Brice Saint-Michel

Valeria Garbin
email: v.garbin@tudelft.nl

Bubble dynamics for broadband microrheology of complex fluids

Keywords: Bubbles, gas diffusion, ultrasound, cavitation, extensional rheology, high-frequency rheology, microrheology

Bubbles in complex fluids are often desirable, and sometimes simply inevitable, in the processing of formulated products. Bubbles can rise by buoyancy, grow or dissolve by mass transfer, and readily respond to changes in pressure, thereby applying a deformation to the surrounding complex fluid. The deformation field around a stationary, spherical bubble undergoing a change in radius is simple and localised, thus making it suitable for rheological measurements. This article reviews emerging approaches to extract information on the rheology of complex fluids by analysing bubble dynamics. The focus is on three phenomena: changes in radius by mass transfer, harmonic oscillations driven by an acoustic wave, and bubble collapse.

These phenomena cover a broad range of deformation frequencies, from 10 -4 to 10 6 Hz, thus paving the way to broadband microrheology using bubbles as active probes. The outstanding challenges that need to be overcome to achieve a robust technique are also discussed.

Introduction

Bubbles are ubiquitous in the processing and structuring of complex fluids [START_REF] Everitt | Bubble dynamics in viscoelastic fluids with application to reacting and non-reacting polymer foams[END_REF][START_REF] Pagani | Competing mechanisms and scaling laws for carbon nanotube scission by ultrasonication[END_REF]. They may be undesirable and lead to poor product performance, for instance contamination of personal care products [START_REF] Lin | Mechanisms and control of gas bubble formation in cosmetics[END_REF], or they may be key to imparting unique properties to advanced materials [START_REF] Fujii | Stimuli-responsive bubbles and foams stabilized with solid particles[END_REF]. Bubbles are extraordinarily dynamic objects [START_REF] Lohse | Bubble puzzles: From fundamentals to applications[END_REF]: they can grow or shrink by gas diffusion, they can rise due to buoyancy, they can undergo breakup and coalescence. Owing to the compressibility of their gas core, they respond to changes in pressure, and can gently oscillate in response to acoustic waves or violently collapse in response to shock waves. Because the control of their number and size distribution is crucial in the processing and structuring of complex flu-ids, bubble dynamics have been studied extensively with a view to improve the stability and performance of formulated products [START_REF] Kloek | Effect of bulk and interfacial rheological properties on bubble dissolution[END_REF].

A useful feature of bubble dynamics is that, provided that the bubble remains spherical, the deformation field in the surrounding fluid is purely extensional [START_REF] Macosko | Rheology: Principles, Measurements and Applications[END_REF]. This feature presents a unique opportunity to gain insights into the rheological properties of the complex fluid surrounding a bubble, because the kinematics of the deformation is completely prescribed, simple and localised -such controlled conditions are necessary to perform a rheological measurement. A measurement of the applied stress, in combination with controlled material deformation, is also required to perform a rheological measurement. Analysis of bubble dynamics provides information to indirectly obtain the stress in the surrounding medium, with some assumptions on the constitutive behaviour [START_REF] Dollet | Bubble dynamics in soft and biological matter[END_REF]. In addition, the deformation time scales of bubble dynamics cover a very broad range that is normally not accessible to a single rheological technique: from the slow dynamics (∼ 10 4 s) driven by gas diffusion [START_REF] Epstein | On the stability of gas bubbles in liquid-gas solutions[END_REF] to the extremely fast dynamics (∼ 10 -6 s) during bubble collapse [START_REF] Brenner | Single-bubble sonoluminescence[END_REF].

The idea of using spherical bubble deformation to probe the rheology of the surrounding material is in fact a classical concept [START_REF] Macosko | Rheology: Principles, Measurements and Applications[END_REF], but its full potential has remained largely unmet due to technical limitations in controlling bubble formation and subsequent deformation. Advances in experimental techniques to generate, control and deform single bubbles, and image their evolution in real time [START_REF] Versluis | High-speed imaging in fluids[END_REF], as well as advances in imaging of the microstructure of complex fluids, are now bringing this new class of methods to the forefront of soft matter and rheology [START_REF] Barney | Cavitation in soft matter[END_REF]. This article focuses on the recent emerging applications of bubble dynamics to the rheology of complex fluids, and highlights the current gaps that still need to be addressed to make it an accessible technique for broadband microrheology of complex fluids.

Controlled deformation by bubble dynamics

Dynamics of spherical bubble deformation

In this paper we focus on three dynamical phenomena that can result in the spherically symmetric deformation of a stationary, isolated bubble: changes in radius by mass transfer, harmonic oscillations of the radius driven by an acoustic wave, and bubble collapse. We do not discuss the the technique called "cavitation rheology", based on bubble formation by injection at the tip of a needle embedded in a soft material, which is in a more advanced state of development and has already been widely adopted, as described in a recent review [START_REF] Barney | Cavitation in soft matter[END_REF]. One advantage of the three phenomena reviewed here is that the bubble is isolated (not attached to a needle) and therefore its spherical deformation results in a purely extensional deformation of the surrounding medium, which will be described in Section 2.2. The deformation is also localised to a region around the probe, decaying quickly away from the surface of the bubble, thus satisfying a necessary criterion for applications in microrheology [START_REF] Furst | Microrheology[END_REF].

The rate of deformation of the material surrounding the bubble covers several orders of magnitude between the three phenomena considered. Firstly, changes in radius by mass transfer, that is bubble growth or dissolution by exchange of gas with the surrounding medium, can occur at rates from 1 s -1 to 10 -4 s -1 or even slower. The analysis of the change in radius of a bubble that is dissolving or growing by mass transfer, to extract information on the rheological properties of the surrounding medium, will be described in Section 3. Secondly, harmonic oscillations of a bubble can be driven by an acoustic field that is close to the resonance frequency of the bubble. In the simplest case of a Newtonian fluid, and if surface tension effects are negligible, this is given by the Minnaert frequency,

ω M = 1 R 0 3κp 0 ρ , (1)
where R 0 is the equilibrium bubble radius, ρ is the fluid density, p 0 the hydrostatic pressure, and κ the polytropic exponent (see Section 4.1). Bubbles with radius ranging from 10 -3 m down to 10 -6 m therefore respond to acoustic waves with frequencies from 10 3 Hz to 10 6 Hz. The methods currently being developed to exploit this phenomenon for rheological measurements are described in Section 4. Finally, bubble collapse is a violent phenomenon where the bubble radius can change by roughly 10 times its equilibrium value on a time-scale of tens of microseconds. The extremely large strain rates associated with bubble collapse, up to 10 6 s -1 , can be used to probe the high-strain rate rheology of soft materials, as will be described in Section 5.

Diffusive transport followed by cavitation have been shown to span 9 orders of magnitude of deformation rate of the surrounding material in a single experiment [START_REF] Bruning | Turning drops into bubbles: Cavitation by vapor diffusion through elastic networks[END_REF]. The range of frequencies accessible by these phenomena therefore paves the way to a new class of techniques for broadband, active microrheology of complex fluids.

How a bubble deforms the surrounding fluid

A spherical bubble at a fixed position and with time-dependent radius R(t) generates a purely radial velocity field. With reference to a spherical coordinate system with origin at the centre of the bubble, where r, θ, and φ are the radial, azimuthal, and polar coordinates respectively, the deformation, or strain, at time t material elements in the surrounding medium. This is in contrast with simple shear flow, which combines pure extension and rotation. In the case of shear flow, the strain rate ˙ is called the shear rate, and it is used to compute the shear stress in liquids; in particular, they are proportional to each other in the case of Newtonian media.

In soft solids and yield-stress fluids, the stress is also a function of the total strain applied to the material elements of the surrounding medium. We can compute such a strain for an element initially at a distance r 0 from the centre of the bubble at rest, i.e. for R = R 0 . This quantity is also a tensor and most continuum mechanics textbooks recommend using the Finger tensor B to quantify it [START_REF] Macosko | Rheology: Principles, Measurements and Applications[END_REF]:

B rr = B -2 θθ = B -2 φφ =       1 + R(t) 3 -R 3 0 r 3 0       -4/3 (4)
In the absence of deformation, the Finger tensor is not zero but rather reduces to the identity tensor I. In the particular case of pure extension, we can relate the components of the Finger tensor B to the linear stretch of the fluid elements. We have in particular, for the material elements of Figure 1(a), B rr = [l(t)/l 0 ] 2 .

Figure 1(a) shows the properties of this strain field respectively for a bubble at rest (left panel), during its expansion (centre) and during its contraction (right panel). Both the particle displacement and the net strain fluid or a soft solid (light blue) when the former is at rest (left), expands (centre) and contracts (right). The bubble surface motion pushes and pulls tracers (small circles) in the radial direction. The tracer displacement is larger close to the bubble surface (as seen on the thick grey streak lines). Material elements (boxes around the tracers) are also strained and undergo biaxial (middle) or uniaxial extension (right) when the bubble grows or shrinks. The colour of the boxes codes the stretch l(t)/l 0 applied to the fluid particles in the radial direction. (b) Time evolution of the radius of a bubble with initial radius R 0 due to mass transfer out of the bubble for a gas-saturated medium (ζ = 1) and into the bubble for a supersaturated medium (ζ = 1.025). (c) Linear oscillations of the bubble radius, R(t), around the equilibrium radius R 0 , under a small-amplitude acoustic excitation at angular frequency ω. (d) A bubble with initial radius 3R 0 undergoes violent collapse and subsequent rebounds, before reaching its equilibrium radius R 0 . In (b-d) the surrounding fluid is assumed to be Newtonian when G = 0 Pa and otherwise combines neo-Hookean elastic and Newtonian viscous elements.

B -I are more sensitive to bubble expansion than compression and decay quickly away from the surface of the bubble.

Bubble dynamics: relating strain and stress fields

Both the local strain and the stress fields need to be known for bubble dynamics studies to qualify as a rheological measurement. While torque is easily measured on a rotational rheometer and readily related to the local stress in the sample, we do not directly measure the stress field around bubbles. We instead rely on the momentum balance of the whole fluid surrounding the bubble in the e r direction, which reads [START_REF] Dollet | Bubble dynamics in soft and biological matter[END_REF]:

ρ R R + 3 2 Ṙ2 = ∞ R (∇ • σ) r dr . (5)
The tensor σ is the Cauchy stress tensor. As it includes pressure terms, its trace is non-zero. Far away from the bubble, we assume the material is at rest, which implies that the diagonal terms of σ are all equal to -p ∞ (t), the total (ambient and acoustic) pressure applied by the operator far away from the bubble. The spherical symmetry of the problem and the absence of torque applied to the bubble implies that σ θθ = σ φφ .

Equation 5 relates an integral of the non-homogeneous stress field (right hand side) to the bubble dynamics (left hand side), yet the local stress distribution σ remains unknown, preventing any direct measurement of the material rheological properties. We therefore need to choose a priori the relation between stress and strain in the surrounding medium and try to accurately model the experimental bubble dynamics R(t). For Newtonian liquids and ideal (neo-Hookean) elastic solids, these relations read:

σ = -pI + η˙ (Newtonian liquid) , (6)
σ = -pI + G (B -I) (neo-Hookean solid) , (7)
where p is a pressure term that depends on r and matches the classical definition of pressure for arbitrary strains in Newtonian liquids, and only for small strains in neo-Hookean solids. The classical Newtonian viscosity η and linear elastic modulus G are considered to be free parameters of the models that are fitted to the experimental data R(t). For more complex surrounding materials, such as visco-elastic liquids or solids, the stress and strain fields are related using more complex constitutive equations relating σ, ˙ and B and additional fitting parameters.

In the specific case of Newtonian liquids and purely elastic solids, all the physical quantities in Equation 5are known or prescribed except for the pressure field p(r). Knowing the whole pressure field is however not needed, as the right hand side term of Equation 5 can be rewritten in the following way:

∞ R (∇ • σ) r dr = -p ∞ (t) -σ rr (R) + 2 ∞ R σ rr -σ θθ r dr . (8)
In Equation 8, the pressure contribution cancels in σ rr -σ θθ and is hence only present in the radial stress at the bubble boundary σ rr (R). This quantity can be independently solved by considering the stress balance at the bubble interface, which reads:

σ rr (R) = -p g + 2γ R , (9)
γ being the surface tension between the gas inside the bubble and the surrounding material, and p g being the gas pressure in the bubble. The final step in solving Equation 5 then consists in choosing an equation of state for the gas pressure as a function of its volume 4πR(t) 3 /3, its temperature and its composition.

Bubbles injected using a syringe and containing air or other non-condensible gases are usually treated as ideal gases with a homogeneous composition and are fairly simple to model. In contrast, bubbles formed through laser vaporisation of the medium are modelled as a mixture of vapour that quickly condenses at the bubble surface and an inert gas, which leads to composition gradients between the centre and the bubble edge, which necessitates additional modelling steps [START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF].

Bubble dissolution or growth by mass transfer

The dissolution or growth of gas bubbles due to mass transfer of gas from the surrounding medium

[Figure 1(b)
] is a relatively slow process for which Equation 5 reduces to the classical isostatic criterion,

∇ • σ = 0.
The saturation concentration of the dissolved gas in the liquid phase, c s , is proportional to its partial pressure in the bubble following Henry's law:

c s = k H p g . (10)
Henry's constant, k H , varies greatly between different gas/liquid pairs and depends on the temperature T .

If we suppose that the liquid has been left long enough in contact with the same gas before conducting the experiments, we can suppose an initial saturation condition, i.e. c(r, t = 0) = c 0 s = k H p 0 , where p 0 is the ambient pressure. For bubbles small enough to be strongly affected by surface tension effects, the pressure in the gas bubble p g is greater than p 0 , and the saturation concentration c s exceeds c 0 s at the bubble interface, driving bubble dissolution.

For a sufficiently slow dissolution rate, the additional dissolved gas simply diffuses into the surrounding medium with a diffusion coefficient D. We then need to write the mass balance of the bubble and provide boundary conditions at the bubble surface and at the outer edge of the surrounding medium to derive the time evolution of the bubble radius. We choose to work with an interface that is saturated with the solute based on the gas pressure in the bubble, i.e. c(R, t) = k H p g . This time, the infinite surrounding medium can be initially under-saturated or super-saturated with the dissolved gas, meaning that c(r → ∞, t) = c(r, t = 0) = ζc 0 s with ζ a dimensionless constant being smaller (respectively greater) than 1 for under-(super-) saturation conditions.

For a neo-Hookean elastic surrounding medium of linear modulus G, the time evolution of the bubble radius then satisfies [START_REF] Kloek | Effect of bulk and interfacial rheological properties on bubble dissolution[END_REF] 1 :

R 2 0 Dk H RT d R dt = ζ -1 -δ/ R + δ G (-5/4 + 1/ R + 1/4 R4 ) 1 + 2δ/3 R + δ G (5/4 -2/3 R + 1/12 R4 ) 1 R + R 0 √ πDt , (11)
in which R = R/R 0 , δ = γ/R 0 p 0 , δ G = G/p 0 , T is the temperature and R is the ideal gas constant.

Equation ( 11) is a non-linear, ordinary differential equation that can be solved numerically. Interestingly, bubble stability against dissolution more simply depends on the sign of the numerator in Equation [START_REF] Versluis | High-speed imaging in fluids[END_REF]. For an initially saturated medium (ζ = 1), the steady-state condition d R/dt = 0 reads:

1 4 R4 + 1 - γ R 0 G 1 R - 5 4 = 0 ( 12
)
For a finite shear modulus G, the left-hand side of Equation ( 12) is negative for R = 1 and tends to +∞ for R → 0 due to the very large positive contribution 1/4 R4 of elastic stresses for small bubbles. Consequently there always exists an equilibrium radius 0 ≤ R * ≤ R 0 in neo-Hookean solids at which dissolution stops.

For R = R * , the gas pressure in the bubble p g becomes equal to p 0 , leading to an equal solute concentration c 0 s at the bubble interface and far away from it. Neo-Hookean solids then always arrest dissolution, as seen in Figure 1(b). In contrast, stresses in a fluid medium do not diverge for R → 0 as they rather depend on the strain rate ˙ in the surrounding material, or, equivalently, in Ṙ. The gas pressure in the bubble may then always exceed p 0 provided that the bubble dissolution rate Ṙ/R is slow enough. Fluids are then incapable of halting bubble dissolution even though very viscous fluids may slow it down [START_REF] Kloek | Effect of bulk and interfacial rheological properties on bubble dissolution[END_REF].

When the surrounding medium is initially super-saturated with the dissolved gas, ζ ≥ 1, bubbles can grow by mass transfer from the surrounding medium. Elastic effects slow down this growth, and the resulting growth dynamics can be fitted to estimate both the super-saturation coefficient ζ and the elastic modulus of the medium, as shown in experiments by Ando and Shirota [START_REF] Ando | Quasistatic growth of bubbles in a gelatin gel under dissolved-gas supersaturation[END_REF]. Bubbles in a soft, viscoelastic solid were found to grow more slowly for increasing G despite similar super-saturation coefficients ζ, as shown in Recent experiments show that bubble dissolution can also be halted in yield-stress materials such as oleogels [START_REF] Saha | Stability of bubbles in wax-based oleofoams: decoupling the effects of bulk oleogel rheology and interfacial rheology[END_REF]. The situation is more complex in this case, as yield-stress materials behave as elastic solids for low applied stresses, yet flow above it. In the context of spherical bubble dynamics, the criterion for the onset of flow behaviour reads (σ rr -σ θθ ) 2 ≥ 3σ 2 Y [START_REF] De Corato | Oscillations of small bubbles and medium yielding in elastoviscoplastic fluids[END_REF], σ Y being a scalar quantity representing the yield strength of the material. Assuming the material behaves as an elastic solid below yielding [Equation 7], all of it remains solid provided that

[σ rr (R) -σ θθ (R)] 2 = G 2        R 0 R 4 - R R 0 2        2 < 3σ 2 Y ( 13
)
Bubble dissolution is then arrested provided that both Equations ( 12) and ( 13) are satisfied. For stiff materials for which γ/R 0 G 1, Equation ( 12) implies R * R 0 , and the no-yield criterion can be derived explicitly, former promoting dissolution by allowing additional material deformation for a given applied stress, and the latter impeding dissolution when the flow is slow enough to result in an increase of σ Y . Surface activity of the fluid, while insufficient to stabilise bubbles by itself in Ref. [START_REF] Saha | Stability of bubbles in wax-based oleofoams: decoupling the effects of bulk oleogel rheology and interfacial rheology[END_REF], may also play a role in slowing down or halting dissolution in conjunction with its bulk rheological properties.

Linear bubble oscillations

Bubbles as harmonic oscillators

The case of linear oscillations of air bubbles [Figure 1(d)] is fairly simple to model. First, on the short time scales involved and the limited oscillation amplitude, we can assume that air does not dissolve into the surrounding medium. Authors then assume [START_REF] Yang | A model for the dynamics of gas bubbles in soft tissue[END_REF][START_REF] Gaudron | Bubble dynamics in a viscoelastic medium with nonlinear elasticity[END_REF][START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF][START_REF] Jamburidze | High-frequency linear rheology of hydrogels probed by ultrasound-driven microbubble dynamics[END_REF] a constant p g R 3κ in the gas phase, choosing a polytropic exponent 1 ≤ κ ≤ 1.4 corresponding to thermodynamic transformations that are neither fully isothermal (κ = 1) nor adiabatic (κ = 1.4 for diatomic gases). We choose to use the standard Kelvin-Voigt model for linear viscoelastic solids, that includes both a viscous and an elastic contribution. Defining

x(t) = R(t)/R 0 -1, we may derive the linear version of Equation 5 [START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF]:

ẍ + 2β ẋ + ω 2 0 x = - 1 ρR 2 0 [p ∞ (t) -p 0 ] (14)
The damping term β includes a viscous term proportional to the surrounding medium viscosity η, a thermal loss term due to thermal gradients present in the bubble, and a term accounting for acoustic damping [START_REF] Prosperetti | Nonlinear bubble dynamics[END_REF].

The natural angular frequency ω 0 also depends on the surrounding medium properties,

ω 2 0 = 3κp 0 + 2(3κ -1)γ/R 0 + 4G ρR 2 0 . ( 15
)
This frequency then deviates from the classical Minnaert resonance frequency (see Equation 1) when bubbles are sufficiently small bubbles: R 0 ∼ γ/p 0 ∼ 1µm in water at ambient pressure, and for elastic moduli G that are comparable with p 0 , i.e. for stiff gels.

Measuring material properties in the Fourier domain

In the time domain, the damping parameter β and the natural oscillation frequency ω 0 can be fitted from the free bubble oscillations, i.e. when p ∞ (t) = p 0 and for initial conditions x(t = 0) 0 or ẋ(t = 0) 0.

An alternative approach is to study the steady-state oscillatory regime at an imposed angular frequency ω.

Equation ( 14) can then be recast in the Fourier domain:

x = 1 ρR 2 0 p ∞ -p 0 (ω 2 -ω 2 0 ) 2 + 4ω 2 β 2 , (16)
Equation 16 gives the resonance curve of an oscillating bubble in the linear regime, where it can be described as a linear harmonic oscillator [START_REF] Dollet | Bubble dynamics in soft and biological matter[END_REF]. It can be measured experimentally by sweeping the forcing frequency around the natural frequency ω 0 , and recording the (maximum) oscillation amplitude x for each value of the frequency. This approach was first demonstrated by Strybulevych et al. by measuring the acoustic response of millimetre-sized bubbles in an agar gel as a function of frequency; the approach was termed "acoustic microrheology" [START_REF] Strybulevych | Acoustic microrheology: Shear moduli of soft materials determined from single bubble oscillations[END_REF]. In a similar approach termed "microbubble spectroscopy", resonance curves can be recorded by directly imaging with a high-speed camera the bubble dynamics during acoustic forcing at different frequencies [START_REF] Van Der Meer | Microbubble spectroscopy of ultrasound contrast agents[END_REF]. Jamburidze et al. [START_REF] Jamburidze | High-frequency linear rheology of hydrogels probed by ultrasound-driven microbubble dynamics[END_REF] also used direct imaging in agarose gels and at moderate pressure amplitudes (p ∞ ≤ 6 kPa), to ensure that the deformation of the material remained in the linear regime (see Figure 3a). They extracted the viscoelastic properties of the gels from the resonance curves (see Figure 3b). Alternatively, Hamaguchi and Ando constructed a resonance curve by keeping the ultrasound frequency constant at 28 kHz, letting the bubble radius increase slowly due to gas transfer into the bubble [START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF] so as to perform a "radius sweep". The radius sweep relies on the inverse proportionality between equilibrium radius R 0 and natural frequency ω 0 when the magnitude of the term 2(3κ -1)γ/R 0 in Eq. 15 is sufficiently small, i.e. for sufficiently large bubbles. Resonance curves measured from radius sweeps do not rely on a prior calibration of the transducer response as a function of the frequency, making them particularly attractive in experiments where the pressure field cannot be measured independently.

In some experiments [START_REF] Jamburidze | High-frequency linear rheology of hydrogels probed by ultrasound-driven microbubble dynamics[END_REF], the viscous contribution to damping β is dominant, and fitting the resonance curve directly provides an estimate of the material parameters of the constitutive model, in our case η and G. In the general case [START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF][START_REF] Saint-Michel | Acoustic bubble dynamics in a yield-stress fluid[END_REF], acoustic and thermal contributions have to be included in the total damping β(ω, R 0 ), which is not a constant in either frequency sweep or radius sweep experiments. In practice, reliable estimates of the viscous and thermal contributions have been determined by Prosperetti [START_REF] Prosperetti | Nonlinear bubble dynamics[END_REF]. Resonance curves are then fitted using the total damping β(ω, R 0 ), prescribing the viscous and thermal terms and letting the medium viscosity η as the adjustable parameter of the fit. and viscous damping also increases, broadening the resonance peak. Reproduced with permission from Ref. [START_REF] Jamburidze | High-frequency linear rheology of hydrogels probed by ultrasound-driven microbubble dynamics[END_REF]

Bubble collapse

Bubble collapse combines the high-frequency description of bubble oscillations of Section 4 with the large deformation framework derived in Section 3. Bubble collapse can be achieved for instance by applying a step change in the external pressure p ∞ (Rayleigh collapse), or by creating bubbles using a laser pulse to vaporise the surrounding medium, and letting them relax (Flynn collapse). Regardless of the preparation protocol and the medium properties, the complex relaxation dynamics of the bubble assumes the shape of an initial, extremely steep decrease in radius followed by a single or multiple rebounds and an eventual relaxation to an equilibrium radius (see, for instance, Figures 1d and4). Conversely, the initial collapse time and the duration, number and amplitude of the rebounds depend on the type of collapse and the surrounding medium rheology, opening the way for high strain-rate characterisation of soft materials [START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF][START_REF] Yang | Extracting non-linear viscoelastic material properties from violently-collapsing cavitation bubbles[END_REF]. Given the violent nature of the process, elasticity effects are only perceptible for stiffer materials (G ≥ 10 3 Pa) whereas viscous effects are almost always noticeable (η ≥ 10 -2 Pa.s) [START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF].

Estrada et al. have studied the collapse dynamics of a relatively stiff 10% polyacrylamide gel and compared it with predictions from various rheological models including or combining neo-Hookean, Newtonian fluid and Maxwell fluid elements. Figure 4 confirms that both elastic and viscous components must be included in the model -resulting in a neo-Hookean Kelvin-Voigt material-to accurately fit the bubble collapse dynamics. Estrada et al. [START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF] have shown that increasing the complexity of the rheological model does not significantly improve the quality of the fit to the experimental bubble collapse data. Nevertheless, Yang et al. [START_REF] Yang | Extracting non-linear viscoelastic material properties from violently-collapsing cavitation bubbles[END_REF] have suggested that including additional strain-stiffening terms in the material elasticity reconciles the classical rheology measurements and the non-linear fits of bubble collapse, in agreement with the strain-stiffening behaviour generally observed in biopolymer gels [START_REF] Storm | Nonlinear elasticity in biological gels[END_REF].

Bubble collapse is a violent, complex process and modelling efforts have been particularly cautious to evaluate all the deviations from the ideal case of adiabatic oscillations in an incompressible model fluid.

Indeed, the Mach number Ṙ/c during collapse is no longer small and first-order compressibility corrections in Equation ( 5) have to be included [START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF][START_REF] Yang | Extracting non-linear viscoelastic material properties from violently-collapsing cavitation bubbles[END_REF]. Barajas et al. has shown that heat transfer between the bubble and the surrounding medium must be taken into account as it noticeably affects the relaxation dynamics and the equilibrium bubble size during collapse [START_REF] Barajas | The effects of heat and mass diffusion on freely oscillating bubbles in a viscoelastic, tissue-like medium[END_REF]. Lastly, Gaudron et al. have studied analytically and numerically the onset of non-spherical bubble oscillations during Rayleigh collapse [START_REF] Gaudron | Shape stability of a gas cavity surrounded by linear and nonlinear elastic media[END_REF]. Their numerical results highlight that such effects should not occur for p ∞ ≤ 15G. They also show that working with a finite deformation elastic framework -the neo-Hookean model-rather than linear elasticity -Hooke's lawstrongly promotes spherical bubble oscillations, in particular for stiff materials and high applied pressures p ∞ ∼ 10 7 Pa.

6. Towards broadband microrheology of complex fluids using bubble dynamics

Current limitations of rheological measurements based on bubble dynamics

Here we identify the main limitations that need to be overcome to achieve a robust technique. The first practical issue is the limited control offered by bubble microrheology in terms of the applied strain or stress, at variance with classical rheometers; the latter are indeed particularly efficient at applying a precise strain or stress history to the sample such as step strain, shear startup, large amplitude oscillatory shear or creep tests. During bubble dissolution, neither the strain rate or the stress at the bubble boundary are fixed, as the dynamics is rather governed by the difference in chemical potential between the bubble and the fluid far from it. Linear bubble oscillations may be viewed as small amplitude oscillatory strain experiments, yet without an explicit control on the oscillation amplitude, which depends on the inertial terms in Equation ( 5)

and the material properties. Lastly, bubble collapse experiments impose a fairly complex strain history to the material which cannot be controlled by the operator. This profile rather results from a balance between the bubble thermodynamics, fluid inertia and material rheology.

The second challenge is related to bubble formation, stability and imaging. Injecting or embedding spherical bubbles of a desired radius in soft materials can be challenging, especially if the material exhibits predominantly solid-like behaviour, or if it is a yield-stress fluid. Bubbles may be injected while a gel is setting [START_REF] Jamburidze | High-frequency linear rheology of hydrogels probed by ultrasound-driven microbubble dynamics[END_REF] but this is not always possible. Generation of bubbles by a focused laser pulse provides the necessary spatio-temporal control on bubble formation [START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF][START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF][START_REF] Ando | Quasistatic growth of bubbles in a gelatin gel under dissolved-gas supersaturation[END_REF], although this is not a widely available technique. Bubble dissolution will become an issue for linear oscillation experiments at f ∼ 1 MHz, corresponding to bubbles with R 0 ∼ 1 µm, as the dissolution time (∼ 10 ms in water) may then fall below the time needed to perform the measurements. Furthermore it is observed that shrinking bubbles in hydrogels can leave behind a pocket of water within a solid-like gel network that remains undeformed [START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF]. Fine-tuning the saturation coefficient ζ by letting samples equilibrate at a different temperature [START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF], or under an excess static pressure, can be used to slow down dissolution sufficiently to allow enough time for experiments, or to drive a slow bubble growth for which solvent pockets are no longer an issue [START_REF] Ando | Quasistatic growth of bubbles in a gelatin gel under dissolved-gas supersaturation[END_REF]. Imaging oscillating or collapsing bubbles around ∼ 1 MHz finally poses the challenge of resolving the radial dynamics, which requires acquisition rates of ∼ 10 7 images per second. Commercial instruments now meet these requirements [START_REF] Kuroda | The Micro-World Observed by Ultra High-Speed Cameras[END_REF],

and will become more widely used as their price becomes more affordable.

A complex issue of linear bubble oscillations is the nature of the quantities G and η obtained from fitting the resonance curve. Even though the two techniques are conceptually equivalent, resonance curves obtained from frequency sweeps obviously contain information at multiple frequencies, whereas resonance curves obtain from radius sweeps are single-frequency measurements. It is therefore tempting to identify the material properties G and η fitted from radius sweep data with the storage modulus G (ω) and loss modulus G (ω) measured from classical oscillatory rheology at the acoustic angular frequency ω. In this case, the effective frequency of quantities fitted to frequency sweep resonance curves become unclear. However, Equation ( 16) is derived assuming a Kelvin-Voigt, linear rheology of the surrounding medium. In classical rheology, this implies that the storage modulus G (ω) = G and loss modulus G (ω) = ηω are prescribed at all frequencies. Fitting the resonance curves then constrains G and G at every frequency based on experimental data covering a limited frequency range -frequency sweep resonance curves-or even covering a single frequency -radius sweeps. This contrasts with classical oscillatory rheology, for which measurements cover three decades in terms of frequency, allowing easy discrimination between different rheological models. Resonance curve data should then ideally be complemented by measurements from another technique to validate (or question) the choice of the rheological model [START_REF] Jamburidze | High-frequency linear rheology of hydrogels probed by ultrasound-driven microbubble dynamics[END_REF], keeping in mind that the rheological models presented here (neo-Hookean, Maxwell, Kelvin-Voigt) are rather crude approximations of the experimental behaviour of real soft materials.

The last limitation is related to the onset of shape oscillations, which break down the spherical symmetry assumption used to derive most of the equations derived previously. Shape oscillations have been observed

both under moderate amplitude bubble oscillations and during cavitation events in both Newtonian fluids and visco-elastic solids [START_REF] Brujan | Stress wave emission and cavitation bubble dynamics by nanosecond optical breakdown in a tissue phantom[END_REF][START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF][START_REF] Saint-Michel | Acoustic bubble dynamics in a yield-stress fluid[END_REF]. Stability of spherical bubbles during during cavitation events and oscillations of moderate amplitude have been studied respectively by Gaudron et al. [START_REF] Gaudron | Shape stability of a gas cavity surrounded by linear and nonlinear elastic media[END_REF] and Murakami et al. [START_REF] Murakami | Shape stability of a gas bubble in a soft solid[END_REF]. The latter article quantifies the shift in shape mode number due to medium elasticity and the impact of viscosity on the critical oscillation amplitude above which shape oscillations occur. Agreement with the available, very limited experimental data [START_REF] Hamaguchi | Linear oscillation of gas bubbles in a viscoelastic material under ultrasound irradiation[END_REF] was found to be good. More experimental data would be valuable to validate the model and eventually use shape oscillation data to obtain rheological information from soft materials.

Opportunities for high-frequency rheology

As complex fluids are structured from intermediate (∼ µm) to molecular (∼Å) sizes, they also show a broad distribution of relaxation time scales which can be probed through linear oscillatory rheology. In polymer solutions and wormlike micelles, these time scales are directly related to spatial physical quantities of the polymers such as the chain persistence length and the entanglement or crosslink density. In particulate suspensions, gels and emulsions, the linear oscillatory spectra contain valuable information on the onset of the glass transition or rigidity percolation [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF]. Commercial rotational rheometers cannot provide high frequency measurements (≥ 50 Hz) due to inertial effects of the rheometer head. Time-temperature superposition has then been widely used in polymer solutions and melts to fill this particular gap in experimental data [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF]; yet, the technique cannot be applied for out-of-equilibrium (arrested or glassy) systems or those undergoing phase transitions. Dedicated piezoelectric and microelectromechanical systems (MEMS) offer direct high-frequency rheological measurements, as recently reviewed by Schroyen et al. [START_REF] Schroyen | Bulk rheometry at high frequencies: a review of experimental approaches[END_REF]. Traditional microrheology techniques, relying on the motion of passive or active tracers embedded in the fluid [START_REF] Mason | Estimating the viscoelastic moduli of complex fluids using the generalized Stokes-Einstein equation[END_REF],

have also been developed to complement relaxation spectra both at very small and large frequencies. Bubble dynamics offer an opportunity to extend the range of available techniques for active microrheology.

Many industrial flows, such as jetting, injection moulding, or lubrication, and natural flows, such as sneezing [START_REF] Scharfman | Visualization of sneeze ejecta: steps of fluid fragmentation leading to respiratory droplets[END_REF], apply strains to complex fluids at high frequency and beyond their linear deformation, that is, at high strain rates. Such transformations break and reorient their microstructure, usually leading to shearthinning behaviour. The associated, very high strain rates ˙ ≥ 10 4 s -1 once again exceed the capabilities of rotational rheometers due to sample expulsion, particle migration and edge fracture. Pressure-driven flows, such as capillary rheometers, are a viable alternative to rotational geometries as they do not present any free surface in the fluid region of interest, yet they require careful pressure corrections due to end effects and potential wall slip [START_REF] Macosko | Rheology: Principles, Measurements and Applications[END_REF]. Bubble microrheology, in particular based on bubble collapse, is unaffected by wall slip, since the strain is applied by a movement normal to the bubble-fluid interface, and meets both the high-frequency and high-strain criteria to provide insights on high-strain rate rheology of many complex fluids.

Lastly, bubble dynamics, collapse and cavitation can generate shear waves in viscoelastic materials when the bubbles are located close to a boundary [START_REF] Montalescot | Electrolysis-induced bubbling in soft solids for elastic-wave generation[END_REF][START_REF] Tinguely | Surface waves on a soft viscoelastic layer produced by an oscillating microbubble[END_REF][START_REF] Rapet | Shear-wave generation from cavitation in soft solids[END_REF], because the strain profile around the bubble is no longer symmetric. These waves have been observed using echography [START_REF] Montalescot | Electrolysis-induced bubbling in soft solids for elastic-wave generation[END_REF], particle tracking [START_REF] Tinguely | Surface waves on a soft viscoelastic layer produced by an oscillating microbubble[END_REF] or birefringence [START_REF] Rapet | Shear-wave generation from cavitation in soft solids[END_REF] techniques. The propagation speed and attenuation of shear waves can provide a direct measurement of the high-frequency material properties G and η of the material [START_REF] Catheline | Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: An inverse problem approach[END_REF], which have been compared

to low-frequency measurements obtained using a rheometer in [START_REF] Tinguely | Surface waves on a soft viscoelastic layer produced by an oscillating microbubble[END_REF].

Opportunities for extensional deformation of complex fluids

Extensional deformation is the main mode of deformation during spinning, rolling, convergent or divergent die injection and spraying in industrial processes [START_REF] Macosko | Rheology: Principles, Measurements and Applications[END_REF] but also in biological flows, e.g. during sneezing [START_REF] Scharfman | Visualization of sneeze ejecta: steps of fluid fragmentation leading to respiratory droplets[END_REF]. Extensional deformation, in contrast with simple shear, strongly imposes the orientation of the fluid elements, leading for instance to the well-known coil-stretch transition and extensional thickening of polymer solutions and the high extensional viscosity of rigid fibre suspensions [START_REF] Macosko | Rheology: Principles, Measurements and Applications[END_REF]. In yield-stress fluids, the yield stress in extensional flow is known to differ from the one obtained in simple shear [START_REF] Niedzwiedz | Extensional rheology of concentrated emulsions as probed by capillary breakup elongational rheometry (CaBER)[END_REF][START_REF] Zhang | Yielding and flow of soft-jammed systems in elongation[END_REF] due to the non-linear normal stress differences arising in the material even below yielding [START_REF] De Cagny | The yield normal stress[END_REF][START_REF] Varchanis | Transition between solid and liquid state of yield-stress fluids under purely extensional deformations[END_REF]. As solid boundaries in a geometry impose shear flows, true extensional flows work with stress-free boundary conditions, for instance by examining the thinning and breakup dynamics of fluid filaments, following either the separation of two end plates [START_REF] Mckinley | How to extract the newtonian viscosity from capillary breakup measurements in a filament rheometer[END_REF] or the Rayleigh-Plateau instability in jet flows [START_REF] Keshavarz | Studying the effects of elongational properties on atomization of weakly viscoelastic solutions using rayleigh ohnesorge jetting extensional rheometry (rojer)[END_REF]. As the strain in the filament leads to an exponential decrease of their cross-section, these techniques are limited in terms of maximum applied strain. Bubble microrheology is then particularly interesting to locally apply very strong extensional strains, given by = B θθ (R 0 ) -1 = (R/R 0 ) 2 -1 at the bubble edge. Such strains easily exceed 10 for both collapsing [START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF] and dissolving bubbles, above the values (typically 6 -7) achieved with capillary breakup rheometry.

Mixed, non-uniform flows containing a strong extensional component have been thoroughly investigated in microfluidic cross-slot [START_REF] Haward | Optimized cross-slot flow geometry for microfluidic extensional rheometry[END_REF][START_REF] Varchanis | Transition between solid and liquid state of yield-stress fluids under purely extensional deformations[END_REF] and hyperbolic [START_REF] Ober | Microfluidic extensional rheometry using a hyperbolic contraction geometry[END_REF] geometries. In such geometries, the ratio between simple shear and extension is not known a priori and depends on the fluid rheology and the magnitude of wall slip. As a consequence, these geometries are usually complemented with local flow birefringence [START_REF] Haward | Optimized cross-slot flow geometry for microfluidic extensional rheometry[END_REF] or microparticle image velocimetry [START_REF] Varchanis | Transition between solid and liquid state of yield-stress fluids under purely extensional deformations[END_REF] to precisely monitor the time-dependent, non-uniform strain applied to the fluid elements. Direct imaging of the deformation field and evolution of the microstructure is also

Conclusion

Spherical bubbles are versatile active probes to characterise the local rheology of soft materials thanks to the purely extensional, localised nature of the deformation field they apply to the material. In this paper we discussed three emerging micro-rheological measurements based on recording the time evolution of the radius of a bubble, during three distinct processes that provide complementary information on the surrounding material. The first process is the slow, quasistatic bubble dissolution dynamics and its potential arrest, which can be used to evaluate the linear elastic modulus G of a material for neo-Hookean or Kelvin-Voigt materials, or to obtain an estimate the critical stress σ Y of yield-stress fluids. A second technique exploits linear bubble oscillations driven by an acoustic wave. Examining the bubble response in the time or in the frequency domain provides an estimate from high-frequency data of both G and the solvent viscosity η, for an equivalent Kelvin-Voigt material. Lastly, the violent bubble collapse process offers additional, deeper insights on the finite-strain rheology of soft materials at extremely high strain rates (typically 10 6 s -1 ). Information is obtained by fitting experimental collapse time series to fairly complex, non-linear, finite strain models of bubble dynamics. These modelling steps are needed to estimate the stress field σ in the material, which is otherwise unavailable, in contrast with classical rheology measurements. Dissolution and collapse processes impose high extensional strains close to the bubble boundary and may be a powerful technique to examine extreme phenomena such as the extensional thickening of polymer solutions due to individual chain uncoiling; in particular since the direct bubble imaging can easily be combined with particle image velocimetry techniques or flow birefringence. We also discussed outstanding challenges of bubble-based microrheology, for instance controlling the stress history and the process of bubble injection. More work is also needed to combine more realistic models to describe soft materials with the governing equations of bubble dynamics, especially under high-frequency, high-strain rate deformation. Under such conditions, the use of simple models, for instance neo-Hookean Kelvin-Voigt solids, should at least be better justified given the numerous deviations from such a behaviour observed in classical rheology [START_REF] Storm | Nonlinear elasticity in biological gels[END_REF].

ζ

  = 1.00, G = 0 Pa ζ = 1.00, G = 200 Pa ζ = 1.025, G = 0 Pa ζ = 1.025, G = 200 Pa

Figure 1 :

 1 Figure 1: Kinematics and dynamics of bubbles in fluids and soft solids. (a) Kinematics of a bubble (in white) surrounded by a

Figure 2 .

 2 Figure 2.

Figure 2 :

 2 Figure2: Effect of medium elasticity on bubble growth by mass transfer. A laser-generated bubble grows in an elastic medium super-saturated with air. The growth dynamics of the bubble radius R is used to estimate the super-saturation coefficient ζ and the elastic modulus G of the surrounding material. Reproduced with permission from Ref.[START_REF] Ando | Quasistatic growth of bubbles in a gelatin gel under dissolved-gas supersaturation[END_REF]

Figure 3 :

 3 Figure 3: Linear high-frequency rheology by acoustic bubble dynamics. (a) Test to confirm the linearity of bubble oscillations and material deformation in agarose gels. The amplitude of oscillations ∆R (normalised by the smallest oscillation amplitude ∆R 1 ) is plotted as a function of the applied pressure amplitude ∆p (normalised by the lowest applied pressure ∆p 1 ). Symbols are experimental data and lines are quadratic fits, with black (respectively red) data series shifted upwards by 1 (respectively 2) for clarity. (b) Impact of the surrounding material properties on the resonance curves. Symbols are experimental data and solid lines represent a fit from Equation (16). As the gel concentration is increased, the gel becomes stiffer and the resonance frequency increases [see Equation[START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF]],

Figure 4 :

 4 Figure 4: Collapse dynamics of a laser-generated bubble in a polyacrylamide gel. The experimental data R(t) (orange squares) are compared to the best fits assuming the gel behaves as a neo-Hookean elastic solid (green dashed line), a Newtonian fluid (purple dashed line) and a neo-Hookean Kelvin-Voigt -viscoelastic solid-model (blue solid line; also see top-right schematic diagram). The fit to the viscoelastic model provides estimates of the its linear elastic modulus G and viscosity η (see right inset). Reproduced with permission from Ref.[START_REF] Estrada | High strain-rate soft material characterization via inertial cavitation[END_REF]

  possible in combination with bubble dynamics experiments. For instance, the reorganisation of a microfibre network around an expanding bubble has been directly visualised for different deformation rates, as shown in Figure5a-d, evidencing a strong reorganisation of the fibre network close to very slowly, quasi-statically expanding bubbles, which progressively disappears as the expansion rate increases. The change in the microstructure of a bicontinuous interfacially jammed emulsion gel (bijel) due to bubble motion during centrifugation has been imaged by confocal microscopy (Figure5e-f). These examples support the notion of simultaneous imaging of the evolution of the microstructure of a complex fluid, in combination with rheological measurements based on the bubble dynamics phenomena described in this paper.

Figure 5 :

 5 Figure 5: Visualisation of bubble-induce deformation of complex fluids. (a-d) Quasi-static expansion of a bubble in a sparse microcellulose yield-stress fluid. Reproduced with permission from Ref. [50]. In (a-b), very slow expansion rates allow a relative motion between the solvent and the fibre network and changes in the microstructure are visible close to the expanded bubble. In (c-d), for higher expansion rates, no relative motion is possible and the microstructure around the bubble remains similar before and after inflation. (e-f) Deformation and destruction of a bijel due to bubble displacement during centrifugation. Reproduced with permission from Ref. [51].

The sign of the elastic component on the numerator has been corrected from Ref. 6.

Acknowledgements

This work was supported by a European Research Council Starting Grant [grant number 639221] (V.G.).

Conflict of Interest Statement

The Authors declare no conflicts.


Philippe Loubaton
email: philippe.loubaton@univ-eiffel.fr

Alexis Rosuel
email: alexis.rosuel@univ-eiffel.fr

Pascal Vallet
email: pascal.vallet@bordeaux-inp.fr

On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime ⋆

Keywords: Spectral Analysis, High Dimensional Statistics, Time Series, Independence Test. 2010 MSC: 62H15, 62H20, 62M15

We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M -variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions M N → 0 and M B → c ∈ (0, +∞). This result is then exploited to build a statistic with controlled asymptotic level for testing independence between the M components of the observed time series. Numerical simulations support our results.

Introduction

The addressed problem and the results

We consider a zero mean M -variate complex Gaussian stationary time series (y n ) n∈Z1 and denote S(ν) and C(ν) its spectral density and spectral coherency matrices defined for each ν ∈ [0, 1] by

S(ν) = u∈Z R(u)e -2iπuν
and C(ν) = dg(S(ν)) -1/2 S(ν) diag(S(ν)) -1/2

where R(u) = E(y n+u y * n ) and dg(S(ν)) = S(ν) ⊙ I M , with ⊙ denoting the Hadamard product (ie. entrywise product) and I M is the M -dimensional identity matrix. Assuming N observations (y n ) n=1,...,N are available, we consider the frequency smoothed estimate Ŝ(ν) defined by

Ŝ(ν) = 1 B + 1 B/2 b=-B/2 ξ y ν + b N ξ y ν + b N * (1)
where B is an even integer, which represents the smoothing span, and

ξ y (ν) = 1 √ N N n=1
y n e -2iπ(n-1)ν

(2)

is the renormalized Fourier transform of (y n ) n=1...,N . The corresponding estimated spectral coherency matrix is defined as:

Ĉ(ν) = dg( Ŝ(ν)) -1 2 Ŝ(ν) dg( Ŝ(ν)) -1 2 (3)
We denote by (y 1,n ) n∈Z , . . . , (y M,n ) n∈Z the M components of y, and s ij (ν), c ij (ν), ŝij (ν), ĉij (ν) the entries i, j of matrices S(ν), C(ν), Ŝ(ν), Ĉ(ν) respectively. We remark that

c ij (ν) = s ij (ν) s ii (ν)s jj (ν)
, ĉij (ν) = ŝij (ν) ŝii (ν)ŝ jj (ν) for each ν.

Under the hypothesis H 0 : (y 1,n ) n∈Z , . . . , (y M,n ) n∈Z are mutually uncorrelated, we evaluate the behaviour of the Maximum Sample Spectral Coherence (MSSC) defined by max

1≤i<j≤M max ν∈G |ĉ ij (ν)|
where

G := k B + 1 N : k ∈ N, 0 ≤ k ≤ N B + 1
is the subset of the Fourier frequencies

F := k N : k ∈ N, 0 ≤ k ≤ N -1
with elements spaced by a distance (B +1)/N . Our study is conducted in the asymptotic regime where M = M (N ) and B = B(N ) are both functions of N such that for some ρ ∈ (0, 1), M ≍ N ρ and B ≍ N ρ as N → ∞2 , while the ratio M/B converges to some constant c ∈ (0, +∞). It is established that, under H 0 and proper assumptions on the time series (y 1,n ) n∈Z , . . . , (y M,n ) n∈Z , for any t ∈ R:

P (B + 1) max (i,j,ν)∈I |ĉ ij (ν)| 2 ≤ t + log N B + 1 + log M (M -1) 2 ----→ N →+∞ e -e -t (4)
where I := {(i, j, ν) : i, j ∈ [M ] such that i < j, ν ∈ G} (5)

with [M ] = {1, . . . , M }.

In other words, under proper normalization and centering, max (i,j,ν)∈I |ĉ ij (ν)| 2 follows asymptotically a Gumbel distribution (see [START_REF] Embrechts | Modelling extremal events: for insurance and finance[END_REF] or [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF] for a general theory of extreme value distributions).

Motivation

This paper is motivated by the problem of testing the independence of a large number of Gaussian time series. Since hypothesis H 0 can be equivalently formulated as

H 0 : max 1≤i<j≤M max ν∈[0,1] |s ij (ν)| 2 = 0,
or by

H 0 : max 1≤i<j≤M max ν∈[0,1] |c ij (ν)| 2 = 0,
this suggests to compute consistent estimators of these quantities, and test their closeness to zero.

Our choice of the high-dimensional regime defined above is motivated as follows. Under mild assumptions on the memory of the time series ((y m,n ) n∈Z ) m≥1 , in the low-dimensional regime where N → +∞ and M is fixed, it can be shown that the sample spectral coherence matrix Ĉ(ν) defined by ( 3) is a consistent estimate (in spectral norm for instance) of the spectral coherence matrix C(ν) as long as B → +∞ and B/N → 0 (up to some additional logarithmic terms). In practice, this asymptotic regime and the underlying predictions are relevant as long as the ratio M/N is small enough. If this condition is not met, test statistics based on Ĉ(ν) may be of delicate use, as the choice of the smoothing span B must meet the constraints B ≫ M (because B is supposed to converge towards +∞) as well as B ≪ N (because B/N is supposed to converge towards 0). Nowadays, for many practical applications involving high dimensional signals and/or a moderate sample size, the ratio M/N may not be small enough to be able to choose B so as to meet B ≫ M and B ≪ N . In this situation, one may rely on the more relevant high dimensional regime in which M, B, N converge to infinity such that M/B converges to a positive constant while B/N converges to zero.

On the literature

Correlation tests using spectral approaches have been studied in several papers, see e.g. [START_REF] Wahba | Some tests of independence for stationary multivariate time series[END_REF], [START_REF] Eichler | Testing nonparametric and semiparametric hypotheses in vector stationary processes[END_REF] and the references therein.

More recently, an approach similar to the one of this paper has been explored in [START_REF] Wu | Asymptotic theory for spectral density estimates of general multivariate time series[END_REF], where the maximum of the sample spectral coherence, when using lag-window estimates of the spectral density, is studied. In the low-dimensional regime where M is fixed and N → ∞, it is proved that the distribution of such statistic under H 0 , after proper centering and normalization, converges to the Gumbel distribution. We also mention other related papers exploring the asymptotic behaviour of various spectral density estimates in the low-dimensional regime: [START_REF] Woodroofe | The maximum deviation of sample spectral densities[END_REF], [START_REF] Rudzkis | On the distribution of the maximum deviation of the gaussian stationary time series spectral density estimate[END_REF], [START_REF] Shao | Asymptotic spectral theory for nonlinear time series[END_REF], [START_REF] Lin | On maxima of periodograms of stationary processes[END_REF] and [START_REF] Liu | Asymptotics of spectral density estimates[END_REF].

In the high-dimensional regime when M is a function of N such that M := M (N ) → +∞ as N → ∞, few results on the behaviour of correlation test statistics in the spectral domain are known. [START_REF] Loubaton | Large random matrix approach for testing independence of a large number of gaussian time series[END_REF] proved that under H 0 and mild assumptions on the underlying time series, the empirical eigenvalue distribution of Ĉ(ν) defined in (3) converges weakly almost surely towards the Marcenko-Pastur distribution, which can be exploited to build test statistics based on linear spectral statistics of Ĉ(ν). In [START_REF] Rosuel | On the frequency domain detection of high dimensional time series[END_REF], a consistent test statistic based on the largest eigenvalue of Ĉ(ν) was derived for the problem of detecting the presence of a signal with low rank spectral density matrix within a noise with uncorrelated components.

In the asymptotic regime where M N → γ > 0, [START_REF] Pan | Testing independence among a large number of high-dimensional random vectors[END_REF] proposed to test hypothesis H 0 when the components of y share the same spectral density. In this case, the rows of the M × N matrix (y 1 , . . . , y N ) are independent and identically distributed under H 0 . Pan et al. (2014) established a central limit theorem for linear spectral statistics of the empirical covariance matrix, and deduced from this a test statistics to check whether H 0 holds or not. We notice that the results of [START_REF] Pan | Testing independence among a large number of high-dimensional random vectors[END_REF] are valid in the non Gaussian case. We note also that in the specific case where the M time series have the same marginal spectral density, it is possible to directly use results on temporally white noise tests, such as those in [START_REF] Li | On testing for high-dimensional white noise[END_REF] and [START_REF] Chang | Testing for high-dimensional white noise using maximum cross-correlations[END_REF]. To explain this, we denote by Y = (y 1 , . . . , y N ) the M × N matrix of observations, and by z T 1 , . . . , z T M the rows of Y, which, in the above particular case, share the same covariance matrix T. Then, the row vectors z T 1 , . . . , z T M are uncorrelated if and only the column vectors z 1 , . . . , z M are extracted from a white noise sequence with covariance T. However, in our more general model, since each time series is generated by a different spectral density, vectors z 1 , . . . , z M are not anymore iid, which prevents to use the results of [START_REF] Li | On testing for high-dimensional white noise[END_REF] and [START_REF] Chang | Testing for high-dimensional white noise using maximum cross-correlations[END_REF].

More results are available in the case where the time series (y m,n ) n∈Z , m ∈ [M ], are temporally white. To test the correlation of the M components, one can similarly consider sample estimates of the correlation matrix, and test whether it is close to the identity matrix. Under the asymptotic regime where M N → γ ∈ (0, +∞), [START_REF] Jiang | The asymptotic distributions of the largest entries of sample correlation matrices[END_REF] showed that the maximum off-diagonal entry of the sample correlation matrix after proper normalization is also asymptotically distributed as Gumbel. The techniques used here for proving (4) are partly based on this paper. Other works such as [START_REF] Mestre | Correlation tests and linear spectral statistics of the sample correlation matrix[END_REF] studied the asymptotic distribution of linear spectral statistics of the correlation matrix, [START_REF] Dette | Likelihood ratio tests for many groups in high dimensions[END_REF] focused on the behaviour of the determinant of the correlation matrix, and Cai et al. ( 2013) considered a U-statistic and obtained minimax results over some class of alternatives. Some other papers also explored various classes of alternative H 1 , among which is [START_REF] Fan | Largest entries of sample correlation matrices from equi-correlated normal populations[END_REF], who showed a phase transition phenomena in the behaviour of the largest off-diagonal entry of the correlation matrix driven by the magnitude of the dependence parameter defined in the alternative class H 1 . Lastly, Morales-Jimenez et al. ( 2018) studied asymptotic first and second order behaviour of the largest eigenvalues and associated eigenvectors of the sample correlation matrix under a specific alternative spiked model.

Main results

Assumptions

Throughout the paper we rely on the following assumptions.

Assumption 1 (Time series). The time series (y m,n ) n∈Z , m ≥ 1, are mutually independent, stationary and zero-mean complex Gaussian distributed3 .

For each m ≥ 1, we denote by r m = (r m (u)) u∈Z (instead of r m,m ) the covariance sequence of (y m,n ) n∈Z , i.e. r m (u) = E[y m,n+u y m,n ], and we formulate the following assumption on (r m ) m≥1 :

Assumption 2 (Memory). The covariance sequences (r m ) m≥1 satisfy the uniform short memory condition

sup m≥1 u∈Z (1 + |u|)|r m (u)| < +∞. We denote by s m (ν) = u∈Z r m (u)e -i2πν (instead of s m,m (ν)) the spectral density of (y m,n ) n∈Z at frequency ν ∈ [0, 1]. Assumption 2 of course implies that the function s m is continously differentiable and that sup m≥1 max ν∈[0,1] s m (ν) < +∞, sup m≥1 max ν∈[0,1] ds m dν (ν) < +∞. ( 6
)
Eventually, as the sample spectral coherence of (y i,n ) n∈Z and (y j,n ) n∈Z involves a renormalization by the inverse of the estimates of the spectral densities s i and s j , we also need that s i , s j do not vanish. This is the substance of the next assumption.

Assumption 3 (Non-vanishing spectrum). The spectral densities are uniformly bounded away from zero, that is

inf m≥1 min ν∈[0,1] s m (ν) > 0. (7)
By Assumptions 2 and 3, there exist quantities s min and s max such that

0 < s min ≤ inf m≥1 min ν∈[0,1] s m (ν) ≤ sup m≥1 max ν∈[0,1] s m (ν) ≤ s max < +∞. ( 8
)
We now formulate the following assumptions on the growth rate of the quantities N, M, B, which describe the high-dimensional regime considered in this paper.

Assumption 4 (Asymptotic regime). B and M are functions of N such that there exist positive constants C 1 , C 2 ∈ (0, +∞) and ρ ∈ (0, 1) such that:

C 1 N ρ ≤ B, M ≤ C 2 N ρ and M B := c N ----→ N →+∞ c ∈ (0, +∞).
Notations. Even if the subscript • N is not always specified, almost all quantities should be remembered to be dependent on N . Moreover, C represents a universal constant (i.e. a positive quantity independent of N, M, B), whose precise value is irrelevant and which may change from one line to another.

Statement of the result

The main result of this paper, whose proof is deferred to Section 4, is given in the following theorem.

Theorem 1. Under Assumptions 1 -3, for any t ∈ R:

P (B + 1) max (i,j,ν)∈I |ĉ ij (ν)| 2 ≤ t + log N B + 1 + log M (M -1) 2 ----→ N →+∞
e -e -t .

Thus, Theorem 1 states that max (i,j,ν)∈I |ĉ ij (ν)| 2 , atfer proper normalization and centering, converges in distribution to a type I extreme value distribution, also known as Gumbel distribution. As it will be clear in the proof, the term log M (M -1) 2 is related to the maximum over (i, j) while the term log N B+1 is related to the maximum over ν ∈ G.

We now illustrate numerically the above asymptotic result. Consider M independent AR(1) processes, driven by a standard Gaussian white noise, i.e.

y n :=    y 1,n . . . y M,n    =    θ 1 . . . θ M       y 1,n-1 . . . y M,n-1    +    ϵ 1,n . . . ϵ M,n    , ϵ m,n i.i.d.
∼ N C (0, 1) (9) with θ 1 , . . . , θ M are iid uniformly distributed on the complex disk of radius 0.9, and (N, M ) = (20000, 500). The smoothed periodogram estimators are computed using B = 1000. We independently draw 10000 samples of the time series (y n ) n∈[N ] and compute the associated MSSC max (i,j,ν)∈I N |ĉ ij (ν)| 2 . On Figure 1 are represented the sample cumulative distribution function (cdf) and the histogram of the MSSC against the Gumbel cdf and probability density function (pdf). We indeed observe that the rescaled distribution of max (i,j,ν)∈I N |ĉ ij (ν)| 2 is close to the Gumbel distribution.

Application to testing

New proposed test statistic

Theorem 1 can be used to design a new independence test statistic with controlled asymptotic level in the proposed high-dimensional regime.

Define q α the α-quantile of the Gumbel distribution: q α = F -1 (α) where .

F (x) = exp(-exp(-x)).

The test statistic T

(MSSC) N defined by

T (MSSC) N = 1 max (i,j,ν)∈I |ĉ ij (ν)| 2 > q 1-α + log N B+1 + log M (M -1) 2 B + 1 (10)
satisfies, as a direct consequence of Theorem 1, lim N →+∞ P[T

(MSSC) N = 1] = α under H 0 .
Note that in the following, the smoothing span B is set according to the dimension M in order to illustrate the asymptotic regime in which B ≍ M . In a practical context, the smoothing span B may be chosen via data-driven methods, such as the one in [START_REF] Ombao | A simple generalised crossvalidation method of span selection for periodogram smoothing[END_REF]. Nevertheless, although it was used in the context of high-dimensional time series in Fiecas and von Sachs (2014); [START_REF] Fiecas | Spectral analysis of highdimensional time series[END_REF], classical span selection procedures such as [START_REF] Ombao | A simple generalised crossvalidation method of span selection for periodogram smoothing[END_REF] usually make sense for low-dimensional time series. The development of a new data-driven span selection method, relevant for high-dimensional time series, is outside the scope of this paper but would be an interesting topic for future research.

Type I error

In order to test the independence of the signals ((y m,n ) n∈Z ) m=1,...,M , we consider the statistic T (MSSC) N defined in (10). In Table 1 are presented the sample type I errors of T (MSSC) N with different combinations of sample sizes and dimensions (ρ = 0.7 and M B+1 = 0.5), when the nominal significant level for all the tests is set at α = 0.05, and all statistics are computed from 30000 independent replications. The time series are still generated according to model ( 9). One can see as expected that the type I error of T (MSSC) N does indeed remain near 5% as M increases.

Power

We now compare the power of our new test statistic against other independence test statistics which are designed to work in the high-dimensional regime. We define the Linear Spectral Statistic (LSS) test from [START_REF] Loubaton | Large random matrix approach for testing independence of a large number of gaussian time series[END_REF] for any ϵ > 0 by

T (LSS) N = 1   sup ν∈[0,1] 1 M tr f ( Ĉ(ν)) - R f dµ (c N ) M P N ϵ (B/N ) > κ 1-α   (11)
where µ

M P (λ) = 1 - 1 c + δ 0 (dλ) + (λ + -λ)(λ -λ -) 2πcλ 1 [λ -,λ + ] (λ) dλ where λ ± = (1 ± √ c) 2 , (•) + := max(•, 0), c N := M B+1
and f is some function defined on R + satisfying regularity assumptions (see more details in [START_REF] Loubaton | Large random matrix approach for testing independence of a large number of gaussian time series[END_REF]). In practice, ϵ will be taken equal to 0.1. It is proven in [START_REF] Loubaton | Large random matrix approach for testing independence of a large number of gaussian time series[END_REF] that under H 0 , T (LSS) N → 0 almost surely in the high-dimensional regime but the exact asymptotic distribution of the LSS test is unknown. Therefore, the detection threshold κ 1-α for this test is based on a sample quantile of T

• the Frobenius test T (FROB) N when f (x) = (x -1) 2 • the logdet test T (LOG) N when f (x) = log x
It remains to define the alternatives. For this, we consider the following multidimensional AR(1) model:

y n+1 = Ay n + ϵ n (12)
where (ϵ n ) n∈Z is a sequence of independent N C M (0, I) distributed random vectors, and A is a bidiagonal matrix. Three choices of A (A (H 0 ) , A (H 1,loc ) , A (H 1,glob ) ) allows us to define two alternatives:

1. H 0 : for |θ| < 1:

A (H 0 ) = θ I M
so the signals ((y m,n ) n∈Z ) m=1,...,M are mutually independent. 2. H 1,loc : for |θ| < 1 and β ∈ R, it exists an unknown pair (i, j), i > j, such that :

A (H 1,loc ) = θ I M + β e i e T j
where e 1 , . . . , e M represents the canonical basis of C M so the couple of time series (i, j) is the unique correlated pair of signals. In the following experiments, we consider the case i = 2 and j = 1. 3. H 1,glob : for |θ| < 1 and β ∈ R: We now fix the value of the parameters involved under the three hypotheses. θ will always be taken equal to 0.5. Under H 1,loc , β = 0.1. Concerning the alternative H 1,glob , more care is required to choose β. Indeed, one can define a measure of total dependence as:

A (H 1,glob ) =          θ 0 . . . .
r := ∥S(ν) -dgS(ν)∥ 2 F dν ∥S(ν)∥ 2 F dν = u∈Z ∥R(u) -dgR(u)∥ 2 F u∈Z ∥R(u)∥ 2 F where R(u) := E[y n+u y * n ], S(ν) = u∈Z R(u)e -i2πuν
and dg denotes the diagonal part operator. Clearly, r = 0 under H 0 , and as r > 0 increases, the M -dimensional time series become correlated. We also see that for any fixed value of β, r is increasing with M . It is therefore more desirable to tune β := β(M ) such that r remains constant as M increases. This will enable our tests to be compared against an alternative which does not become asymptotically trivial.

The two alternatives H 1,loc and H 1,glob are useful to measure the performance of the independence tests under two different setups. Under H 1,loc , each pair of time series are independent except the pair (y 1,n ) n∈Z ,(y 2,n ) n∈Z , whereas under H 1,glob each time series has a small correlation with every other time series.

In Table 2 and Table 3 are presented the sample powers when the type I error is fixed at 5% for the considered tests and the two alternatives. The asymptotic regime is the same as the one considered for Table 1: ρ = 0.7 and M B+1 = 0.2. All statistics are computed from 30000 independent replications. We observe that under H 1,glob , with r = 0.01, all the tests asymptotically detect the alternative, however with different performances. The LSS test statistics show better power which indicates that they may be more suited to detect alternative under H 1,glob than the MSSC test statistics. Under H 1,loc the results are opposite: the power of T (MSSC) N rapidly increases to 1 as M increases. These results are not surprising since the MSSC test statistic is designed to detect peaks in the off-diagonal entries of Ĉ(ν) which is exactly the class of alternative considered in H 1,loc . However, when the correlations are spread among all pairs of time series under H 1,glob , the test statistics based on the global behaviour of the eigenvalues of Ĉ(ν) seem more relevant.

On Figure 2 are represented the ROC for each test under both alternatives. We observe that T for the alternative H 1,glob , while T (MSSC) N has better performance for H 1,loc .

To conclude this section, we mention that the work of [START_REF] Fan | Largest entries of sample correlation matrices from equi-correlated normal populations[END_REF], which extends the study of [START_REF] Jiang | The asymptotic distributions of the largest entries of sample correlation matrices[END_REF] on the asymptotic distribution of the largest entry of the sample covariance matrix under a specific H 1 scenario, shows that a phase transition phenomenon occurs, as the asymptotic distribution may take three different forms depending on the behaviour of the distance between the H 0 and H 1 sample distributions. Extending such results to the MSSC test statistic for a general class of alternative hypotheses H 1 would be a deep and interesting perspective requiring a significant work. .

Proof of Theorem 1

We will detail in this section the main steps prove Theorem 1, while some details will be left in the Appendix.

General approach

First, we notice that the frequency smoothed estimate ŝi,j (ν) can be written as ŝij (ν

) = 1 B + 1 ξ y j (ν) * ξ y i (ν) (13)
where

ξ y i (ν) = ξ y i ν - B 2N , . . . , ξ y i ν + B 2N T .
This is a sesquilinear form of the finite Fourier transform of the M time series samples (y i,1 , . . . , y i,N ) i∈[M ] . To handle the statistical dependence between the components of ξ y i (ν), we use the well-known Bartlett decomposition (see for instance [START_REF] Walker | Some asymptotic results for the periodogram of a stationary time series[END_REF]) whose procedure is described hereafter.

From Assumptions 2 and 3, the spectral distribution of (y m,n ) n∈Z is absolutely continuous with density s m being uniformly bounded and bounded away from 0. Therefore, from Wold's Theorem (Brockwell and Davis, 2006, Th. 5.7.1, Th. 5.7.2), each time series (y m,n ) n∈Z admits a causal and causally invertible linear representation in terms of its normalized innovation sequence:

y m,n = +∞ k=0 a m,k ϵ m,n-k , (14)
where (ϵ 1,k ) k∈Z , . . . , (ϵ M,k ) k∈Z are mutually independent sequences of N C (0, 1) i.i.d. random variables, and (a 1,k ) k∈N , . . . , (a M,k

) k∈N ∈ ℓ 2 (N) such that if h m (ν) = +∞ k=0 a m,k e -2iπkν (15)
then |h m (ν)| 2 = s m (ν) and h m (ν) coincides with the outer causal spectral factor of s m (ν). Define now sij (ν), an approximation of ŝij (ν), as:

sij (ν) = 1 B + 1 B/2 b=-B/2 h i ν + b N h j ν + b ξ ϵ i ν + b N ξ ϵ j ν + b N or equivalently sij (ν) = ξ ϵ j (ν) * Π ij (ν) B + 1 ξ ϵ i (ν) (16)
where

Π ij (ν) = dg h i ν + b N h j ν + b N b=-B/2,...,B/2 (17)
and

ξ ϵ i (ν) = ξ ϵ i ν - B 2N , . . . , ξ ϵ i ν + B 2N T .
Instead of working directly with

|ĉ ij (ν)| 2 = |ŝ ij (ν)| 2
ŝi (ν)ŝ j (ν) , it turns out that it is more convenient to show the limiting Gumbel distribution for

|s ij (ν)| 2 σ 2 ij (ν) where σ 2 ij (ν) = 1 B + 1 B/2 b=-B/2 h i ν + b N 2 h j ν + b N 2 = 1 B + 1 B/2 b=-B/2 s i ν + b N s j ν + b N := tr Σ ij (ν) B + 1 ( 18
)
where

∆ 1 = α∈I β∈Iα P (X α = 1) P (X β = 1) ∆ 2 = α∈I β∈Iα\{α} P (X α = 1, X β = 1) ∆ 3 = α∈I E P X α = 1| (X β ) β∈I\Iα -P(X α = 1)
In particular, if for each α ∈ I, X α is independent of {X β :

β ∈ I \ I α }, then ∆ 3 = 0.
Lemma 2 is the keystone for the proof of Proposition 1, and is a standard tool for analyzing distributions of maxima of dependent random variables. We now prove Proposition 1.

Proof. We start by proving (20). Define

t N = x + log M (M -1) 2 + log N B + 1 (21)
and for (i, j, ν) ∈ I (recall that I is defined in (5), and that it depends on N , but in order to avoid cumbersome notations we do not recall this dependency) the Bernoulli random variables X ij (ν) as

X ij (ν) := 1 (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2 N . (22)
Define the set I (i,j,ν)

I (i,j,ν) = {(i ′ , j ′ , ν) : 1 ≤ i ′ < j ′ ≤ M, i = i ′ or j = j ′ }. (23)
From ( 16) and under Assumption 1, if (i ′ , j ′ , ν ′ ) ∈ I\I (i,j,ν) , then si ′ j ′ (ν ′ ) is independent from sij (ν) since we have either

(1) i ′ ̸ = i, j ′ ̸ = j, ν ′ = ν;

(2) i ′ = i or j ′ = j, and

ν ′ ̸ = ν (implying |ν -ν ′ | > B N by assumption), in which case ξ ϵ i ′ (ν ′ ), ξ ϵ j ′ (ν ′ ) is independent from ξ ϵ i (ν), ξ ϵ j (ν) .
From the definition of X ij (ν) in ( 22),

P (B + 1) max (i,j,ν)∈I |s ij (ν)| 2 σ 2 ij (ν) ≤ t 2 N = P   (i,j,ν)∈I X ij (ν) = 0  
which can be estimated by Lemma 2 as:

P   (i,j,ν)∈I X ij (ν) = 0   -e -λ ≤ ∆ 1 + ∆ 2 + ∆ 3 where λ = (i,j,ν)∈I P (X ij (ν) = 1) = (i,j,ν)∈I P (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2 N and ∆ 1 = (i,j,ν)∈I (i ′ ,j ′ ,ν)∈I (i,j,ν) P (B + 1) |s i,j (ν)| 2 σ 2 ij (ν) > t 2 N (B + 1) |s i ′ j ′ (ν)| 2 σ 2 i ′ j ′ (ν) > t 2 N (24) ∆ 2 = (i,j,ν)∈I (i ′ ,j ′ ,ν)∈I (i,j,ν) (i ′ ,j ′ )̸ =(i,j) P (B + 1) |s i,j (ν)| 2 σ 2 ij (ν) > t 2 N , (B + 1) |s i ′ j ′ (ν)| 2 σ 2 i ′ j ′ (ν) > t 2 N (25) ∆ 3 = (i,j,ν)∈I E P (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2 N | (s i ′ j ′ (ν ′ )) (i ′ ,j ′ ,ν ′ )∈I\I (i,j,ν) -P (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2 N . (26)
We now have to control the four quantities λ, ∆ 1 , ∆ 2 and ∆ 3 , which requires studying moderate deviations results for

P (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2
N as well as

P (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2 N , (B + 1) |s i ′ j ′ (ν)| 2 σ 2 i ′ j ′ (ν) > t 2
N for all (i ′ , j ′ , ν) ∈ I (i,j,ν) . The following Proposition 3, proved in Appendix C from [START_REF] Rosuel | On the asymptotic distribution of the maximum sample spectral coherence of gaussian time series in the high dimensional regime[END_REF], provides exactly this.

Proposition 3. Under Assumptions 1 -3, there exists a constant η > 0 such that for any C > 0, we have

max t∈[0,CB η ] max (i,j,ν)∈I P (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2 e t 2 -1 ---→ N →∞ 0 (27) and max t,s∈[0,CB η ] max (i,j,ν)∈I (i ′ ,j ′ ,ν)∈I (i,j,ν) P (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2 , (B + 1) |s i ′ j ′ (ν)| 2 σ 2 i ′ j ′ (ν) > s 2 × e t 2 +s 2 -1 ---→ N →∞ 0. (28)
First, concerning exp(-λ), since t N as defined in ( 21) is O(log N ), one can use Proposition 3 to get exp

(-λ) = exp   - (i,j,ν)∈I P (B + 1) |s ij (ν)| 2 σ 2 ij (ν) > t 2 N   = exp - N B + 1 M (M -1) 2 e -t 2 N (1 + o(1)) ---→ N →∞ exp (-exp(-x)) .
We now turn to the control of ∆ 1 , ∆ 2 , and ∆ 3 . Regarding ∆ 3 , since under Assumption 1 the random variables sij (ν) and si ′ j ′ (ν ′ ) for (i ′ , j ′ , ν ′ ) ∈ I\I (i,j,ν) are independent, we clearly have ∆ 3 = 0. Consider now ( 24) and ( 25). = O 1 N .

∆ 2 is handled similarly with equation ( 28) from Proposition 3:

∆ 2 = (i,j,ν)∈I (i ′ ,j ′ ,ν)∈I (i,j,ν)

P (B + 1) |s ij (ν)| 2 σ 2 ij (ν)
> t 2 N , (B + 1) P (B + 1) |s i,j (ν)| 2 σ i,j (ν) 2 > t 2 N , (B + 1)

|s i ′ j ′ (ν)| 2 σ 2 i ′ j ′ (ν) > t 2 N ≤ |I| max
|s i ′ j ′ (ν)| 2 σ 2 i ′ j ′ (ν) > t 2 N e 2t 2 N =1+o(1) = O 1 N .
The proof of (20) is complete.

Proof of Proposition 2

To prove Proposition 2, ie. the fact that max (i,j,ν)∈I

|ŝ ij (ν)| 2
ŝi (ν)ŝ j (ν) and max (i,j,ν)∈I |s ij (ν)| 2 σ 2 ij (ν) are close enough in probability, we work separately on the numerator and the denominator. This constitutes the statement of the two following propositions.

Proposition 4 (Change of numerator). Under Assumptions 1 -3, there exists δ > 0 such that as N → ∞, √ B + 1 max

(i,j,ν)∈I |ŝ ij (ν) -sij (ν)| = O P (N -δ ). ( 29
)
The proof is deferred to Appendix B from [START_REF] Rosuel | On the asymptotic distribution of the maximum sample spectral coherence of gaussian time series in the high dimensional regime[END_REF]. A consequence of Proposition 4 and Proposition 1 is that √ B + 1 max

) 33
The proof is deferred to Appendix A in [START_REF] Rosuel | On the asymptotic distribution of the maximum sample spectral coherence of gaussian time series in the high dimensional regime[END_REF]. We recall that for any sequences (a n ) and (b n ), the following inequality holds: This result could be proved by writing the following decomposition:

(B + 1) max

(i,j,ν)∈I |s ij (ν)| 2 σ 2 ij (ν) - |ŝ ij (ν)| 2 ŝi (ν)ŝ j (ν) ≤ Ψ 3 (Ψ 1 + Ψ 2 ).

Figure 1 :

 1 Figure 1: sample cdf and histogram of the MSSC as defined in Theorem 1 vs Gumbel distribution.

  under H 0 computed from Monte-Carlo simulation. For fairness comparison, we also use this procedure for the new test statistic T (MSSC) N . More precisely, we compute the sample (1-α)-quantile κ 1-α of a test statistic T (LSS) N from samples under H 0 , and then reject the null hypothesis under H 1 if T (LSS) N > κ 1-α . It remains to choose a test function f , and we again follow Loubaton and Rosuel (2021) by considering

Figure 2 :

 2 Figure 2: ROC associated to each test under H (glob) 1 with r = 0.01 (left) and H (loc) 1 with β = 0.1 (right) when (N, M, B) = (2846, 290, 580))

  ,j ′ ,ν)∈I(i,j,ν)

  Change of denominator). Under Assumption 2, for any ϵ > 0, as N → ∞,max (i,j,ν)∈I ŝi (ν)ŝ j (ν) -σ 2 ij (ν) = O P

  sup n a n -sup n b n ≤ sup n |a n -b n |.Therefore, to show that Proposition 2 holds, it is enough to show that max |ĉ ij (ν)| 2 = o P (1).

Table 1 :

 1 Sample type I error at 5%

				(MSSC) N T
	N	B	M
	42	20	10	0.012
	316	100 50	0.035
	659	180 90	0.037
	1044 260 130	0.045
	1459 340 170	0.046
	1901 420 210	0.048
	5623 1000 500	0.048
	13374 2000 1000	0.049

Table 2 :

 2 Power comparison under H 1 global, type I error = 5%

				T N (FROB)	T N (LOG)	(MSSC) N T
	N	M	B
	42	10 20	0.050	0.049	0.052
	316 50 100	0.036	0.042	0.067
	659 90 180	0.067	0.065	0.086
	1044 130 260	0.142	0.122	0.133
	1459 170 340	0.339	0.255	0.214
	1901 210 420	0.601	0.462	0.328
	2364 250 500	0.836	0.682	0.503
	2846 290 580	0.960	0.852	0.672

Table 3 :

 3 Power comparison under H 1 local, type I error = 5%

				T N (FROB)	T N (LOG)	(MSSC) N T
	N	M	B
	42	10 20	0.049	0.049	0.061
	316 50 100	0.038	0.044	0.352
	659 90 180	0.038	0.041	0.881
	1044 130 260	0.034	0.038	0.999
	1459 170 340	0.034	0.038	1.000
	1901 210 420	0.035	0.039	1.000
	2364 250 500	0.031	0.039	1.000
	2846 290 580	0.032	0.036	1.000
	(MSSC) N outperform T

  The aim is to show that ∆ 1 = o(1) and ∆ 2 = o(1) when t N is defined by (21). Using the moderate deviation result (27) from Proposition 3, and recalling that C represents a universal constant independent of N whose value can change from one line to another, we get:

	∆ 1 ≤ |I|	max (i,j,ν)∈I	|I (i,j,ν) |	max (i,j,ν)∈I	P (B + 1)	N σ i,j (ν) 2 > t 2 |s i,j (ν)| 2	2
	O( N B M 2 )	O(M )
	≤ C	N B	M 3 e -2t 2 N O( 1 M 4 B 2 N 2 )	max (i,j,ν)∈I	P (B + 1) =1+o(1) |s i,j (ν)| 2 σ i,j (ν) 2 > t 2 N e t 2 N	2

any finite linear combination x of the components of (y n ) n∈Z is a complex Gaussian random variable, i.e. Re(x) and Im(x) are independent zero-mean Gaussian random variables having the same variance

For two sequences (x n ) n≥1 , (y n ) n≥1 , we denote byx n ≍ y n if there exists k 1 , k 2 > 0 such that k 1 |y n | ≤ |x n | ≤ k 2 |y n | for all large n.

A complex random variable Z is zero-mean complex Gaussian distributed with variance σ 2 , denoted as Z ∼ N C (0, σ 2 ), if Re(Z) and Im(Z) are i.i.d. N (0, σ 2 2 ) random variables.

⋆ This work is funded by ANR Project HIDITSA, reference ANR-17-CE40-0003.

and where

This is the aim of Proposition 1 below.

Proposition 1 (Gumbel limit for max (i,j,ν)∈I |s ij (ν)| 2 ). Under Assumptions 1 -3, for any t ∈ R, we have

Once equipped with Proposition 1, it remains then to show that max (i,j,ν)∈I

prove that these quantities have the same limiting distribution. This result is given by the following Proposition.

As Theorem 1 is directly obtained by Proposition 1, Proposition 2 and an application of Slutsky's lemma, the two remaining subsections are devoted to the proofs of Proposition 1 and Proposition 2.

Proof of Proposition 1

To prove Proposition 1, the main tool is Lemma A.4 from [START_REF] Jiang | The asymptotic distributions of the largest entries of sample correlation matrices[END_REF], which is a special case of Poisson approximation from [START_REF] Arratia | Two moments suffice for poisson approximations: the chen-stein method[END_REF]. We rewrite it here for the sake of completeness.

Lemma 2. Let (X α ) α∈I be a finite collection of Bernoulli random variables, and for each α ∈ I, let I α ⊂ I such that α ∈ I α . Then,

.

It is clear by ( 20) that max

Combining this with Proposition 5 and equation ( 29) from Proposition 4, there exists δ > 0 such that (B + 1) max

which is O P ( √ log N N -δ ). Using (33), this implies that

Similarly, using Proposition 5, for any ϵ > 0,

Combining the estimates of Ψ 1 , Ψ 2 and Ψ 3 we get that for any ϵ > 0:

This quantity is o P (1) if N ϵ √ B = o(1) which is satisfied by choosing ϵ < ρ 2 from Assumption (4).