphone position, Eq. 5 is used to estimated the transfer function, H n m , between the control voltage U and the acoustic pressure at the position x n = x m + 0.02 (x m has been corrected of the thickness of the sliding system) corresponding to p me .

We then used the procedure described in [START_REF] Poirion | Simulation numérique de champs vectoriels stochastiques gaussiens homogènes et non homogènes[END_REF] to generate time trajectories with a power spectral density function defined by |H n m | 2 on the frequency range [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. ii. experiment[END_REF]200] Hz and zero outside. These trajectories, corrected by the corresponding excitation level U RMS , were used as time trajectories of p me . Eq. ( 10) was then solved numerically using the Newmark method.

The following values of the parameters were chosen corresponding to the experimental set up: ρ me = 980 kg.m -3 , ν = 0.49, E = 12.10 5 Pa, h me = 1.8.10 -4 m, R me = 0.03 m and m a = 2.97 10 -4 kg. Two parameters f 1me (≥ f 0me ) and η(> 0) were adjusted as f 1me = 40 Hz and η = 8.10 -4 -corresponding to moderate pre-tension (as f 0me = 3.8 Hz) and light damping.

The resulting impedance functions Z me brought back to the microphone position are plotted in Fig. 6 and Fig. 7 for the five excitation levels and compared to the corresponding ones identified experimentally by the proposed method. Also reported are the associated reflection coefficients. The viscoelastic model leads to a very good approximation of the measured data, except at the lowest amplitude (0.03 V) for which the impedance curve in Fig. 6 presents an anti-resonance peak close to 150 Hz which does not fit well with the experimental data. This might come from a coupling between the NES and the source setup, which would apparently shift the anti-resonance peak towards low frequencies. For all other excitation levels, the viscoelastic model seems quite adequate to reproduce the acoustic behavior of the NES. It describes correctly the shift of the resonance frequencies (hardening) and the spectral broadening which occurs around the resonance frequencies when the excitation level increases as estimated through the impedances and associated reflection coefficients.

The NES as a family of Helmholtz resonators

The impedances plotted in Fig. 6 have a shape suggesting that the system behaves like a resonant one degree of freedom system in the frequency range studied, however with level dependent parameters. An attempt is thus made to identify these parameter values expressed in quantities that permit comparisons with Helmholtz resonators, which are often used for sound control in this frequency range. At each excitation level, the behavior of the NES is then approximated by the impedance Z H of a linear Helmholtz resonator :

Z H = jωM a + 1 jωC a + R a (14)
which is transformed at the microphone position as (Eq. ( 4)) : where M a is the acoustic mass, C a is the acoustic compliance, R a is the acoustic resistance. These parameters are then estimated from the measured data by solving the following minimization problem : min

Z th H = Z H + jZ c tan(kx m ) j(Z H /Z c ) tan(kx m ) + 1 (15)
(Ma,Ca,Ra)∈R 3 + F l=1 |R th H (ω l ) -R T (ω l )| 2 (16)
where R th H is the reflection coefficient deduced from the Helmholtz model Eq. ( 15), R T is the estimated reflection coefficient (Eq. 3) and ω l = 2πf l , with f l the frequency of the experiment number l.

This minimization problem has been solved for each excitation level using Matlab c unconstrained nonlinear optimization solver "fminsearch". The results are reported in Table 1. The efficiency of the optimization process can be observed on Figure 8, where are compared R th H and R T and on Figure 9, where are compared the corresponding Z th data, except at the lowest amplitude as mentioned earlier.

Looking to Table 1, the acoustic mass M a is not much affected by the change of excitation level (values between 114 and 135 kg m -4), compared to the large variations exhibited by the compliance C a (180%) and the resistance R a (55%). This is coherent for linear resonators, because the acoustic mass M a of the Helmholtz model can be related to a physical mass m me described by the viscoelastic model by the relationship M a = 4m me /S 2 m . The parameters used for the simulations in section 3.3 correspond to M a = 133. kg m -4 , which is close to its values in Table 1.

The compliance C a and the resistance R a are however much more level dependent.

U Their variations with respect to U RMS can be analyzed considering the evolution of the resonance frequency and the quality factor defined as, respectively :

f H = 1 2π √ M a × C a and Q H = M a /C a R a . (17)
Variations of f H are plotted in Figure 10(a) and illustrate the hardening behavior of the NES mentioned previously. This hardening is not linear as it would be for an uncoupled purely cubic resonator excited at its resonance frequency. Figure 10(b) shows that the apparent quality factor is rather small, around 3 and is relatively constant after a sharp decrease at low excitation levels, until about 3 V. The NES quality factors estimated here are realistic for actual Helmholtz resonators [START_REF] Herzog | Passive and active control of the low-frequency modes in a small room[END_REF], although NES and Helmholtz resonators differ in their action. It means that Helmholtz resonators could be built with such quality factors. A small quality factor indicates a high dissipation, interpreted here as large extraction of energy from the system by several mechanisms: thermal dissipation in the viscoelastic membrane, acoustic radiation, and conversion to other frequencies than the excitation frequency [START_REF] Côte | Experimental evidence of simultaneous multi-resonance noise reduction using an absorber with essential nonlinearity under two excitation frequencies[END_REF].

Energy extraction can be seen more clearly on Fig. 10(c) where the ratio E of extracted 14 energy over the frequency range of the source is defined by :

E = (f F -f 1 ) -1 f F f1 (1 -|R T (2πf )| 2 )df . (18)
is plotted, with f 1 = 10 Hz and f F = 200 Hz. At low levels the energy extraction E is quite small because the NES behaves like a linear Helmholtz resonator: it is efficient only close to its resonant frequency, with low damping. At higher levels E however reaches 0.35, meaning that more than one third of the total incoming energy (spread over a wide frequency range) is extracted by the NES. It should however be noted that most of the energy extracted at these high levels corresponds to frequencies over 150 Hz, part of which might be converted toward frequencies outside the analysis band. The shape and magnitude of the three curves Figure 10(c) are close. The Helmholtz model agrees best with the measurements in the upper half of the excitation level. The worse agreement between measurements and the viscoelastic model may come from the fact that it does not include the acoustic source in the model or that the effective NES stiffness doesn't fit well with the viscoelastic model. It illustrates the relevance of the Helmholtz model for the NES characterization. 17))versus excitation level. (c) Extracted energy E (Eq. ( 18)) versus excitation level obtained from the measured reflection coefficient (dot markers), the viscoelastic model (square markers) and the Helmholtz model(circle markers).

This study shows that the NES may be described as a parametric Helmholtz resonator which parameters can be estimated with realistic values depending on the excitation level. It may prove useful for nonlinear absorber design, since the resonance frequency and attenuation as a function of sound level are key parameters [START_REF] Bellet | Experimental study of targeted enrgy transfer from an acoustic system to a nonlinear membrane absorber[END_REF]. In the framework of linearisation procedure [START_REF] Socha | Linearization methods for stochastic dynamic systems[END_REF], the amplitude level of the NES response could be also used as a parametric variable. This will be considered in future work.

Conclusions

The acoustic properties of a nonlinear vibroacoustic absorber have been investigated at low frequencies and high sound levels.

A measurement system based on impedance tube with a linear response up to 300 Pa in the [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. ii. experiment[END_REF]200] Hz frequency range has been built. It was tested and used with a bandlimited white source signal. In addition a multi-load technique has been developed to extract an apparent acoustic reflection coefficient. The method only requires a single microphone after a source characterization using a set of known loads.

The apparent reflection coefficient of the nonlinear vibroacoustic absorber was measured for increasing excitation levels. The results confirm a nonlinear behavior of the absorber. It is able to remove up to 90% of the incoming acoustic wave, around a frequency which is dependent on the source level. Unlike linear systems a fraction of the energy removed is probably converted into sound at frequencies above the excitation frequency band, in addition to the usual thermal dissipation mechanism. This frequency conversion has not been investigated yet, and is the focus of ongoing work.

A nonlinear viscoelastic model previously published was found adequate to describe the nonlinear characteristics of the NES measured here, based on a simple latex membrane. From the same acoustic measurements, a family of equivalent Helmholtz resonators with level-dependent parameters was identified. At each excitation level, the corresponding model captures most of the action of the nonlinear absorber with realistic parameter values: the identified mass is almost constant with the excitation level and close to the value used in the nonlinear viscoelastic model.

This study showed the good efficiency of the method and setup proposed for high level, low frequency measurements of non-linear loads such as the one considered here. It paves the way for future works about other NES devices.

where Z rad is approximated by a flanged plane piston of radius R 7 ([21] Eq. (12.126)), as: Using Eq. ( 4), the reference load impedance at the microphone location x = 0 is obtained for the known loads. Using H ae and Z s fitted to the the six reference loads measurements, we measure the reflection coefficient R T of the test load.

Z rad = Z c 1
In our experiments, the worst error on R T among all the test frequencies is below 5%. We checked that any of the reference load relative error is below this value. We consider this worst error to be representative of the uncertainty of the measurements.

Figure 1 :

 1 Figure 1: Scheme of (a) the experimental set-up and (b) the equivalent electroacoustic circuit of the one-microphone identification method.

Figure 2 :

 2 Figure 2: (a) Front view of the NES (sliding system and membrane). (b) NES mounted on the source (with a sight of the sliding system).

Figure 3 :

 3 Figure 3: Measured NES transfer function Hm for several excitation levels: modulus(up) and phase (down). (a) Low excitation level (linear behavior) ; (b) High excitation level (nonlinear behavior).

Figure 3

 3 Figure 3 shows modulus and phases of the transfer function H m measured at the seven RMS values of the control voltage U .At low excitation levels, U RMS = 0.01 V and 0.02 V, the system shows a linear behavior as observed in Fig.3(a). The transfer function exhibits a minimum near 30 Hz. This minimum of pressure within the tube corresponds to a maximum of the membrane motion characterizing the linear resonance frequency of the NES (at low excitation levels). A nonlinear behavior appears when increasing the excitation level, see Fig.3(b). The resonance frequency is shifted toward higher frequencies with a broadening resonance dip. This is characteristic of a system hardening with level.For each excitation level, the acoustic impedance Z T , Eq. (3), and the corresponding reflection coefficient R T , Eq. (3), can be deduced from the measured transfer function H m and the estimated source characteristics H ae and Z s (calibration process). The calibration process was carried out before the measurement series of the NES and using the six reference loads (see Appendix C). The results are reported Fig.4showing a smooth and flat behavior of H ae and Z s over the considered frequency range. The source is linear without obvious resonances over the frequency range of the study. As a consequence, H ae and Z s curves should be continuous and smooth. On the actual measurements, each point of these quantities has been estimated independently of the others, and the smoothness of the actual curves is an indication about the quality of the measurements. The flatness of the curves can be viewed as a property of the source. It means that the response of the system has no sharp differences along the frequency range, that resonances of the source do not screen DUT characteristics, or in other words that the sensitivity to measurement errors is similar along the frequency range.Figure5(a) shows modulus and phases of Z T for the five excitation levels considered in Fig.3(b). A similar nonlinear behavior as the transfer function H m is observed for Z T . The reflection coefficient R T (modulus and phases) is shown by Fig.5(b). It allows to assess the amount of energy removed out of the system by the termination, compared to the case of a total energy reflection |R T | 2 = 1. Here, the energy is extracted by the NES over a frequency range widening with the increase of the excitation level. It is important to note that only a part of extracted energy is dissipated (absorbed) by the NES while another part is radiated by the outward face of the membrane or converted to different frequencies[START_REF] Côte | Experimental evidence of simultaneous multi-resonance noise reduction using an absorber with essential nonlinearity under two excitation frequencies[END_REF].The maximum of extraction of energy (|R T | ≈ 0.05) is observed for U RMS = 5.84 V at a frequency f 80 Hz. Considering these observations, up to 90% energy can be extracted by the NES near its resonance frequency. Further increase of U RMS however tends to slightly reduce the energy extracted by the NES.

Figure 4 :

 4 Figure 4: Estimated source parameters Hae and Zs using the six reference loads (see Appendix C). Modulus (top) and phase (bottom).

[ 1 ]

 1 , a 1/2 coefficient was included in the expression of k 3 .

Figure 5 :

 5 Figure 5: Identified acoustic characteristics of the NES for the excitation levels used in Fig. 3(b): (a) Impedance (Z T ) and (b) associated reflection coefficient (R T ). Modulus (top) and phase (bottom).

Figure 6 :

 6 Figure 6: Viscoelastic model : Impedance (Z T ) identified from the measurement method (thick lines) and resulting from the viscoelastic model (lines and circle markers) for the excitation levels used in Fig. 3(b). Modulus (top) and phase (bottom).

H

  and Z T .The parametric Helmholtz model leads to a very good approximation of the measured

Figure 7 :

 7 Figure 7: Viscoelastic model : Reflection coefficient (R T ) identified from the measurement method (thick lines) and resulting from the viscoelastic model (lines and circle markers) for the excitation levels used in Fig. 3(b). Modulus (top) and phase (bottom).

Figure 8 :

 8 Figure 8: Helmholtz model: Reflection coefficient (R T ) identified from the measurement method (thick lines) and resulting from the Helmholtz model (lines and circle markers) for the excitation levels used in Fig. 3(b) . Modulus (top) and phase (bottom).

Figure 9 :

 9 Figure9: Helmholtz model: Impedance (Z T ) identified from the measurement method (thick lines) and resulting from the Helmholtz model (lines and circle markers) for the excitation levels used in Fig.3(b). Modulus (top) and phase (bottom).

Figure 10 :

 10 Figure 10: Helmholtz model fitting: Evolution of (a) the resonance frequency and (b) the quality factor (Eq. (17))versus excitation level. (c) Extracted energy E (Eq. (18)) versus excitation level obtained from the measured reflection coefficient (dot markers), the viscoelastic model (square markers) and the Helmholtz model(circle markers).

Table 1 :

 1 Estimated Helmholtz resonator parameters: Ma, Ca, Ra as a function of U RMS .

	RMS [V]	0.03 0.70 1.5	2.9	4.4	5.8	7.3	8.8 10.2 11.7
	M a	135 127 128 128 129 126 123 118 116 114
	Ca[×10 -7 ] 1.83 1.11 0.74 0.46 0.39 0.28 0.24 0.22 0.20 0.18
	R a [×10 4 ]	0.16 0.57 0.93 1.42 1.82 2.23 2.49 2.70 2.89 2.96

Table 2 :

 2 The first term corresponds to the acoustic mass of the fluid within the cylindrical aperture of the termination, of thickness e 7 . The second term ([START_REF] Chaigne | Acoustics of Musical Instruments[END_REF] Eq. (7.163)) corresponds to the discontinuity in cross section between the aperture and the tube, with α = R 7 /R t .The first six known loads, named reference loads, are employed to estimate Z s and H ae as described in Appendix A and one, number 7, is a test load. Characteristics of the known loads.

	2	(kR 7 ) 2 + j	8 3π	kR 7	(23)

Acknowledgements

This works has been done in the framework of the labex MEC, part of the Idex A*MIDEX. The authors acknowledge SATT Sud-EST (France) for its financial support.

Appendix A

In this appendix, the principle of the method used to estimate the source characteristics H ae and Z s from N (≥ 2) known loads is briefly summarized.

Following Desmons [START_REF] Desmons | A least squares method for evaluation of characteristics of acoustical sources[END_REF], a least square method is developed. The target of the minimization is the reflexion coefficient R T because it is the quantity of interest for the characterization of the energy extracted by NES. For this purpose, the following cost function is used :

where R th Tn is the theoretical reflection coefficient deduced from the impedance model of the known load number n and R Tn is the reflection coefficient deduced from Eq. (3) with the measured transfer function H mn . The dependency of R Tn with respect to H ae and Z s appears from Eq. (3).

Separating the characteristics of the source into real and imaginary parts :

the optimum values of H ae and Z s are obtained solving the minimization problem expressed as : min

This non-quadratic minimization problem is solved for each frequency with Matlab c unconstrained nonlinear optimization solver "fminsearch" (based on the Nelder-Mead simplex algorithm). At the smaller frequency, the starting point of the algorithm is obtained from Eqs. [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. i. theory[END_REF]. The frequencies are then considered in increasing order, with a starting point defined by the optimum obtained at the previous frequency.

Appendix B

In this appendix, a brief description of the source design is given. The air-tightness of the source is critical and has been carefully checked.

Appendix C

In this appendix, the calibration process is presented. Seven known loads are used (given Table 2). They are made of PVC tubes having the same diameter as the source tube, with a first side end open (for the connection with the source) and the second side end depending on the load number. The second side end can be open, or closed, or terminated by a drilled lid. They are described by means of their theoretical impedances at x = L+x m . Theoretical impedances are set to zero for a closed tube, and for an unflanged open tube, we adopt the classical approximation ( [START_REF] Chaigne | Acoustics of Musical Instruments[END_REF] Eq. (12.133)) recalled Table 2. For the last load, number 7, terminated by a drilled lid, we express the theoretical impedance at the microphone position as:


T Krajewski

A Tanasa

D L Vu

Wigner law for matrices with dependent entries -a perturbative approach

Keywords: Wigner law, replica trick, Feynman diagrams, zero-dimensional quantum field theory

We show that Wigner semi-circle law holds for Hermitian matrices with dependent entries, provided the deviation of the cumulants from the normalised Gaussian case obeys a simple power law bound in the size of the matrix. To establish this result, we use replicas interpreted as a zerodimensional quantum field theoretical model whose effective potential obey a renormalisation group equation.

Introduction

In the last decade or so, several extensions of Wigner law for matrices with dependent entries have been considered, see for instance [START_REF] Schencker | Semicircle law and freeness for random matrices with symmetries or correlations[END_REF]. In this paper, the authors impose bounds on the number of entries of the matrices that are correlated. In [START_REF] Hofmann-Credner | Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices[END_REF], a generalisation of this result was proven. Our approach is complementary: we do not impose such restrictions but we assume that the size of the correlations go to zero as N , the size of the matrix, becomes large. Another result concerning Wigner law for matrices with dependent entries have been obtained in [START_REF] Friesen | The Semicircle Law for Matrices with Dependent Entries[END_REF], where the matrices considered are real-valued, symmetric and have stochastically independent diagonals.

To establish our result we use the replica method. For a standard reference on the use of replica techniques in the context of random matrix theory, we refer the interested reader to [START_REF] Kanzieper | Replica Approach in Random Matrix Theory[END_REF], see also [START_REF] Zee | Law of addition in random matrix theory[END_REF] or [START_REF] Cicuta | Real symmetric random matrices and replicas[END_REF]. In this paper we interpret replicas as fields of a zero-dimensional quantum field theoretical model and use an analogue of the renormalisation group equation. We give conditions on the joint cumulants of the entries of the matrix, see Theorem 1, under which the moments of the eigenvalue distribution converges towards the Wigner semi-circle law.

For the sake of completeness, let us also mention that Wigner law has been studied by mathematical physicists using the so-called supersymmetric technique (see, for example, [START_REF] Disertori | Random Matrices and the Anderson Model[END_REF] and [START_REF] Gurau | The Wilson loop in the Gaussian Unitary Ensemble[END_REF]), technique which uses calculus over commuting and anti-commuting variables.

In the first part of the paper, we present our result and illustrate it for Wigner matrices and for invariant matrices. The second part is devoted to the proof.

2 Statement of the main result; a few illustrations

Semi-circle law from a condition on cumulants

Let us consider a probability law on Hermitian N × N matrices given by the joint probability density ρ N for the real diagonal elements and the complex upper diagonal ones, the lower diagonal ones being recovered by complex conjugation. We assume that the joint cumulants exist for all N and collect them in their generating function defined as

log exp Tr(M J) = log dM ρ N (M ) exp Tr(M J) (1) i j k l i j (M ij ) 2 M jk M ll c M ij M ji c
Figure 1: Oriented graphs associated to some cumulants

where dM is the product of the Lebesgue measures on the entries of M . The source J is another Hermitian N × N matrix and the cumulants are obtained by derivation with respect to J at the origin,

∂ ∂J j 1 i 1 . . . ∂ ∂J j k i k log exp Tr(M J) J=0 = M i 1 j 1 • • • M i k j k c , (2)
where we used the subscript c because in field theory they correspond to connected correlation functions. Alternatively, the moments are denoted by

M i 1 j 1 • • • M i k j k .
Let us associate to each cumulant an oriented graph G constructed as follows. The vertices of G are given by the distinct indices appearing in the cumulant. Let us emphasize that it is fundamental that the indices attached to the vertices are all different, this can be achieved by inserting 1 = δ ij + 1 -δ ij for every pair of indices. We draw an arrow oriented from the vertex associated to i to the vertex associated to j if the cumulant involves a matrix element M ij , see figure 1 for a few examples. Then, we consider the cumulants as functions C G on the vertex indices, labelled by the graphs

G, i 1 , ..., i v(G) ∈ {1, . . . , N } v(G) → C G (i 1 , ..., i v(G) ), with v(G)
the number of vertices of G. The generating function of the cumulants can be written as [START_REF] Friesen | The Semicircle Law for Matrices with Dependent Entries[END_REF] where s(e) and t(e) are the source and the target vertices of e. For a fixed graph, we consider C G as a function of the indices attached to the vertices. For example, in the Gaussian case ρ(M ) ∝ exp -1 2α 2 Tr M 2 , the only non vanishing cumulant is M ij M kl c = α 2 δ il δ jk and corresponds either to an oriented cycle with two vertices (when i = j) or to a graph with a single vertex and two edges (when i = j).

log exp Tr(M J) = G oriented graph 1 |Aut(G)| 1≤i 1 ,...,i v(G) ≤N all different C G (i 1 , . . . , i v(G) ) e edge J i s(e) i t(e)
In this letter, we are interested in the normalised density of eigenvalues of a random N × N Hermitian matrix

ρ N (λ) = 1 N 1≤i≤N δ(λ -λ i ) , (4)
where λ i are the eigenvalues of the random matrix M √ N and δ the Dirac distribution. The average is computed using a probability density ρ N (M ). Note that we have used the same letter to denote the density of eigenvalues ρ N (λ) and the probability law on the space of matrices ρ N (M ) in order to simplify the notations.

To state our main result, we impose different conditions on the cumulants, depending on whether G is Eulerian or not. Recall that an oriented graph G is Eulerian if every vertex of G has an equal number of incoming and outgoing edges Theorem 1. Let ρ N be a probability law on the space of Hermitian N × N matrices M such that its cumulants can be decomposed as C G = C G + C G , with C G a Gaussian cumulant and C G a perturbation such that, uniformly in the vertex indices i 1 , ..., i v(G) ,

• lim

N →∞ N v(G)-c(G)-e(G)/2 C G (i 1 , ..., i v(G) ) = 0 if G is Eulerian, • N v(G)-c(G)-e(G)/2 C G (i 1 , ..., i v(G) ) is bounded if G is not Eulerian,
where v(G), e(G), c(G) are the number of vertices, edges and connected components of G. Then, the moments of the eigenvalue distribution of the matrix M √ N converge towards the moments of the semi-circle law, with α given by the Gaussian cumulant

M ij M kl c = α 2 δ il δ jk , lim N →∞ R dλ λ k ρ N (λ) = 1 2πα 2 2α -2α dλ λ k √ 4α 2 -λ 2 if k is even, 0 if k is odd. (5)
The conditions are uniform in the sense that they must not depend on the vertex indices but can depend on the graph. These conditions can be understood heuristically using the moment method proof of Wigner's law (see for instance [START_REF] Anderson | An Introduction to Random Matrices[END_REF])

R dλ λ k ρ N (λ) = 1 N 1+k/2 Tr M k = 1≤i 1 ,...,i k ≤N M i 1 i 2 M i 2 i 3 • • • M i k i 1 . (6)
Next, we express the moments in terms of the cumulants and label the latter by the oriented graphs G. The trace imposes the existence of an Eulerian cycle in these graphs, which always exists for a connected graph such that all its vertices have an equal number of incoming and outgoing edges. If not, some identifications of vertices are necessary, i.e. the vertex indices i 1 , . . . , i k cannot all be distinct. Then, the conditions on the cumulants imply that only the contribution of the Gaussian one survives in the limit N → ∞. Also note that in the case of real symmetric matrices, the condition for Eulerian graphs apply to all graphs.

Illustration 1: Semi-circle law for Wigner matrices

Recall that for Wigner matrices the diagonal elements are independent and identically distributed (iid =)and the real and imaginary parts of the upper diagonal elements are also iid, independent from the diagonal ones but possibly with a different law, such that the expectation value of the off diagonal elements vanish. We further assume that all moments (thus also cumulants) remain finite as N becomes large. Under these assumptions, Wigner's seminal result follows from Theorem 1.

Corollary 1. For Wigner matrices with finite moments, the eigenvalue distribution converges in moments towards the semi-circle law (5) when N → ∞, with α given by the Gaussian cumulant.

Proof. Since the matrix elements are independent, all cumulants with v ≥ 3 vanish. Therefore, we have to check the bounds of Theorem 1 for v = 1, 2 only. For v = 1 the condition is obviously satisfied because of the factor N e/2 and the fact that moments and therefore also cumulants are bounded. Furthermore, if v = 2 and e ≥ 3, the condition is also satisfied, for the same reason.

The case v = 2 and e = 1 corresponds to the expectation values of the off-diagonal terms and vanish identically. The case v = 2 and e = 2 remains to be studied:

• With c = 2, C i j (i, j) = M ii M jj c = M ii M jj -M ii M jj = 0, because diagonal matrix elements are independent. • With c = 1, non Eulerian, C i j (i, j) = M ij M ij c = M ij M ij -M ij M ij = ( M ij ) 2 - ( M ij ) 2 + 2i M ij M ij = 0,
since the real and imaginary parts of the off diagonal elements are independent and identically distributed with mean 0.

• With c = 1, Eulerian, C i j = α 2 is the Gaussian cumulant leading to the semi-circle law. Note that only the part i = j contributes at large N .

Illustration 2: Unitary invariant potential

Let us consider a single trace unitary invariant potential. In this case, the probability law is

ρ(M ) = exp -Tr V (M ) Z with Z = dM exp -Tr V (M ) (7)
and V (M ) a polynomial potential

V (M ) = 1 2 M 2 + p≥3 g p N 1-p/2 p M p . (8)
Recall that here we study the eigenvalue distribution of the matrix M/ √ N , the usual formulation employed in the physics literature being recovered after the rescaling M → √ N M . Cumulants of order k can be computed as sums of connected ribbon graphs Γ (not to be confused with the graphs G appearing in the cumulants) with k univalent vertices corresponding to the insertions of the source J. Its dependence on N reads N f (Γ)+ p (1-pvp(Γ)/2) where f (Γ) is the number of closed faces of the graph (closed cycles in the double line representation) and v p (Γ) the number of vertices of degree p.

Denoting by e(Γ) the number of internal edges of the graph Γ (edges not attached to the sources), we have 2e(Γ) + k = p pv p (Γ). Moreover, the Euler characteristics of the surface with boundary in which the graph is embedded reads 2

-2g(Γ) -b(Γ) = f (Γ) -e(Γ) + v(Γ), with v(Γ) = p v p (Γ)
the total number of vertices (not including the sources), g(Γ) the genus of the surface and b(Γ) its number of boundary components (open faces including insertions of the source). Combining these identities together, the power of N in a graph contributing to a cumulant reads

N 2-2g(Γ)-b(Γ)-k/2 . (9)
The leading order contribution is obtained for planar graphs (g = 0) with all sources in the same open face (b = 1). These cumulants correspond to graphs G that are oriented cycles with e(G) = v(G) = k and c(G) = 1. According to [START_REF] Gurau | The Wilson loop in the Gaussian Unitary Ensemble[END_REF], they scale as N v(G)-c(G)-e(G)/2 for large N , thus violating the first condition on the cumulants in Theorem 1. Since it is known that such random matrices do not obey the semi-circle law if V is not Gaussian, we conclude that the conditions in Theorem 1 are optimal, at least when formulated using vertices, edges and connected components of G.

3 Replica proof of the main result

An expression of the Green function using replicas

Using ρ(λ) = -1 π G(λ + i0 + ), the density of eigenvalues is determined by the Green function

G(z) = 1 N Tr z - M √ N -1 = 1 N dM ρ N (M ) Tr z - M √ N -1 (10)
To compute the resolvent, first notice that Tr z -

M √ N -1 = ∂ ∂z log det z -M √ N .
Then, the resolvent is computed using bosonic replicas

G(z) = - 1 N ∂ ∂z dX † dX exp -zTr(X † X) exp Tr X † M √ N X order 1 in n ( 11
)
where X is a N × n complex matrix and dX † dX is the product of Lesbegue measure over real and imaginary parts of X divided by a factor of π nN . Although the use of the replica technique is common in physics, let us give some explanations. We first express det -n z -M √ N using a Gaussian integral over n complex vectors, each with N components that we collect in the Then, its logarithm is computed using the identity A n = 1 + n log A + O(n 2 ) for n → 0. In our context, we evaluate the integral [START_REF] Zinn-Justin | Quantum field theory and critical phenomena[END_REF] as a perturbation of a Gaussian integral using Feynman diagrams, thus leading to a power series in 1/z. Because of the U(n) invariance X → XU of the integral in [START_REF] Zinn-Justin | Quantum field theory and critical phenomena[END_REF], each Feynman amplitude is a polynomial in n and we retain only the term of order n. In perturbation theory, this is nothing but a convenient substitute for the power series of log det z -M √ N . Here, we stick to the perturbative approach but it is worthwhile to mention that, beyond perturbation theory, one encounters the phenomenon of replica symmetry breaking, ruining the simple polynomial dependence on n, see for [START_REF] Kamenev | Level correlations in disordered metals: The replica sigma model[END_REF].

Diagrammatic approach for the Gaussian case

Before proceeding to the general case and establish Theorem 1, let us consider the Gaussian case ρ(M ) ∝ exp -Tr(M 2 ) 2α 2 . Performing the integral over M leads to a quartic interaction for the replicas

exp Tr X † M √ N X = exp α 2 2N Tr(X † XX † X). (12)
Then, we expand the integral in (11) using Feynman diagrams. The latter are ribbon graphs with double lines made of a solid line for the matrix indexes i, j, k, ... ∈ {1, ..., N } and a dotted line for the replica indexes a, b, c, ... ∈ {1, ..., n}. The graphs that correspond to the terms of order n are those with a single dotted face. Moreover, when we take the large N limit, we only retain the planar graphs. Each solid face yields a factor of N which is cancelled by the factor of 1/N in the vertices and in front of the integral [START_REF] Zinn-Justin | Quantum field theory and critical phenomena[END_REF]. Note that it is crucial to select the term of order 1 in n before taking the large N limit. In this example, it is also easy to see that at each order in 1/z we have only a finite number of graphs, thus yielding a polynomial in n. They consist in several solid faces such that each two faces share at most one vertex. Moreover, these faces are enclosed in a single dotted face, in particular all these graphs are connected. Furthermore, taking the derivative with respect to z inserts a cilium on one of the edges and removes all symmetry factors. The sum over graphs with a cilium is nothing but G(z) and obeys the equation, in the large N limit,

G(z) = ∞ k=0 α k G k (z) z k+1 = 1 z -αG(z) . ( 13
)
Indeed, if we remove the solid face containing the cilium, we get k copies of G(z), where k is the number of vertices of that face, see figure 3. The solution that behaves as 1/z for large z is

G(z) = z 2α 2 1 -1 - 4α 2 z 2 . ( 14
)
Finally, from the cut of the square root on the negative real axis, we obtain Wigner semi-circle law (5) in the large N limit.

→

Figure 3: A graph to G(z) in the large N limit and its decomposition

Cumulants and the replica effective action

When we substitute J = XX † √ N , the generating function of the cumulants plays the role of an interacting potential for the replicas V 0 (X,

X † ) = log exp Tr M XX † √ N
, so that [START_REF] Zinn-Justin | Quantum field theory and critical phenomena[END_REF] writes

G(z) = - 1 N ∂ ∂z dX † dX exp -zTr(X † X) + V 0 (X, X † ) order 1 in n (15)
Although the expansion of V 0 (X, X † ) in powers of X, X † (see ( 3)) involves a power series, at any order in 1/z, only a finite number of cumulants C G appear. This expansion is based on Feynman diagrams, treating V 0 (X, X † ) as a perturbation. Motivated by the quantum field theory analogy, we introduce the effective potential

V (t; X, X † ) = log dY † dY exp - Tr(Y † Y ) t + V 0 (X + Y, X † + Y † ) -N n log t (16)
where we have set t = 1/z for later convenience. The normalisation of the measure only contains a factor of π N n and we added an extra contribution of -N n log t in such a way that the Gaussian integral is fully normalized. At t = 0 (equivalently for z → ∞), we have V (t = 0; X, X † ) = V 0 (X, X † ), since there is no integration over Y and Y † . The effective potential obeys the Gaussian convolution identity, see [START_REF] Zinn-Justin | Quantum field theory and critical phenomena[END_REF],

V (t + s; X, X † ) = log dY † dY exp - Tr(Y † Y ) s + V (t; X + Y, X † + Y † ) -N n log s (17)
Expanding to first order in s yields the differential equation, see also [START_REF] Zinn-Justin | Quantum field theory and critical phenomena[END_REF],

∂V ∂t = i,a ∂ 2 V ∂X i,a ∂X i,a + ∂V ∂X i,a ∂V ∂X i,a , (18)
where 1 ≤ i ≤ N is a matrix index and 1 ≤ a ≤ n a replica index. This is nothing but a zero dimensional analogue of the Polchinski exact renormalisation group equation [START_REF] Polchinski | Renormalization and Effective Lagrangians[END_REF] that governs the flows of effective actions in quantum field theory. Equivalently, (18) can be written in integral form,

V (t; X, X † ) = V 0 (X, X † ) + t 0 ds i,a ∂ 2 V (s; X, X † ) ∂X i,a ∂X i,a + ∂V (s; X, X † ) ∂X i,a ∂V (s; X, X † ) ∂X i,a . (19)
Solving this integral equation iteratively proves to be helpful to establish results order by order in powers of t = 1 z . The Green function G(z) and thus also the eigenvalue density can be expressed in terms of the effective potential. Indeed, let us rewrite (16) as

dY † dY exp -zTr(Y † Y ) + V 0 (Y, Y † ) = z -nN exp V (t = 1/z; X = 0, X † = 0). ( 20
)
Deriving this equation with respect to z and using ( 18) and ( 11), we obtain

G(z) = 1 z + 1 N z 2 i,a ∂ 2 V (1/z; 0, 0) ∂X i,a ∂X i,a + ∂V (1/z; 0, 0) ∂X i,a ∂V (1/z; 0, 0) ∂X i,a order 1 in n (21)
In order to compute the RHS of (15), let us expand it in powers of X, X † . The effective potential can be developed on oriented graphs G as in (3), which we recover for t = 0,

V (t; X, X † ) = G oriented graph 1 |Aut(G)|N e(G)/2 1≤i 1 ,...,i v(G) ≤N all different C G (t; i 1 , . . . , i v(G) ) e edge (XX † ) i s(e) i t(e) . (22)
The oriented graphs G in the previous equation should not be confused with Feynman diagrams appearing in a perturbative computation, they are merely labels for the terms in the generating functions of the cumulants and in the effective potential.

In particular, only the first non trivial term in this expansion contributes to the Green function, as seen from ( 15),

G(z) = 1 z + 1 N 3/2 z 2 1≤i≤N C i (1/z; i) order 0 in n . ( 23
)
Notice that only the order 0 in n is necessary since an extra power of n is created by the summation over a.

Inductive bounds on Green functions

In order to establish Theorem 1, let us recall that the cumulants are written as a sum of a Gaussian term and a perturbation. This translates into a similar decomposition for the potential V (t = 0; X, X † ) = V (t = 0; X, X † ) + V (t = 0; X, X † ), with V (t = 0; X, X † ) = α 2

2N Tr(X † XX † X) the quartic interaction derived from the Gaussian cumulant and V (t = 0; X, X † ) a perturbation. Then, solving iteratively the integral equation (19) as a power series in t yields a similar decomposition for the effective potential V (t; X, X † ) = V (t; X, X † ) + V (t; X, X † ), where V (t; X, X † ) only involves the quartic interaction V (t = 0; X, X † ) whereas V (t; X, X † ) contains at least one occurrence of the perturbation, in particular, it vanishes if V (t = 0; X, X † ) = 0.

Furthermore, let us decompose both terms at order 0 in n using oriented graphs as in ( 22), so that, as a power series in t,

C G (t; i 1 , . . . , i v(G) ) = ∞ k=0 t k C (k) G (t; i 1 , . . . , i v(G) ) + C (k) G (t; i 1 , . . . , i v(G) ) . (24)
Using (19), we show by induction on k that the perturbation obeys a bound identical to the assumption of Theorem 1.

Proposition 1. The coefficients of the development of the effective potential over graphs satisfy, uniformly in i 1 , ..., i v(G) ,

• N v(G)-c(G)-e(G)/2 C (k) G (i 1 , ..., i v(G) ) order 0 in n is bounded for any G, • lim N →∞ N v(G)-c(G)-e(G)/2 C (k) G (i 1 , ..., i v(G) ) = 0 order 0 in n if G is Eulerian, • N v(G)-c(G)-e(G)/2 C (k) G (i 1 , ..., i v(G) ) order 0 in n is bounded if G is not Eulerian. ∂ 2 V ∂X i,a ∂X i,a : v G → v G ∂V ∂X i,a ∂V ∂X i,a : v 1 G 1 v 2 G 2 → v 1 = v 2 G Figure 4: Graphical interpretation of the RHS of (19)
Proof. At order 0, the conditions are satisfied by the Gaussian cumulant [START_REF] Polchinski | Renormalization and Effective Lagrangians[END_REF] and the non Gaussian ones since these are just the assumptions of Theorem 1. Let us assume that the conditions hold up to order k -1 and use (19) to show that they also hold at order k.

The derivative with respect to X i,a (resp. X i,a ) acting the graph expansion (24) removes an outgoing (resp. incoming) half line attached to a vertex labelled i. This operation is performed either on a single graph (first term in (19)) or on two independent graphs (second term in (19)). The subsequent summation over i and a reattaches the remaining half lines, see picture 4. Collecting all contributions to a graph appearing on the LHS of (19) (order k) to those appearing on its RHS (order < k) allows us to express an order k term using order < k terms, all to order 0 in n. In the sequel, we denote by d + (v) (resp. d -(v)) the number of incoming (resp. outgoing) edges to a vertex v.

Let us consider the first operation on a vertex v of a graph G on the RHS of (19) and denote by G the resulting graph. First note that this operation does not change the nature of the graph (Gaussian or not, Eulerian or not). If d + (v) = 0 or d -(v) = 0, then the derivation yields 0. Moreover, we always have v( G) -c( G) ≤ v(G) -c(G). This can be compensated by a summation over i if d + (v) = d -(v) = 1 and both ends belong to the same edge.However, in this case the replica index a is free and the summation over it yields a factor of n. Since we are only concerned with the order 0 in n, such a term does not contribute. Therefore, we conclude that the contribution of the first term of the LHS of (19) to C In the case of the second operation, let us denote by G 1 and G 2 the two graphs on which the operation is performed and by v 1 the vertex of G 1 (resp. v 2 the vertex of G 2 ) where we derive with respect to X i,a (resp. X i,a ) and by G the result of this operation. If d -(v 1 ) = 0 or d + (v 2 ) = 0. the derivative vanish. In the remaining cases, let us first observe that v(G) -c(G) ≤ (G 1 ) -c(G 1 ) + v(G 2 ) -c(G 2 ). Then, the following combinatorial lemma, whose proof is elementary, is helpful. As a consequence of this lemma, G is Eulerian if and only if G 1 and G 2 both are, since the operation only modifies the valence of the vertices V 1 and V 2 . Moreover, in the particular case d -(v 1 ) = d + (v 2 ) = 1, the inequality is strict and the extra power of N cancels the power arising from the summation over i.

However, it is not the graph G that contributes directly to the LHS of (19) to C Finally, Theorem 1 follows from (23) since in the large N limit the non Gaussian contribution to the Green function vanishes in the large N limit, N -1/2 C i (1/z, i) order 0 in n → 0, as a consequence of proposition 1, order by order in 1/z.

2 δFigure 2 :

 22 Figure 2: Feynman rules in the Gaussian case

  , ..., i v(G) ) and C (k) G (i 1 , ..., i v(G)) also obey the bound.

Lemma 1 .

 1 If G is a connected oriented graph, then G is either Eulerian or has at least two vertices such that d + (v) = d -(v).

G(i 1

 1 , ..., i v(G) ) since some indices on the vertices of G 1 may be equal ton some indices on the vertices of G 2 , leading to the identification of some vertices in G. If we denote the resulting graph by G, thenv( G) -c( G) ≤ v( G) -c( G).Moreover, an application of lemma 1 shows that the inequality is strict if G is Eulerian and G is not. Then, we may conclude that the contribution of the second operations to to C (k) G (i 1 , ..., i v(G) ) and C (k) G (i 1 , ..., i v(G) ) also obey the bound.

Acknowledgements

The authors are partially supported by the grant ANR JCJC "CombPhysMat2Tens". AT is partially supported by the grant PN 16 42 01 01/2016.