R Bhar

A Jardy

P Chapelle
email: pierre.chapelle@univ-lorraine.fr

V Descotes

3D NUMERICAL SIMULATION OF THE VAR CONSUMABLE ELECTRODE MELTING PROCESS

Keywords: VAR process, consumable electrode, dripping, liquid metal film, melt rate

A 3D numerical model was set-up to simulate the formation and dynamics of the liquid metal film under the consumable electrode during VAR process. In the present paper the implementation of this model is described. It was developed using the open source computational fluid dynamics (CFD) software OpenFOAM. The model solves coupled momentum and energy equations combined with a volume-of-fluid (VOF) method to track the liquid metal free surface. The melting of the electrode material is modelled with an enthalpyporosity approach. The electric power supplied by the arc is supposed to be uniformly distributed over the surface of the electrode tip. For a given electric arc power, the model enable to quantitatively predict the dripping rate, hence the overall melt rate. Besides the thermal behavior of the electrode, simulation results illustrate the dynamics of the liquid film and the transfer mechanisms of the liquid metal during VAR melts performed with short and long interelectrode gaps.

I. Introduction

Vacuum arc remelting (VAR) is a secondary remelting process used to improve cleanliness as well as chemical and mechanical homogeneity of metal ingots. VAR was the first remelting process to be used commercially for superalloy processing. It is also typically the final stage in the melting cycle of reactive metals such as titanium and zirconium alloys. [START_REF] Yu | Modeling for Casting and Solidification Processing[END_REF] The process consists of melting a consumable electrode under vacuum (see Fig. 1). The heat source is a DC electric arc of low voltage and high current. The arc is created between the electrode (cathode) and the base plate of a water-cooled copper crucible at the beginning of the melt, then between the electrode and the secondary ingot (anode) forming in the crucible. The melting of the tip of the electrode generates a liquid metal film under the electrode, from which metal drops are produced that fall under the action of gravity into the crucible and progressively solidify to form the secondary ingot. At any instant, the ingot is composed of three zones: the fully solidified metal, the liquid pool fed by metal drops and an intermediate mushy zone. The quality of the produced ingots strongly depends on the operating conditions of remelting. Among them, the melt rate and the interelectrode gap play a key role, since they have significant effects on heat transfer conditions at the free surface of the liquid pool, which have important implications on the ingot structure and chemical homogeneity [START_REF] Yu | Metallurgy and Applications[END_REF] .

The VAR process has been investigated previously with both experimental and numerical approaches. On the experimental side, some studies were devoted to establish various correlations between the operating parameters [START_REF] Zanner | [END_REF]4] , whereas some research work focused on the electric arc behavior and metal transfer mechanisms in the interelectrode region, which were observed using high speed video cameras in specifically instrumented VAR furnaces [5,6] . Modelling of the VAR process is a difficult task, because the process involves a wide range of coupled complex physical and chemical phenomena, such as fluid flow, heat and mass transfer, solidification (macro and microsegregation), electromagnetic forces… In the literature, most modelling works deal with the development of Computational Fluid Dynamics (CFD) models of the ingot growth and solidification. The majority of authors considers a 2D axisymmetric geometry of the ingot and solves the conservation equations of mass, momentum and energy, accounting for turbulence phenomena and electromagnetic forces in the liquid pool as well as the solidification of the metal. Examples of such models are the SOLAR code [START_REF] Jardy | Xiyou Jinshu Cailiao yu Gongcheng[END_REF] and the MeltFlow-VAR code [START_REF] Kelkar | 11th World Conference on Titanium (Ti-2007)[END_REF] . More recently, a multiscale 3D numerical model of VAR was developed by Pericleous et al. [START_REF] Pericleous | [END_REF] , which deals also with the ingot behavior. Contrary to the ingot, the consumable electrode has received relatively little attention, with very few modelling works reported, all restricted to thermal phenomena in the electrode. Numerical studies on the formation and dynamics of the liquid film under the electrode are in particular missing up to now. Bertram and Zanner [START_REF] Bertram | Electrode Tip Melting Simulation during Vacuum Arc Remelting of Inconel 718[END_REF] described a transient and one-dimensional model of the heat transfer in the electrode, which was applied to study the effect of the melting current and the gap length on the electrode melting. A similar model, including an explicit account of radiative losses from the lateral walls of the electrode, was presented by Jardy et al. [START_REF] Jardy | [END_REF] . Lately, a step forward was made by El Mir et al. [12] and Jardy et al. [13] , who reported an unsteady model of heat transfer in the electrode, considering respectively 2D and 3D geometries. Besides the electrode melt rate, these latter models enable to predict, contrary to previous studies, the evolution of the shape of the electrode tip throughout the melt.

II. Numerical model

The formation and dynamics of the liquid film under the consumable electrode during the VAR process is simulated with a multiphase CFD approach. The metal phase change is accounted for using the enthalpy-porosity method [14] and the shape and position of the free surface of the liquid film are calculated using the volume of fluid (VOF) interface capturing method [15] .

The developed model is based on the following assumptions.

(1) The heat flux provided by the arc to the electrode is considered to be uniformly distributed at the base of the electrode. The influence on the arc heat flux distribution of the motion of individual cathode spots and of the possible existence of a relatively slow ensemble motion of the arc (see e.g. [16] [17] ) is not examined in this study.

(2) As a first step towards a complete description of the liquid film behavior, magnetohydrodynamic effects produced by the arc current (i.e. electromagnetic forces acting on the liquid metal) are not taken into account.

(3) In a VAR furnace, the liquid metal film is exposed to a low pressure arc plasma. The present model does not deal with the description of this complex latter phase, which is represented here as a neutral gas phase.

(4) All thermophysical properties are considered to be independent of temperature and identical in the solid and liquid phases. The metal density is made temperature dependent only in the buoyancy term in the momentum equation.

(5) As explained later in section II.A, when computing the flow, a specific procedure was introduced to eliminate non-physical spurious velocities generated near the interface in the gas region. This procedure is applied at the end of each time step after calculating the flow in both the liquid and gas regions. It is based on a simple filtering scheme designed to set to zero the velocity calculated in all computational cells of the gas phase. Note that this filtering was observed to have negligible influence on the flow calculated in the melt pool.

A. Governing equations

Fluid flow and behavior of the free surface

The computed behavior of the free surface of the liquid metal film is based on the VOF approach. In order to identify the metal-gas interface, the model solves an advection equation of the volume fraction of the metal α m (either solid or liquid) present in each computational cell:

+ . + . - = (1)
α m = 0 if no metal is present in the cell, α m =1 for a cell completely filled with metal, and 0 < α m < 1 if the interface is present in the cell. In the VOF formulation implemented in OpenFOAM, the volume fraction transport equation contains an additional convective term (third term in Eq. 1), referred to as "compression term". This term compresses the interface by minimizing the numerical diffusion of the volume fraction while ensuring its boundedness. It can be noticed that this additional compression term acts only at the vicinity of the interface region where α 1 -α ≠ 0. Therefore, it does not affect the fluid flow outside this region [18] . The parameter U r in the compressive term is the relative velocity. Since the VOF method considers only a single velocity for the liquid and gas phases, the relative velocity cannot be evaluated directly. The evaluation of U r is performed at cell faces using the following approximation:

, = | | !" ! , #$ % | | !" ! &'
where the max operation is carried out over the entire domain. is the face centered interface normal vector, φ is the face volume flux and S ) is the cell face area vector. The parameter C + is a user specified constant, which controls the contribution of the compressive term: the higher this value, the sharper the interface. Recommended values are in the range 2 ≤ C + ≤ 4. This constant has been set to 4 in all the simulations presented in this paper.

The gas-liquid metal flow is governed by a single set of continuity and Navier-Stokes equations. Turbulent phenomena in the liquid phase are modeled through the realizable k-ε model (the well-known equations of this model may be found in [19] ). This model was selected, since it provides better predictions of flows including streamline curvature and recirculation.

• Continuity equation

, + . , = (2)
• Navier-Stokes equation -.

-/ + 0. . = -01 + 0. 2 + 2 3 0 + 4 + . 567 1 -8 9 : -: ;<= > -2 9 ?

The thermophysical properties are calculated as volume fraction weighted averages of the properties of the phases present in the computational cell:

ξ = A 9 ξ 9 + 1 -A 9 ξ BCD (4)
The turbulent viscosity is computed as a function of the turbulent kinetic energy k and its dissipation rate ε and is expressed as:

μ F = . 9 G H I J K (5)
where C µ is a coefficient (whose expression can be found in [19]) computed as a function of the main strain and rotation rates and the turbulence fields (k and ε).

Surface tension effects are taken into account through a continuous equivalent volumetric force f σ (Continuum Surface Force model) as proposed by Brackbill et al. [20] . This force, acting only in the vicinity of the interface and directly linked to the interface curvature k c , is defined as:

4 = L 2. . 9 + . BCD I N 0A 9 (6)
A problem encountered in this formulation is the presence of numerical parasitic velocity vortices known as "spurious currents" [21] . These "spurious velocities" appear in the gas phase near the interface despite the absence of any external forces and can be significant when the capillary effects are predominant. These high velocities force to use very small time steps to maintain the stability of the simulation (CFL criterion), which increases the computational time. In addition, their magnitude does not decrease with neither mesh refinement nor smaller computational time steps [22] . In the current study, a filtration method consisting in setting to zero the velocities of the gaseous phase was implemented to eliminate those spurious velocities. This method was not restricted to the interface region, but it was applied to all computational cells of the gas region.

The investigation and modelling of the solid to liquid (melting) process has received much less attention in the literature than that of the liquid to solid (solidification) process. In particular, it remains unclear whether the melting process involves the existence of a mushy region similarly to what it is observed during solidification. To the best of the authors knowledge, existing attempts to model melting (e.g. [34,35]) are all based on the assumption of the existence of a mushy region and treats this region using the classical enthalpy-porosity technique originally developed to deal with solidification. A similar approach is used in the present study, considering a dendritic morphology of the mushy region. It involves the implementation of the last source term on the right hand side of Eq. 3, which is derived from the Darcy law. The mushy zone is treated as a porous medium, whose permeability is calculated using the well-known Kozeny-Carman model, which was extended in the present work to deal with a three phase system (solid-liquid-gas).

? = O J J 1 -P Q 180 P² (7)
In this equation, λ 2 denotes an order of magnitude of the secondary dendrite arm spacing. γ is the solid volume fraction in the cell which may be expressed as a function of the metal volume fraction and the liquid volume fraction g l : γ = α m (1-g l ). In practice, the effect of this term is as follows. In the fully liquid cells (γ = 0), the source term is zero and the "classical"

Navier-Stokes equation is solved. In the cells where a phase change occurs, the source term dominates over the transient, convective and diffusive terms, thereby forcing to imitate the Kozeny-Carman model [14] . In the fully solid cells (γ = 1), the source term swamps out all other terms in the momentum equation, which forces velocities to zero [START_REF] Brent | Numerical Heat Transfer[END_REF] . Strictly speaking, the velocity appearing in the last source term in Eq. 3 should be the relative velocity between the liquid and the solid. Since the solid phase is kept at rest in our simulations, this velocity reduces here to the fluid velocity.

Heat transfer with phase change

The enthalpy-porosity method is based on the enthalpy formulation of the energy conservation equation:

-ℎ -/ + 0. ℎ = 0. I + A 9 G1 9 2 3 0 :

The enthalpy may be computed as the sum of sensible enthalpy and latent heat absorbed when the metal changes from solid to liquid:

ℎ = U .G V W: + . 9 A 9 X ; Y Z Z [\] (9)
The energy conservation may thus be expressed as follows:

-.G V : -/ + 0. .G V : = 0. I + A 9 G1 9 2 3 0 : -. 9 Y ^-A 9 X ; -/ + 0. A 9 X ; _ + `C5N -`5Ca

(10) The second term on the right hand side of Eq. 10 accounts for the evolution of the latent heat during phase change. The liquid-solid interface is not tracked explicitly. Instead, the melting zone is treated as a porous zone as described previously, in which the liquid fraction lies between 0 and 1. The liquid fraction is a function of temperature and can be, for example, calculated as follows:

X ; = 0 bc : < : De; The third term on the right hand side of Eq. 10 is the thermal power provided by the electric arc to the metal. In fact, this power corresponds to a boundary condition of the problem, which is treated here as a volumetric source term applied only in the cells crossed by the metal/gas interface (defined as the cells where α m = 0.5). The total power delivered by the arc to the electrode P tot is assumed to be uniformly distributed at the electrode base, which leads to the following expression of P arc :

`C5N,< = `3e3 ∑ h < h < i N< (12)
where V ci is the volume of the cell i (crossed by the interface), S i is the area of the portion of the interface contained in cell i and ∑S i is the total area of the interface. The interfacial area is approximated by the following expression:

h b = 2A j |∇A j | i lb .
The last term on the right hand side of Eq. 10 represents thermal losses by radiation at the electrode lateral wall towards the crucible wall. The radiative heat fluxes at the electrode base can be neglected, given the small difference between the electrode and ingot diameters and the similarity of the metal temperatures at the electrode base and on the ingot top. Our evaluation of the radiative heat fluxes between the electrode and the crucible is based on a simplified approach, considering radiative exchanges between two infinite parallel planes.

The energy flux density radiated by a surface element of the electrode lateral wall may thus simply be expressed as:

m 5Ca,< = L D3 K 6= : < n -: 9e;a n ) (13)
The equivalent emissivity is defined by:

K 6= = 1 1 K 6;6N35ea6 + 1 K 9e;a -1 (14)
Similarly to the treatment of the heat flux provided by the arc to the electrode base, this radiative flux density is converted into an equivalent volumetric source term.

`5Ca,< = m 5Ca,< h < i N< (15)

B. Boundary conditions

Figure 2 shows a vertical cross-section of the 3D computational domain and illustrates the boundary conditions used for the simulation. The red lines represent the surface of application of the arc power while the purple lines represent the surface associated to radiative losses. In order to get a realistic simulation of the formation of molten metal bridges (i.e. drip-short)

between the electrode and the ingot, the domain includes a liquid region under the electrode representing a part of the molten metal pool present at the ingot top.

The main boundary conditions of the model, for each of the dependent variables under consideration (i.e. metal volume fraction, velocity, pressure, turbulent kinetic energy and its dissipation rate, temperature), are as follows:

• Metal volume fraction: A fixed value (α m = 1) is specified at the top wall, while a zero normal gradient is imposed at all other walls.

• Velocity: A no-slip boundary condition is used at walls.

• Pressure: The pressure is specified at the outlet.

• Turbulent kinetic energy and its dissipation rate: The normal gradients are set to zero at all boundaries.

• Temperature: The walls as well as the outlet are treated as adiabatic, which results in a zero temperature normal gradient condition.

C. Computational details

The pressure-velocity coupling is handled by the PIMPLE algorithm (combination of the PISO [24] and SIMPLE [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF] algorithms). The convective terms in the conservation equations are discretized with a second order VanLeer scheme, while the diffusion terms are central differenced. The transport equation of the metal volume fraction is solved using the OpenFOAM solver called InterFoam, which is based on the MULES limited interface compression method. Temporal integration is done using a first order implicit Euler method.

The order of magnitude of the time step is 10 -4 s and is controlled via a CFL condition (set to 0.5). The simulations are parallelized on 256 processors (DELL C6320, Intel Xeon 2.1 GHz CPU). The computation time of the melt of a 10 cm high large diameter electrode is about 3 weeks.

III. Application of the model

298 299

The numerical model described above was applied to simulate two different melt 300 configurations. The first one is the melt of a small diameter Ti-6Al-4V electrode with a long 301 interelectrode gap performed experimentally by Chapelle et al. [6] , while the second one is [START_REF] Mills | Recommended Values of Thermophysical Properties for Selected Commercial Alloys[END_REF] Maraging Steel [START_REF] Descotes | [END_REF] A. Small diameter electrode with a long gap Fig. 3 shows a vertical cut of the computational domain with the initial metal represented in yellow. The electrode diameter is 160 mm, its height is 150 mm and the interelectrode gap is 66 mm. The height of the electrode considered here is much smaller than that of the actual electrode, in order to limit the computational time. The total electric arc power is 183.6 kW.

Gas

The electrode and the metal bath at the ingot top are considered to receive respectively 60% and 40 % of this power [START_REF] Jardy | [END_REF] . Initially, the electrode is in solid state with a temperature of 25 °C.

The liquid bath is at the liquidus temperature of the alloy. The computational grid is a uniform structured mesh consisting of around 3.5 million cells.

Liquid film behavior

At the initial state, the electrode is isothermal. The thermal power provided by the electric arc heats the electrode. At this stage, the heat transfer is purely conductive. As soon as the temperature of the tip reaches the solidus temperature, the electrode starts to melt and a distorted liquid film is formed. The film thickness is about 1 mm, which is agreement with the estimates given in the literature [START_REF] Mitchell | [END_REF]29] .

Owing to the Rayleigh-Taylor instability which occurs when a heavy fluid is placed above a lighter one in a gravitational field [START_REF] Chandrasekhar | Hydrodynamic and Hydromagnetic Stability[END_REF] , small protuberances appear and are spatially distributed in lattices as shown in Fig. 4, which gives the depth reached by the protuberances in the liquid film. Continuous feeding of the film due to the metal melting eventually destabilizes these structures. The protuberances are subject to horizontal movements. Once they get close to each other, they coalesce, grow and form drops. Such a behavior is qualitatively consistent with the observations made by Limat et al. [START_REF] Limat | [END_REF] in a model experiment.

Although the mechanism of formation of the protuberances seems random, the drops appear to detach in specific locations, namely either at the center of the electrode or at its periphery.

The dripping sites tend to form a square network with a centered face. This can be clearly seen on the right image of figure 4, where a drop is about to form in the center due to the coalescence of two protuberances, while four protuberances located at the periphery of the electrode will give rise later on to four drops due to the drainage of the liquid film. al. [6] . On the left image, the protuberance which grows by drainage of the liquid metal has a cylindrical shape. Then it stretches resulting in the formation of a drop. The drop remains attached to the liquid film by a filament which gradually becomes thinner. This latter process eventually triggers the break-up of the filament and subsequent detachment of the drop before touching the molten pool. The dimension of the simulated drop is of the same order of magnitude (around 1 cm) as that of the actual drop. It is also interesting to note that the characteristic times for each stage are of the same order of magnitude, even if the "numerical" drop takes less time to detach than in reality. Although this comparison can only be qualitative since the simulation does not take into account the electromagnetic effects related to the presence of the arc, the simulated behavior remains close to the experimental visualizations.

Fluid flow inside a protuberance

As an example of the typical fluid flow present in a protuberance, Fig. 6.a shows a 3D representation of the computed velocity vectors in a protuberance. The fluid flow is fed by the drainage of the liquid film and is governed by buoyancy forces. It is characterized by a large recirculation loop that occupies the whole volume of the protuberance. The velocity magnitude reaches a value of around 8 cm.s -1 at the central axis of the protuberance. The maximum velocity is reached in the vicinity of the lateral edge of the protuberance and is about 14 cm.s -1 . Fig. 6.b illustrates the turbulence level of the metal flow in the film and in a protuberance. In the protuberance, the turbulence level is moderate, with a maximum value of the ratio of the turbulent to molecular viscosities of about 180. In the liquid film, the viscosity ratio is much lower, which corresponds to a flow regime remaining laminar.

Heat transfer in the electrode

The computed temperature field of the electrode at t = 22.2 s (quasi-stationary regime reached) is presented in Fig. 7.a, while the temperature profile calculated at the same time along a vertical line cutting through a protuberance (cf. Fig. 7a) is plotted in Fig. 7.b. Due to the consumption of the electrode, the power supplied by the electric arc affects mainly a small region at the electrode tip, usually referred to as the "heat affected zone". Accordingly, figure 7.b shows a sharp exponential decay in the axial direction of the temperature from the base of the electrode toward the upper part of the electrode. In the present case, the heat affected zone is about 3 cm high, which is good agreement with the experimental observations of El Mir et al. [12] and the results provided by previous models [10][11,12] . As observed in Fig. 7a, the variations in the azimuthal direction of the electrode lateral surface temperature are negligible.

Also, the temperature distribution over the surface of the liquid film (excluding the drops) is relatively homogeneous, which is partly related to the uniform arc heat flux distribution considered here.

The superheat of the metal in the liquid film remains small (not exceeding 50 °C). On the other hand, a much higher superheat is calculated inside the protuberance, which reaches about 400 °C at the tip of the protuberance. Such higher superheat may be caused by the recirculating flow of the metal in the protuberance, which contributes to hold the metal inside the protuberance, thus enabling the metal to reach higher temperatures. The predicted value of the superheat of the metal at the free surface of the liquid film is consistent with values reported in the literature for different materials (namely about 100 °C for IN718 [START_REF] Bertram | Electrode Tip Melting Simulation during Vacuum Arc Remelting of Inconel 718[END_REF] and between 150 °C and 200 °C for a zirconium alloy [12] ).

B. Large diameter electrode with a short gap

Simulation results are presented for an industrial VAR melt of a maraging steel electrode.

The electrode considered is 610 mm in diameter and 150 mm high (figure 8). Similarly to the previous case; the height of the electrode is significantly reduced compared to that of an actual full-scale electrode, so as to keep the computational time reasonable. The gap length is 10 mm. Again, it is considered that 60 % of the total electric arc power (300 kW) is transferred to the electrode, while the remaining part (40 %) of this power goes to the metal bath at the ingot top [START_REF] Jardy | [END_REF] . Initially, the electrode is in solid state with a temperature of 25 °C and the metal bath is at the liquidus temperature. In order to reduce further the computational time, an irregular mesh was used. The mesh is refined inside the gap and in its vicinity and gets progressively coarser moving further away from this area. The number of cells is 2.9 millions.

Liquid film behavior

Similarly to the observations made above for the melt of a Ti-6Al-4V electrode with a short gap, the destabilization of the liquid film formed at the electrode tip gives rise again to the growth of liquid metal protuberances. However, as illustrated in figure 9a, due to the short gap considered here, the growing protuberances do not evolve into detaching drops but end up forming transient capillary bridges (known as drip-shorts) between the electrode and the metal bath at the ingot top. The mass of the drip-short shown in the figure 9 is 16 g. A part of this amount of metal is eventually transferred to the metal bath as a result of the break-up of the bridge. Such a drip-short based metal transfer mechanism is consistent with the observations made during the actual VAR melt. Indeed, although drip-shorts could not be visualized during the melt, the occurrence of drip-shorts could be monitored through their signatures left on the arc voltage signal. Note that the precise mechanisms of the bridge break-up observed in the present simulation results are different from those described in the literature [6] , due to the neglect of electromagnetic phenomena in our model. Protuberances tend to form near the center of the electrode and migrate to the periphery of the electrode forming approximately concentric circles. This arrangement in concentric circles was previously observed for a 440 mm superalloy electrode in an experimental study carried out by Wadier et al. [START_REF] Wadier | Influence de La Refusion d'électrodes Consommables Sur La Propreté Inclusionnaire[END_REF] , which consisted in abruptly switching off the arc current and freezing the liquid film with a gas stream . Such a behavior has been attributed by some authors to Lorentz forces [START_REF] Andreev | Titanium Alloys. Melting and Casting of Titanium Alloys[END_REF] . According to our simulations, this phenomenon is purely thermohydrodynamic and is likeky to be linked to the concavity of the electrode tip. Due to the cooling resulting from the radiation losses at the lateral surface of the electrode, the melting of the electrode tip is greater at the electrode center than at its periphery, which leads (as will be illustrated below) to a slightly concave shape of the electode tip. Because of gravitational effects, the protuberances appearing near the electrode center tend then to "roll" towards the periphery of the electode, leading to the outward radial motion of the protuberances observed in figure 10. Note that this outward radial motion of the protuberances tends to be periodic, repeating every about 40 s.

Heat transfer in the electrode

Predicted melt rate

The time evolution of the electrode melt rate calculated (with a sampling rate of 10 s) from the evolution of the electrode total mass (including both solid and liquid regions) is plotted in Fig. 12. The electrode begins effectively to melt after a pre-heating stage of about 7 min. The variation of the melt rate is characterized by a sharp increase until reaching a quasi-stationary regime. In the current model, it should be noted that the process starts at full power, contrary to the real process in which the electric current is gradually increased. The average calculated melt rate is 7.2 kg.min -1 with fluctuations between 6 and 8 kg.min -1 . This value is in good agreement with that monitored during the actual melt which quantitatively validates the present model.

Conclusion

A 3D numerical model was developed in order to simulate the melting of a VAR consumable electrode. This work couples the enthalpy-porosity approach to simulate the melting process and the volume of fluid method to model the deformation of the liquid film formed under the electrode.

The model was applied to simulate the melt of a small diameter Ti-6Al-4V electrode with a long interelectrode gap and that of a large diameter maraging steel electrode with a short interelectrode gap. The model was shown to be able to reproduce two different modes of electrode consumption in accordance with experimental observations. For a long gap, liquid metal protuberances formed at the liquid film surface evolve into drops that eventually detach and fall into the metal bath at the ingot top, while for a short gap (10 mm in the present work) the metal protuberances evolve until forming intermittent bridges (drip-shorts) between the electrode and the metal bath. Simulation results have shown that the protuberances tend to form circle-shaped patterns appearing near the electrode center and expanding towards the electrode periphery. The metal flow in the liquid film was found to be laminar with a maximum velocity of a few cm/s. On the other hand, a turbulent flow has been observed in the metal protuberances.

The metal temperature is relatively homogeneous over the free surface of the liquid film, with a superheat of about 50 °C and about 60°C for the Ti-6Al-4V and maraging steel electrodes, respectively. The metal temperature is larger inside the protuberance than that in the liquid film, which may be caused by the flow structure of the metal in the protuberance.

The model enables also to predict the electrode overall melt rate, which compares favorably to the value measured during the melt of the large diameter electrode.

In the future, the present model will be further developed by improving the description of the interactions between the electric arc and the liquid film. Two points need in particular to be addressed. The first one deals with the repartition of the arc power over the electrode tip, considering the actual dynamics of the arc. The second one is the consideration of the influence of electromagnetic forces on the dynamics of the liquid metal.

Figure 1 :

 1 Figure 1: Schematic representation of the vacuum arc remelting process.

  The present work focuses on the interelectrode gap of the VAR process. The aim is to numerically study the formation and deformation of the liquid film under the electrode and the transfer mechanisms of the liquid metal in the interelectrode gap. A further objective is to predict the melt rate of the electrode for a given electric arc power. For this purpose, a 3D model describing the melting of the consumable electrode and the dynamics of the liquid film formed at the electrode tip was developed using the CFD open source software OpenFOAM.The model considers fuid flow under turbulent regime, heat transfer with phase change and the deformation of the free surface of the liquid film. The model is concerned with both large interelectrode gaps, for which the metal transfer results from the formation of molten metal drops from the electrode and their detachment before contacting the ingot, and short interelectrode gaps, for which the metal transfer involves the formation of intermittent molten metal bridges (drip-shorts) between the electrode and the ingot. In section 2, the model is described, including physical and mathematical issues, constitutive equations, boundary conditions and the numerical procedure. In section 3, examples of model results detailing the computed dynamics of the liquid film and thermal behavior of the electrode during the VAR melt of a small-scale electrode and a fully-scale one are presented. Finally, conclusions of the present study are drawn in section 4.

  De; : ;<= -: De; bc : De; < : < : ;<=

Figure 2 :

 2 Figure 2: Cross-section of the computational domain.

Figure 3 :

 3 Figure 3: Vertical cross-section of the computational domain used for the simulation of the Ti-6Al-4V electrode.

Figure 4 :

 4 Figure 4: Depth of protuberances formed from the liquid film.

Figure 5 :

 5 Figure 5: Formation and detachment of a drop observed experimentally [6] (top row) and predicted by the numerical model (bottom row).

Figure 6 :

 6 Figure 6: (a) Velocity vectors (3D) inside a protuberance. (b) Turbulent to molecular viscosity ratio in a vertical cross-section of a protuberance.

Figure 7 :

 7 Figure 7: (a) Temperature field of the outer surfaces of the electrode, (b) Temperature profile along a vertical line cutting through a protuberance. Results obtained during the quasistationary regime at t=22.2 s.

Figure 8 :

 8 Figure 8: Vertical cross-section of the computational domain used for the simulation of the maraging steel electrode.

Fluid

  flow inside a metal bridgeAs shown in figure9.b, the fluid flow inside a bridge is characterized by an important drainage of the metal from the liquid film to the metal bath. The velocity magnitude reaches a maximum value of around 22 cm.s -1 located closed to the central axis of the bridge. The order of magnitude of the ratio of the turbulent to molecular viscosities is about 500, which indicates a significant turbulence level of the metal flow inside the bridge.

Figure 9 :

 9 Figure 9: (a) Example of a calculated capillary bridge (drip-short). (b) Velocity vectors (3D) in the bridge.(c) Turbulent to molecular viscosity ratio in a vertical cross-section of the bridge.

Figure 10 :

 10 Figure 10: Temporal evolution of the protuberance distribution at the tip of the electrode.

Figure 11

 11 Figure11illustrates the electrode temperature distribution and the temperature profile along the axis (white dashed line)at 662 s. It can be seen that the electrode tip has a slightly concave shape due to the cooling effect caused by radiation at the electrode lateral surface. Because of the much smaller electrode height to diameter ratio considered here compared to the previous simulation, a quasi-stationary regime is not reached. The heat affected zone is much more important than previously and extends almost all the way to the top of electrode. Moreover, the temperature profile has not a such prononced exponential shape as in the previous simulation (Fig.7b).. Similarly to the observations made in figure7, the variations of the electrode lateral surface temperature in the azimuthal direction are negligible and the temperature distribution over the electrode tip is relatively homogeneous. The superheat of the metal at the free surface of the liquid film is 60 °C, while that at the tip of a protuberance is much more important and reaches 225 °C.

Figure 11 :

 11 Figure 11: (a) Temperature field of the outer surfaces of the electrode, (b) Temperature profile along the electrode central axis. Results obtained at t = 662 s

Figure 12 :

 12 Figure 12: Time evolution of the predicted melt rate of the maraging steel electrode.

Table 1 :

 1 Thermophysical parameters used for the simulations.

	Density , Dynamic viscosity t Surface tension 4 Specific heat { Emissivity Thermal conductivity } Dilatation coefficient • Liquidus temperature ' ƒ " Solidus temperature ' † ‡ƒ Reference temperature ' ˆ Latent heat of melting ‰ Secondary dendrite arm spacing λ 2 323	3925 2.4 × 10 sQ 1.525 831 0.43 33.4 6.7 × 10 s• 1625 1595 1625 2.86 × 10 • 5 × 10 sŠ	7933 5 × 10 sQ 1.93 554 0.5 23 2.2 × 10 s• 1450 1413 1450 2.61 × 10 • 5 × 10 sŠ	0,2 2.7 × 10 sn -0 -9,55.10 sJ 0 -----	IX. j sQ `w. x y. j sz |. IX sz ? sz ~. j sz . ? sz ? sz °G °G °G |. IX sz m
	324
	325
	326
	327
	328

Acknowledgments

The authors would like to thank Y. Millet and J. Jourdan from TIMET Savoie, France, who have supported the acquisition of the experimental data shown in figure 5 of the present work.


Joe Èlle

Pe Ârard-Viret

Thierry Prange

Alain Tomas

Jacques Royer
email: jacques.royer@pharmacie.univ-paris5.fr

A simple and ef®cient asymmetric synthesis of 3-alkyl-isoindolin-1-ones

Keywords: isoindolin-1-one, alkylation, carbanions, (R)-phenylglycinol, amide bases. p

AbstractÐA simple asymmetric access to 3-alkyl-isoindolin-1-ones was investigated through the diastereoselective alkylation of 2-[(1R)-2hydroxy-1-phenylethyl]-2,3-dihydro-1H-isoindolin-1-one 5. High diastereoselectivities were observed with LDA or LiHMDS while the isolated yields were modest (about 50%). In contrast the use of NaHMDS gave good isolated yields (up to 85%) but lowered diastereoselectivities. This methodology offers an ef®cient asymmetric synthesis of 3-alkylated isoindolin-1-ones.

Numerous synthetic and natural products possessing the 3-alkyl-isoindolin-1-one substructure were described. The compounds isolated from natural sources possess more or less complex structures 1 as for example indolocarbazoles 1 (staurosporine 2 and derivatives) or tetrahydro-isoindolobenzazepines (lennoxamine (2)). 3 On the other hand, synthetic 3-substituted isoindolin-1-ones were studied as anxiolytics (pazinaclone (3)), 4 5-HT 1A antagonists, 5 reverse transcriptase inhibitors, 6 or vasodilatators. 7 More generally, the easily metabolised benzylic alcohol can be replaced by the bioisosteric isoindolinone. 8 Several synthetic pathways were known to synthesize 3-alkyl-isoindolin-1-ones. 9 Indeed a limited number of methods towards the preparation of the heterocyclic skeleton were reported and only very recent works were devoted to develop a general synthetic strategy mainly through the alkylation of isoindolinones at C-3 (Fig. 1). This alkylation was described using both carbanionic 10 and carbocationic 11 strategies. Carbanionic methods may utilise a stabilizing group, 10a that was introduced at the carbon ipso. Quite recently, the group of A. Couture 10b was the ®rst to propose the substitution of isoindolinones through the formation of an unstabilized carbanion. This very simple method was used with success by Guo, 10c but was not found to be generally applicable by Enders 10d in the case of N-dimethyl amino isoindolinones. This latter author used benzotriazole stabilized carbanion as an ef®cient but multistep methodology. The carbocationic approach was reported in some instances 11 from which the work developed by Allin et al. 11a,b constitutes the unique asymmetric synthesis of 3-alkyl-isoindolin-1-ones. Modest to good diastereoselectivities were obtained by Allin and co-workers.

We became interested to try and propose a new and asymmetric approach since we anticipated that an asymmetric version of the unstabilized carbanion method could be developed using isoindolidinone bearing a chiral appendage Pergamon TETRAHEDRON 0040±4020/02/$ -see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0040-4020(02)00462-3 at nitrogen atom. This was inspired from a methodology we developed for some years on chiral g-lactam. 12 This method was based on the preparation of the chiral non-racemic g-lactam 4 in a single step from dimethoxydihydrofuran and a chiral amine, followed by the asymmetric alkylation of this chiral lactam (Scheme 1). For this alkylation step, it was necessary to use a chiral appendage bearing a free alcohol function able to strongly coordinate the lithium enolate: 12 then the use of (R)-phenylglycinol as the starting chiral amine was found to be essential in order to get high diastereomeric excesses. This paper will present the asymmetric alkylation of 2-[(1R)-2-hydroxy-1phenylethyl]-2,3-dihydro-1H-isoindol-1-one 5 through the formation of an unstabilized carbanion.

Results and discussion

The preparation of 5 was envisaged through the condensation of the primary amine function of (R)-phenylglycinol with phthalic dialdehyde following the condensation method discovered in 1909 by Thiele. 13 This method was widely used for the preparation of isoindolinone skeleton but claimed to give modest or low yields with some primary amines and several modi®cations were proposed giving improvement in speci®c cases. 9,[START_REF] Dominh | For various examples of the Thiele's condensation and related procedures see[END_REF] In our hands, the treatment of a 1/1 mixture of (R)-phenylglycinol and phthalic dialdehyde in isopropanol and in the presence of HCl gave the isoindolinone 5 in 52% yield after puri®cation (Scheme 2). Despite the modest yield, we found this method convenient for our purpose, since it is a very simple one-step reaction. Indeed, the isoindolinone 5 is a known product prepared through the condensation of phenylglycinol and 2-formylbenzoic acid, followed by reduction (59% overall yield). 15 With isoindolinone 5 in hand, we considered its alkylation through the formation of carbanion. In the ®rst attempt we chose to use LDA following the procedure described by Couture. 10b The deprotonation was conducted at 2788C, using 2.2 equiv. of base in THF and the so-formed carbanion was trapped with methyl iodide to give the 3-methyl isoindolinone 6a (Scheme 2) in 40% yield along with the starting material that was recovered in about 40% yield. This alkylation was proved to be highly diastereoselective: compound 6a was isolated as a unique diastereomer. Careful examination of the 1 H and 13 C NMR spectra of the crude reaction mixture revealed the presence of less than 5% of epimeric material. Further studies were undertaken in order to improve the yield of this very simple process by varying mainly the nature of the base and the nature of the metal. Methyl, propyl, allyl and benzyl halides were used in this study. All the reactions were conducted under the above conditions. From Table 1 it can be seen that excellent diastereoselection was obtained with all the alkyl halides used when the base was LDA or LiHMDS. Unfortunately, in each case, the yield was always modest and less than 50%. Using a large excess (10 equiv.) of halide did not give any improvement. Raising the temperature up to rt or using large excess of base were without effect upon the yield of the reaction, and about 40% of starting material were recovered. It must be noticed that, even in this large excess procedure, no bis-alkylated product was obtained.

We eventually obtained an improved alkylation yield by changing the counterion of the base. The use of NaHMDS gave the alkylation products 6a±d in yields improved up to 80%. Unfortunately, in these cases, the diastereoselectivity was dramatically lowered, mainly for the more electrophilic compounds: benzyl and allyl bromides. In order to con®rm this assumption, we tried an alkylation reaction with benzyl chloride: if NaHMDS was used as base, the yield was low (25%) but the diastereoselectivity excellent, while KHMDS underwent an acceptable 50% yield and a low dr In most cases deprotonation with KHMDS gave worse diastereoselectivity. These different diastereoselectivities observed through the use of different bases suggested the formation of a chelated intermediate. The chelation would be better with lithium than with sodium or potassium.

Nevertheless, the results obtained with propyl bromide were particularly astonishing since they did not follow the same tendency.

An another proof for an intramolecular chelation was given by the experiments we conducted with the O-tert-butyl derivative 7 (prepared from 5 in 65% yield, Scheme 3). Using LiHMDS as base, compound 7 was alkylated with propyl iodide to give 8 (85% yield) with only 74:26 dr (the diastereomeric ratio was lowered to 69:31 by the use of KHMDS as base).

A3 R con®guration was determined for compound 6b through an X-ray analysis of suitable crystals. The ORTEP representation is given in Fig. 2. The same con®guration was expected for all alkyl compounds 6 obtained through the deprotonation of 5 by LDA or LiHMDS.

Furthermore, this was con®rmed for compound 6c, [START_REF]The spectral data for this compound[END_REF] since its hydrogenation (H 2 , Pd/C in MeOH) gave, in 95% yield, the propyl derivative identical in all aspects with 6b. The different experimental results obtained along this work clearly proved the existence of an intramolecular chelation.

The con®guration of the newly formed chiral centre was consistent with two possible models for the carbanionic species depicted in Fig. 3. In model (A), we assumed a con®gurationally stable anion which is stabilised through a intramolecular chelation by the alcoholate. The expected chair like conformation and the retention of con®guration would lead to the observed 3R con®guration. In the enolate model (B), an intramolecular chelation is also expected and the attack of the electrophile from the less sterically hindered face (anti to the phenyl group) would also provide the 3R alkylated derivative.

At this stage we have no experimental evidence to support either model A or B, however further work is underway to elucidate the most favorable conformation.

In conclusion we developed a rapid and ef®cient process for the asymmetric synthesis of 3-alkyl-isoindolinones. Excellent diastereoselectivities were obtained and render the process particularly ef®cient despite the modest yield. Cleavage of the chiral appendage in this 2-hydroxy-1phenylethyl isoindolinone series was reported by Fains et al. 15 and by Allin 11b to occur in good yield and without loss of stereochemical integrity. 123.7, 127.2, 127.9, 128.2, 128.8, 131.9 (7CH), 131.9, 137.9, 147.0 (3C) The crystal dimensions (0.03 mm£0.10 mm£0.04 mm) being too small, diffraction data were recorded using the LURE Synchrotron facility in Orsay (France). Diffraction data were collected at wavelenth of 0.964 A Ê using a 345 mm diameter MAR Research Image Plate system connected to the W32 beam line; distance crystal-detector110 mm, total rotation v 3168. A total of 4241 re¯ections with a resolution range 5.2±1.03 A Ê were measured. This set was processed using the hkl suite of programs. [START_REF] Ottwinosky | Academic[END_REF] This led to 785 independent re¯ections with I.2 s (I), R sym 5.7%. They were kept in the re®nement calculations.

The structure was solved by direct methods using SHELXS-97. [START_REF] Sheldrick | SHELXLS 97: program for solution of crystal structures[END_REF] Re®nement, based on F 2 , was carried out by full matrix least squares with SHELXL-97 [START_REF] Sheldrick | High Resolution Re®nement[END_REF] software. An ORTEP [START_REF] Johnson | ORTEP: a thermal ellipsoid plotting program[END_REF] diagram is given in Fig. 2. Non-hydrogen atoms were re®ned anisotropically. Hydrogen atoms were positioned geometrically and re®ned riding on their carrier atom with isotropic thermal displacement parameters ®xed at 1.2 times those of their parent atoms. Convergence was reached at R w 0.034 for 785 re¯ections (I.2 s (I), R w2 0.092 for all data and S1.165 for 262 parameters. The residual electron density in the ®nal difference Fourrer does not show any feature above 0.269 e A Ê 23 and below 20.298 e A Ê 23 .

Lists of the fractional atomic coordinates, thermal parameters and bond distances and angles have been deposited with Cambridge Crystallographic Data Centre, UK as Supporting Information (CCDC reference number 176074).

Table 1 .

 1 Yield and diastereoselectivity in the alkylation of 5 (THF, 2788C) with different primary alkyl halides (1.2 equiv.) and with different bases(2.2 equiv.)

	Scheme 1.

a (%) dr b (%) Yield a (%) dr b (%) Yield a (%) dr b (%)

a Yield of isolated alkylated product.

b

Diastereoselective ratio for the crude reaction mixture.

  The reaction was quenched by 2 mL of saturated aqueous NH 4 Cl solution, water was added and the crude product was extracted with CH 2 Cl 2 , dried over magnesium sulfate, ®ltered and evaporated under reduced pressure. The residue was puri®ed by ¯ash column chromatography on SiO 2 using CH 2 Cl 2 /MeOH (97.5:2.5) as eluent.

	2. Experimental	2.2.1. (3R)-Methyl-2-[(1R)-2-hydroxy-1phenylethyl]-2,3-
	2.1. General	dihydro-1H-isoindol-1-one (6a). Oil; [a ] D 184 (c0.9, CH 2 Cl 2 ) (lit. 11b 121.3, c2.34, CH 2 Cl 2 ); ( 1 H NMR d 1.44
		(d, J6.6 Hz, 3H), 4.1 (ddd, J15.75, 6, 3.6 Hz, 1H), 4.35
	All solvents were dried by standard methods. Melting points	(q, J6.6 Hz), 4.48 (ddd, J15, 7.5, 6.5 Hz, 1H), 4.78 (dd,
	were determined on a Leica melting point microscope and are uncorrected. IR spectra were obtained using a Nicolet	J8, 3.5 Hz, 1H), 4.90 (t, J6.9 Hz, 1H (OH)), 7.26±7.35 (m, 6H), 7.45±7.57 (m, 2H), 7.87 (d, J7.45 Hz, 1H); 13 C
	Scheme 3. 205-FT infrared spectrophotometer. Only noteworthy IR absorptions are listed (cm 21 ). 1 H and 13 C NMR spectra (d	NMR d 18.3 (1CH 3 ), 56.9, 62.2 (2CH), 64.5 (1CH 2 ), 121.9,
	(ppm), J (Hertz)), solvent CDCl 3 ) were recorded with a
	Bruker AC-300 (300 and 75.5 MHz) instrument. Elemen-
	tary analyses were performed at the Microanalysis Labora-
	tory of the Pierre et Marie Curie University (Paris). Mass
	spectra were recorded with a Nermag R10-10C instrument.
	HRMS: Micromass LCT (positive Electrospray mode with
	internal standard `lock-mass'). Optical rotations were
	measured with a Perkin±Elmer 241 polarimeter. Analytical
	TLC was carried out using aluminum-backed plates coated
	with 0.2 mm silica gel 60 F 254 , Merck. Preparative ¯ash-
	column chromatography was carried out using SDS silica
	gel 60 (35±70 mm).
	2.1.1. 2-[(1R)-2-Hydroxy-1phenylethyl]-2,3-dihydro-1H-
	isoindol-1-one (5). A solution of (R)-phenylglycinol (5 g,
	36 mmol) in 150 mL of water and 47.2 mL of HCl 1N was
	added dropwise to a solution of 1,2-phthalic dicarboxalde-
	hyde (4.8 g, 36 mmol) in 2-propanol (20 mL) at room
	temperature. The solution was stirred for 4 h, and then
	made alkaline with solid sodium hydrogenocarbonate, and
	extracted with dichloromethane. The extracts were
	combined, dried over magnesium sulfate and evaporated
	under vacuum. The oily residue was puri®ed by precipi-
	tation with a mixture of diethyl ether and a small quantity
	of dichloromethane, ®ltration gave 1.7 g (18.7%) of a
	slightly yellow powder. The ®ltrate was evaporated and
	puri®ed by ¯ash column chromatography on SiO 2 using
	CH 2 Cl 2 /MeOH (97.5:2.5) as eluent to afford 3.1 g (34%)
	of 5 (total yield 52%).
	Mp 1158C; [a ] D 219 (c0.6, CH 2 Cl 2 ); 1 HNMRd 3.84 (t,
	J6.25 Hz, 1H), 4.12 (d, J17.1 Hz, 1H), 4.15 (m, 1H),
	4.42 (d, 17.1 Hz, 1H), 5.32 (dd, J8.48, 4.64 Hz, 1H), 7.2±7.5 (m, 8H), 7.8 (d, J7.4 Hz, 1H); 13 CN M Rd 48.5
	(1CH 2 ), 59.3 (1CH), 63.0 (1CH 2 ), 122.5, 123.6, 127.4,
	127.8, 127.9, 128.8, 131.3 (7CH), 132.4, 137.4, 141.4
	(3C), 169.7 (CO); IR (nujol) 3440, 1664, 1455, 1213,
	1070, 696; Anal. calcd for C 16 H 15 NO 2 : C, 75.87; H, 5.97;
	N, 5.53. Found: C, 75.27; H, 5.89; N, 5.27; m/z (CI) 254 [MH 1 , 100%].
	2.2. General procedure for the alkylation of 5
	Isoindolinone 5 (126 mg, 0.5 mmol) was dissolved in anhy-
	drous THF (10 mL) under argon, cooled in a dry ice-acetone
	bath; then a solution of base (1.1 mmol) was added drop-
	wise. After 20 min, the mixture was treated with the appro-
	priate alkyl halide (1.2±4 equiv.), stirred for 2 h at 2788C.
	Figure 2.	Figure 3.

Acknowledgements

The authors want to thank Professor S. M. Allin for furnishing NMR spectra of 3-allyl-isoindolinone 6c and for stimulating discussions. J. P. and J. R thank V. Jenkin and K.

Baghi, two undergraduate students who participated in this work.

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Raphaël Lachièze-Rey

Variance linearity for real Gaussian zeros

Keywords: Gaussian fields, point processes, crossings, nodal set, excursion, hyperuniformity, rigidity, chaos decomposition, linear statistics AMS Classification: 60G10, 60G15, 60G55

We investigate the zero set of a stationary Gaussian process on the real line, and in particular give lower bounds for the variance of the number of points and of linear statistics on a large interval, in all generality. We prove that this point process is never hyperuniform, i.e. the variance is at least linear, and give a necessary condition to have linear variance, which is close to be sufficient. We study the class of symmetric Bernoulli convolutions and give an example where the zero set is maximally rigid, weakly mixing, and not hyperuniform.

Background and motivation

We study here the zero set of a Gaussian stationary process X:

Z X := {t ∈ R : X(t) = 0}.
The study of Gaussian zeros has emerged in the fifties, with pioneering works of Kac and Rice [START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF], and subsequent applications in the fields of telecommunications and signal processing, ocean waves, and random mechanics. Rigorous results can be found in the book of Cramer and Leadbetter [START_REF] Cramér | Stationary and Related Stochastic Processes[END_REF], along with the first second order results. In the following decades, most significant second order analyses have been made by Cuzick [START_REF] Cuzick | A central limit theorem for the number of zeros of a stationary Gaussian process[END_REF], Slud [START_REF] Slud | Multiple Wiener-Ito integral expansions for level-crossing-count functionals[END_REF], Kratz & Léon [START_REF] Kratz | Central limit theorems for level functionals of stationary gaussian processes and fields[END_REF], see the survey [START_REF] Kratz | Level crossings and other level functionals of stationary Gaussian processes[END_REF] and references therein. The weakest available condition for non-degenerate asymptotic variance is the square integrability of the covariance function, in [START_REF] Slud | Multiple Wiener-Ito integral expansions for level-crossing-count functionals[END_REF]. We give here a variance lower bound, under virtually no hypotheses, that implies that the variance is always at least linear, and give a necessary condition very close to known sufficient conditions for it to be actually linear. The result is actually more general, as it concerns linear statistics on Gaussian nodal sets, which have also been the topic of many recent results, see for instance [START_REF] Wigman | Fluctuations of the nodal length of random spherical harmonics[END_REF] on the sphere, or [START_REF] Hough | Zeros of Gaussian Analytic Functions and Determinantal Point Processes[END_REF][START_REF] Nazarov | Fluctuations in random complex zeroes: Asymptotic normality revisited[END_REF] for zeros of Gaussian entire functions; these statistics can be exploited in particular to prove the rigidity of a point process, see [START_REF] Ghosh | Rigidity and tolerance in point processes: Gaussian zeros and ginibre eigenvalues[END_REF]. We also explore the class of symmetric Bernoulli convolutions and focus on an example whose spectral measure has a Cantor nature.

Feldheim [START_REF] Feldheim | Variance of the number of zeroes of shift-invariant Gaussian analytic functions[END_REF] has studied zeros of Gaussian analytic functions (GAFs) whose law is invariant only under horizontal shifts, reduced to a horizontal strip of the form R × [a, b], with -∞ < a < b < ∞. In a related work, Buckley and Feldheim [START_REF] Buckley | The winding of stationary Gaussian processes[END_REF] study the winding number of a GAF X : R → C. A strong motivation of their work is the analogy between 1 the winding number and the number of zeros. Some convenient identities coming from complex analysis can provide explicit expressions for these indexes and their moments, whereas the number of zeros of a real Gaussian process is mostly studied in the literature through its Wiener-Ito expansion, which is sometimes not amenable to analysis; one can also use directly Kac-Rice formula [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] but this does not ease the task of estimating the variance. In particular, we cannot obtain an expression as explicit as [START_REF] Chatterjee | Rigidity of the three-dimensional hierarchical coulomb gas[END_REF] in [START_REF] Buckley | The winding of stationary Gaussian processes[END_REF] or Section 3.2 in [START_REF] Feldheim | Variance of the number of zeroes of shift-invariant Gaussian analytic functions[END_REF] for the number of zeros. It will be instructive to observe in Section 2.1 that the results about GAFs are similar to the results presented here. Very recently, Azaïs, Dalmao and Letendre [START_REF] Azaïs | Studying the winding number of a gaussian process: the real method[END_REF] have studied the winding number of planar Gaussian fields on the real line, realizing the Wiener-Ito counterpart to the work [START_REF] Buckley | The winding of stationary Gaussian processes[END_REF], obtaining similar results without analyticity assumptions.

Another motivation is to explore the behavior of Gaussian zeros in the light of the concepts of hyperuniformity and rigidity. Let us introduce more formally point processes before going further. Let N (R) be the space of locally finite subsets of R, endowed with the σ-algebra B(N (R)) generated by the mappings

ϕ A : Z ∈ N (R) → #Z ∩ A
for A a Borel subset of R, where # denotes the cardinality of a set. A point process Z is a measurable mapping from an underlying probability space (Ω; A , P) to (N (R), B(N (R)));

Z is furthermore stationary if Z + x (d)
= Z for x ∈ R, and we denote by V Z (T ) = Var(#Z ∩ [-T, T ]). In physics of condensed matters and statistical physics, a great attention has been given recently to the phenomenon of fluctuation suppression of particle systems, which can be incarnated by the property of hyperuniformity: a stationary point process Z on R is hyperuniform, or superhomegeneous, if lim inf

T →∞ V Z (T ) T = 0.
A striking property of many systems with suppressed fluctuations is their rigidity, i.e. the complete determination of a feature of the point process in a bounded domain given its configuration outside the domain: for a measurable mapping ϕ : N (R) → R, we say a point process Z is ϕ-rigid if for all T > 0,

ϕ(Z ∩ [-T, T ]) ∈ σ(Z ∩ [-T, T ] c ).
In particular, we say it is number-rigid if it is ϕ-rigid for ϕ = #, and maximally rigid if it is ϕ-rigid for every bounded measurable

ϕ, i.e. if Z ∩ [-T, T ] is completely determined by Z ∩ [-T, T ] c for T > 0.
This phenomenon has been much documented, with many rigorous results proven in dimension 1 and 2, and recently in dimension 3 by Chatterjee on hierarchical Coulomb gases [START_REF] Chatterjee | Rigidity of the three-dimensional hierarchical coulomb gas[END_REF]. The link between rigidity and hyperuniformity is the topic of many works in the recent years. Most notable examples of stationary hyperuniform models are determinantal processes whose kernels are projection operators [START_REF] Ghosh | Fluctuations, large deviations and rigidity in hyperuniform systems: A brief survey[END_REF], zeros of planar Gaussian analytic functions [START_REF] Ghosh | Rigidity and tolerance in point processes: Gaussian zeros and ginibre eigenvalues[END_REF], some Coulomb systems [START_REF] Ghosh | Fluctuations, large deviations and rigidity in hyperuniform systems: A brief survey[END_REF], the β-sine processes for β > 0 [START_REF] Dereudre | DLR equations and rigidity for the sine-beta process[END_REF], perturbed lattices [START_REF] Peres | Rigidity and tolerance for perturbed lattices[END_REF], and also many non-stationary models, with a more general definition of hyperuniformity. In all these models, the point process is hyperuniform and ϕ-rigid, at least for ϕ(Z) = #Z. It is widely believed that hyperuniformity is required to have rigidity [START_REF] Ghosh | Fluctuations, large deviations and rigidity in hyperuniform systems: A brief survey[END_REF], and the usual method of proof of rigidity, introduced in [START_REF] Ghosh | Rigidity and tolerance in point processes: Gaussian zeros and ginibre eigenvalues[END_REF], involves linear statistics and implicitly assumes hyperuniformity (see [START_REF] Dereudre | DLR equations and rigidity for the sine-beta process[END_REF] for an alternative method involving DLR equations). The phenomenon of rigidity, although mathematically intriguing, has also practical uses in some percolation models related to the underlying point process, see [START_REF] Ghosh | Continuum percolation for gaussian zeroes and ginibre eigenvalues[END_REF].

We exploit the variance results derived in this paper to exhibit a non artificial example of a point process in dimension 1 that experiences the strongest possible form of rigidity, without being hyperuniform. Such examples are actually not so hard to build on R d , take for instance the grid

Z d = L(Z d + U )
where L is some non-trivial non-negative L 2 real random variable and U is independent and uniformly distributed on [0, 1) d . It is also clear that there is absolutely no form of asymptotic independence in Z d , in the sense that given two events A, B, the events Z d ∈ A and Z + t ∈ B have no reason to be uncorrelated for large T . For this reason, we might ask for additional desirable properties of such counter-examples. Definition 1.1. We say that a stationary point process Z of R is weakly mixing if for all events A, B ∈ B(N (R)), lim

T →∞ T -1 T 0 |P(Z ∈ A, (Z + t) ∈ B) -P(Z ∈ A)P(Z ∈ B)|dt = 0.
The randomly scaled grid Z 1 does not satisfy this property because it is not even ergodic. We prove the following result: Corollary 1.2. Let X be the centered stationary Gaussian process with reduced covariance function

C(t) = +∞ k=1 cos((k!) -1 t), t ∈ R.
Then Z X is maximally rigid, weakly mixing, and for ε > 0 there is c > 0 such that Var(#Z X ∩ [-T, T ]) cT 2-ε . The process X is furthermore a.s. unbounded.

The proof is a direct consequence of Theorem 2.1, Propositions 3.2 and 4.2. Regarding rigidity and hyperuniformity, stationary point processes obtained as the zeros of a random Gaussian function have mainly been studied for Gaussian analytic functions in the complex plane, see [START_REF] Hough | Zeros of Gaussian Analytic Functions and Determinantal Point Processes[END_REF]. The joint requirements of complex Gaussianity, analycity, and stationarity have reduced the class of zeros to essentially one model, up to rescaling, which happens to be rigid and hyperuniform [START_REF] Hough | Zeros of Gaussian Analytic Functions and Determinantal Point Processes[END_REF]. The links between Gaussian zeros and rigidity is yet essentially to discover.

We present in Section 2 some results and definitions about Gaussian processes and their zeros, variance lower bounds, and we give a condition for maximal rigidity. We also properly explore the analogies with variance results for GAFs. In Section 3 we introduce the class of symmetric Bernoulli convolutions, give some properties and examples, in particular an example yielding maximally rigid non hyperuniform zeros. In Section 4 we prove the lower bounds on the variance by using chaotic decompositions of the number of zeros.

Gaussian zeroes

We introduce here basic notions about Gaussian processes. See the monograph of Azaïs & Wschebor [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] for background material and advanced results about Gaussian processes and their level sets. A real Gaussian process is a random function X : R → R for which any finite dimensional distribution (X(t 1 ), . . . , X(t n )), t 1 , . . . , t n ∈ R, is a Gaussian vector with covariance matrix (C(t i , t j )) 1 i,j n , where C is the covariance function of the process X.

The law of X is entirely determined by C, and the stationarity of X, defined by X(•+t)

(d)
= X for t ∈ R, is characterised by the existence of a function C X , called reduced covariance function, such that C(t, s) = C X (t -s), t, s ∈ R. Such a function is known to admit a spectral representation of the form

C X (t) = R e ixt µ X (dx) = 2 ∞ 0 cos(xt)µ X (dx), t ∈ R, (1)
where µ X is the (symmetric) spectral measure of X on R. We say that X is degenerate if C X (x) = A cos(ϕx) for some A 0, ϕ ∈ R, or equivalently if µ X is formed by two symmetric atoms with same mass.

Zero set

We are primarily interested in the variance V Z X (T ) of #(Z X ∩ [-T, T ]), denoted by V X (T ) for short, where X is a Gaussian stationary process. The Kac-Rice formula, established in the forties [START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF], is the starting point of most rigorous subsequent works. Cramer and Leadbetter [START_REF] Cramér | Stationary and Related Stochastic Processes[END_REF] proved that it is necessary for V X (T ) to be finite for every T > 0 that C X is twice differentiable in 0 and for some δ > 0,

δ 0 1 t 2 (C X (t) -C X (0)t)dt < ∞. (2)
Geman [START_REF] Geman | On the variance of the number of zeros of a stationary Gaussian process[END_REF] then proved that the condition is actually necessary on any non-negligible interval, and this condition is thus referred to as Geman's condition. When this condition is satisfied, we often use without loss of generality the convention C X (0) = -1, equivalent to replacing X(t) by X(αt) with α = (-C X (0)) -1/2 . By using approximation by m-dependent fields, Cuzick [START_REF] Cuzick | A central limit theorem for the number of zeros of a stationary Gaussian process[END_REF] proves that the variance is not super-linear, that is lim T →∞ T -1 V X (T ) exists and is finite, if Geman's condition holds, as well as

C X (t) 2 dx < ∞, C X (t) 2 dx < ∞. (3)
He actually proves a central limit theorem under the additional assumption that lim

T →∞ T -1 V X (T ) > 0.
Computing the decomposition of the number of zeros with respect to Wiener-Ito integrals based on the field X, Slud [START_REF] Slud | Multiple Wiener-Ito integral expansions for level-crossing-count functionals[END_REF] proves that (3) is sufficient for the asymptotic variance to be finite and positive, and for the central limit theorem to hold. Kratz & Léon [START_REF] Kratz | Central limit theorems for level functionals of stationary gaussian processes and fields[END_REF] compute the chaos decomposition with respect to the joint process formed by the field X and its derivative X and ease some computations related to the decomposition, allowing them to generalise the results to other crossing problems; this is the approach we will use at Section 4.

We present here our results regarding the variance lower bound for the number of zeros. The method extends to linear statistics

N X (ϕ) := z∈X ϕ(z)
for ϕ in the space C b of compactly supported piecewise continuous functions ϕ such that R ϕ(x)dx = 0. We are interested in the asymptotics of

N X (ϕ T ) for ϕ T (•) = ϕ(T -1 •) as T → ∞. We abuse the notation V X (ϕ) = Var(N X (ϕ)), with V X (T ) = V X (ϕ T ) if ϕ is the indicator function of [-1, 1]. Say that µ X is regular in x ∈ R if it admits a L 2 density on some neighbourhood of x.
Theorem 2.1. Let X be a non-degenerate stationary Gaussian field on R with covariance C, and let ϕ ∈ C b . Assume either that ϕ is non-negative, or that Geman's condition is satisfied, in the latter case denote by x 0 = (-C (0)) 1/2 . Then (i) The variance is at least linear, i.e. lim inf

T >0 T -1 V X (ϕ T ) ∈ (0, ∞]. (ii) The existence of C satisfying C + C ∈ L 2 (R) is necessary for linear fluctuations,
i.e. for lim inf

T →∞ T -1 V X (ϕ T ) < ∞. If C exists and µ X is regular in ±x 0 , the condition C + C ∈ L 2 (R) is equivalent to (3), which is sufficient for linear fluctuations of V X (T ) ([9]). (iii) If µ X has an atom in R \ {-x 0 , x 0 }, inf T >0 T -2 V X (ϕ T ) > 0.
The proof is at Section 4. We also give an example of a nonatomic spectral measure µ X for which T 2-ε = o(V X (T )) for ε > 0. Regarding the quadratic variance, as soon as the variance is finite on some interval with non-empty interior (Geman's condition (2)), then the variance cannot be more than quadratic, see [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF].

Related works. Even in the case V X (ϕ T ) = V X (T ), it is not clear which of these results already appeared in full generality, and in particular when µ does not have a density. We could not locate a necessary condition other than (3) for linear variance in the literature, see also Remark 1.2 in [START_REF] Feldheim | Variance of the number of zeroes of shift-invariant Gaussian analytic functions[END_REF]. It remains open to find a necessary and sufficient condition. During the revision of the present work, Assaf, Buckley and Feldheim [START_REF] Assaf | An asymptotic formula for the variance of the number of zeroes of a stationary gaussian process[END_REF] have given an upper bound for the variance, and a mixing condition under which the condition (C, C ∈ L 2 ) is necessary and sufficient. Regarding point (iii), they also investigated the behaviour of the variance when there is an atom at ±x 0 , that they call a special atom.

In many practical studies of Gaussian zeros, the covariance is assumed to be square integrable ( [START_REF] Ancona | Zeros of smooth stationary gaussian processes[END_REF][START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF][START_REF] Cuzick | A central limit theorem for the number of zeros of a stationary Gaussian process[END_REF][START_REF] Kratz | Central limit theorems for level functionals of stationary gaussian processes and fields[END_REF][START_REF] Slud | Multiple Wiener-Ito integral expansions for level-crossing-count functionals[END_REF]), it is in particular square integrable in the neighbourhood of ±1, hence the variance linearity is equivalent to C, C ∈ L 2 , and to

C + C ∈ L 2 .
For positive asymptotic variance to hold ((i) with ϕ = 1 [-1,1] ), we can notice that Cuzick [START_REF] Cuzick | A central limit theorem for the number of zeros of a stationary Gaussian process[END_REF] needs

(C (t)) 2 1 -C(t) 2 dt < π 2 x 2 µ(dx)
for lim T →∞ T -1 V X (T ) to be strictly positive, a requirement still present in [START_REF] Kratz | Level crossings and other level functionals of stationary Gaussian processes[END_REF]. Other works aimed at proving central limit theorems with linear variance give various types of conditions:

C ∈ L 2 in [28], C ∈ L 1 , C ∈ L 2 in [21].
Linear statistics are classical functionals of interest for point processes, see for instance [START_REF] Nazarov | Fluctuations in random complex zeroes: Asymptotic normality revisited[END_REF] which focuses on complex Gaussian zeros, and references therein, as well as the variance computed in [START_REF] Ghosh | Rigidity and tolerance in point processes: Gaussian zeros and ginibre eigenvalues[END_REF] aimed at proving rigidity results. For linear statistics of real Gaussian zeros, we were notified while revising the current work that Ancona and Letendre [START_REF] Ancona | Zeros of smooth stationary gaussian processes[END_REF] obtained asymptotic results under some integrability hypotheses on the reduced covariance function. The condition R ϕ(x)dx = 0 does not appear in their results, but it is necessary in the current general framework, and in particular in the periodic case (point (iii)).

The assumption that X is not degenerate cannot be removed, for if C X (t) = cos(t), the variance is bounded (see the proof of Theorem 2.1). The presence of a singularity at ±x 0 muddies the waters anyway, for instance if C X (t) = cos(t)+εe -t 2 /2 1+ε for some ε > 0, the variance is at least linear.

Comparison with zeros of GAFs. It is interesting to compare the results above to those obtained by Feldheim [START_REF] Feldheim | Variance of the number of zeroes of shift-invariant Gaussian analytic functions[END_REF], who studied the zero set Z for some -∞ < a < b < ∞, and those of Buckley and Feldheim [START_REF] Buckley | The winding of stationary Gaussian processes[END_REF] who study the winding number W T of a GAF X : [0, T ] → C. A GAF X is also characterised by its reduced covariance C X and its spectral measure ρ X along the real line:

C X (t -s) = E(X(t)X(s)) = R e -2πi(t-s)x ρ X (dx), t, s ∈ R.
Let us reproduce some of their results regarding the variances

V [a,b] X (T ) = Var(#Z [a,b] X ∩ ([-T, T ] × [a, b])), V X (T ) = Var(W T ).
The field X is called degenerate if ρ X consists of a single atom.

Theorem 2.2 ( [START_REF] Feldheim | Variance of the number of zeroes of shift-invariant Gaussian analytic functions[END_REF][START_REF] Buckley | The winding of stationary Gaussian processes[END_REF]). Let X be a non-degenerate GAF.

(i) If the differentiability condition (3) in [START_REF] Buckley | The winding of stationary Gaussian processes[END_REF] holds, for T > 0

T -1 V X (T ) C for some C > 0.
The left hand member has a limit in (0,

∞) as T → ∞ if C X , C X ∈ L 2 .
(ii) The limit lim

T →∞ T -1 V [a,b] X (T )
exists and is strictly positive. It is finite if

|C X (t + iy)| 2 dt < ∞, |C X (t + iy)| 2 dt < ∞, y ∈ {a, b}
where C X is analytically continued to S a,b , (iii) the limits lim

T →∞ T -2 V [a,b] X (T ), lim T →∞ T -2 V X (T )
exist and are finite, and are strictly positive iff ρ X has an atom.

Hence the results are similar to ours regarding the quadratic variance and the linear variance. Feldheim [START_REF] Feldheim | Variance of the number of zeroes of shift-invariant Gaussian analytic functions[END_REF] also gives a lower bound which yields a sufficient condition for asymptotic super linear variance, but leaves linearity undecided for some measures with a singularity, such as for ρ X (dx) = c1 {|x| 1} |x| -1/2 dt, x ∈ R. This is analogous to our "dark spot" regarding linear variance, at the difference that we allow singularities as long as they are not located at ±x 0 .

Remark 2.3. A degenerate GAF is, up to rescaling and multiplication, X(t) = γe 2iπt , where γ = α + ıβ is a Gaussian standard complex variable (i.e. α, β are i.i.d real Gaussian variables), whereas a degenerate real Gaussian process in the sense of this paper is of the form X 0 (t) = α cos(t) + β sin(t), up to rescaling and multiplication, and its reduced covariance function is C 0 (t) = cos(t).

It is on the other hand difficult to infer results for zeros of Gaussian processes on R from results about zeros of GAF because a GAF will typically take non-real values on the real line, and hence have no real zeros (this is why the winding number might be seen as a generalisation of the number of zeros). For instance, the unique (up to rescaling/multiplication) GAF X such that for t ∈ R, E(X(0)X(t)) = 2 cos(t) is X = X 0 + iX 0 where X 0 is an independent copy of X 0 .

Completely predictable real zeros

We will consider a property stronger than maximal rigidity. Definition 2.4. Say that a point process Z is completely predictable if

σ(Z) = ∩ t∈R σ(Z ∩ (-∞, t)).
This property yields that the knowledge of Z on any half line determines the whole process. We have the obvious fact that if Z is completely predictable, it is maximally rigid.

The problem of whether the data of a stationary Gaussian process on a half line {X(t); t 0} determines the whole process has been widely investigated, see Szegö's alternative ([11, Section 4.2]). Such a behaviour does not imply that the knowledge of the zeros on a half line determines all the zeros. One might also try to reconstruct the whole process from the zeros, but Weierstrass factorisation theorem yields uncountably many functions with a given infinite set of zeros (without accumulation point). It would still be interesting to try to characterise processes whose zero set is completely predictable, or exhibit a weaker form of rigidity. We give here a simple condition on C X for the predictability of Z X .

Proposition 2.5. Assume X is almost surely of class C 1 , and there is a sequence

t n → ∞ such that |C X (t n )| → 1. Then Z X is predictable.
Proof. Let us denote the shifted process by X n = X(t n + •). We have for t ∈ R, ±Cov(X(t), X n (t)) → 1, hence ±X n (t) → X(t) a.s. since these are Gaussian variables.

Let now I be a bounded interval of R. Denote by İ the interior of I and by Ī its closure. Since Z X , Z Xn are stationary point processes, with probability 1, they don't have zeros on ∂I = Ī \ İ. Hence a.s., for simultaneously all n,

(Z Xn ∩ Ī = ∅) = (Z Xn ∩ İ = ∅); (Z X ∩ Ī = ∅) = (Z X ∩ İ = ∅).
Since (X(0), X (0)) is not degenerate, the zeros of X are almost all regular, i.e. X and X don't vanish simultaneously (see [START_REF] Kratz | Level crossings and other level functionals of stationary Gaussian processes[END_REF]Th. 1.1]). Hence for each zero z of X we have X(t)X(s) < 0 for some s, t arbitrarily close from z.

Then we have almost surely

Z X ∩ İ = ∅ ⇒X(t)X(s) < 0 for some t, s ∈ İ ∩ Q ⇒∃t, s ∈ İ ∩ Q : X n (s)X n (t) < 0 for n sufficiently large
because we have a.s. the simultaneous convergences X n (t) → X(t), t ∈ Q, hence we can conclude that if X vanishes in İ, so does X n for n sufficiently large.

Let us prove the converse statement. We have by Fatou's lemma, and using the stationarity again,

E(lim inf n X n I ) lim inf n E( X n I ) = E( X I )
and the last quantity is finite since X is a.s. bounded on I (see [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]). Hence

P(lim inf n |X n | = ∞) = 0
and there is a.s. L < ∞ and N ⊂ N infinite such that X n I L for n ∈ N.

So let us assume that X n (t n ) = 0 with t n ∈ İ for n sufficiently large. Then it also vanishes for n ∈ N sufficiently large, hence there is a subsequence n ∈ N such that X n (t n ) = 0. Let us take another subsequence n such that t n → t ∈ Ī. We have

|X(t)| |X(t) -X n (t)| + |X n (t) -X n (t n )| |X(t) -X n (t)| + L|t -t n |,
hence the latter a.s. converges to 0, and X vanishes in t ∈ Ī. since X does not vanish on ∂I a.s., we have proved the a.s. equivalence,

Z X ∩ I = ∅ ⇔ Z Xn ∩ I = ∅ for n sufficiently large. In particular, Z X ∩ I = ∅ is completely determined by n σ(Z Xn ∩ I) = n σ((Z X + t n ) ∩ I) = n σ(Z X ∩ (I -t n )) ⊂ t σ((Z X ∩ [t, ∞))),
which concludes the proof.

Let us give simple examples of Gaussian processes that satisfy this assumption, see [START_REF] Corduneanu | Almost periodic functions[END_REF] for the details. Say that a continuous function f : R → R is (Bohr)-almost periodic if for each ε > 0 there exists T 0 (ε) > 0 such that every interval of length T 0 (ε) contains a number τ with the following property: |f (t + τ ) -f (t)| < ε for each t ∈ R. Bochner came up with the following characterisation: f is almost periodic iff for every sequence t n → ∞, there is a subsequence t n such that

f -f (t n + •) → 0.
Such functions are equivalently characterised as uniform limits of trigonometric polynomials on the real line.

Proposition 2.6. Let a = (a i ) be a summable sequence of non-negative numbers, ϕ = (ϕ i ) a sequence of real numbers, and

C(x) = i a i cos(ϕ i x). ( 4
)
Then C is almost periodic, and it is the reduced covariance C = C X of a Gaussian stationary process X which is a.s. bounded, almost periodic, and whose zero set Z X is completely predictable.

Proof. C is obviously the uniform limit of trigonometric polynomials. Then X can be represented by

X(x) = i a i (α i cos(ϕ i x) + β i sin(ϕ i x))
where the α i , β i form a sequence of independent standard Gaussian variables, hence X is a.s. periodic for the same reasons as C. Finally, Z X is completely predictable because C(x) gets arbitrarily close to 1 as x → ∞, hence the previous proposition applies.

Even if such processes could be useful for modelling pseudo-crystalline arrangements, these periodicity properties prevent them from having the slightest form of asymptotic independence, ergodicity, let alone mixing properties. We shall give in the next section examples of fields whose zero set is weakly mixing and completely predictable.

The excursion volume of almost periodic fields have interesting properties. It is proved in [START_REF] Lachièze-Rey | Diophantine gaussian excursions and random walks[END_REF] that V X (T ) = Var(|[-T, T ] ∩ {X > 0}|) strongly depends on the diophantine properties of the ϕ i , where | • | denotes Lebesgue measure. If for instance C X (x) = cos(x) + cos(ϕx), V X (T ) can either be bounded, or grow as an arbitrarily high power of T , depending on weather ϕ has "good" or "bad" approximations by rational numbers.

Symmetric Bernoulli convolutions

Let λ = (λ k ) be a square summable family of R + , and

C λ (t) = ∞ k=1 cos(λ k t), t ∈ R.
This function converges for every t ([23, (3.7.9)]), and it is the reduced covariance of some stationary Gaussian field denoted by X λ . The order of multiplication does not matter in the value C λ (t), hence we can assume throughout that λ k λ k+1 . By studying the uniform convergence on every compact, it is clear that X λ is analytic, with an analytic continuation to C.

The spectral measure µ λ of X λ is called a symmetric Bernoulli convolution because it is the law of the random variable

Y λ := ∞ k=1 λ k ε k
where the ε k are i.i.d. Rademacher variables. Infinite products of characteristic functions have been studied by Lukacs [START_REF] Lukacs | Characteristic functions[END_REF]Chapter 3.7], where basic results stated here are proved. The main focus of study in the literature is the nature of µ λ . It is known that µ λ is pure, i.e. either continuous, purely discrete, or purely singular [START_REF] Lukacs | Characteristic functions[END_REF]Th. 3.7.7]. Bernoulli convolutions with geometric progression are a very active research subject, see for instance [START_REF] Varjú | On the dimension of Bernoulli convolutions for all transcendental parameters[END_REF], they are one of the most studied examples of self-similar measures, and they are objects of great interest in fractal geometry.

Regarding ergodicity, let {X(t); t ∈ R} be a Gaussian stationary process with almost surely continuous paths, and B = σ(X) the σ-algebra generated by the random variables X(t), t ∈ R. For B ∈ B, t ∈ R, let B + t be the event B translated by t. The process X is called weakly mixing if lim

T →∞ T -1 T 0 |P(A ∩ (B + t)) -P(A)P(B)| dt = 0. Proposition 3.1. Let R n = k>n λ k .
Assume λ n > R n > 0 for every n. Then µ λ has no atoms, its support has Cantor type (compact, completely disconnected and without isolated points), and lim

T →∞ 1 T T 0 C λ (t)dt = 0,
hence X λ and its zero set Z λ are weakly mixing.

Proof. The support of µ is

Σ = { k λ k ε k : ε = (ε k ) ∈ {-1, 1} N }. Hence it is clear that given t = k λ k ε k ∈ Σ, η = λ n for some n ∈ N, the set (Σ \ {t}) ∩ [t -η, t + η] is infinite because it contains { n k=1 λ k ε k + k>n λ k ε k-n : ε ∈ {-1, 1} N } using R n = k n λ k < λ n . It follows that t is an accumulation point of Σ. To prove that Σ is completely disconnected, let t = k ε k λ k , s = k ε k λ k ∈ Σ with t < s, and k 0 = min{k : ε k = ε k }. Then let t + = k<k0 ε k λ k -λ k0 + k>k0 λ k , s -= k<k0 λ k ε k + λ k0 - k>k0 λ k .
We have t t + < s - s, where the strict inequality comes from R k0 < λ k0 . Noticing that there is no point of Σ in (t + , s -), t and s are not connected through Σ.

To prove that there is no atom, symmetry considerations yield that µ({t}) = µ({s}) for every t, s ∈ Σ; since µ is finite, it follows that µ(t) = 0 for every t ∈ Σ. Hence according to Maruyama's theorem [START_REF] Maruyama | The harmonic analysis of stationary stochastic processes[END_REF], X is ergodic, and according to Zak & Rosiński [START_REF] Rosinski | The equivalence of ergodicity and weak mixing for infinitely divisible processes[END_REF], it is hence weakly mixing.

Since X is a.s. continuous, for

I ∈ B(R) open, we have a.s. #Z X ∩ I = lim n 1 ∞ k=-∞ 1 {X(k/n)X((k+1)/n) 0} ,
hence any event A ∈ σ(Z X ) is σ(X)-measurable. It readily follows that since X is weakly mixing, so is Z X .

There are also many cases where C λ has stronger mixing properties. Most available results concern coefficients of the form λ k = a k for some a ∈ (0, 1). The case a = 1/2, i.e. λ k = 2 -k , actually yields the sine process thanks to Vieta's formula:

sin(t) t = 2 sin(t/2) cos(t/2) t = 4 sin(t/4) cos(t/4) cos(t/2) t = . . . = lim n 2 n sin(2 -n t) t k cos(2 -k t) = C λ (t).
This example is central, as it is the only a for which R n = λ n , and it is the smallest a for which

µ λ (dx) = 1 {[-1,1]} (x)dx is continuous. Since C λ (t) → 0 as t → ∞, X λ
and its zero set Z λ are (strongly) mixing. For a < 1/2, the previous considerations yield that µ λ is Cantor-like with no atoms, hence weakly mixing. If a is not of the form m -k for some

m ∈ N, C λ (t) = O(| log(t)| -γ
) for some γ > 0 as t → ∞, whereas for a = m -k , m 3, lim sup t→∞ C λ (t) > 0, see [START_REF] Lukacs | Characteristic functions[END_REF].

There are many other examples of such products, with diverse behaviours. If for instance

λ k = k -1 , C λ (t) = k cos t k = n sinc t 2n + 1
decreases faster than any polynomial. Let us give now an example of symmetric Bernoulli convolution which has rigid zeros. As a further proof of asymptotic independence, we show that this process is a.s. unbounded, at the contrary of almost periodic Gaussian fields mentioned in the previous section.

Proposition 3.2. Assume λ k = k! -1 π. Then (-1) n C λ (n!) → 1 as n → ∞.
In particular, the zero set Z of the Gaussian process X with reduced covariance C λ is completely predictable, hence maximally rigid. Furthermore, µ λ is purely singular without atoms, hence X and Z are weakly mixing. Also, sup t∈R |X λ (t)| = ∞ a.s.

Remark 3.3. It is proved at Proposition 4.2 that the variance of #Z ∩ [0, T ] dominates T 2-ε as T → ∞, for any ε > 0. Proof. Let t n = n!. We have n k=1 cos(t n λ k ) = n k=1 cos(πn!/k!) = (-1) n . For k n + 1, cos πn! k! 1 - πn! k! 2 1 - π 2 k 2
hence the first result is proved with

1 (-1) n C(t n ) ∞ k=n+1 1 - π k 2 1 -c/n.
To prove the unboundedness of X λ , consider the pseudo-metric on R d(t, s) = 1 -C(t -s).

We will prove that (R, d) cannot be covered by finitely many d-balls of some radius ε > 0, which implies that the sample paths of X λ are a.s. unbounded (see for instance [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]Th. 1.19]). Let t ∈ B d (0, ε), and for k ∈ Z, let n k = n k (t) be the integer such that for some

ε k = ε k (t) ∈ [-1/2, 1/2) k! -1 t = n k + ε k .
Then for some c > 0

d(t, 0) 2 1 -exp( k ln(| cos(ε k π))|) c k ε 2 k hence c k ε 2 k < ε 2
, and in particular each ε k is smaller than δ := √ c -1 ε. Up to diminishing ε, assume δ < 1/4. Let us now cover R + by the sets

A m,k = {t = mk! + k!ε k (t) : ε k (t) ∈ [-1/2, 1/2]} for k ∈ N, 0 < m k.
In the rest of the proof, we show that

|B d (0, ε) ∩ A m,k | (4δ) l * +1
where l * = [k -δ -1 ], which implies that the sum of the |B d (0, ε) ∩ A m,k | is finite and hence that B d (0, ε) has finite measure. By stationarity, all B d (t, ε) have same measure, hence R cannot be covered by finitely many such balls.

We have for t ∈ A m,k , 0 l k, with k (l) = k(k -1) . . . (k -l + 1),

t (k -l)! = mk (l) + k (l) ε k (t) = n k-l (t) + ε k-l (t), hence ε k (t) ∈ C k (l)
, 0 l k, where

C K = {t ∈ [-1/2, 1/2] : d(Kt, Z) δ}, K > 0.
Let us then define E 0 = [-δ, δ] and for 0 l < k,

E l+1 = E l ∩ C k (l+1) .
Let n l be the number of maximal segments forming E l . We prove by induction that

n l (2δ)(4δ) l-1 k (l) for k -l δ -1 , i.e. l l * := [k -δ -1 ]
. Indeed, through the intersection with C k (l) , E l is made up of segments of length at most a l := 2δk -(l) , and each segment of length a l is intersected by at most

a l k (l+1) + 2 = 2δ(k -l) + 2 4δ(k -l) segments of C k (l+1) . It follows that n l+1 n l 4δ(k -l) 2δ(4δ) l k (l) (k -l);
which proves the induction. Hence the Lebesgue measure of E l for l l * is smaller than 2δ(4δ) l k (l) 2δk -(l+1) (4δ) l+1 . We have for l l * , E l ⊂ E l * . It follows that

|B d (0, ε) ∩ A m,k | |E k | |E l * | (4δ) l * +1 .
It follows that for δ < 1/4

|B d (0, √ ε)| k m k (4δ) k-δ -1 -1 < ∞.
By stationarity, |B d (t, ε)| = |B d (0, ε)| < ∞ for t ∈ R, hence R cannot be covered by a finite number of such balls.

4 Variance lower bounds (Proof of Theorem 2.1) If Geman's condition is not satisfied (and ϕ 0), the variance is infinite because the support of ϕ has non-empty interior, and the result of the theorem holds true.

In the rest of this section we take X as a stationary Gaussian process whose reduced covariance function C satisfies Geman's condition [START_REF] Assaf | An asymptotic formula for the variance of the number of zeroes of a stationary gaussian process[END_REF]. In particular, C and C (0) exist, hence Fatou's lemma yields that the spectral measure µ has a finite second moment, and

C (t) = -t 2 e itx µ(dx), t ∈ R. (5)
We work under the convention C (0) = 1, i.e. x 0 = 1, which amounts to a temporal rescaling of the field. Let us start by the case where C(τ ) = ±1 for some τ > 0. Recalling [START_REF] Ancona | Zeros of smooth stationary gaussian processes[END_REF], it implies that e ixτ = ±1 µ-a.s., hence µ is supported by 2π τ Z and has (at least) an atom, and X is a.s. 2τ -periodical. The behaviour of N X (ϕ T ) as T → ∞ is determined by the zeros z 1 , . . . , z Q ∈ Z X ∩ [0, 2τ ), for some random Q ∈ N. We have

T -1 N X (ϕ T ) = Q i=1 T -1 k∈Z ϕ(T -1 (z i + 2kτ )) = Q i=1 (2τ ) -1 ϕ(t)dt + o(1)
due to the hypotheses on ϕ. If we upper bound ϕ by ϕ on its support, it easily follows by Lebesgue's Theorem that T -2 V X (ϕ T ) converges to Var(Q)(2τ ) -2 ϕ 2 . The only case where Var(Q) = 0 is when C(t) = cos(t) up to rescaling and multiplication, which is the degenerate case excluded from the theorem. Hence in all other cases we have indeed a quadratic variance, and point (iii) is proved in this case.

Let us now assume that C(t) = ±1 for t = 0. Following Kratz & Léon, it is proved in [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] that we have the decomposition

N X (T ) = E(N X (T )) + ∞ q=1 N X,q (T ) with N X,q (T ) = q k=0 a k d q-k T 0 H k (X(t))H q-k (X (t))dt
where H m , m ∈ N is the m-th Hermite polynomial, and a m , d m , m ∈ N are coefficients that vanish for m odd. In particular,

a 0 = 1 √ 2π , a 2 = 1 2 √ 2π d 0 = 1, d 2 = - 1 2
and N X,q (T ) vanishes for odd q. (The decomposition also holds if we only assume that C is four times differentiable in 0, see [START_REF] Kratz | Level crossings and other level functionals of stationary Gaussian processes[END_REF]). By linearity, the decomposition still holds if ϕ is a piecewise constant compactly supported function, and N X (ϕ) has the L 2 -orthogonal decomposition ∞ q=0 N X,q (ϕ), with

N X,2 (ϕ) = -1 2 √ 2π R ϕ(t) H 0 (X(t))H 2 (X (t))dt - T -T H 2 (X(t))H 0 (X (t))dt .
Then, with

H 0 = 1, H 2 (x) = x 2 -1, since for two standard Gaussian variables α, β with correlation ρ, Cov(α 2 , β 2 ) = 2ρ 2 , V X (ϕ) Var(N X,2 (ϕ)) = 1 4π R 2 ϕ(t)ϕ(s) Cov(X(t) 2 , X(s) 2 ) + Cov(X (t) 2 , X (s) 2 ) -2Cov(X(t) 2 , X (s) 2 ) dtds.
We can show that this bound can be extended to any ϕ ∈ C b by using Fatou's lemma and an approximating sequence of piecewise constant functions ϕ n , n 1. Then, with z = t -s, w = -s,

Var(N X (T )) 1 4π R 2 [C(z) 2 + C (z) 2 -2C (z) 2 ] R ϕ(z -ω)ϕ(ω)dw dz = 1 4π R 2 [C(z) 2 + C (z) 2 -2C (z) 2 ]ϕ 2 (z)dz
where denotes the convolution product. Denote by

ψ(x) = R e ixt ψ(t)dt, x ∈ R, the Fourier transform of a L 2 function ψ. By (5), R C(z) 2 ϕ 2 (z)dz = R ϕ 2 (z) R 2
e ixz e iyz µ(dx)µ(dy)dz

= R φ(x + y) 2 µ(dx)µ(dy) R C (z) 2 ϕ 2 (z)dz = R ϕ 2 (z) R 2 (ix)(iy)e ixz e iyz µ(dx)µ(dy)dz = - R xy φ(x + y) 2 µ(dx)µ(dy) R C (z) 2 ϕ 2 (z)dz = R ϕ 2 (z) R 2
x 2 y 2 e ixz e iyz µ(dx)µ(dy)dz = R

x 2 y 2 φ(x + y) 2 µ(dx)µ(dy)

whence finally,

V X (ϕ) 1 4π R 2 (1 + xy) 2 φ(x + y) 2 µ(dx)µ(dy).
Then remark that φT (x) = T φ(T x). Since ϕ = 0, φ(0) = 0, by continuity there is α > 0 such that | φ(x)| | ϕ|/2 > 0 for |x| α. It simplifies the argument (without loss of generality) to rescale ϕ so that α = 1. Then, for some c ϕ > 0,

Var(N X (ϕ T )) c ϕ T 2 R 1 {|T (x+y)|<1} (1 + xy) 2 µ(dx)µ(dy). (6)
If µ has an atom x 0 / ∈ {-1, 1}, the right hand side is larger than c ϕ T 2 (1 -x 2 0 ) 2 µ({x 0 }) 2 , hence Theorem 2.1-(iii), is proved.

Let now ε > 0 and a set A being either

(-∞, -1 -ε], [-1 + ε, 1 -ε] or [1 + ε, ∞),
the choice being made so that, with µ ε = µ1 {A} : µ ε = 0 (recall that µ was assumed to not be concentrated on {±1}), and if µ does not have a density on R \ {-1, 1}, µ ε does not have a density either. Also define

C ε (t) = R e ixt µ ε (dx).
For T > 2ε -1 , we have for some c ε > 0, for x, y ∈ A,

(1 + xy) 2 1 {|T (x+y)|<1} c ε 1 {|T (x+y)|<1} c ε ∆ T (x + y) where ∆ T (x) := 1 {|T x|<1} (1 -|T x|). The Fourier inversion formula yields for t ∈ R ∆ T (x) = 2T -1 √ 2π sinc(T -1 t) 2 e 2itx dt, hence (6) yields for some c ϕ > 0 Var(N X (ϕ T )) c ϕ T 2 ∆ T (x + y)µ ε (dx)µ ε (dy) =c ϕ T e 2ixt e 2iyt sinc(T -1 t) 2 µ ε (dx)µ ε (dy) =c ϕ T R sinc 2 (T -1 t)C ε (2t) 2 dt c ϕ sin(1) 2 T R 1 {|t|<T } C ε (2t) 2 dt (7)
and we have used the finiteness of µ ε and the integrability of sinc 2 . Since by construction µ ε has positive mass, C ε is not identically zero and Theorem 2.1-(i) is proved.

In view of proving (ii), we assume that the variance is linear, hence we need to show that C + C ∈ L 2 (R). By [START_REF] Corduneanu | Almost periodic functions[END_REF], C ε is L 2 , hence µ ε has a L 2 density, and µ also by construction of µ ε . Let f be the density of µ. By ( 6), (after changing y to -y and using the symmetry of µ) The integrands are square integrable on I, so we need to prove the equivalence on B := R \ I. We have

C ∈ L 2 (B) ⇔ f ∈ L 2 (B), C ∈ L 2 (B) ⇔ B x 4 f (x) 2 dx < ∞, C +C ∈ L 2 (B) ⇔ B (1-x 2 ) 2 f 2 (x)dx < ∞,
hence the equivalence stems from the existence of c > 0 such that for x / ∈ I c(1 + x 4 ) (1 -x 2 ) 2 1 + x 4 .

Remark 4.1. One might think that going for higher order chaoses could help replace (ii) by the sufficient condition C, C ∈ L 2 , but the gap present at ±1 is necessarily still present at higher order chaoses because the zeros of C(t) = cos(t) satisfy the same chaotic decomposition (see [START_REF] Kratz | Level crossings and other level functionals of stationary Gaussian processes[END_REF]Prop. 2.2]), hence the variance of every chaos is bounded in T.

It remains open to find a necessary and sufficient condition for linear variance.

  [a,b] X of a Gaussian analytic function (GAF) X on the infinite strip S a,b = {x + ıy : x ∈ R, a y b},

TT - 1 R\{( 1 -

 11 Var(N X (ϕ T )) c ϕ T 2 (1 -xy) 2 1 {|x-y|<T -1 } f (x)f (y)dxdy lim inf Var(N X (ϕ T )) c ϕ xy) 2 f (y)dy f (x)dx =c ϕ (1 -x 2 ) 2 f (x) 2 dx =c ϕ (C + C) 2 by Parseval's identity, hence C + C ∈ L 2 .To conclude the proof of (ii), let us assume now that µ has a L 2 density f on a bounded neighbourhood I of {-1, 1}, and show that (3) is equivalent to C + C ∈ L 2 . If µ does not have a density on all R, then C / ∈ L 2 , otherwise C would be a L 2 density of µ. Then, if C + C ∈ L 2 , Fourier's inversion formula and (5) yieldC(t) + C (t) = 1 √ 2π (C + C )(x)e itx dx = (1 -x 2 )e itx µ(dx), t ∈ R, hence µ has density C + C (x)(1 -x 2 ) -1, which leads to a contradiction. Hence none of the conditions hold if µ does not have a density. Let us assume finally that µ's density f can be extended to all R. Then C(t) = e ixt f (x)dx, C (t) = -x 2 f (x)e ixt dx, (C + C )(t) = (1 -x 2 )f (x)e ixt dx.

Proposition 4 . 2 . 2 . 1 4T 2 )

 42212 If C X (t) = k cos(t/k!),then for all ε > 0,T ε-2 Var(N X (T )) → ∞ as T → ∞. Proof. Let R N = k>N k! -1 .Recall that the spectral measure µ is the law ofY = lim N Y N where Y N = N k=1 ε k k! , for random i.i.d. Rademacher variables (ε k ). In particular, |Y -Y N | < R N a.s. Two consecutive points of the support of Y N have distance 2/n! > 2R N (proved by induction) and they all have mass 2 -N . Let Y , Y N , N ∈ N be i.i.d. copies of Y, Y N . Using (6), there is c > 0 such that for T > 4, Var(N X(T )) cT 2 P |Y -Y | < 1/T, |Y | < x 0Let N = N T be the integer such that R N -1 > R N . For T sufficiently largeP(|Y -Y | < 1/T, |Y | < x 0 P(Y N = Y N = 0) = 2 -2N .Then we have by convexityln(4T ) ln(R -1 N -1 ) > ln((N -1)!) = N -1 k=1 ln(k) > (N -1) ln((N -1)/2).It follows that (4T ) ε-2 Var(N X(T )) c exp[ε(N -1) ln((N -1)/2))]2 -2N → ∞.

Acknowledgements

I am indebted to M. A. Klatt, who introduced to me the fascinating topic of hyperuniformity and with whom I had many fruitful discussions on this subject. I warmly thank Anne Estrade, with whom I had many discussions along the elaboration of this work, about Gaussian processes and their zeros. I also wish to thank Jose Léon for a discussion around the variance of Gaussian zeros.

This study was supported by the IdEx Université de Paris, ANR-18-IDEX-0001