H Li

R Pöschl

F Richard

HZ Recoil Mass and Cross Section Analysis in ILD

This note describes the details of a simulation study of the Higgs boson production for processes in which the Higgs is produced together with a well measurable di-lepton system using the current proposal of the ILD detector. The analysis is optimised for the measurement of the Higgs-strahlung process, i.e. e + e -→ HZ. The cross section can be determined with a precision of 2-3% and by combining the decay channels a precision of ∼30 MeV is obtained for the mass of the Higgs boson. The background can be largely reduced and the analysis exhibits a sensitivity to the configuration of the accelerator.

Introduction

The understanding of electro-weak symmetry breaking is intimately coupled to the study of the Higgs boson. It arises as a consequence of the observation of massive gauge bosons which can be generated by the spontaneous breaking of the SU(2) × U(1) symmetry of the electroweak Lagrangian.

If existing, a Higgs boson with a mass M H of 120 GeV as favoured by recent analyses of electro-weak data [1] will be discovered at the LHC or even at the TEVATRON. The ILC will allow for the detailed investigation of the nature of the Higgs boson as has been demonstrated in [START_REF]The Higgs Working Group at Snowmass '05[END_REF][START_REF] Lohmann | Prospects to Measure the Higgs Boson Mass and Cross Section in ee → ZH Using the Recoil Mass Spectrum[END_REF][START_REF] Ruan | A precision determination of Higgs mass using the fully simulated Higgs-strahlung process e + e -→ hZ → hµµ at ILC[END_REF] and references therein. The relevant processes for the present study are the recoil reaction e + e -→ HZ → Hf f (where f =leptons and quarks), also called Higgs-strahlung, or e + e -→ He + e -, also called ZZ fusion. The Feynman diagrams are shown in Figure 1. Please note that the cross section of the Higgs-strahlung dominates largely over that of the ZZ fusion. Hence, the analysis will be optimised for the measurement of the Higgs-strahlung process.

By detecting the decay products of the Z boson, the introduced processes and in particular the Higgs-strahlung process allow for the search of Higgs signals without any further assumption on its decay modes. In contrast to the LHC, the initial state is very well known at the ILC. These two items together allow for an unbiased search for the Higgs boson also called Model Independent Analysis which is only possible at a Lepton Collider such as the ILC. The presumably cleanest way to study the Higgs is given by the process e + e -→ HZ and the subsequent decay Z → µ + µ -or Z → e + e -, i.e. searching for the decay leptons of the well known Z boson. These channels, also named µµX-channel and eeX-channel hereafter, will be examined in detail in this note for a centre-of-mass energy of √ s = 250 GeV as proposed in the definition of the benchmark scenarios for the Letter of Intent Studies for ILC detectors [START_REF] Battaglia | [END_REF].

ILD Detector

A detailed description of the current model of the ILD detector can be found elsewhere [START_REF] The | ILD -Letter of Intent[END_REF].

The z-axis of the right handed co-ordinate system is given by the direction of the incoming electron beam. Polar angles given in this note are defined with respect to this axis. The most important sub-detectors for this study are described in the following.

• The vertex detector consists of three double layers of silicon extending between 16 mm and 60 mm in radius and between 62.5 mm and 125 mm in z direction. It is designed for an impact parameter resolution of σ rφ = σ rz = 5 ⊕ 10/(psin 3 2 θ) µm.

• The measurement of charged tracks is supported by an inner Silicon Tracker (SIT) in the central region and by a set of silicon disks in forward direction, i.e. towards large absolute values of cosθ.

• The ILD detector contains a large Time Projection Chamber (TPC) with an inner sensitive radius of 395 mm and an outer sensitive radius of 1743 mm. The half length in z is 2250 mm. Recent simulation studies confirm that the momentum of charged particle tracks can be measured to a precision of δ(1/P T ) ∼ 2 × 10 -5 GeV -1 . Here P T denotes the transverse component of the three momentum P of the particles.

• The electromagnetic calorimeter is a SiW sampling calorimeter. Its longitudinal depths of 24 X 0 allows for the complete absorption of photons with energies of up to 50 GeV as relevant for the studies here. The simulated energy resolution of the electromagnetic calorimeter is ∆E E = 15%/ E [GeV] • The hadronic calorimeter surrounds the electromagnetic calorimeter and comprises 4. [START_REF] Battaglia | [END_REF] interaction length λ I . Two proposals exist for the hadronic calorimeter. A digital variant consisting of steel absorbers and gas RPC chambers with a pixel size of 1×1 cm 2 as active material. The second one features scintillating tiles with size of 3 × 3 cm 2 as active material. The latter option is employed in the present work.

In the current design of the ILC the initial beams enter with a crossing angle of 14 mrad. This crossing angle is not taken into account in the present study.

Event Generation, Detector Simulation and Event Reconstruction

All data analysed for this note have been centrally produced by the ILD Group in autumn/winter 2008/09 based on generator files known as SLAC samples. For the event generation the version 1.40 of the event generator WHIZARD [START_REF] Kilian | WHIZARD: Simulating Multi-Particle Processes at LHC and ILC[END_REF] has been used. The incoming beams have been simulated with the GUINEA-PIG package [START_REF] Schulte | GUINEA-PIG -An e + ebeam simulation program[END_REF]. The energy of the incoming beams is smeared with an energy spread of 0.28% for the electron beam and with 0.18% for the positron beam. In addition the energy is modulated by beamstrahlung. The impact on the precision of the physics result of this uncertainly will be discussed below. The generated signal and background samples are given in the Table 1 for the beam polarisation mode e - L e + R : P e -= -80% and P e + = +30% and in Table 2 for the beam polarisation mode e - R e + L : P e -= +80% and P e + = -30%. The initially generated samples of the signal are combined such that they yield L = 10 ab -1 in each of the polarisation modes. For background samples the integrated luminosity is mostly larger than 250 fb -1 . Where it is smaller, it is still provided that the samples contain considerable statistics. Note, that in Tables 1 and2

- L e + R .
The signal is indicated by bold face letters; the cross-section in the parentheses of e + e -and µ + µ -are that after Pre-Cuts, see Table 3 for the Pre-Cuts definition.

Due to the large cross section of the Bhabha Scattering, i.e. e + e -→ e + e -and muon pair production, i.e. e + e -→ µ + µ -, pre-cuts have been applied in order to reduce the simulation time. These cuts are given in Table 3 and will be later on referred to as Pre-cuts.

Here, M e + e -and M µ + µ -, respectively, are the invariant mass of the di-lepton system for signal events, while P T e + e -and P T µ + µ -denote the transverse momentum calculated from the vectorial sum of the two leptons. The generated events are subject to a detailed detector simulation. The simulation is performed with the MOKKA [10] software package which provides the geometry interface to the GEANT4 [START_REF]The GEANT4 collaboration[END_REF] simulation toolkit. The event reconstruction is performed using the MarlinReco framework. For this study the versions as contained in the Software Package ILCSoft v01-06 [10] are employed. The main output of this framework for the present study are the so-called LDC Tracks which is a combination of track segments measured in the vertex detector, the Silicon Inner Tracker and the TPC or Forward Tracking Disks. Their momenta are compared with the energy of calorimeter clusters composed from signals in the electromagnetic and hadronic calorimeter for the particle identification.

Signal Selection and Background Rejection

The signal is selected by identifying two well measured leptons in the final state which yield the mass of the Z boson. The mass M recoil of the system recoiling to the di-lepton system is computed using the expression:

M 2 recoil = s + M 2 Z -2E Z √ s
Here M Z denote the mass of the Z boson as reconstructed from the di-lepton system and E Z its corresponding energy. A number of background processes have to be suppressed. Techniques of background suppression similar to those presented in this note were already introduced in [START_REF] Brient | Measurement of the Higgs Mass and e + e -→ ZH cross sections at Linear Colliders[END_REF]. This section firstly defines the criteria of lepton identification and then addresses the means to suppress the background. This will be done under two assumptions: 1) model independent 2) model dependent, i.e. assuming a Higgs boson as predicted by the Standard Model. The latter excludes decay modes in which the Higgs boson decays invisibly.

Lepton Identification

The task is to identify the muons and electrons produced in the decay of the Z boson. In a first step, the energy deposition in the electromagnetic calorimeter (E ECAL ), the total calorimetric energy E total and the measured track momentum P track are compared accordingly for each final state particle. The lepton identification is mainly based on the assumption that an electron deposits all its energy in the electromagnetic calorimeter while a muon in the considered energy range, see Figure 2, passes both the electromagnetic and the hadronic calorimeter as a minimal ionising particle. The observables and cut values are summarised in Table 4.1. The motivation of the cut values can be inferred from Figure 3 where the spectra for the corresponding lepton type in the relevant momentum range P > 15 GeV are compared with those from other particles are displayed. Here N true defines the generated number of the corresponding lepton type and N iden defines the reconstructed number of the corresponding lepton type according to the selection criteria. For electrons and muons with P > 15 GeV in the signal samples the obtained values are listed in Table 4. The efficiencies and purities are well above 95% except for the purity of the muon identification. This is caused by final state charged pions which pass the detector as minimal ionising particles and which are indistinguishable from muons with the applied selection criteria. This deficiency is partially balanced by the fact that two leptons of the same type with opposite charge are required for the reconstruction of the Z boson and that they should yield the mass of the Z boson. Indeed, using the above selection cuts, the efficiency to identify both leptons from the Z boson decay is 95.4% for the case Z → µµ and 98.8% for the case Z → ee. Note, that the cut on P > 15 GeV has been omitted in this case.

P (GeV)

µ

Track Selection

As the invariant mass of the Z boson and thus the recoil mass will be calculated from the four momenta of the LDC Tracks, badly measured LDC Tracks need to be discarded from the analysis. The track quality can be estimated by the ratio ∆P/P 2 where the uncertainty ∆P is derived from the error matrix of the given track by error propagation. The Figures 4 and5 show, for muons and electrons separately, the dependency of ∆P/P 2 on the polar angle cosθ and on the track momentum P . For reasons discussed in the following the latter has been restricted to |cosθ| < 0.78, i.e. the central region. For both variables the distributions exhibit for muon tracks a narrow band with well measured momenta equivalent to small ∆P/P 2 . The track quality decreases as expected towards large |cosθ|, i.e. towards the acceptance limits of the TPC which motivates the restriction to the central region when displaying ∆P/P 2 versus P . These distributions show a decrease in track quality towards small particle momenta as expected from multiple scattering effects. Beyond that, it is clearly visible that for electrons the situation is much more diluted and the number of badly measured tracks is significantly higher than that for muons. This can be explained by the Bremsstrahlung of the electrons in the detector material.

The procedure for track rejection is developed for the better measured muon induced tracks:

• For |cosθ| < 0.78 the shape of ∆P/P 2 versus P is approximated by: δ(1/P ) = ∆P/P 2 = a⊕b/P = c(P ); with a = 2.5×10 -5 GeV -1 and b = 8×10 -4 . (1) Tracks are rejected if δ(1/P ) > 2c(P ).

• For |cosθ| > 0.78 tracks are rejected if ∆P/P 2 > 5 × 10 -4 GeV -1 .

The cuts are indicated in Figure 4 and 5 and underline that tracks created by electrons are rejected considerably more often which will reduce the number of reconstructed Z bosons in the corresponding channel.

Background Rejection

The recoil analysis is based on the identification of the di-lepton system as produced by the decay of the Z boson. It is thus necessary to distinguish the processes which lead to two leptons in the final state as given in Table 1 and Table 2 from the ones produced in the Higgs-strahlung process.

For the Higgs-strahlung process the invariant mass of the di-lepton system M dl should be equal to the Z boson mass while the invariant mass of the recoiling system, M recoil is expected to yield the introduced mass of the Higgs boson of 120 GeV. It is unlikely that combinations of background processes fulfil both conditions at once. This argumentation is supported by Figures 6 and7 which show the invariant mass distributions for the di-lepton system and the recoil mass for both, the di-lepton system consisting of muons and the di-lepton system consisting of electrons. These distributions suggest to restrict the analysis to the following mass ranges:

• 80 < M dl < 100 GeV • 115 < M recoil < 150 GeV
In a next step the selection is to be made by means of the different kinematic properties. In the following the variables used to distinguish signal events from background events will be introduced.

• Acoplanarity acop, see Figure 8: As for e + e -collisions with beams of equal energy the centre-of-mass system is at rest, it is expected that in processes in which the leptons are produced at the Z * vertex these two leptons are back-to-back in azimuth angle.

The distance in azimuth angle is expressed by the acoplanarity acop, defined as acop = |φ +φ -|, where φ ± is the azimuth angle of the an individual lepton of the dilepton system. If the particles are produced from a intermediate particle with a given transverse momentum, the exact back-to-back configuration is modulated. Therefore a cut on 0.2 < acop < 3 is applied.

• Transverse Momentum P T dl of the di-lepton system, see Figure 9: As the Higgsstrahlung process can be interpreted as a two body decay, both bosons gain equal transverse momentum which is conserved by their decay products. The total final state for muon pair production or Bhabha Scattering has in first approximation no transverse Here, τ τ refers to the µµ or ee created in the decay of τ τ . Note that the Pre-cuts defined in Section 3 have been applied to the µµ background sample. momentum. In order to suppress this background, a cut P T dl > 20 GeV of the di-lepton system is applied. This cut cannot suppress events in which initial state radiation of the incoming beams leads to a transverse momentum of the colliding system. This case will be discussed separately.

• cosθ missing : this cut discriminates events which are unbalanced in longitudinal momentum, essentially, those of the type e + e -→ l + l -γ. The distributions in Figure 10 motivate a cut on |cosθ missing | < 0.99.

The last introduced cut also suppresses events with initial state radiation happening approximately collinear with the incoming beams. The final state in e + e -→ µ + µ -(e + e -) can, however, gain sizeable transverse momentum by initial state radiation of a high energetic photon. Figure 11 shows the correlation between the transverse momentum P T γ of a detected high energetic photon, assumed to be created by initial state radiation, and the transverse momentum P T dl of the di-lepton system for both, events in which only a muon pair is created at the Z * -Boson vertex and signal events. The first type shows a clear correlation in transverse momentum. In order to suppress this background the variable ∆P T bal. = P T dl -P T γ is introduced which is shown in Figure 12 for signal events and background events superimposed with each other. By selecting events with ∆P T bal. > 10 GeV, a considerable fraction of background can be suppressed. It should finally be noted that background events of type e + e -→ µ + µ -(e + e -) which are undergoing final state radiation are suppressed by the requirement that the lepton should yield the Z boson mass.

The number of events remaining after each of these cuts for signal and backgrounds are given in Tabs. 5 through 8 for two beam polarisations and the different compositions of the di-lepton system. The combination of cuts will be later referred as MI Cuts. Please note that the cut variables f L and |∆θ 2tk | will be introduced later.

From the tables the following conclusions can be drawn

• The requirement to have two well measured leptons retains always more than 95% of the signal while it suppresses in most of the cases the largest part of background events.

• The requirement of a minimum P T dl of the di-lepton system is very efficient for events in which the di-lepton system is produced directly at the Z * vertex, see Figure 1. This type of background is further reduced by comparing the transverse momentum of the di-lepton system with the transverse momentum of a radiative photon. The cut is particulary efficient to suppress background events generated by Bhabha Scattering.

• Although largely suppressed, the number of events generated by Bhabha background still exceeds the number of signal events. This remains an irreducible background.

• The acoplanarity acop is particularly efficient against background in which the di-lepton system is composed by τ -Leptons. The larger mass of this particle reduces the phase space for radiative processes. Hence this lepton type is more often produced in a backto-back configuration than the lighter lepton types.

The tables demonstrate that mostly events in which the di-lepton system is produced at the Z * vertex can be efficiently rejected by the defined cuts. The background by events in which two bosons are produced, i.e. e + e -→ ZZ/γγ or e + e -→ W + W -, is less well distinguishable from the signal events. As these events however have slightly different spectra. Further rejection can be achieved by a multi-variate analysis of a set of suited discriminative variables. These variables are introduced in the following. • The γ-pair production leads to a flat distribution in the di-lepton mass spectrum in the Z-mass region The shape of the invariant mass M dl of the di-lepton system and hence also that of the transverse momentum P T dl of the di-lepton system can be employed to suppress background from γ-pair production.

• The production of Z boson and W boson pairs happens predominantly via exchange reactions which lead to a strong increase of the differential cross section towards large absolute values of the cosine of the polar angle. On the contrary, the Higgs-strahlung process is expected to decrease towards the forward and backward direction. Therefore the polar angle spectrum as shown in Figure 13 of the di-lepton system is expected to discriminate between signal events and background from Z and W pair production.

• The acollinearity, defined as acol = acos(P + P -/|P + ||P -|), is sensitive to the boost of the di-lepton system. In case of Z pair production the decay products are expected to be boosted more strongly than in the case of Higgs-strahlung. This results in a different position of the Jacobian Peak in the dσ/d(acol) differential cross section as demonstrated in Figure 14.

The likelihood of an event to be the signal is defined as L S = P S i , where the P S i is the probability of the event to be the signal according to the PDF of the signal of the ith selection variable. Similarly, the likelihood of an event to be the background is defined as Here τ τ refers to the µµ or ee created in the decay of τ τ . Note that the Pre-cuts defined in Section 3 have been applied to the µµ background sample.

L B = P B i . dl θ cos -1 -0.
Hereafter, the Likelihood Fraction is defined as f L = L S /(L S + L B ), which is within (0, 1). The Figures 15 through 18 outline the optimisation procedure in the likelihood analysis separately for the two analysis channels and polarisation modes using the four variables introduced above, for details see [START_REF] Li | [END_REF]. It is clearly visible that the separation between signal and background improves towards small values of f L . The cut on f L is optmised according to the maximum in the significance S/ √ S + B where S and B are the number of remaining signal and background events, respectively. The cut on f L is adjusted for each polarisation mode of the incoming beams and the type of the di-lepton system under study. The final number of events also included in Tables 5 through 8 shows that with the multivariate analysis the number of background events are further reduced by roughly 50% while the major part of the signal events is kept.

Model Dependent Analysis

If the analysis of the Higgs-strahlung process is restricted to modes in which the Higgs can solely decay into charged particles as e.g. suggested by the Standard Model, hence introducing a Model Dependency, the different track multiplicities can be used for the separation of signal and background events. The Higgs boson decays into oppositely charged particles such that events with less than four tracks can be considered as background. Figure 19 shows the number of reconstructed tracks beside the ones from the di-lepton system for final states of the types µµX, µµ, τ τ and µµνν.

As expected, the Higgs-strahlung process leads to a considerable amount of charged particles while processes with a low multiplicity of charged particles also create only a small number of tracks. The distributions tell that a large fraction of events have exactly two additional tracks beside those of the di-lepton system. The two additional tracks originate from two sources.

• Tracks created by charged particles by H → τ + τ -and the subsequent decays of the τ -Leptons into charged particles.

• Tracks created by photon conversion. This photon may be created by initial state radiation.

The first type of events need to be kept in the signal as the τ -Leptons constitute an important analyser to determine e.g. quantum numbers like CP of the Higgs boson [START_REF]The Higgs Working Group at Snowmass '05[END_REF]. The second type of events can be rejected by taking into account that the opening angle of e + e -pair created by photon conversion is expected to be very small. This is underlined by Figure 20 which shows the angular difference ∆θ 2tk between the two additional tracks. While the signal events result in a flat distribution, the background events exhibit a strong maximum around ∆θ 2tk = 0. This observation motivates the a cut |∆θ 2tk | > 0.01. The di-lepton system of a given type might be contaminated from particles of the other type. Therefore, the polar angle of each of the two particles of the di-lepton system is also compared with the polar angle of the additional tracks, defining the observable ∆θ min as shown in Figure 21. Again, a strong maximum around ∆θ min = 0 can be observed which suggests the cut ∆θ min > 0.01 Tables 9 through 12 give the resulting number of events after each cut applied under the assumption that the Higgs boson decays into Standard Model particles. Note, that the cut on the transverse momentum of the di-lepton system has been omitted in order to maximise the number of signal events. The combination of cuts will be later referred as MD Cuts. The numbers in the tables show that the cuts introduced for the additional tracks allow for an entire suppression of backgrounds with a small multiplicity of charged particles in the final state. It has to be pointed out that in particular the background from Bhabha events can be removed almost completely. On the other hand at least 50% of the signal is retained by the cuts.

Again the remaining major background is given by events in which vector bosons pairs are produced. This background is further reduced by a likelihood analysis as described above. The results of this likelihood analysis is also given in Tables 9 through 12. From these tables, it can be deferred that the f L cuts reject the background from Z pair production by a factor of two, and reduce the signal by only 10%. At the same time, the background µµ, ee, τ τ , µµνν and eeνν is entirely suppressed.

Tables of Background Rejection

MI, µ + µ -X, e - L e + R N evts Remained µ + µ -X µ + µ -τ + τ -µ + µ -νν µ + µ -ff
- L e + R . MD, µ + µ -X, e - R e + L N evts Remained µ + µ -X µ + µ -τ + τ -µ + µ -νν µ + µ -ff

Extraction of Higgs Mass and the Higgs production Cross Section

In the previous section the criteria to select the signal events and to suppress the background from various sources have been introduced and applied. The remaining spectra are a superposition of signal and background events convoluted with beam effects. In the following, the relevant observables as the Higgs boson mass M H and the total Higgs-strahlung cross section σ are extracted. Note in passing, that the results for the eeX-channel will always contain a small admixture of the ZZ fusion process.

As indicated above, the resulting spectrum is composed by several components. This motivates to approximate this spectrum in a non-parametric way using a Kernel Estimation as introduced in [START_REF] Cranmer | Kernel estimation in High-energy physics[END_REF] and applied in [START_REF] Cranmer | Kernel Estimation for Parametrization of Discriminant Variable Distributions[END_REF][START_REF]Search for Neutral Higgs Bosons in e + e -Collisions at √ s ≈ 192-202 GeV[END_REF]. In order to reduce the effort of finding a parent function using either the already simulated data set or by simulating another independent set of data a so-called Simplified Kernel Estimation is proposed.

The signal spectrum is approximated by the following function:

F S (x) = 1 N m j=1 n j G(x; t j ; h j ) (2)
with

h j = 4 3 1/5 N -1/5 ∆x N n j , (3)
Here G is a Gaussian with the parameters µ = t j where t j is the center of the jth bin of a histogram with m bins and σ = h j where h j is the smoothing parameter of bandwidth of the individual Gaussians placed around the bin centers. The parameter ∆x is assumed to the the standard deviation in each bin and n j ∆xN is an estimate for the parent distribution. By the transformation x → x = xm H , the approximated function becomes sensitive to the value of the Higgs-Mass.

The background is approximated by a second order Chebyshev polynomial. By this statistical fluctuations in the remaining background events are smoothened. Using this polynomial as input the background is generated again with 40 times higher statistics. Therefore, statistical uncertainties are nearly excluded. The combination of signal and background is finally fitted by the sum of the signal and the background functions.

The simulated signal sample is separated into two sets of data. One of them, the Reference Sample, is employed to determine all fit parameters except the normalisation N of the signal signal and the actual Higgs mass, M H . The normalisation N and the Higgs mass M H enter as free parameters of the fit to the second sample, the Result Sample. The spectra of the Result Sample, scaled to a luminosity of L=250 fb -1 , including the defined fitting function are displayed in Figures 24 and25 for the Model Independent Analysis and in Figures 26 to 27 for the Model Dependent Analysis. The fit based on the Kernel estimation for the signal part describes the shape of the mass spectra very well and are therefore suited for the extraction of the relevant parameters which are listed in Table 13 for the Model Independent Analysis and in Table 14 for the Model Dependent Analysis. In [START_REF] Li | [END_REF]

Discussion of the Results

The Higgs mass can be determined to a precision of the order of 0.03% when the eeX channel and the µµX are combined. Regarding the individual results, it can be deferred that the precision in the µµX channel is more than two times smaller than that of the eeX Channel This increase of the error is mainly induced by bremsstrahlung of the electrons in the detector material. This can be concluded by comparing the results between Tables 13 and14 as for the latter the background of Bhabha scattering is suppressed while the difference in the precision between the two decay modes of the Z bosons remains the same. The precision on the cross section is less sensitive to this experimental drawback as it is derived within a basically arbitrary mass window. The derived values for the Model Dependent Analysis are consistently slightly more precise. The relatively small difference of the results confirms that the methods employed for background suppression in the Model Independant Analysis are already very efficient. Table 15: Mass Resolution with contributions by machine (∆M mac. ) and detector (∆M det. ) separated.

The width of the Higgs boson mass as shown before is mainly given by a convolution of detector uncertainties and uncertainties on the energy of the incoming beams. Uncertainties on the energy of the incoming beams are imposed by accelerator components such as the initial linac, the damping rings or, in the case of electrons, by a tentative undulator in the electron beam line. Another source of uncertainty is the beamstrahlung when particles of a beam bunch interact in the electromagnetic field of the opposite one. The Figure 22 shows the Higgs mass spectrum before and after full detector simulation. The detector response leads only to small additional widening of the maximum of the recoil mass distribution. Using a Gaussian fit to the left side of the recoil mass distribution, the width before detector simulation can be quantified to be 560 MeV while it increases to 650 MeV for the µµX channel after detector simulation, see Table 15. For the given configuration, the uncertainty on the incoming beams remains the dominant contribution to the observed width even for the eeX channel.

On the Control of Systematic Errors

Naturally, the measurement of the Higgs mass is sensitive to the calibration of the detector and the beam energies as the Higgs mass is directly computed from the four momenta of the particles composing the di-lepton system and the centre-of-mass energy. Both uncertainties can be controlled by the measurement of the e + e -→ ZZ process as the Z mass is known to a few MeV and the cross section for Z pair production is approximately 40 times higher than that of the Higgs-strahlungs process. Once the detector is calibrated the Higgs-strahlung process can be used to determine, within reasonable limits, arbitrary Higgs masses. The algorithms presented in this note are also suited for the quantification of the systematic error. Note, that the systematic error of the cross section determination might be easier to control by using a smaller set of cut variables than those presented above. Such a set could comprise only the invariant mass and the transverse momentum of the di-lepton system, M dl , P T dl or, in case of the model dependant analysis, the number of additional tracks, N add,T K . The expected increase of the statistical error is only about 10%.

Recovery of Bremsstrahlungs Photons

The lower precision obtained in the eeX-Channel is due to the Bremsstrahlung of the final state electrons in the passive material of the detector. In the following an attempt is made to improve the precision in that channel by recovering the bremsstrahlungs photons [START_REF] Thomson | Private Communication[END_REF]. The four momenta of the selected electrons are combined with those of photons which have a small angular distance to the electrons. If these combined objects together with the corresponding other electron candidate form the Z mass, they are included in the Z reconstruction. The inclusion of low energetic photons leads to a penalty in the momentum reconstruction due to the poor energy resolution of the electromagnetic calorimeter for low energetic particles. This drawback might get counterbalanced by the gain in statistics due to the described recovery of the energy loss.

Figure 23 shows the recoil mass spectrum after the recovery of the Bremsstrahlungs photons. The worse resolution around the mass maximum is clearly visible. The corresponding results are given Tables 16 through 17 and the fitted spectra in Figures 28 through 29. The numbers show that the mass resolution is improved by 10% while the precision in the cross section is improved by 20%. The cross section benefits directly from the gain in statistics while the determination of the recoil mass suffers from the reduced momentum resolution.

Conclusion and Outlook

Using mainly the Higgs-strahlung process with the Z boson decaying leptonically and a Higgs boson mass of 120 GeV as input, the current design of the ILD detector promises to determine the mass of the Higgs boson to a precision of the order of 30 MeV. According to [START_REF] Bambade | Strategies to measure the Higgs mass, width and invisible decays at ILC[END_REF] Ana. Pol. and references therein, such a precision renders sensitivity to effects from super-symmetric extensions to the Standard Model. Assuming a heavier Higgs, the precision might allow also for the determination of the Higgs boson mass width at centre-of-mass energies higher than 250 GeV. Staying with small Higgs masses, it has been demonstrated semi-analytically [START_REF] Bambade | Strategies to measure the Higgs mass, width and invisible decays at ILC[END_REF] and confirmed with full simulation studies [START_REF] Li | Precision Measurements of SM Higgs Recoil Mass and Cross Section for Ecm of 230 GeV and 250 GeV at ILC[END_REF] that the precision can be further increased by working at a centre-of-mass energy close to the HZ production threshold, i.e. at √ s=230 GeV.

In the present study, the cross section and therefore the coupling strength at the HZZ vertex is determined to a precision of the order of 2-3% which might already be sufficient to get sensitive to contributions to this coupling from physics beyond the Standard Model.

The signal to background ratio can be enhanced to a value of at least 30% even if the cross sections of the background processes are several orders of magnitudes higher. Note, that this ratio is way higher in the region around the signal maximum. The background suppression exploits the considerable capabilities of track recognition as allowed by the current design of the ILD detector. The precision of the measurement can be improved by a better muon recognition by e.g. including a muon system in the analysis which has not been done so far. The precision obtained in the branch in which the Z boson decays into electrons might gain considerably from a revision of the amount of passive material in the detector. Both decay modes may gain also from an exploitation of the particle identification of the ILD detector by means of a dE/dx measurement in the TPC. For this, further studies are needed. A future study clearly will have to quantify the systematic uncertainties and to identify those which have the largest influence on the systematic errors. This would give important directions on the final detector layout and the precision needed for e.g. alignment systems. For such a study realistic inputs on e.g. the uncertainty of drift times in the TPC or residual misalignments after detector movements are needed.

The analysis has proven that the results are sensitive to details of the accelerator configuration. Using the set of parameters as has been chosen for the SLAC samples which in turn correspond to the current best knowledge of the beam parameters, approximately half of the statistical error is generated by uncertainties caused by beamstrahlung and the energy spread of the incoming beams. The Higgs-strahlung process constitutes an important benchmark for the optimisation of the accelerator performance. The lines show the fits using the Simplified Kernel Estimation fitting formula to the signal and a polynomial of second order to the background as explained in the text.

Figure 1 :

 1 Figure 1: Higgs boson production via the Higgs-strahlung process (left) and ZZ fusion (left) and associated final state fermions with opposite charge at e + e --colliders.

Figure 2 :Figure 3 :

 23 Figure 2: Momentum range of the final state leptons as produced in Z → µ + µ -decays from e + e -→ HZ events at √ s = 250 GeV.

Figure 5 :

 5 Figure 5: 2D ∆P/P 2 distribution vs. cosθ (left) and ∆P/P 2 distribution vs. track momentum (right) of electron candidates

Figure 6 :

 6 Figure6: Normalised signal and background distributions of the invariant mass of the dilepton system M dl for the µ + µ -X (top) and e + e -X Channel (bottom). Here, τ τ refers to the µµ or ee created in the decay of τ τ . Note that the Pre-cuts defined in Section 3 have been applied to the µµ background sample.

Figure 7 :Figure 8 :

 78 Figure7: Normalised signal and background distributions of the recoil mass M recoil distributions for the µ + µ -X (top) and the e + e -X Channel (bottom). Here, τ τ refers to the µµ or ee created in the decay of τ τ . Note that the Pre-cuts defined in Section 3 have been applied to the µµ background sample.

Figure 9 :

 9 Figure 9: Normalised signal and background distribution of the transverse momentum P Tdl of the dilepton system for the µ + µ -X Channel (top) and the e + e -X-Channel (bottom).Here τ τ refers to the µµ or ee created in the decay of τ τ . Note that the Pre-cuts defined in Section 3 have been applied to the µµ background sample.

Figure 10 :

 10 Figure 10: Normalised signal and background distributions of the |cosθ missing | of the system of undetected particles for the µ + µ -X-Channel (top) and e + e -X-Channel (bottom).

Figure 11 :Figure 12 :

 1112 Figure 11: P T dl vs. P T γ for background by e + e -→ µ + µ -(left) and for signal (right) in the µµX-channel.

Figure 13 :Figure 14 :

 1314 Figure13: Normalised signal and background distribution of the cosine of the polar angle cosθ dl of the di-lepton system for the µ + µ -X Channel (top) and the e + e -X-Channel (bottom). Here τ τ refers to the µµ or ee created in the decay of τ τ . Note that the Pre-cuts defined in Section 3 have been applied to the µµ background sample.

Figure 15 :Figure 16 :

 1516 Figure 15: The distributions of the Likelihood Fraction f L (left), the number of remaining events versus the cut on f L (middle), and the significance versus f L cuts (right). The distributions are shown for the µµX-channel in the Model Independent Analysis and for the polarisation mode e - L e + R .

Figure 17 :Figure 18 :

 1718 Figure 17: The distributions of the Likelihood Fraction f L (left), the number of remaining events versus the cut on f L (middle), and the significance versus f L cuts (right). The distributions are shown for the µµX-channel in the Model Independent Analysis and for the polarisation mode e - R e + L .

Figure 19 :

 19 Figure 19: Number of additional tracks (N add.T K ) for µµX, µµ, τ τ and µµνν final states.

Figure 20 :

 20 Figure20: Distribution of ∆θ 2tk , which is the ∆θ between two additional tracks for N add.T K = 2, for signal events (µµX) and background by muon pair production (µµ).

Figure 21 :

 21 Figure 21: Distribution of ∆θ between muon candidates and additional tracks, for the signal events (µµX) and background by muon pair creation (µµ).

  ∆M tot. (MeV) ∆M mac. (MeV) ∆M dec. (MeV)

Figure 22 :

 22 Figure 22: Comparisons of recoil mass spectra in generator level and after full simulation, for the µµX-channel (left) and the eeX-channel (right).

Figure 24 :

 24 Figure24: Reconstructed Higgs mass spectrum together with the sum of underlying background for the Model Independent Analysis for the µµX-channel (top) and eeX-channel (bottom). The polarisation mode is e - L e + R . The lines show the fits using the Simplified Kernel Estimation fitting formula to the signal and a polynomial of second order to the background as explained in the text.

Figure 25 :

 25 Figure25: Reconstructed Higgs mass spectrum together with the sum of underlying background for the Model Independent Analysis for the µµX-channel (top) and eeX-channel (bottom). The polarisation mode is e - R e + L . The lines show the fits using the Simplified Kernel Estimation fitting formula to the signal and a polynomial of second order to the background as explained in the text.

Figure 26 :

 26 Figure 26: Reconstructed Higgs mass spectrum together with the sum of underlying background for the Model Dependent Analysis for the µµX-channel (top) and eeX-channel (bottom). The polarisation mode is e -L e + R . The lines show the fits using the Simplified Kernel Estimation fitting formula to the signal and a polynomial of second order to the background as explained in the text.

Figure 27 :

 27 Figure 27: Reconstructed Higgs mass spectrum together with the sum of underlying background for the Model Dependent Analysis for the µµX-channel (top) and eeX-channel (bottom). The polarisation mode is e - R e + L . The lines show the fits using the Simplified Kernel Estimation fitting formula to the signal and a polynomial of second order to the background as explained in the text.

Figure 28 :

 28 Figure 28: Reconstructed Higgs mass spectrum after the recovery of Bremsstrahlungs photons together with the sum of underlying background for the Model Independent Analysis for eeXchannel. The polarisation mode is e - L e + R (top) and e - R e + L (bottom).The lines show the fits using the Simplified Kernel Estimation fitting formula to the signal and a polynomial of second order to the background as explained in the text.

Figure 29 :

 29 Figure 29: Reconstructed Higgs mass spectrum after the recovery of Bremsstrahlungs photons together with the sum of underlying background for the Model Dependent Analysis for eeXchannel. The polarisation mode is e - L e + R (top) and e - R e + L (bottom).The lines show the fits using the Simplified Kernel Estimation fitting formula to the signal and a polynomial of second order to the background as explained in the text.

Table 1 :

 1 the background samples have been grouped by the resulting final state 1 . Processes and cross sections for polarisation mode e

	Process	Cross-Section	Process	Cross-Section
	µµX	11.67 fb	eeX	12.55 fb
	µµ	10.44 pb (84.86 fb)	ee	17.30 nb (357.14 fb)
	τ τ	6213.22 fb	τ τ	6213.22 fb
	µµνν	481.68 fb	eeνν	648.51 fb
	µµf f	1196.79 fb	eef f	4250.58 fb

Table 2 :

 2 Processes and cross sections for polarisation mode e - R e + L . The signal is indicated by bold face letters; the cross-section in the parentheses of e + e -and µ + µ -are that after Pre-Cuts, see Table3for the Pre-Cuts definition. e + e -→ e + e - e + e -→ µ + µ - |cosθ e + /e -| < 0.95 M e + e -∈ (71.18, 111.18) GeV M µ + µ -∈ (71.18, 111.18) GeV P T e + e

	Process	Cross-Section	Process	Cross-Section
	µµX	7.87 fb	eeX	8.43 fb
	µµ	8.12 pb (58.26 fb)	ee	17.30 nb (335.47 fb)
	τ τ	4850.05 fb	τ τ	4814.46 fb
	µµνν	52.37 fb	eeνν	107.88 fb
	µµf f	1130.01 fb	eef f	4135.97 rb

-> 10 GeV P T µ + µ -> 10 GeV M recoil ∈ (105, 165) GeV M recoil ∈ (105, 165) GeV

Table 3 :

 3 List of Pre-cuts applied to Bhabha scattering and muon pair production in order to reduce the simulation time.

Table 4 :

 4 Lepton ID Efficiency and Purity for reconstructed particles with P > 15 GeV.

		ID in µµX e ID in eeX
	N true	31833	34301
	N true∩iden 31063	33017
	N iden	33986	34346
	Efficiency 97.6%	96.3%
	Purity	91.4%	96.1%

|<0.78 θ vs. P, for |cos 2 P/P Δ

	Δ	2 P/P	θ vs. cos
	(1/GeV)	0.0008 0.001					(1/GeV)	0.001 0.0008		Δ	2 P/P	parameterization
	2 P/P	0.0006		Δ	2 P/P	cut	2 P/P	0.0006		Δ	2 P/P	cut
	Δ						Δ
			0.0004						0.0004
			0.0002						0.0002
			0						0
			-1	-0.5	0		0.5	1	0	20	40	60	80	100
								θ cos				P (GeV)
	Figure 4: 2D ∆P/P 2 distribution vs. cosθ (left) and ∆P/P 2 distribution vs. track momen-
	tum (right) of muon candidates
	Δ	2 P/P	θ vs. cos
	(1/GeV)	0.0008 0.001					(1/GeV)	0.001 0.0008		Δ	2 P/P	parameterization
	2 P/P	0.0006		Δ	2 P/P	cut	2 P/P	0.0006		Δ	2 P/P	cut
	Δ						Δ
			0.0004						0.0004
			0.0002						0.0002
			0						0
			-1	-0.5	0		0.5	1	0	20	40	60	80	100
								θ cos				P (GeV)

|<0.78 θ vs. P, for |cos 2 P/P Δ

Table 5 :

 5 Number of events left after each cut for the µ + µ

	Before any restriction	2918 (100.0%) 2.6M 1.6M	111k	317k
	+ Lepton ID
	+ Tightened Pre-Cuts	2472 (84.72%)	9742 4582	9268	8175
	+ P Tdl > 20GeV	2408 (82.50%)	7862 3986	8462	7222
	+ M dl ∈ (80, 100)GeV + acop ∈ (0.2, 3.0) + ∆P Tbal. > 10GeV	2292 (78.54%) 2148 (73.61%) 2107 (72.20%)	6299 2679 5182 112 335 80	5493 5179 4705	5658 5083 4706
	2104 (72.11%) 2046 (70.09%) + M recoil ∈ (115, 150)GeV 2028 (69.48%) + |∆θ 2tk | > 0.01 + |cosθ missing | < 0.99 + f L > 0.26 1596 (54.68%)	149 82 75 41	80 80 80 0	4647 4647 3642 1397	4676 3614 2640 1125
	Before any restriction	3138 (100.0%)	4.3G 1.6M	147k	110k
	+ Lepton ID
	+ Tightened Pre-Cuts	2019 (64.33%) 43607 6422	13196	12548
	+ P Tdl > 20GeV	1962 (62.50%) 39152 5551	12054	10583
	+ M dl ∈ (80, 100)GeV + acop ∈ (0.2, 3.0) + ∆P Tbal. > 10GeV	1755 (55.93%) 25501 3806 1645 (52.41%) 23228 245 1606 (51.16%) 1725 157	7786 7239 6286	7509 6739 5904
	1603 (51.09%) 1564 (49.83%) + M recoil ∈ (115, 150)GeV 1539 (49.04%) + |∆θ 2tk | > 0.01 + |cosθ missing | < 0.99 + f L > 0.28 1153 (36.74%)	990 679 576 243	157 157 41 29	6150 6149 4824 2019	5844 4643 3335 1217

-X channel in the MI Analysis. Fractions of number of events remained of the Higgs-Strahlung process are given inside parentheses, the last one gives the efficiency of signal selection. The polarisation mode is e - L e + R .

MI, e + e -X, e - L e + R N evts Remained e + e -X e + e -τ + τ -e + e -νν e + e -ff

Table 6 :

 6 Number of events left after each cut for the e + e -X channel in the MI Analysis. Fractions of number of events remained of the Higgs-Strahlung process are given inside parentheses, the last one gives the efficiency of signal selection. -τ + τ -µ + µ -νν µ + µ -ff

	The polarisation mode is

Table 7 :

 7 Number of events left after each cut for the µ + µ

	Before any restriction	2107 (100.0%)	4.3G 1.2M	17k	1.1M
	+ Lepton ID
	+ Tightened Pre-Cuts	1352 (64.16%) 40896 5257	1469	10198
	+ P Tdl > 20GeV	1313 (62.33%) 36742 4546	1351	8430
	+ M dl ∈ (80, 100)GeV + acop ∈ (0.2, 3.0) + ∆P Tbal. > 10GeV	1177 (55.88%) 23993 3051 1103 (52.36%) 21846 107 1077 (51.11%) 1612 92	943 881 805	5909 5266 4517
	1076 (51.05%) 1050 (49.82%) + M recoil ∈ (115, 150)GeV 1033 (49.04%) + |∆θ 2tk | > 0.01 + |cosθ missing | < 0.99 + f L > 0.16 909 (43.14%)	927 638 539 326	92 92 12 4	799 799 586 368	4465 3484 2521 1294

-X channel in the MI Analysis. Fractions of number of events remained of the Higgs-Strahlung process are given inside parentheses, the last one gives the efficiency of signal selection. The polarisation mode is e - R e + L .

MI, e + e -X, e - R e + L N evts Remained e + e -X e + e -τ + τ -e + e -νν e + e -ff

Table 8 :

 8 Number of events left after each cut for the e + e -X channel in the MI Analysis. Fractions of number of events remained of the Higgs-Strahlung process are given inside parentheses, the last one gives the efficiency of signal selection. The polarisation mode is e - -τ + τ -µ + µ -νν µ + µ -ff

	R e + L .

Table 9 :

 9 Number of events left after each cut for the µ + µ -X channel in the MD Analysis. Fractions of number of events remained of the Higgs-Strahlung process are given inside parentheses, the last one gives the efficiency of signal selection. The polarisation mode is e -

	Before any restriction	3138 (100.0%)	4.3G 1.6M	147k	110k
	+ Lepton ID
	+ Tightened Pre-Cuts	2019 (64.33%) 43607 6422	13196	12548
	+ N add.TK > 1	2004 (63.87%)	3136 1740	374	10202
	2001 (63.77%) 1969 (62.75%) 1840 (58.62%) 1792 (57.11%) + M recoil ∈ (115, 150)GeV 1731 (55.16%) + |∆θ 2tk | > 0.01 + |∆θ min | > 0.01 + acop ∈ (0.2, 3.0) + |cosθ missing | < 0.99 + f L > 0.27 1378 (43.90%)	655 1073 155 128 134 0 91 0 73 0 27 0	79 6 6 6 1 0	10095 9271 8366 6696 4950 1652

L e + R .

MD, e + e -X, e - L e + R N evts Remained e + e -X e + e -τ + τ -e + e -νν e + e -ff

Table 10 :

 10 Number of events left after each cut for the e + e -X channel in the MD Analysis. Fractions of number of events remained of the Higgs-Strahlung process are given inside parentheses, the last one gives the efficiency of signal selection. The polarisation mode is e

Table 11 :

 11 Number of events left after each cut for the µ + µ -X channel in the MD Analysis. Fractions of number of events remained of the Higgs-Strahlung process are given inside parentheses, the last one gives the efficiency of signal selection. The polarisation mode is e - R e + L .

	Before any restriction	1967 (100.0%) 2.0M 1.2M	9k	291k
	+ Lepton ID
	+ Tightened Pre-Cuts	1667 (84.73%)	6696 3471	1048	5324
	+ N add.TK > 1	1654 (84.07%)	415	391	9	4160
	1651 (83.93%) 1629 (82.81%) 1522 (77.34%) 1476 (75.03%) + M recoil ∈ (115, 150)GeV 1453 (73.85%) + |∆θ 2tk | > 0.01 + |∆θ min | > 0.01 + acop ∈ (0.2, 3.0) + |cosθ missing | < 0.99 + f L > 0.17 1289 (65.53%)	41 22 20 11 10 8	379 105 0 0 0 0	0 0 0 0 0 0	4108 3739 3312 2438 1803 875
	Before any restriction	2107 (100.0%)	4.3G 1.2M	17k	1.1M
	+ Lepton ID
	+ Tightened Pre-Cuts	1352 (64.16%) 40896 5257	1469	10198
	+ N add.TK > 1	1342 (63.69%)	2935 1500	22	8227
	1340 (63.60%) 1319 (62.59%) 1232 (58.47%) 1201 (57.00%) + M recoil ∈ (115, 150)GeV 1161 (55.10%) + |∆θ 2tk | > 0.01 + |∆θ min | > 0.01 + acop ∈ (0.2, 3.0) + |cosθ missing | < 0.99 + f L > 0.32 889 (42.20%)	617 146 125 84 67 20	859 57 0 0 0 0	4 0 0 0 0 0	8133 7388 6651 5265 3886 1119

MD, e + e -X, e - R e + L N evts Remained e + e -X e + e -τ + τ -e + e -νν e + e -ff

Table 12 :

 12 Number of events left after each cut for the e + e -X channel in the MI Analysis. Fractions of number of events remained of the Higgs-Strahlung process are given inside parentheses, the last one gives the efficiency of signal selection. The polarisation mode is e -

R e + L .

Table 13 :

 13 alternative fit methods are discussed which lead to nearly identical results. Resulting Higgs mass M H and cross section σ of the MI Analysis using Kernel Estimation.

	Pol.	Ch.	M H (GeV)	σ (fb)
	e -R e + L L = 250 fb -1 e + e -X µ + µ -X 120.006 ±( 0.039 ) 7.89 ± 0.28 ( 3.55 %) 120.005 ±( 0.092 ) 8.46 ± 0.43 ( 5.08 %) merged 120.006 ±( 0.036 ) 8.06 ± 0.23 ( 2.91 %) e -L e + R µ + µ -X 120.008 ±( 0.037 ) 11.70 ± 0.39 ( 3.33 %) L = 250 fb -1 e + e -X 119.998 ±( 0.085 ) 12.61 ± 0.62 ( 4.92 %) merged 120.006 ±( 0.034 ) 11.96 ± 0.33 ( 2.76 %)
	Pol.	Ch.	M H (GeV)	σ (fb)
	e -R e + L L = 250 fb -1 e + e -X µ + µ -X 120.008 ± 0.037 7.88 ± 0.26 ( 3.30 %) 120.001 ± 0.081 8.46 ± 0.38 ( 4.49 %) merged 120.007 ± 0.034 8.06 ± 0.21 ( 2.66 %) e -L e + R µ + µ -X 120.009 ± 0.031 11.68 ± 0.32 ( 2.74 %) L = 250 fb -1 e + e -X 120.007 ± 0.065 12.58 ± 0.46 ( 3.66 %) merged 120.009 ± 0.028 11.97 ± 0.26 ( 2.19 %)

Table 14 :

 14 Resulting Higgs mass M H and cross section σ of the MD Analysis using Kernel Estimation.

Table 16 :

 16 Comparison of the Higgs recoil mass distributions of e + e -X channel with and without the bremsstrahlung photons recovery. Resulting Number of Signal (S) and Number of Background (B), and the efficiencies of signal selection (in the parentheses) after background rejection, for eeX channel with Bremsstrahlung Photons Recovery

	recoil M	(GeV)

Table 17 :

 17 Resulting Higgs mass M H and cross section σ for the Model Independent Analysis and Model Dependent Analysis of the Higgs-Strahlung pr ocess of e + e -X channel with Bremsstrahlung photons Recovery.

		M H (GeV)	σ (fb)
	MI MD e -e -R e + L e -L e + R R e + L e -L e + R	120.003 ± 0.081 8.41 ± 0.36 ( 4.28 %) 119.997 ± 0.073 12.52 ± 0.49 ( 3.91 %) 119.999 ± 0.074 8.41 ± 0.31 ( 3.69 %) 120.001 ± 0.060 12.51 ± 0.38 ( 3.04 %)

Please note, that due to a bug in the luminosity spectrum in the initial generation, the background has been re-weighted to the correct spectrum according to[START_REF] Berggren | Talk in ILD Optimisation Meeting 13[END_REF].


Tellier Marion
email: mariontellier@yahoo.com

How do teacher's gestures help young children in second language acquisition?

Keywords: teaching gestures, foreign language teaching, lexicon, memorisation, children

This article aims to discus what we know about the impact of gestures on memorisation of items. There have been few studies on that subject so far. I am specifically working on the impact of teaching gestures (very iconic movements used by teachers in classroom, especially in second language teaching) on young children (aged 5 years old). This article describes an experiment in which young subjects had to memorise words in L1. Theses words were heard on a video and were illustrated by gestures. Some of the children just had to look at the gestures and repeat the words heard, others had also to reproduce the gestures they saw. Results show that in a free recall task, children who have reproduced the gestures memorise better than those who only looked at them and than those of the control group. These results concern short term memorisation of L1 words but will help us to conduct further experiments on long term memorisation of L2 lexical items.

In the second language classroom, teachers behave in a specific way: the way they speak and the way they gesture are altered by the classroom situation. They slow down their speech, and they intensify the articulation of every word and of the prosodic parameters to make sure that the students will understand them better. For the same reason, they use helping gestures. We are not dealing here with typical everyday communicative gestures but with specific ones, which have to be clear and unambiguous in order to help the students understand the verbal input the gestures illustrate. This is particularly true when teaching to young children. Official and scientific texts dealing with the teaching of foreign languages to children advise teachers to use gestures to illustrate their speech and thus to improve the children's understanding and memorisation of the foreign language. This is given as an affirmation and it appeared to us that nobody has ever tried to assess how much the teacher's gestures could help young children understand and memorize the foreign language lexicon. Therefore, the past few years we have been trying for to elaborate different methods and experiments aimed at assessing the impact of teacher's gestures on the learning process. We have first worked on their impact on the understanding of language (Tellier, 2004[START_REF] Tellier | L'utilisation des prototypes gestuels comme support pour la compréhension en langue étrangère[END_REF], and to be published) and we are currently analyzing the effect of gestures on the process of short term memorisation. The following article will present an experiment conducted on this latest premise.

Theoretical basis

What is a teaching gesture?

In the field of second language teaching, a teaching gesture is, as we have already stated, different from everyday communicative gesture. The reason is that, in ordinary communication, gestures and words work together, and one relies on the other to have meaning; in short, they are complementary. In usual communication, where all the participants have the same level of fluency in the vehicular language, speakers produce gestures rather unconsciously to help them structure their thought and organize their discourse. But this is not the case in foreign language teaching, because the participants do not share the same level of fluency in the target language: on the one hand we have the teacher who has great command of the language, and on the other hand we have learners who understand more or less what is being said. When the learners are beginners and/or children, the linguistic relationship is highly asymmetric. Thus, gestures are what the learners rely on to understand what the teacher says. This means that the gestures need to convey enough meaning to be understood alone (without verbal language), and have to help one to infer the meaning of the words they are associated with. We, therefore, believe that teaching gestures are produced more consciously than usual communicative gestures and that they are specifically addressed to the learners.

We have listed different types of teaching gestures which appear in class and we have discovered that they serve various functions. So far we have found three main roles for teaching gestures: management of the class (to start/end an activity, question students, request silence, etc.), evaluation (to show a mistake, correct, congratulate, etc.), and explanation (give indication on syntax, underline specific prosody, explain new vocabulary, etc.). In our study, we are particularly interested in the gestures which occur in the process of explaining new or unknown vocabulary.

They appear in various shapes: hand gestures, facial expressions, pantomime, body movements, etc. They can either mime or symbolize something and they truly help the learners to infer the meaning of the verbal, providing that they are unambiguous and easy to understand (Tellier, 2004[START_REF] Tellier | L'utilisation des prototypes gestuels comme support pour la compréhension en langue étrangère[END_REF], and to be published).

However, teaching gestures do not only help in the process of understanding. We can indeed suppose that they have an impact on the memorisation of the vocabulary. Therefore, in the following study we will try to assess the impact of these gestures on the memorisation of words.

Previous work on the impact of gestures on memorisation

We have specifically relied on the following studies to elaborate the present experiment.

The first experiment is one elaborated by [START_REF] Cohen | The mnemonic effect of speech gestures: Pantomimic and non-pantomimic gestures compared[END_REF]. They worked with three groups of adult subjects. The subjects had to watch a video containing several different sentences, then write down as many sentences as they could remember. Each group had the same sentences but the videos were slightly different: one just exposed the sentences, another one showed somebody illustrating each sentence with illustrative/pantomimic gestures and in the last one, sentences were also accompanied with gestures but they were non-pantomimic.

Cohen and Otterbein wanted to assess what modality would help the subjects remember the greatest number of words. The illustrative gestures proved to be the most significant helping modality in short term memorisation.

Linda Quinn Allen (1995) explored the same objective but in the field of second language learning. She worked with 112 university students in French. A non-treatment group and a comparison group were shown 10 French sentences and their English equivalent on a screen and they also heard a teacher pronouncing them 3 times. They were told to repeat them. The experimental group's procedure differed only in that the students were also provided with an illustrative gesture for each sentence, which they saw three times (with the three repetitions of the sentence) and had to reproduce. However, they did not repeat the sentences, only the gestures. Then, immediately after all 10 sequences, a posttest was given: the teacher said the had to write down the English equivalent. The comparison group and the experimental group were given the gestures as well. There were 5 sessions of this kind with different groups of 10 French expressions. The analysis of the results shows that the students presented with illustrative gestures recalled more sentences than the others. The experimental group who reproduced the gestures did better than the comparison group who just saw them during the posttest.

Both experiments dealt with adult subjects. In our study on second language teaching to young children and the role of teaching gestures in lexical memorisation, it seemed interesting to elaborate similar experiments with younger subjects. Our first work in this field (Tellier, 2005a) was with 32 children age 5 who were divided into 2 groups (control and experiment) and had to watch 3 videos which contained a list of 10 words each. The words were in French, the subjects' mother tongue, for we did not want to work with foreign language yet. The children watched the video individually and had to do a free recall task immediately after. The three videos watched by the control group only presented them with words pronounced by a person on the film. The first video watched by the experiment group was the same as the control group, the second video was illustrated with gestures and the third had pictures to illustrate the words pronounced. The experiment group had significant better results with video 2 and 3. This enabled us to infer that the use of visual modalities improved short term memorisation.

In this article we will still work on short term memorisation in a free recall task. We will thus deal with the notion of mnemonic span (Milner, 1956;[START_REF] Baddeley | Human memory, theory and practice[END_REF] which refers to the quantity of items a subject can memorize from a list. The average score is 7 items plus/minus 2 for an adult, but is less important for children age 5, which we have assessed around 3 (Tellier, 2005a) and which is likely to grow with age and cognitive development.

Also, in the present study, we will not deal with foreign language. Since we were interested in short term memory, we did not teach new words to children but worked on words they are familiar with in their mother tongue. Thus, we will be able to assess the real effect of gestures.

Method

Design

The experiment was a 3 x 3 x 4 factorial design. There were 3 groups: control, comparison and experiment. They were working on 3 different videos named Video 1, Video 2 and Video 2G. There were 4 different tasks involved: listen and recall, repeat words and recall, view gestures then repeat words and recall, and finally, reproduce gestures and repeat words then recall.

Sample

Forty-two (42) French children from the same school were involved in the experiment. The age mean was 5 years 9 months old, and the range was 5 years 3 months old to 6 years 3 months old. They were divided into 3 groups of 14 children for the purpose of the experiment: a control group, a comparison group and an experiment group.

2.3.

Materials

Two (2) lists of 10 French words were elaborated. They were designed for children, based on everyday vocabulary. They were built on the same pattern: a fruit, an aspect, a drink, a school tool, a place, an animal, etc. They were recorded on video tape. For Video 1, a person pronounced every word with an interval of 2 seconds between words. For Video 2, there were two versions: one was like video 1 (to be watched by the control group) and the other was illustrated with gestures and which we will call Video 2G. The person on the video did an illustrative gesture for each word. The gesture slightly preceded the pronunciation of the word so that the children would focus on the gesture first. The videos assigned for each group are present in Table 1, below.

Tab. 1: Videos assigned for each group

G rou p V id eo 1 V id eo 2 V id eo 2 G C o ntrol X X C o m p ariso n X X E x perim en tal X X

Procedure

The subjects were tested individually in a quiet room of the school. The children sat in front of a TV set and the experimenter sat a bit apart: not too close to the child, to avoid disturbing his/her concentration, and not too far to avoid formal atmosphere and to prevent the child from moving his/her chair to get closer to the experimenter. The sessions were also recorded with a video camera placed next to the TV set.

To avoid problems of fatigue and weariness, we did two sessions: one for each video. The second session was done a week after the first one.

The children were introduced to the experiment as a game to play. At the beginning of each session, the memory task was explained and children were given time to practice with another and shorter word list to make sure they understood the nature of the task. The task depended on the group the children were in and the video they were working with (1 or 2).

In the first session, with Video 1, the children of the control group were told to listen to the list of words and then recall as many items as they could and in a free order. The other groups were told to repeat each word in the 2 seconds blank between each item and then proceed to a free recall of the list.

In the second session, the children of the control group had to repeat the words of Video 2 the same way the other groups did with Video 1. The comparison group had to repeat the words and look at the gestures that were shown with them. However we did not want them to imitate the gestures. We did not want to tell them "Don't reproduce the gestures" for we thought it might confuse them, so we placed a small teddy bear in their hands which they were told to hold. Thus their hands were busy and they were not tempted to gesture. As far as the experiment group was concerned, subjects were asked to reproduce the gestures and repeat the words after the person on the video, in short to imitate her.

The following Does the repetition of words associated with the visualization of an illustrative gesture improve the number of items recalled?

Does the repetition of words associated with the reproduction of illustrative gestures improve the number of items recalled? What impact can these results have on second language teaching to young children?

Analysis and results

Does the repetition of words improve the number of items recalled?

For the task of Video 1, the mean of words recalled by the control group was 2.786; it was 2.929 for the comparison group; and 3 for the experiment group. Analysis of variance (ANOVA) revealed that the repetition had no significant effect on the short term memorisation of words: F(2,39) = .144, p<.05, even if the means of the lasts two groups seem higher.

For the task of Video 2, the control group who just had to repeat words heard on the video recaled a mean of 3.071 words which is very close to the means the other groups obtained in the same condition with Video 1. We may thus infer that the difficulty of both word lists was similar.

Does the repetition of words associated with the visualization of an illustrative gesture improve the number of items recalled?

For Video 2, the comparison group visualized an illustrative gesture with each word they heard and had to repeat this word. They recalled 3.357 words. Did the fact of seeing the gestures help to improve memorisation? Impaired t-test comparing the number of words recalled by the control group (3.071) and the comparison group (3.357) revealed that the visualization of gestures had no significant effect on short term memorisation: t(26)= -.702 with p< .4889. This result is clearly different from what we found in a previous research (Tellier, 2005a) in which children who had seen gestures did significantly better than those who did not. However we may explain this phenomenon by the fact that the task was not exactly the same since the children in the previous experiment did not have to repeat the words, we may infer that in the present experiment the children's attention might have been caught more by the repetition than by the gesture. We will come back to that question later on.

3.3.

Does the repetition of words associated with the reproduction of illustrative gestures improve the number of items recalled?

For Video 2, the experimental group had to repeat the words and reproduce the gestures seen on screen. Did the reproduction of gestures enable these children to memorize more words than those of the two other groups? The control group recalled a mean of 3.071 words, the comparison group: 3.357 words and the experiment group 4.143 words.

Impaired t-test comparing the number of words recalled by the control group and the experiment group revealed that the reproduction of gestures had significant effect on short term memorisation: t(26)= -2.912 with p< .0073. Impaired t-test comparing the number of words recalled by the comparison group and the experimental group also revealed that the reproduction of gestures had significant effect on short term memorisation: t(26)= -2.317 with p< .0286.

3.4.

Impact these results can have on second language teaching to young children.

This experiment only involved children memorizing words in their first language and on a short term memorisation basis so one might think it can have little to do with second language acquisition. However, we think that if reproducing gestures enables children to improve memorisation in their mother tongue, it will probably have an impact on the memorisation of second language lexicon. Moreover, short term memorisation was analyzed in this experiment but we may infer that reproducing gestures may help for long term memorisation as well.

Our next step will be to experiment the impact of gestures on the learning of foreign language words: that means work on both an unknown language and long term memory.

Also, we have noticed that the repetition of words did not really contribute to improve memorisation. On the contrary, compared to the results we had obtained in the previous experiment which did not imply repeating, it seems that the repetition of words has weakened the process of short term memorisation. It appeared to us that when the children repeated the words of the list after the person on the video, they did it in a very mechanic and passive way. Whereas in the previous experiment where they had to remain silent during the listening task, we can assume that they did a mental repetition of words in order to remember as many as they could. In short, their listening was more active and concentration was strengthened. The fact that they had to repeat the words out loud in the present experiment killed in a way the mental repetition and the concentration. Nevertheless, the reproduction of gestures had the reverse effect: memorisation significantly improved. We can suppose that having to reproduce a gesture implied more attention to the visualization of gestures and to the items, and the repetition became more active.

Conclusion

We, therefore, assume that the use of teacher's gestures in the learning of foreign vocabulary can have an effect on memorisation; but to make the most of this effect, teachers should make sure that the children reproduce the gestures while repeating the words. Thus, they will be more active in their repetition and reinforce its trace in memory. Also, the three ways of learning will be solicited: auditory modality will be provided by the teacher's voice and the repetition, visual modality will be exposed through the visualization of gestures, and kinesthetic modality will appear thanks to the reproduction of gestures. All this will enable every child to make the most of the three modalities proposed to reinforce their memorisation.

We consider important that the impact of gestures in second language acquisition should be promoted in teacher's training. Reflection on gestures and second language learning as well as development of teaching gesture techniques should be part of teacher's training so that the majority of gestures can be made by learners, especially young ones.


Cyrille Lucas
email: cyrille.lucas@u-paris10.fr

Asymptotics of the odometer function for the internal Diffusion Limited Aggregation model

Keywords: asymptotic shape, divisible sandpile, Green's function, internal diffusion limited aggregation

We present precise asymptotics of the odometer function for the internal Diffusion Limited Aggregation model. These results provide a better understanding of this function whose importance was demonstrated by Levine and Peres [3]. We derive a different proof of a time-scale result by Lawler, Bramson and Griffeath [2].

Introduction

The internal Diffusion Limited Aggregation model was first introduced by Diaconis and Fulton in [START_REF] Diaconis | A growth model, a game, an algebra, Lagrange inversion, and characteristic classes[END_REF] gives a protocol for building a random set recursively. At each step, the first vertex visited outside the cluster by a simple random walk started at the origin is added to the cluster. The resulting limit shape is the euclidian ball, as proved in 1992 by Lawler, Bramson and Griffeath in [START_REF] Lawler | Internal diffusion limited aggregation[END_REF].

More recently, however, Levine and Peres [START_REF] Levine | Scaling limits for internal aggregation models with multiple sources[END_REF], [START_REF] Levine | Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile[END_REF] have shown that this model is related to the rotor-router and divisible sandpile models. In the former, random walkers are replaced by eulerian walkers. In the latter, each vertex can hold 1 unit of mass, and the excess is divided equally among its neighbors when the vertex topples. Thus an initial mass at the origin becomes a stable shape after a suitable infinite series of topplings.

For each of these three models, one can define an odometer function, which will be the total number of times a walker passes through a given point (counting multiple passages of the same walker) for the internal DLA and rotor-router model, and the total mass emitted from a given point in the construction of the cluster for the divisible sandpile model. This function plays a comparable role in all three models and turns out to be instrumental in their relation. While the limiting shape of these models is known to be the Euclidian ball, the behavior of their odometer functions in the particular case where all the mass is started at the origin remains to be studied.

This paper provides a closer look at the odometer function in the case of the internal DLA model, with an almost sure convergence of the normalized functions and asymptotics of this function near the origin. These results provide in turn a new proof of the time scale of the cluster introduced in [START_REF] Lawler | Internal diffusion limited aggregation[END_REF].

Definitions, main results

Let (S j ) j∈N a sequence of independent simple random walks on Z d , and let us define the cluster A(n) and stopping times (σ k ) k∈N recursively in the following way:

σ 0 = 0, A(0) = {0} = {S 0 (σ 0 )},
and for all j > 0, ∀j > 0, σ j = inf{t ≥ 0 : S j (t) ∈ A(j -1)}, A(j) = A(j -1) ∪ {S j (σ j )}.

Let ||.|| denote the Euclidean norm on R d , B r the Euclidean ball of radius r of Z d , and ω d the volume of the unit ball of R d . We will consider the cluster A(ω d n d ), which has the same volume as B n . Lawler, Bramson and Griffeath proved in [START_REF] Lawler | Internal diffusion limited aggregation[END_REF] that the normalized cluster 1 n A( ω d n d ) converges to the Euclidean unit ball with probability one. Lawler then improved this result, defining the inner and outer errors as follows:

δ I (n) = n - inf z ∈A(⌊ω d n d ⌋) ||z||, δ O (n) = sup z∈A(⌊ω d n d ⌋) ||z|| -n.
to get the following bounds, with probability 1:

lim n→∞ δ I (n) n 1/3 (ln n) 2 = 0, and lim n→∞ δ O (n) n 1/3 (ln n) 4 = 0.
(2.1)

An important part in the study of this model is played by Green's functions for the simple random walk, which we introduce now, as defined in [START_REF] Lawler | Intersection of random walks[END_REF]. Let S x be a simple random walk started from a point x ∈ Z d , and ξ n be the time at which S exits B n , that is :

ξ n = inf{t : S t / ∈ B n }.
Then we define for all y ∈ Z d

G n (x, y) = ξn t=0 1 S x (t)=y .
This is zero if either x or y lies outside B n . For d ≥ 3 we define, for all

x, y ∈ Z d , G(x, y) = ∞ t=0 1 S x (t)=y . Let z ∈ R d be a non-zero point in the open unit ball. Let us write z = (x 1 , • • • , x d ) Let us define z n := (⌊nx 1 ⌋, • • • ⌊nx d ⌋) .
The odometer function u n (z) at rank n measures the number of walkers passing through z n in the process of building the cluster A(⌊ω d n n ⌋). It is defined as follows:

u n (z) := ⌊ω d n d ⌋ i=0 σ i t=0 1 S i (t)=zn
Our results are as follows:

Theorem 2.1 For all non-zero points of the open unit ball z, when n goes to infinity,

u n (z) n 2 →||z|| 2 -1 -2 ln(||z||) almost surely if d = 2, u n (z) n 2 →||z|| 2 + 2 (d -2)||z|| d-2 - d d -2 almost surely if d ≥ 3. (2.2)
Remark: In [START_REF] Lawler | Internal diffusion limited aggregation[END_REF], the authors estimate the time it takes to build a cluster of radius n, that is to say the total number of steps done by random walks during the construction of the cluster. The authors count these steps by estimating the number of steps for each given random walk. Our convergence result 2.1 allows us to take a different perspective on the problem, and count the total number of steps as the sum over all points in the cluster of the number of steps through this point.

This leads to a different proof of the result of Lawler, Bramson and Griffeath on the time scaling of the cluster, which is presented in section 5.

The functions in 2.2 vanish when ||z|| tends to 1, and tend to infinity when ||z|| tends to 0. To understand this behavior around the origin, we can use a different scaling. Consider a sequence of non-zero points y n such that yn n converges to 0 as n tends to infinity. If we now consider the following sequence of values takes by the n-th odometer at point yn n :

u n y n n = ⌊ω d n d ⌋ i=0 σ i t=0 1 S i (t)=yn
We get the following result:

Theorem 2.2 Let y n be a sequence of non-zero points such that yn n converges to 0.

• If y n converges to a point a ∈ Z d , d ≥ 3, when n tends to infinity, u n ( y n n ) ∼ ω d n d G(0, a) almost surely • If ||y n || tends to infinity, when n goes to infinity, u n ( yn n ) n 2 ∼ 4 ln n ||y n || almost surely, if d = 2, u n ( yn n ) n 2 ∼ 2 d -2 ||y n || n 2-d almost surely, if d ≥ 3.
Remark: The question of the asymptotics of the odometer function near the boundary remains open, and is presumably linked to the difficult problem of the fluctuations of the cluster around its limiting shape.

Let us comment further on Theorem 2.1, and give a heuristic of it when d ≥ 3 based on the work of Levine and Peres ( [START_REF] Levine | Scaling limits for internal aggregation models with multiple sources[END_REF]) (the two-dimensional case is similar if a little more technical). In the case of the divisible sandpile model, each site of Z d contains a continuous amount of mass. A site with an amount greater than 1 can topple, that is to say keep mass 1 and distribute the rest equally between its neighbors. With any sequence of topplings that topples each full site infinitely often, the mass approaches a limiting distribution, which does not depend on the sequence.

The odometer function u for this model is defined as the total mass emitted from a given point. Since each neighbor y of a given point x divides mass equally between its 2d neighbors, the total mass received by x is 1 2d y∼x u(y). If we define the discrete Laplacian ∆ as ∆f (x) =

1 2d y∼x f (y) -f (x), we get that ∆u(x) = ν(x) -σ(x),
where σ and ν are the initial and final amounts of mass at x, respectively.

In our case, the initial mass ω d n d is concentrated at the origin, and the final mass is 1 in each fully occupied site of the cluster. Hence:

∆u n (x) =      n -1 if x = 0, -1
if x is a non-zero point in the cluster, 0 outside the cluster.

To solve this equation, we first introduce a function γ n that has discrete Laplacian σ n -1, in our case:

γ n (z) = -||z|| 2 -ω d n d G(0, z).
Then the odometer function u n is given by lemma 3.2 of [START_REF] Levine | Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile[END_REF] as the difference between the least superharmonic majorant of γ n and γ n :

u n = s n -γ n ,
where s n = inf{f (x) such that f is superharmonic and f ≥ γ n }.

Recalling the notations z

= (x 1 , • • • , x d ) and z n = (⌊nx 1 ⌋, • • • ⌊nx d ⌋), we get: 1 n 2 γ n (z n ) → γ(z) := -||z|| 2 - 2 d -2 ||z|| 2-d
The function γ is radial, and its particular shape makes it easy to determine its superharmonic majorant s, as one can see on Figure 1:

s(z) = -2 d-2 if ||z|| ≤ 1, γ(z) if ||z|| ≥ 1.
Since γ n converges to γ, s n should converge to s. This would mean that u n converges to s -γ. This method is in fact made rigorous in [START_REF] Levine | Scaling limits for internal aggregation models with multiple sources[END_REF], with the noticeable difference that in the case considered the starting mass has a bounded density, as opposed to our case where all of the mass is started at the origin. The authors then prove that the odometer function for the divisible sandpile model is the expected value of the odometer function for the internal DLA model, which gives a heuristic proof of the following result:

E u n (z) n 2 →||z|| 2 -1 -2 ln(||z||) if d = 2, E u n (z) n 2 →||z|| 2 + 2 (d -2)||z|| d-2 - d d -2 if d ≥ 3.
This analytic method could probably be rendered rigorous, but we prefer a more probabilistic approach based on the convergence of the cluster, which gives us an almost sure convergence result. 3 Proof of Theorem 2.1

The main idea of this proof is to use the fact that the cluster is very similar to a disc when n is large enough to get two inequalities framing u n (z) between random valuations which do not depend on the shape of the cluster.

Lawler's theorem proves that the following set inequality holds for n large enough with probability one:

B n-n 1/3 (ln n) 2 ⊂ A( ω d n d ) ⊂ B n+n 1/3 (ln n) 4 ,
Let us define:

η I (n) = n 1/3 (ln n) 2 η O (n) = n 1/3 (ln n) 4
Let us define the stopping time ξ j k as the time at which S j leaves a ball B k . As a consequence of the set inequality, we get the following inequality on stopping times for k large enough, for all j such that ω d (k -1) d ≤ j ≤ ω d k d , with probability one:

ξ j (k-1)-η I (k-1) ≤ σ j ≤ ξ j k+η O (k) (3.1)
We get the following framing of u n (z):

n-1 k=0 ⌊ωd(k+1) d ⌋ j=⌊ω d k d ⌋ ξ k-η I (k) t=1 1 S i (t)=zn ≤ u n (z) ≤ n k=1 ⌊ωd(k+1) d ⌋ j=⌊ω d k d ⌋ ξ j k+η O (k) t=1 1 S i (t)=zn
This inequality holds for n large enough with probability one because ||z n || -n||z|| is smaller than a constant. This ensures that the inequality (3.1) will start to hold as soon as k becomes greater than n||z|| 2 for instance. All the other terms of the sum are asymptotically zero.

We will now show that the left and right bounds on u n (z), which we will call u n (z) -and u n (z) + , once normalized, converge almost surely to the same function. In order to do this, we take a look at the following family of random variables:

χ j k,n (z) = ξ k t=1 1 S j (t)=zn
A direct application of the Markov property shows that for i ≥ 1, there exists p 0 and p depending on k, n and z such that:

P(χ j k,n (z) = i) = p 0 p i-1 (1 -p)
Since we know that P(χ j k,n (z) = 0) = P(τ zn > ξ k ) where τ zn is the reaching time of z n , and that E(χ j k,n (z)) = G k (0, z n ), where G n is Green's stopped function, a simple computation shows that:

p 0 = P(ξ k < τ zn ) p = 1 - P(ξ n < τ zn ) G k (0, z n )
This determines the parameter of the geometric distribution χ j k,n (z) follows.

We will need the following lemma which gives an estimate of G n (0, z):

Lemma 3.1 If z ∈ B n , z = 0, we have G n (0, z) = 2 ω 2 ln n ||z|| + O( 1 ||z|| ) + O( 1 n ) if d = 2, and
G n (0, z) = 2 d -2 1 ω d (||z|| 2-d -n 2-d ) + O(||z|| 1-d ),
where the O are uniform on the unspecified variables.

This lemma is due to Lawler and can be found in section 1.6 of [START_REF] Lawler | Intersection of random walks[END_REF].

We will now prove that 1 n 2 u n (z) converges almost surely to the limit of its mean value. We first compute the mean value of u n (z) -, then we will bound its variance.

Consider the left side of (3):

u n (z) -= n-1 k=0 ⌊ω2(k+1) 2 ⌋ j=⌊ω d k 2 ⌋ ξ k-η I (k) t=1 1 S j (t)=zn u n (z) -= n-1 k=0 ⌊ω2(k+1) 2 ⌋ j=⌊ω d k 2 ⌋ χ j k-η I (k),n (z) E(u n (z) -) = n-1 k=0 ⌊ω2(k+1) 2 ⌋ j=⌊ω d k 2 ⌋ G k-η I (k) (0, z n ) = n-1 k=⌊||z||n⌋ ( ω d (k + 1) 2 -ω 2 k 2 )G k-η I (k) (0, z n ) (3.2)
Let us consider separately the case d = 2. In that case, we have:

E(u n (z) -) = n-1 k=⌊||z||n⌋ ( ω 2 (k + 1) 2 -ω 2 k 2 ) 2 ω 2 ln k -η I (k) n||z|| + O( 1 n ) E(u n (z) -) = n-1 k=⌊||z||n⌋ 4k ln k n||z|| + O(n 5/3 ) 1 n 2 E(u n (z) -) = 1 n n-1 k=⌊||z||n⌋ 4 k n ln k n||z|| + O(n -1/3 ) (3.3)
We recognize a Riemann sum, thus :

1 n 2 E(u n (z) -) → 1 ||z|| 4t ln( t ||z|| )dt = ||z|| 2 -1 -2 ln(||z||)
When d ≥ 3, we get instead the following estimate for E(u n (z) -) :

n-1 k=⌊||z||n⌋ ( ω d (k + 1) d -ω d k d ) 2 d -2 1 ω d ((n||z||) 2-d -(k-η I (k)) 2-d )+O(n 1-d )
This formula is the same as in the estimation of E(u n (z) -) and yields:

Var u n (z) - n 2 ≤ g(||z||, d) n 2 ,
where g is a function of ||z|| and d only.

When d = 2, taking a suited constant K 2 gives us:

Var u n (z) - n 2 ≤ 1 n 4 n-1 k=0 K 2 kG k (0, z n ) ln(n) 2 ,
which yields:

Var u n (z) - n 2 ≤ ln(n) 2 h(||z||) n 2 ,
where h is a function of ||z||.

In both cases, the sum Var un(z) - n 2

is finite, which means un(z) - n 2

converges almost surely to lim n→∞

E(un(z) -) n 2
. The same set of arguments can be used to prove that un(z) + n 2 converges almost surely to lim n→∞

E(un(z) + ) n 2 . Since lim n→∞ E(un(z) -) n 2 = lim n→∞ E(un(z) + ) n 2
and u - n (z) ≤ u n (z) ≤ u + n (z) for all n large enough, we have proved our result.

Proof of Theorem 2.2

The proof of the first assertion relies only on the fact that the following framing holds for n large enough:

n-1 k=⌊ln n⌋ ⌊ωd(k+1) d ⌋ j=⌊ω d k d ⌋ ξ k-η I (k) t=1 1 S i (t)=yn ≤ ⌊ωdn d ⌋ i=⌊ω d (ln n) d ⌋ σ i t=0 1 S i (t)=yn ≤ n k=1+⌊ln n⌋ ⌊ωd(k+1) d ⌋ j=⌊ω d k d ⌋ ξ j k+η O (k) t=1 1 S i (t)=yn
This is true because the equation (3.1) holds as soon as k is large enough, which happens eventually since k ≥ ln n.

The difference to the relevant quantity can be bounded using this inequality which holds for n large enough:

⌊ω d (ln n) d ⌋ i=0 σ i t=0 1 S i (t)=yn ≤ ⌊ωd(ln n) d ⌋ i=0 ξ ln n-η I (ln n) t=1 1 S i (t)=yn
In both cases, the sum Var un( yn n ) n d is finite. Since the same calculation can be applied to u + n , we can use the same argument as in the proof of Theorem 2.1 to say that almost surely, as n tends to infinity,

u n ( yn n ) n d → ω d G(a).
The proof of the second assumption is similar to that of the first one and relies on the following estimate:

for d = 2, E    n-1 k=0 ⌊ω2(k+1) 2 ⌋ j=⌊ω 2 k 2 ⌋ ξ k-η I (k) t=1 1 S i (t)=yn    = n k=0 4k ln ||y n || d-2 k -η I (k) + O 1 ||y n || + 1 n = 4n 2 ln n ||y n || + O 1 ||y n || + 1 n And for d ≥ 3, E    n-1 k=0 ⌊ωd(k+1) d ⌋ j=⌊ω d k d ⌋ ξ k-η I (k) t=1 1 S i (t)=yn    = n k=0 2d d -2 k d-1 1 ||y n || d-2 - 1 (k -η I (k)) d-2 + O n d-1 ||y n || d-1 = 2n 2 d -2 n ||y n || d-2 - 1 2 + O n d-1 ||y n || d-
n 2 ln n ||yn||   ≤ J 2 ln 2 (n) n 2 ln n ||yn|| Var   u n yn n n 2 ln n ||yn||   ≤ J d n 2 ||y n || n d-2
These quantities have finite sums, which proves our result.

Back to a result of Lawler, Bramson and Griffeath

The result in Theorem 2.1 can be used to derive a different proof of an existing time-scale result by Lawler, Bramson and Griffeath.

In [START_REF] Lawler | Internal diffusion limited aggregation[END_REF], the authors are interested in the time it takes to build a cluster of radius n, that is to say the total number of steps done by random walks during the construction of the cluster. They define Ã(t) as the internal DLA cluster on the time scale of individual random walks. Namely, if χ(t) = max{k such that σ 1 + • • • + σ k ≤ t} then Ã(t) = A(χ(t)). While the original proof of this theorem studies the time spent by a given random walk inside the cluster, we have studied the time spent in a given point by all the random walks. This leads naturally to a different approach of the problem, in which we will sum the odometer function over all the points in the cluster.

Let us call t ′ n the time taken to build the cluster A(n). Then

t ′ n = z:nz∈Z d ∩A(n) u n (z)
Then for all n, z, nz ∈B n-η I (n)

u n (z) ≤ t ′ n ≤ z, nz ∈B n+η O (n) u n (z)
We can compute the following asymptotics, as n tends to infinity:

Figure 1 :

 1 Figure 1: The functions s and γ as radial functions of z.

Theorem 5 . 1 (

 51 [START_REF] Lawler | Internal diffusion limited aggregation[END_REF]) Let Ã(t) be the internal DLA cluster on the time scale of individual random walks, then for all ǫ > 0, almost surely,B n(1-ǫ) ⊂ Ã(t n ) ⊂ B n(1+ǫ) , where t n = n d+2 dω d d + 2 .

z 1 0

 1 , nz ∈B n-η I (n) u n (z) = n k=1 dω d k d-1 u n k n + o(n d+2 ) = d ω d n 2 n k=1 k d-1 f k n + o(n d+2 ), almost surely.where f is defined by (2.2). Hence:z:nz∈B n-η I (n) u n (z) = d ω d n d+2 x d-1 f (x)dx + o(n d+2 ) = n d+2 d ω d d + 2 + o(n d+2 )The same asymptotics hold for z, nz ∈B n+η O (n) u n (z), so that we have,

The same arguments show that 1 n 2 E(u n (z) + ) has the same limit as its counterpart.

We will now give a bound on the variance of these variables. Let us first state the following lemma: Lemma 3.2 The random valuations χ j k,n (z) have the following variance, for all j, k, n ∈ N, z ∈ Z d provided that ||z||n ≤ k:

The proof of this lemma is a straightforward calculation which relies only on the knowledge of the law of χ j k,n (z). We will need bounds for G k (z n , z n ). Provided that ||z||n ≥ k, z n is in the ball of radius k centered at the origin, and it has at least one neighbor in this same ball, which proves that G k (z n , z n ) ≥ 1 + 1 4d 2 . When d ≥ 3, we can use the simple upper bound given by G k (z n , z n ) ≤ G(0, 0) < ∞.

However, when d = 2, we will use the following bound: G k (z n , z n ) ≤ G n+k (0, 0). This bound comes from the fact that a random walk started in z n will exit the ball of radius k centered in 0 before it exits the ball of radius n + k centered in z n . When k ≤ n, we will even use G k (z n , z n ) ≤ G 2n (0, 0).

We are now ready to bound the variance of u n (z) -.

Var

When d ≥ 3, naming K d a suited constant depending only on the dimension d, we get:

It is a consequence of the calculations in the proof of Theorem 2.1 that the right side tends to zero almost surely once divided by n 2 .

Just like in the proof of Theorem 2.1, we estimate the expected value of our lower bound:

The calculation for the upper bound yields the same result. To finish the proof, we just need to bound the variance of these two random variables, using lemma 3.2:

Var(χ j k-η I (k),n (

where C ′ 2 is a suited constant.


Nathalie Lambeng

Yann Wallez

Christine Rampon

Francine Cand

Georges Christé

Danielle Gulino-Debrac

Isabelle Vilgrain

Philippe Huber

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