I Zaytseva

O Abal'oshev

P Dłużewski

W Paszkowicz

L Y Zhu

C L Chien

M Kończykowski

Marta Z Cieplak

Negative Hall coefficient of ultrathin niobium in Si/Nb/Si trilayers

Keywords: number(s): 74, 62, En, 74, 25, F-

Structural and transport properties of thin Nb layers in Si/Nb/Si trilayers with Nb layer thickness d from 1.1 nm to 50 nm have been studied. With decreasing thickness, the structure of the Nb layer changes from polycrystalline to amorphous at d 3.3 nm, while the superconducting temperature T c monotonically decreases. The Hall coefficient varies with d systematically but changes sign into negative in ultrathin films with d < 1.6 nm. The influence of boundary scattering on the relaxation rate of carriers, and band broadening in the amorphous films, may contribute to this effect.

The properties of ultrathin superconducting films differ from those of the bulk material. One of the best known examples is the decrease of the superconducting transition temperature T c with decrease of film thickness, accompanied by the increase of the normal-state film resistance [1][2][3][4][5][6][7][8][9][10][11][12]. Many factors may influence thin-film properties. The enhanced density of defects or large boundary scattering may reduce the mean-free path of the carriers, the electron-electron and electron-phonon interactions may become different from those in bulk materials, and the structural properties may be affected by the strain or interdiffusion, thus modifying the electronic band structure. The establishing of which of these factors plays a major role is important both for the understanding of superconductivity in the ultrathin limit and for the successful application of these films in novel devices. Many studies have been devoted to this subject. However, in most of them the transport evaluation is limited to the resistance or magnetoresistance measurements, while the Hall effect is rarely investigated.

In this work we study ultrathin niobium (Nb) films, and we show that the measurements of the Hall coefficient provide unexpected insights into ultrathin film properties. Nb has the highest T c among elemental superconductors and is widely used in many investigations and applications. The properties of thin Nb films have been addressed by numerous studies, with particular emphases on the dependence of the T c on the film thickness [2,[6][7][8][9][10][11][12]. These studies already show that the thin film fabrication process and the substrate strongly influence the film properties. The Nb films grown on sapphire at elevated temperatures often have superior superconducting properties [8,9,12], but may suffer from interdiffusion between the substrate and the film [9,12]. On the other hand, in Nb films made by magnetron sputtering at room temperature [10,11] interdiffusion should be absent, and such films are an excellent choice for many applications [13]. Here we focus on the films prepared by the second method.

In addition to superconducting properties, we show that there are other subtle changes in Nb thin films as one systematically decreases film thickness, including structure transformation and the evolution of the resistance per square (R sq ). Most surprisingly, there is also a change of sign in the Hall coefficient (R H ) from positive to negative in ultrathin Nb films, suggesting two types of carriers contributing to conduction. This raises the question of its role in the disappearance of superconductivity in the thinnest films.

The Nb films of thickness d, from 1.1 nm to 50 nm, sandwiched between two Si layers of 10 nm to prevent oxidation, have been made by magnetron sputtering at room temperature on glass substrates. We have verified by secondary ion mass spectrometry that the Si buffer layer between the substrate and Nb is necessary to avoid diffusion of oxygen from glass into Nb. We have also determined that the Si overlayer does not affect the T c of Nb, indicating negligible proximity effect between Nb and Si, as has been reported [14]. The Nb film thickness was controlled by deposition time after the deposition rate has been established from low-angle x-ray reflectivity measurements. In addition, a control set of Si/Nb/Si trilayers with different d has been deposited on the Si wafers, for imaging by the high-resolution transmission electron microscopy (HRTEM). From these images, small differences between the nominal and actual d values have been found for ultrathin films. The actual d values are used here.

In addition to the HRTEM imaging, we use x-ray diffraction to evaluate the structural properties of the trilayers. The resistance per square, R sq , as a function of temperature is measured on a lithographically patterned resistance bridge using a standard four-probe method. The Hall coefficient R H is measured on the patterned "Hall bar" structure, using a physical property measurement system (PPMS; Quantum Design). In addition, the measurements of R H using the van der Pauw method have been done for some of the (unpatterned) samples. The results obtained by two methods are found to agree within experimental accuracy.

Figure 1 shows the x-ray diffraction patterns of several trialyers with different d, and of a reference sample with two buffer Si layers but without the niobium layer in between (d = 0). A single broad maximum situated at low angle, characteristic for an amorphous material, is visible in the pattern of the reference sample. On the other hand, the pattern for film with d = 11.3 nm reveals diffraction peaks consistent with the bcc crystal structure and space group I m3m of Nb with the lattice parameter of 3.36 ± 0.01 Å, slightly larger than the value of 3.294 Å for bulk Nb. With d decreasing to 5 nm the intensity of the diffraction peaks decreases, and the peaks become substantially broadened. At d = 3.9 nm only one peak is still faintly visible. This suggests that the film still contain some polycrystalline grains, but a substantial portion of the film is amorphous. Finally, at d = 3.3 nm there is no evidence of any crystalline phase. enclosed by the white square. The Fourier filtered HRTEM image of this grain, shown in the bottom inset, confirms the bcc structure with 111 orientation. With further deposition of Nb, the grains coalescence, forming continuous polycrystalline film with good structural alignment [15]. Similar formation of amorphous Nb layer at the Si/Nb interface, about 1-2 nm thick, has been reported in case of Si/Nb superlattices [16].

The dependence of R sq on d at T = 10 K is shown in Fig. 3(a). The data follow approximately the power law R sq ∼ d -2 , as indicated by a straight line fitted to the data. Such dependence has been predicted for the size effect, resulting either from film-boundary scattering in single-crystalline thin films [17,18] or from a combination of grain-boundary and film-boundary scattering in polycrystalline films [19], and it has been observed before in both types of Nb films [2,7,9]. Thus, it appears that in our polycrystalline films the boundary scattering may be an important scattering mechanism. Interestingly, same dependence seems to extend to amorphous films, in which grain boundaries are absent and the mean-free path is short. It is possible that in this case some other factors, dependent on the film thickness, affect the R sq , such as the enhanced defect density. We note also that the T dependence of R sq changes with d, so that we observe dR sq /dT > 0 for thick films, and dR sq /dT < 0 for ultrathin films. The transition between these two limits, indicated in the figure by dashed area at d 4 nm, correlates approximately with the transition region between the polycrystalline and the amorphous films.

The influence of d on the T c is shown in Fig. 3(b). Here T c is the midpoint of the superconducting transition and the vertical error bars reflect 90% to 10% transition width. The T c decreases to zero for d 1.3 nm when R sq exceeds about 1800 , but in the film with d = 1.2 nm we still see the evidence of superconducting fluctuations. The dependence of T c (R sq ) is similar to the one reported for Nb films prepared by other methods [7,11], as will be discussed later.

Figure 3(c) shows the R H , measured at T = 60 K, versus d. We observe that in the thickest films the R H is close to the value for the bulk, equal to 0.9 × 10 -10 m 3 /C [START_REF] Hurd | The Hall Effect in Metals and Alloys[END_REF]. With decreasing d the R H initially increases above the bulk value. According to theories, the boundary scattering should result in the increase of the Hall coefficient [18]. However, when d decreases below 6 nm, the R H starts to decrease. Eventually, it changes sign into negative for very thin films, when d decreases below 1.6 nm.

The change of sign in the thinnest samples is further enhanced when temperature is decreased. To emphasize this we show in Fig. 4 the evolution the Hall resistance R xy with increasing B, measured at various temperatures for samples with different d. We observe several features. First, the slope of the R xy (B) dependence changes dramatically, from positive in the thicker sample shown in (a) to negative in the thinnest amorphous sample shown in (c). Second, with the decrease of temperature the slope increases in both cases, but while in (a) it becomes more positive, in (c) it becomes more negative; this feature will be discussed in more detail below. Finally, in the thinnest sample the R xy (B) dependence becomes slightly nonlinear at the lowest temperatures. This nonlinearity is even more visible in the case of the intermediate sample, d = 2 nm, shown in (b). The R xy (B) is linear with positive slope at high T , but as T is lowered nonlinearity gradually develops, until at T = 2 K a negative slope appears at small B.

To illustrate better the second feature described above, we extract the R H from the linear R xy (B) dependencies. The results are shown in Fig. 5, where we plot the variation with T of the quantity R H /R 300 H , where R H (T ) ≡ R H (T ) -R 300 H is defined as a relative change of the R H with respect to the room-temperature value, R 300 H . We observe that in all samples the absolute value of R H increases with the decreasing T . However, while R H is positive in the thick samples, it is negative in the ultrathin films. The change of sign from positive into negative R H occurs for d 2.8 nm, indicating that for all thinner samples the Hall coefficient is reduced with decreasing T .

All the features listed above strongly suggest that the conduction in the ultrathin Nb layers involves contribution from two types of carriers, holes and electrons. While holes dominate the transport in thicker, polycrystalline samples, similar as it is in bulk niobium, the negative electron contribution becomes increasingly important as d decreases; it is also more pronounced at low T . Due to small electron effective mass the negative contribution dominates at low magnetic fields, while in the limit of large fields the positive hole contribution prevails. The exception is the thinnest film, with d = 1.3 nm, for which the magnetic field used in our experiment may be too small to observe hole contribution.

The electron contribution to the conduction in the thin-film limit may have several origins. One possibility is that in the thin amorphous films the Nb band structure is modified, so that the character of the dominating carriers changes. However, this effect alone cannot be the cause, because the films become purely amorphous for d = 3.3 nm, while R H is still positive and comparable to the bulk value, so there is no strict correlation between these two effects.

Another possible origin is the formation of silicide niobium layer at the Nb/Si interface [START_REF] Wiech | [END_REF], which may exhibit negative R H [22]. Such a layer may be very thin and amorphous, and therefore undetectable by our experimental methods. The conduction could proceed via two parallel conduction channels with carriers of different signs, with the effect becoming stronger as d decreases.

Still another possibility is the effect described for polycrystalline films of indium, in which a change of sign of R H , from positive in thick films to a negative in the low-field limit in thin films, has been reported [23][24][25]. It is explained by the influence of boundary scattering on the relaxation times of the two types of carriers which exist in In, i.e., the heavy holes in the second Brilloiun zone and light electrons in the third zone [START_REF] Hurd | The Hall Effect in Metals and Alloys[END_REF]. In the bulk the R H is positive, because the relaxation time for holes is larger than for electrons, which are more strongly scattered by impurities. The boundary scattering in thin films affects similarly both types of carriers, tending to equalize the relaxation rates, and electron contribution becomes evident at low magnetic fields.

Trying to assess whether such mechanism may exist in Nb, we note that the Fermi surface in bcc niobium consists of closed-hole surface in the second zone, while in the third zone there are distorted hole ellipsoids and an open multiply connected surface known as the "jungle gym," which may support both hole and electron orbits [26,27]. In the bulk the electron orbits do not affect R H , but when d decreases, the boundary scattering may gradually uncover the electron contribution. While this mechanism is likely to be present in polycrystalline films, in amorphous films, without grain boundaries and with short mean-free path, it is probably less efficient. It is possible that some other, d-dependent scattering mechanism [evident from R sq (d) dependence] may additionally affect the relaxation rates. Finally, the band broadening in the amorphous films may play some role.

It is intriguing to ask to a what extent our finding of electron contribution to transport may modify the understanding of the suppression of superconductivity in ultrathin Nb films. To comment on this we examine in Fig. 6 the dependence of T c /T c0 on the normal-state sheet resistance R sq for the whole set of trilayers with different d. Here T c0 = 9.22 K is the T c of the bulk niobium, and the R sq is measured at the onset of superconducting transition. We also plot by continuous lines the predictions of theoretical model proposed by Finkel'stein [28] for homogeneously disordered films, which takes into account the fact that disorder diminishes screening of the Coulomb interactions between carriers. The model includes parameter γ = ln ( h k B T c0 τ ), where k B is the FIG. 6. (Color online) The ratio T c /T c0 versus normal-state sheet resistance R sq for Si/Nb/Si trilayers with different d (full magenta squares) and for several different sets of Nb films prepared by other methods, Ref. [12] (open black squares), Ref. [11] (open blue triangles), Ref. [7] (open green diamonds), and Ref. [8] (full cyan triangle). The dashed magenta line is a guide to the eye. The predictions of the Finkel'stein model are shown by continuous lines, for γ = 9 (black) and γ = 15 (blue).

Boltzmann constant and τ is the elastic diffusion time. The plots of two curves, for γ equal 15 and 9, indicate that the theoretical predictions with a single value of γ cannot describe correctly the data in the full range of R sq . While reasonable description is obtained for samples with R sq 200 , for thinner samples the decrease of the T c with increasing of R sq slows down. The important observation is that the limiting value of R sq 200 occurs for d 5 nm, below which the electron contribution to the conductance begins to affect the Hall coefficient. Thus, a possibility exists that in thinner samples the scattering of electron carriers contributes strongly to the increase of R sq , but it does not contribute (or contributes weakly) to the pair breaking. Such situation may occur if the electron carriers are not involved in superconductivity.

Finally, the question arises of whether our findings are limited to the particular set of Nb layers studied here. We cannot answer this definitely, because the Hall coefficient has not been evaluated in previous studies of thin Nb films. However, we may compare the T c (R sq ) dependence for our samples and similar data for different sets of Nb films deposited on various substrates using different technological methods [7,8,11,12]. As seen in Fig. 6, most of the data follow the trend similar to that found present study, i.e., initial fast decrease of the T c for small R sq , followed by long "tail" for samples with large R sq . The only exception is the set studied in Ref. [12], which appears to be distinctly different, and it is well described by theory using γ = 9. The smaller value of γ implies larger elastic diffusion time, which is reasonable, because these are the single-crystalline films grown at much higher temperature (660 • C) than all the remaining film sets. On the other hand, the similarity between the results for our trilayers and the other previously studied Nb films may suggest that the electron contribution is a general phenomenon which appears in sufficiently disordered films.

In conclusion, we have studied the properties of ultrathin Nb films in the Si/Nb/Si trilayers. We show that Nb films are amorphous for d < 3.3 nm. In the thinnest amorphous films we observe a change of the sign of the Hall coefficient into negative, possibly due to the influence of boundary scattering and other d-dependent scattering mechanisms, on the relaxation rate of carriers. The band structure modification in the amorphous films may also be important.

FIG. 1 .

 1 FIG. 1. (Color online) X-ray diffraction spectra for the Si/Nb(d)/Si trilayers with different thickness d indicated in the figure. The consecutive curves are shifted vertically for clarity.

Figure 2 shows

 2 FIG. 2. HRTEM images for Si/Nb(d)/Si trilayers, with d equal to 11.3 nm (a), 3.3 nm (b), and 1.2 nm (c). The layers in images are (from the top) amorphous Si, niobium, and amorphous Si. The inset at the bottom shows the enlarged Fourier-filtered part of image (a) indicated by the white square.

FIG. 3 .

 3 FIG. 3. (Color online) R sq (a), T c (b), and R H at T = 60 K (c) vs film thickness d. In (a) the continuous line shows the fitted dependence R sq ∼ d -2 , and the hatched area indicates the transition region between the polycrystalline and amorphous films. In (b) and (c) the dashed lines are guides to the eye.

FIG. 4 .

 4 FIG. 4. (Color online) Hall resistance xy versus B at different temperatures for samples with d = 3.9 nm (a), d = 2 nm (b), and d = 1.3 nm (c).

FIG. 5 .

 5 FIG. 5. (Color online) R H /R 300 H versus T for trilayers with different d. The data for d = 1.3 nm are multiplied by 0.5.

We are grateful to A. Malinowski and V. Bezusyy for their involvement in setting up the PPMS apparatus. This work has been supported by Polish NSC Grant No. 2011/01/B/ST3/00462, by NSF Grant No. DMR 1262253, and by the French-Polish Bilateral Program PICS 2012. The research was partially performed in two laboratories co-financed by the ERDF Projects POIG.02.01-00-14-032/08 and NanoFun POIG.02.02.00-00-025/09.


C P Strehlow

M Konczykowski

J A Murphy

S Teknowijoyo

K Cho

M A Tanatar

T Kobayashi

S Miyasaka

S Tajima

R Prozorov

M Kończykowski

Comparative study of the effects of electron irradiation and natural disorder in single crystals of SrFe 2 (As 1-x P x ) 2 superconductor (x = 0.35)

Keywords: number(s): 74.70.Xa, 74.20.Rp, 74.62.Dh

The pairing mechanism in Fe-based superconductors has been the focal point of many theoretical and experimental works [1,2]. The proximity to magnetism and high superconducting transition temperatures T c in the upper 50 K range prompted the search for a nonphonon mechanism of superconductivity [3]. Based on early experiments, Mazin et al. [3,4] suggested an unconventional superconducting state with interband pairing in which the superconducting gap function changes sign between different sheets of the Fermi surface, but remains full (without line nodes) on each sheet. Experimental verification of this so-called s ± pairing mechanism quickly became a main point of superconducting gap structure studies in iron-based materials.

The verification of the k-space sign-changing gap in iron-based superconductors turned out to be more difficult than it was in the cuprates, in which sign change along a single Fermi surface was proven by directional phase sensitive experiments [5,6]. It was suggested that impurity scattering can be used as a probe of a sign-changing gap [7]. This approach becomes significantly more powerful when, in addition to the suppression of T c , other thermodynamic quantities, such as the London penetration depth, are studied on the same samples [8,9]. At low temperatures, the temperature dependence of the superconducting gap becomes negligible and the total variation is determined by the temperature-induced population of quasiparticles. In the clean limit, λ(T ) is exponentially attenuated in a full gap superconductor (including multiband s ++ , such as MgB 2 [10]), but has a T -linear behavior in superconductors with line nodes [8,11]. In the dirty limit, λ(T ) in conventional s and s ++ superconductors remains exponential, but evolves into a T 2 dependence in both line-nodal and * Corresponding author: prozorov@ameslab.gov nodeless s ± cases [8,11]. However, in the s ± superconductor with accidental nodes, this convergence is nonmonotonic. First, the nodes are lifted and the behavior becomes close to exponential. Upon a further increase of scattering, more additional in-gap states are created and a gapless T 2 dirty limit is eventually reached [8,12]. In nodeless s ± this limit is reached from the exponential behavior [13,14]. A convenient way to describe all possible scenarios of low-temperature evolution of the penetration depth is to use the power-law function λ(T ) = AT n . Exponential behavior is seen as large values of the exponent n. However, if the nodes are symmetry imposed, as in d-wave materials, the exponent grows monotonically from n = 1 to n = 2 and never exceeds this terminal value.

Artificial disorder in superconductors can be introduced in a controlled way by irradiation. Depending on the irradiation type and energy, the induced defects have different characteristics. Early studies of T c [9,15,16] and λ(T ) in Ba(Fe 1-x Co x ) 2 As 2 (BaCo122) [9] and Ba(Fe 1-x Ni x ) 2 As 2 (BaNi122) [9,16] used heavy ion irradiation that produces one-dimensional columnar defects [15]. The analysis of T c and of the exponent n of the power-law function used to fit the temperature-dependent London penetration depth λ = AT n was consistent with the predictions of the s ± model. A later study in the optimally hole-doped (Ba 1-x K x )Fe 2 As 2 (x = 0.4) [17] found a crossover of λ(T ) from exponential to a T 2 at very high irradiation doses, but virtually no change in T c , most probably indicating a dominant intraband pairing interaction in this compound [8]. Irradiation with 2 MeV α particles [18] and 3 MeV protons [19][20][21], both creating clusterlike defects [START_REF] Damask | Point Defects in Metals[END_REF], found a much faster suppression of T c than in the case of columnar defects, but still much slower than originally predicted for a simplified "symmetric" s ± scenario [START_REF] Onari | [END_REF]. More recently, the predictions for the s ± scenario were significantly relaxed in a realistic "asymmetric" model [8]. These calculations were used to fit a significant variation of T c 1098-0121/2014/90(2)/020508 (5) 020508-1 ©2014 American Physical Society induced by 2.5 MeV electron irradiation in BaRu122 [24]. A similar suppression rate was found in other 122 compounds, including BaCo122 and Ba(AsP)122 [25]. In the material of this study, SrFe 2 (As 1-x P x ) 2 , the effect of postgrowth disorder was studied previously by measuring both T c and λ(T ) in samples before and after annealing [26]. It was known that annealing of optimally (x = 0.35) substituted samples leads to an enhancement of T c typically from 25 to 27 K to almost 35 K [27,28]. The analysis of the low-temperature behavior of the penetration depth was unambiguously consistent with the presence of line nodes in the gap [26], very similar to another material with isovalent P substitution, BaFe 2 (As 1-x P x ) 2 [29].

In this Rapid Communication we report a comparative study of the effects of artificial and natural disorder on T c and the quasiparticle excitations in single crystals of SrFe 2 (As 1-x P x ) 2 with an optimal level of isovalent phosphorus substitution, x = 0.35. The natural defects include screw dislocations and related residual after-growth strain and other faults in the crystal structure. These defects are typically extended in one or two dimensions. The electrons, on the other hand, introduce a pointlike disorder. Our main observation is that electron irradiation and natural defects change T c and the London penetration depth λ(T ) in significantly distinct ways. We relate this dissimilarity to a possible difference in the scattering amplitude and characteristic spatial range of the scattering potential.

Single crystals of SrFe 2 (As 1-x P x ) 2 with an optimal isovalent substitution level, x = 0.35, were grown from stoichiometric mixtures of Sr, FeAs, and FeP powders. Details of the growth and postgrowth annealing are described elsewhere [27,28]. The samples studied were from two different batches, A and B. They were cleaved with a razor blade from the inner parts of larger single crystals and had two shiny cleavage surfaces, and thicknesses of about 60-70 μm. The side surfaces of the samples were also cleaved along the a axes in the plane, and the samples were close to 0.6 × 0.6 mm 2 in the surface area. Prior to the penetration depth measurements, the same samples were measured using a magneto-optical technique [30] to check for possible cracks and macroscopic inhomogeneity. We found a typical Bean profile of the magnetic induction distribution with no visible anomalies, reflecting the high sample quality and good magnetic uniformity.

Measurements of the in-plane London penetration depth were performed by using a self-resonating tunnel diode resonator (TDR), which is essentially a radio-frequency (14 MHz) magnetic susceptibility measurement [31,32]. In the experiment, the sample was mounted at the end of a high-purity sapphire rod whose temperature can be precisely controlled. The rod with the sample was inserted into a coil of the tank circuit with an excitation ac magnetic field of H ac ≈ 20 mOe. As the sample was cooled through the superconducting transition, the changes of the sample magnetic susceptibility χ (T ) changed the inductance of the coil and led to a shift of f (T ) in the characteristic resonant frequency of the LC oscillator. The susceptibility χ (T ) is related to f (T ) via f (T ) = -G4πχ(T ), where G is a geometric calibration factor, G = f 0 V s /2V c (1 -N ), N is the demagnetization factor, V s is the sample volume, and V c is the coil volume [32]. The calibration factor G was determined by physically pulling the sample out of the inductor coil at the lowest temperature and measuring the full frequency shift. The magnetic susceptibility can be written in terms of the London penetration depth λ and the characteristic sample dimension R, 4πχ = (λ/R) tanh(R/λ) -1, from which λ may be extracted [31,32]. The 2.5 MeV electron irradiation was performed at the SIRIUS Pelletron linear accelerator operated by the Laboratoire des Solides Irradiés (LSI) at the Ecole Polytechnique in Palaiseau, France. The irradiation dose is represented here in C/cm 2 . To convert to electrons per cm 2 , this number needs to be divided by the electron charge e. The sample of batch A was exposed to a dose of 2.2 C/cm 2 and the sample of batch B was exposed to 1.1 C/cm 2 . After the irradiation, the samples were warmed up to room temperature, which results in up to 30% partial annealing of the defects [24]. Importantly, the comparative measurements of the effect of irradiation were conducted on physically the same samples before and after treatment.

Figure 1 shows the variation of the London penetration depth in samples A and B over the whole temperature range from the base temperature of 0.5 K to above T c . Open and solid symbols show the data for the same samples before and after irradiation. The inset in Fig. 1 shows that the two studied samples exhibit a similar slope of T c versus the irradiation dose. Also, both samples show sharp superconducting transitions before and after the irradiation, suggesting a spatially homogeneous distribution of the induced defects. This is not strange considering that the electrons at an energy of 2.5 MeV have a stoppage distance of more than 500 μm, which is significantly longer than sample thickness. the pristine state (before irradiation) are in agreement with the results of our previous study (see Fig. 3 below [26]). The electron irradiation significantly decreases the total variation of λ(T ) reflected in an increased exponent n. In sample B irradiated with 1.1 C/cm 2 , the exponent n = 1.8. In sample A, exposed to 2.2 C/cm 2 electron irradiation, we find n = 2.26. Plotting the data versus (T /T c ) 2 clearly shows that the sample as grown

T c =29K n=2.2 2.2C [A] T c =31K n=1.8 1.1C [B] T c =32.5K n=1.5 T c =34 K n=1.1 T c =34.7K n=1.2 T c =34.8K n=1.2 Δ (nm) (T/0.3T c ) n FIG.
3. (Color online) λ(T ) vs (T /0.3T c ) n , with n as a fitting parameter selected to linearize the data. The data for pristine annealed samples (top four curves, T c 32.5 K) are well described with a single power law each over the whole range. The data for samples with growth defects show n = 1.5, while the samples subjected to electron irradiation show a rapid increase of n with T c suppression. Note that for all samples λ(0.3T c ) is about the same. with a larger dose reveals an upward curvature, suggesting that n > 2 at low temperatures.

It is generally accepted that the exponent n > 2 cannot be explained by the effect of disorder in superconductors with symmetry-imposed line nodes. There is, however, a caveat, that in a multiband superconductor the range over which a characteristic T 2 dependence is observed can be significantly smaller that in a single gap superconductor. To study this possibility, in Fig. 3 we performed a different analysis of the functional form of λ(T ). Here λ(T ) for each sample was plotted versus (T /0.3T c ) n , where n was chosen to produce the closest to linear dependence. Previously, we showed that the data for both as-grown low T c samples and annealed high T c samples can be well described by using the Hirschfeld-Goldenfeld interpolation formula [11,26], with the effective temperature T * increasing with the amount of disorder. The data for these samples can be actually linearized using an exponent n close to 1, for all samples with T c > 34 K, clearly suggesting line nodes in the superconducting gap. The exponent n increases for samples with lower T c . However, this linearization procedure does not produce nonmonotonic dependencies as expected for pronounced multigap effects. Therefore our data are best described by a true power law, with n > 2 in the most irradiated sample.

To summarize our findings, Fig. 4 shows the exponent n plotted versus T c , controlled either by the growth defects or by the defects induced by electron irradiation. The data for the two types of disorder reveal a striking dissimilarity. While the variation of both n and T c in the samples with growth disorder is consistent with an impurity effect in superconductors with symmetry-imposed line nodes, irradiation brings the exponent n above the range allowed for such superconductors, despite a significantly milder suppression of T c . Furthermore, we do not observe any increase of λ(T ) on cooling [33], which could be suggestive of paramagnetic effects induced by the irradiation, so the difference between the effects is unlikely to be nonmagnetic versus magnetic scattering. The irradiation with electrons at energies between 1 and 10 MeV is known to create primarily Frenkel pairs of vacancies and interstitial ions [START_REF] Damask | Point Defects in Metals[END_REF]. Interstitials tend to migrate and disappear at surfaces and other sinks in the crystal structure, leaving vacancies as pointlike disorder [START_REF] Damask | Point Defects in Metals[END_REF]. On the other hand, disorder in as-grown samples mostly appears as dislocations, which have a long-range elastic strain field, and therefore these two types of defects have significantly different characteristics. It is therefore conceivable that scattering on these defects is characterized by a notably different momentum transfer. In other words, the strength and characteristic range and dimensionality of the scattering potentials corresponding to these defects are quite different.

In a generic s ± multiband scenario of superconductivity, applicable to iron pnictides, the interplay between intraband and interband pairing channels leads to a diverse phase diagram of the possible gap structures. A sign-changing gap is locked by the interband pairing, while the ratio between the interband to intraband pairing potentials determines gap amplitudes and anisotropies [1,2]. These considerations helped to explain the strong doping evolution of the superconducting gap observed experimentally in pnictides that were fully gapped at optimal doping [34,35]. Our present and previous studies show that the superconducting gap of SrFe 2 (As 1-x P x ) 2 has line nodes [26,36,37], similar to BaFe 2 (As 1-x P x ) 2 [29]. The nature of these nodes can be probed by studying the effect of a controlled disorder. Scattering in interband and intraband channels has a very different effect on superconductivity [1,2,8,38]. Scattering with a small moment transfer would transfer the electrons only within the same Fermi surface sheets (intraband scattering), while scattering between different sheets of the Fermi surface necessitates a large momentum transfer. Since point defects have a characteristic size of a unit cell or less, the characteristic wave vector is of the order of Q ∼ 2π/a, a sizable fraction of the Brillouin zone. For this type of disorder we would naturally expect a strong contribution to the interband scattering channel. By the same logic, extended defects would be characterized by a much smaller Q, and contribute mostly to the intraband scattering. This and previous studies show that SrFe 2 (As 1-x P x ) 2 has a superconducting gap with line nodes [26,36,37]. Small Q scattering within each sheet of the Fermi surface in this situation will be quite similar to the usual effect of disorder in nodal superconductors, as we observed here for the growth defects. Large Q scattering on point defects, however, will lift the nodes and drive the gap structure towards the full gap, if the nodes are accidental [8]. Our observation of n > 2 supports this scenario. An alternative scenario by Korshunov et al. suggests a transition from s ± to a conventional s ++ state as a function of disorder [38].

In conclusion, we found dramatically different effects of artificial and natural disorder of as-grown samples on the superconducting transition temperature T c and on quasiparticle excitations measured by a variation of the London penetration depth in isoelectron-substituted SrFe 2 (As 1-x P x ) 2 . The response to the postgrowth disorder is similar to the usual effect of disorder in nodal superconductors. The response to electron irradiation is notably different, suggesting the evolution of the superconducting gap structure from nodal to nodeless, which is expected for accidental nodes. We relate the difference between the two types of disorder to the difference in the characteristic scattering wave vector Q, with small Q for postgrowth disorder and large Q for a point-type disorder.

Note added. Recently a related work appeared where the authors studied the effects of electron irradiation on both T c and penetration depth in another isovalent-substituted compound, BaFe 2 (As 1-x P x ) 2 [12]. Their results support the accidental node scenario in this class of nodal iron pnictides.

FIG. 1 .

 1 FIG. 1. (Color online) Full temperature range variation of the London penetration depth, λ(T ), in two single crystals of SrFe 2 (As 1-x P x ) 2 , x = 0.35, A (black triangles) and B (blue circles) before (open symbols) and after (solid symbols) irradiation with doses of 2.2 and 1.1 C/cm 2 , respectively. The superconducting transition temperature T c was defined as the peak in the temperature-dependent derivative of the signal dλ/dT , roughly corresponding to the midpoint of the transition. The inset shows the change of T c as a function of the irradiation dose.

Figure 2 FIG. 2 .

 22 FIG. 2. (Color online) Low-temperature variation of the London penetration depth in single crystals of SrFe 2 (As 1-x P x ) 2 , x = 0.35, plotted vs reduced temperature T /T c (left panel) and vs (T /T c ) 2 (right panel). The data for sample A in the pristine annealed state are shown by open black triangles, and after the irradiation with 2.2 C/cm 2 by solid black triangles. The data for sample B before (blue open circles) and after 1.1 C/cm 2 irradiation (blue solid circles) are offset by 20 nm to avoid overlapping. The line in the right panel is a guide to the eye to show a slight upward curvature suggesting n > 2 after electron irradiation with 2.2 C/cm 2 .

FIG. 4 .

 4 FIG. 4. (Color online)The exponent n of the power-law fit of λ(T ) from Fig.3, vs T c . Note the significantly smaller exponents for as-grown and annealed compared to the samples with irradiation defects. The dashed lines are guides for the eye.

The work in Ames was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Ames Laboratory is operated for the U.S. DOE by Iowa State University under Contract No. DE-AC02-07CH11358. We thank the SIRIUS team, B. Boizot, V. Metayer, and J. Losco, for running electron irradiation at Ecole Polytechnique (supported by EMIR network, proposal 11-11-0121.) Work at Osaka was partly supported by a Grant-in-Aid IRONSEA from the Japan Science and Technology Agency (JST).


Damien Breck
email: damien.breck@univ-lorraine.fr

Jean-Philippe Georges

Thierry Divoux

Delay and backlog bounds for an aggregation system in wireless networks

Keywords: performance evaluation, aggregation system, network calculus, IEEE 802.11n

The IEEE 802.11 standard is one of the most used wireless local area network technology. Even if it is not a deterministic protocol, it is possible to bound its performances, and thus to provide some quality of service. Its evolution 802.11n integrates an aggregation system which improves its efficiency under specific conditions. Particularly, such a system could be interesting in a wireless sensor and actuator network, where data are gathered before being forwarded to a sink. We propose to model this aggregation system since it significantly impacts the frames delay. The network calculus theory is used to compute worst-case performances. Rather than considering the aggregates in a macroscopic way and providing global performances indicators, this paper aims at defining the service offered to each incoming flow by evaluating backlogs and end-to-end delays.

INTRODUCTION

Different ways exist to carry goods: individually (which implies as many travel as goods) or grouped. This is true in logistic as in communication networks for data transmission. Let us consider a truck with a half full trailer. An opposition appears between carrier and customer requirements. The first one wants to wait in order to fill the trailer. The second one prefers the truck to leave immediately in order to be delivered as soon as possible. Information is needed to make a good choice. What delay can the customer tolerate? When do the next goods arrive? What kind of goods (size, . . . )? There is a compromise to find between trailer filling and customer requirements. Two thresholds can be defined: a maximum leaving date and a minimum filling level. We name aggregation system, a system which stores things before sending them according to these two thresholds. Our purpose is to evaluate its performances from a specific customer point of view. This kind of system exists in some communication networks protocols. It is used in the Synchronous Digital Hierarchy (SDH) and in the Asynchronous Transfer Mode (ATM) for example. This system could also be used in a wireless sensor network, where measurements done by several nodes are gathered by a cluster head before being together forwarded to a sink. Aggregating offers two advantages: improve the network lifetime by reducing the number of transmissions, and minimize the consumed bandwidth. IEEE-802.11n (2012) standard includes an aggregation system which is our case study. The 802.11 Working Group initial purpose was to reach a 100 Mbps data throughput. Many improvements have been made on the physical layer to achieve this goal. Nevertheless, frames aggregation remains one of the major enhancements made on the Medium Access Control (MAC) layer that enables to reach this bit rate. This system is located at the input of the MAC layer where there is the most to gain, because of collisions or variable bit rate which is a function of the distance to the access point. Therefore, when the medium is won, it is interesting to keep it as long as possible in order to transmit the maximum amount of data. Aggregation is a way to reduce the number of transmissions and thus the number of competitions for the medium access. Some authors have studied the 802.11n aggregation performances by using various theories:

• Queuing theory (throughput, channel utilization, aggregate latency) [START_REF] Lin | WSN01-1: frame aggregation and optimal frame size adaptation for[END_REF]; [START_REF] Kuppa | Modeling and analysis of frame aggregation in unsaturated WLANs with finite buffer stations[END_REF]. • Analytic study (throughput) [START_REF] Ginzburg | Performance analysis of a-MPDU and a-MSDU aggregation[END_REF]. • Simulations (channel utilization, aggregate latency) [START_REF] Skordoulis | IEEE 802.11n MAC frame aggregation mechanisms for next-generation highthroughput WLANs[END_REF]; [START_REF] Wang | IEEE 802.11n MAC enhancement and performance evaluation[END_REF].

These works suffer from several drawbacks. First, chosen performances indicators are macroscopic. They mainly deal with the delay for an aggregate to cross a given topology. It is then impossible to quantify the impact of the aggregation system on one of the aggregated flows. Aggregating isn't always appropriate, especially when considering critical applications. Indeed, some frames are voluntarily delayed, in order to wait for others with which they will be aggregated. Furthermore, many authors provide average values for performance indicators (delay, throughput), when worst case values may be required. Obtaining a priori bounds is necessary for critical applications. Some authors computed them like Ginzburg and Kesselman (2007). However, their results need a detailed knowledge of incoming traffics which is often impossible to get in practice. Indeed, even in a simple case study, the number of scenarios increases quickly without any hypothesis on the input flows (packets size, inter-arrival, jitter, . . . ) and on the system thresholds.

To overcome this multitude of cases, we propose to use the network calculus theory [START_REF] Cruz | A calculus for network delay. 1 network elements in isolation[END_REF]; [START_REF] Chang | Performance guarantees in communication networks[END_REF]; Le [START_REF] Boudec | Network calculus: a theory of deterministic queuing systems for the Internet[END_REF]. The network calculus enables to compute delay and backlog bounds by using a worst case analysis. Our study can help to decide if using an aggregation system is compliant with the applications constraints.

Section 2 defines precisely the aggregation system in 802.11n. In section 3 we remind the fundamentals of the network calculus, we propose a model of the service provided to a particular flow, and we discuss its relevance. Section 4 gives delay and backlog bounds. We also compare these bounds with the values obtained by using the method promoted by [START_REF] Ginzburg | Performance analysis of a-MPDU and a-MSDU aggregation[END_REF].

AGGREGATION SYSTEM

The aggregation system collects frames issued from different flows and creates a new "super flow". Cumulated data are sent in the network according to two criteria. On one hand if cumulated data reaches a size threshold. On the other hand if the time spent in the system by the first collected frame reaches a temporal threshold. The Fig. 1 represents the aggregation system with the size threshold s and h the temporal one. It aggregates several Mac Service Data Units (MSDU) to form an Aggregated-MSDU (A-MSDU) encapsulated in a 802.11n frame (Fig. 2). One of the purpose of the aggregation was to provide better compatibility between IEEE 802.3 (Ethernet) and IEEE 802.11n (Wi-Fi n) standards. There are Ethernet pseudoheaders in the aggregation headers which have been preserved in order to reduce switching latencies.

It is important to note that our study is restricted to an aggregation system which only considers the maximal size threshold S.

MODELLING OF AGGREGATION SYSTEMS

Introduction to network calculus

The network calculus fundamentals can be found in [START_REF] Cruz | A calculus for network delay. 1 network elements in isolation[END_REF], [START_REF] Chang | Performance guarantees in communication networks[END_REF] and Le [START_REF] Boudec | Network calculus: a theory of deterministic queuing systems for the Internet[END_REF].

The network modelling is achieved by non-negative and non-decreasing functions which characterize an amount of data at a time t.

The first use of such functions is for the input R(t) and the output R * (t), which cumulatively count the number of bits that respectively enter and go out of a system. As R(t) is unknown, the network calculus introduces the concept of arrival curve as follows. Given a flow with an input function R(t), a wide-sense increasing function α is an arrival curve for R(t) if and only if:

∀t, s ≥ 0, t ≥ s : R(t) -R(t -s) ≤ α(s)
In the same way, the service offered by a system to a flow is modelled by a (non-negative and non-decreasing) minimum service curve β if β (0) = 0 and:

∀t ≥ 0, R * (t) ≥ inf s≤t {R(s) + β(t -s)}
In addition, β is called a strict service curve for the system if during any backlogged period ]s, t], at least β (t -s) data is served. These arrival and service curves allow to compute the following bounds. Theorem 1. [START_REF] Chang | Performance guarantees in communication networks[END_REF]; Le [START_REF] Boudec | Network calculus: a theory of deterministic queuing systems for the Internet[END_REF]). Assume a flow, constrained by arrival curve α, traverses a system that offers a service curve of β (α ≤ β) and that stores input data in a FIFO-ordered queue. The backlog b and the delay d are defined as:

b(t) ≤ sup {s ≥ 0 | α(s) -β(s)} (1) d(t) ≤ sup {s ≥ 0 | inf {τ ≥ 0 | α(s) ≤ β (s + τ )}} (2)
Lemma 2. (Residual service: Le [START_REF] Boudec | Network calculus: a theory of deterministic queuing systems for the Internet[END_REF]; [START_REF] Schmitt | Delay bounds under arbitrary multiplexing: When network calculus leaves you in the lurch[END_REF]). Let us consider a server node offering a strict service curve β to two input flows, with respective arrival curves α 1 and α 2 . A service curve for the flow 1 is:

β 1 = (β -α 2 ) + = max (0, β -α 2 ) (3)
In this section, our purpose is to compute a service curve for a 802.11n aggregation system. This curve represents the service provided to a particular input flow. Our study highlights limitations of the residual service (lemma 2).

Aggregate departure

Let us consider the aggregation system of Fig. 1. Assume first that the incoming traffic is merged in a "super flow" formed by the two flows such as

R (t) = R 1 (t) + R 2 (t).
Assumption 3. The aggregation system sends an aggregate with a throughput C when its backlog is greater than or equal to a maximal size threshold S. The aggregate's size varies between S-l max and S, with l max the maximum packet size at the system input.

As shown in Fig. 3 the backlog varies with frames arrival and decreases only when reaching the critical size S, i.e R(t) -R * (t) ≥ S. Consequently, this system is non work conserving and its service varies with data arrival (unlike the systems with a constant bit rate service

β (t) = Ct). S b i (t) bits t R i (t) R i (t) α i (t) α i (t) t τ i σ i ρ i ρ i

Fig. 3. Backlog and incoming traffic envelopment

As sending an aggregate depends on the amount of input traffic, we propose to low-bound the input flows with a minimal arrival curve like in Real-Time Calculus [START_REF] Thiele | Realtime calculus for scheduling hard real-time systems[END_REF]). In this study, we assume to consider only affine arrival curves (Fig. 3). Thus ∀0 ≤ s ≤ t:

α i (t -s) ≤ R i (t) -R i (s) ≤ α i (t -s) ρ i (t -s -τ i ) + ≤ R i (t) -R i (s) ≤ σ i + ρ i (t -s) (4)
The arrival curves parameters α i and ρ i may be deduced from the output of some traffic regulators such as a leaky bucket. In the same way, α i and ρ i are deduced from the traffic generation characteristics. The model of the service offered to the "super flow" is given below. Proposition 4. Given an aggregation system with two input flows. Each of them is constrained by two arrival curves α k (t) and α k (t) with k = 1, 2. The service curve β (t) becomes:

β (t) = R (t -∆) +
(5) with a rate R = ρ = ρ i + ρ j with i = arg min i=1,2 τ i and j = arg max j=1,2 τ j , and a latency ∆ =

τ i + S/ρ i if S/ρ i ≤ τ j -τ i or else ∆ = τ i + S + ρ j (τ j -τ i ) / ρ i + ρ j .
Two variables must be computed in this model, the service rate R and the latency ∆. Our reasons for choosing an rate-latency service curve and the computation of R can be found in appendix A. The computation of the maximal time between the departure of two aggregates, ∆ can be found in appendix B.

The minimal service rate R is expressed as the sum of the minimal arrival rates. This is one of the particularity of the aggregation systems, because the offered service depends on the incoming traffic. Thus, the other flows which are usually considered as rivals may be now recognized as friends, since they enable the aggregate to be sent earlier.

It is also interesting to note that the service rate and the temporal performances are bad for a flow if it is not "helped" by an other incoming flow. This is an example of the opposition between customer and provider interests.

Service dedicated to a flow

After having formulated the service offered to R 1 (t)+R 2 (t) (proposition 4), we will now identify a service curve noted β 1 (t) for R 1 (t) only. As our service curve from proposition 4 is strict by definition of ∆, applying the residual service lemma (equation ( 3)) leads to: It means that the maximal delay before the flow 1 begins to be served is:

β 1 (t) = (β(t) -α 2 (t)) + = (ρ 1 + ρ 2 )(t -∆) -(σ 2 + ρ 2 t) ∀t > σ 2 + ∆(ρ 1 + ρ 2 ) ρ 1 + ρ 2 -ρ 2 (6) α 2 (t)β(t)β 1 (t) ∆ 1 t ∆ σ 2 ρ 2 ρ ρ 1 + ρ 2 -ρ 2
∆ 1 = σ 2 + ∆(ρ 1 + ρ 2 ) ρ 1 + ρ 2 -ρ 2
We can also identify the service rate for the flow 1:

β 1 (t) = (ρ 1 + ρ 2 )(t -∆) -(σ 2 + ρ 2 t) = ρ 1 + ρ 2 -ρ 2 (t -∆) -σ 2 When t > ∆ 1 , the service offered to the flow 1 is ρ 1 + ρ 2 -ρ 2 .
The Fig. 4 shows the graphical resolution of the equation ( 6). Below is the proposition arising from these developments. Proposition 5. The aggregation system defined in the proposition 4 offers to the flow R 1 (t) a minimal service curve:

β 1 (t) = (ρ 1 + ρ 2 -ρ 2 ) (t -∆ 1 ) + (7)
Since the forwarding rate might be inferior to the arrival rate (i.e ρ 1 +ρ 2 -ρ 2 ≤ ρ 1 ) a stability condition is necessary:

ρ i = ρ i .
So, the service rate for the flow 1 becomes ρ 1 . It means that the worst service rate is obtained when an flow is alone. Besides, the delay ∆ 1 doesn't ensure that data from flow 1 was arrived. Furthermore, ∆ 1 is increased by a flow 2 burst which doesn't appear in the expression of β(t).

To avoid the pessimism introduced by these observations we propose to replace ∆ 1 in equation ( 7).

We search ∆ ′ 1 ≥ ∆, the maximal inter-leaving time between two aggregates that contain data from flow 1. Following the FIFO delivery, if the flow 1 arrives before the flow 2 (τ 1 < τ 2 ), then ∆ ′ 1 = ∆. Otherwise, it means that we can have ∆ ≤ τ 1 and it is necessary to wait for the next aggregate (sent after τ 1 ). This new inter-leaving time ∆ ′ 1 consists in two parts: the delay time τ 1 and the departure date of the first aggregate including at least a packet of the flow 1. This last part depends on the threshold S, on the maximum number of aggregates only based on the flow 2 ⌊ τ1-τ2 S/ρ2 ⌋, and on the quantity of data of the flow 2 continuously arriving ρ 2 (τ 1 -τ 2 ). It gives:

∆ ′ 1 = τ 1 + S -ρ 2 (τ 1 -τ 2 ) + ⌊ τ1-τ2 S/ρ2 ⌋S ρ 2 + ρ 1
We have all the elements for a new proposition. Proposition 6. The aggregation system defined in the proposition 4 offers to the flow R 1 (t) a minimal service curve:

β ′ 1 (t) = ρ 1 (t -∆ ′ 1 ) + with ∆ ′ 1 = max (∆, δ ′ 1 )

Discussion

Let us consider the case study of the Fig. 5. Two traffics Fig. 6. β ′ 1 (t) and β 1 (t) compared to a simulation trace are characterized with a 1 ms period, 1000 bytes input packets, and a size threshold of 3839 bytes (defined in 802.11n). We will compare the efficiency of propositions 5 and 6 with values provided by a very simple simulator we have developed which reproduces the aggregation system behaviour. This tool generates a trace of the aggregates (departure time and size).

Arrival curves are calculated according to packet size and minimal inter-arrival time as mentioned in [START_REF] Chakraborty | A general framework for analysing system properties in platform-based embedded system designs[END_REF]. The minimal arrival rate is ρ i = ρ i = 8 M bps and the maximal burst size is σ i = 1000 bytes.

The Fig. 6 pictures the service dedicated to a flow following the propositions 5 and 6 compared to the results given by the trace. It shows that for this scenario, the proposition 6 is the closest to the simulation. Residual service leads here to a modeling pessimism. System performances for an user-flow are better when other flows give traffic. The residual service only considers other flows as rivals. That leads to a pessimistic approximation of the service for one flow. This is why we propose an upgrade of the residual service which reduces the pessimism for such systems.

DELAY AND BACKLOG BOUNDS

QoS bounds computation

Let us now compute the backlog and delay bounds by using equations ( 1) and ( 2). The Fig. 7 shows that they correspond respectively to the vertical and the horizontal maximal distance between the arrival and service curves. The vertical distance is maximal at t = T as shown in Fig. 7. Hence T = ∆, and the backlog bound is then,

b = α(∆) -β(∆) = σ 1 + σ 2 + ∆(ρ 1 + ρ 2 ) (8)
To return to the introduction analogy, the backlog bound computation sizes the trailer filling. In 802.11n, this value sizes the transmission buffers. We want to compute the delay bound crossing the aggregation system. It is computed for each sub-frame of the aggregate using the service curve dedicated to a flow. As shown in Fig. 7, the delay bound is:

α(t)β(t) b t T σ ρ ρ α 1 (t)β 1 (t) d1 t T 1 σ 1 ρ 1 ρ 1
d 1 (t) ≤ sup {s ≥ 0 | inf {τ ≥ 0 | ∀t ≥ 0, β 1 (s + τ ) ≥ σ 1 }} d 1 = σ 1 ρ 1 + ∆ 1

Discussion

We consider the same scenario as in section 3.4 to illustrate the bounds pessimism. That is why we need a reference value which is computed by summing the elementary latencies of the system like done by [START_REF] Ginzburg | Performance analysis of a-MPDU and a-MSDU aggregation[END_REF]. As shown in Fig. 8, we sum the time to receive all packets required to create an aggregate (P + T R ), the time to send the aggregate (T E ) and the MAC layer latency (T M AC ) set to 2 µs (IEEE-802.11n (2012)).

There are 14 bytes of MSDU headers and 18 bytes of MAC headers. Our reference value is then,

D = 2P + T R + T E + T M AC = 2.52ms
Let us now compute d 1 . Here s/ρ i > τ j -τ i , then,

∆ = τ 2 + (s + ρ 1 (τ 1 -τ 2 )) / (ρ 1 + ρ 2 ) .
We obtained d 1 = 3.91 ms, with τ 1 = τ 2 = 1 ms, ρ 1 = ρ 2 = 1000 bytes/ms and σ 2 = 1000 bytes. The delay d 1 well bounds the reference value with an overestimation of about 35 % with D. This might indicate that our hypothesis are too pessimistic or that our arrival model is not precise enough. However, the residual service gives d 1 = 7.82 ms which is more than 3 times superior to D. In fact, most of our pessimism is due to the maximal arrival latency τ 1 used in our model while in the scenario, the flow 1 arrived immediately (τ 1 = 0).

Moreover, this result applies only on this particular scenario. Let us now consider the input frames defined in the table bellow (compatible with the minimal arrival curves previously defined).

time (ms) 0 1 1.5 2.1 2.5 2.85 flow 1 2 2 1 1 2 size (bytes) 1000 500 1300 250 650 139

β s (t) β s-lmax (t) β(t) t ∆ ∆ + ∆ s-lmax s -l max s 2s C Fig. A.1. Service curves for different aggregate size
The reference value becomes D = 3.16 ms which reduces the overestimation to 20 %. It is mostly due to the traffic modeling quality and to the method for the delay computation. Indeed, the delay bound is the horizontal distance between the service curve based on the minimal and maximal arrival curves. There is 0.75 ms between the computed bound and the trace which is not so much compared to the 2 ms gap between the maximal and minimal arrival curves (the uncertainty on the frame arrival time). Furthermore, the error on the burst for the considered flow can impact the pessimism. The lower the gap between maximal and minimal arrival curves is, the greater the precision of the delay bound is.

The backlog bound computation gives 7839 bytes. That is more than twice the simulated backlog of 3000 bytes. For flow 1, the backlog bound is 3919 bytes compared to the 2000 bytes given by the simulation. It is not very precise, but when using the residual service curve we obtain 7839 bytes. All these values come to support the conclusions made on the delay bound.

CONCLUSION AND OUTLOOK

We have proposed a first model of the service dedicated to one flow in a 802.11n aggregation system. We have identified two service curves for this system: a service curve for all input flows enabling to compute a backlog bound, and a service curve for a particular flow enabling to compute a delay bound. The residual service defined in the network calculus was not compliant with our expectations. We have then refined it to significantly reduce the pessimism introduced when modeling the aggregation system. We have compared our delay bound with two reference values issued from our scenario. Using simulations tools is interesting to improve our service curves.That is why we have developed a very simple one in the expectation of the implementation of the 802.11n aggregation mechanism now available in the Riverbed Opnet Modeler version 17.5 simulator. Future works will take into account the temporal threshold. We will evaluate the bounds pessimism evolution when the system has n input flows. This generalization could introduce a difficulty caused by the chosen scheduling for the n -1 rival flows.

Appendix This proposition assumes that the following stability condition is respected:

ρ i + ρ j ρ i α(t) Case 2 t τ i τ j ρ i + ρ j ρ i α(t) Case 3 t τ j ρ i + ρ j
C ≫ i ρ i . Appendix B. MAXIMAL DEPARTURE TIME BETWEEN TWO AGGREGATES, ∆
Remember that the backlog is defined as b(t) = R(t) -R * (t). Assume that during the interval [t, t + ∆] there is no offered service, the value ∆ corresponds to the maximum value of δ such that we have (with k ∈ N): , different formula could be considered for α k (t) for a given t. We have hence:

∆ ≤ sup t≥0 inf δ≥0 j k=i (R k (t + δ) -R * k (t + δ)) ≥ S ∀0 ≤ δ ≤ ∆,
Case 1: ∀0 ≤ t < τ i , there may be no traffic

∆ ≤ sup 0≤t<τi inf δ≥0 j k=i ρ k (t + δ -τ k ) + ≥ S
Case 1.1: ∀S/ρ i ≤ τ j -τ i , flow i has enough traffic to reach the threshold Here, α i τ i + S/ρ i = S whereas α j τ i + S/ρ i = 0.

Then, the threshold is reached for ρ

j (t + δ -τ j ) + = 0. ∆ ≤ sup 0≤t<τi inf δ≥0 ρ i (t + δ -τ i ) ≥ S ≤ S ρ i + τ i
Case 1.2: ∀S/ρ i > τ j -τ i , flow i hasn't enough traffic to reach the threshold It means that α i (τ j ) < S and thus δ exists only if For the first supremum in equation B.2, we have τ jt > S/ρ i . Applying α i to this inequality gives α i (τ j )α i (t) > S. As α j (τ j ) = 0, δ exists ∀t < τ j -S/ρ j and ρ j (t + δ -τ j ) + = 0.

ρ j (t + δ -τ j ) + = ρ j (t + δ -τ j ). ∆ ≤ sup 0≤t<τi inf δ≥0 j k=i ρ k (t + δ -τ k ) ≥ S ≤ S ρ i + ρ j + ρ i τ i + ρ j τ j ρ i + ρ j = τ i + S + ρ j (τ j -τ i ) ρ i + ρ j
Contrariwise if τ j -t ≤ S/ρ i , then α i (τ j ) -α i (t) ≤ S and δ exists if ρ j (t + δ -τ j ) + = ρ j (t + δ -τ j ). If flow i has enough traffic to reach the threshold, (S/ρ i ≤ τ j -τ i ):

∆ ≤ sup

∆ ≤ τ i + S/ρ i if 0 ≤ t < τ i ∆ ≤ S/ρ i if τ i ≤ t < τ j -S/ρ i ∆ ≤ S + ρ j S/ρ i / ρ i + ρ j if τ j -S/ρ i ≤ t < τ j ∆ ≤ S/ ρ i + ρ j else (t ≥ τ j ) else if both flows are needed to reach the threshold (S/ρ i > τ j -τ i ):

∆ ≤ τ i + S + ρ j (τ j -τ i ) / ρ i + ρ j if 0 ≤ t < τ i ∆ ≤ S + ρ j (τ j -τ i ) / ρ i + ρ j if τ i ≤ t < τ j ∆ ≤ S/ ρ i + ρ j else (t ≥ τ j )

The maximal value of ∆ is obtained when 0 ≤ t < τ i : ∆ = τ i + S/ρ i if S/ρ i ≤ τ j -τ i = τ i + S + ρ j (τ j -τ i ) ρ i + ρ j else (S/ρ i > τ j -τ i )

  Fig. 1. Functional model of aggregation system

Fig. 4 .

 4 Fig. 4. Service curve dedicated to flow 1

  Fig. 5. The topology of our case study

Fig

  Fig. 7. Backlog and delay bounds computation

Fig

  Fig. B.1. Considered cases packet is then excluded from the aggregate. The two curves in Fig. A.1 represent alternatively worst case and best case. That is why we propose to use the rate-latency curve β (t) = R(t -∆), with the expression of ∆ explained in appendix B and R formulated below: R = S -l max S-lmax ρ = S S ρ = ρ = ρ 1 + ρ 2

  not enough traffic has been received to send an aggregate, we have thus R * k (t + δ) = R k (t) and from equation (t + δ) -α k (t) ≥ S (B.1) As shown by Fig. B.1

Case 2 :

 2 ∀τ i ≤ t < τ j , there may be traffic from flow i only∆ ≤ sup τi≤t<τj inf δ≥0 ρ i δ + ρ j (t + δ -τ j ) + ≥ SCase 2.1: ∀S/ρ i ≤ τ j -τ i , flow i has enough traffic to reach the + ρ j (t + δ -τ j ) + ρ j (t + δ -τ j )

  + ρ j (t + δ -τ j ) ∀S/ρ i > τ j -τ i , flow i hasn't enough traffic to reach the threshold Here δ exists if ρ j (t + δ -τ j ) + = ρ j (t + δ -τ j ). Hence:∆ ≤ sup τi≤t<τj inf δ≥0 ρ i δ + ρ j (t + δ -τ j ) ≥ S ≤ S ρ iCase 3: there is traffic from both flows (t≥ τ j ) t + δ) -α k (t) ≥ S ≤ inf δ≥0 δ ρ i + ρ j ≥ S ≤ S ρ i + ρ jResults can be summarized as follow:

  A. MINIMAL SERVICE RATE R Consider that aggregates are transmitted at a constant throughput C and at each ∆ time units (pictured in Fig. A.1). Without fragmentation (like in 802.11n), aggregate's size can vary in ]s -l max , s]. Indeed, the worst case occurs if last packet has a maximal length l max and if this packet leads backlog to exceed S of one bit. The

	Case 1	α(t)
	τ i τ j	t


Asymptotic description of stochastic neural networks. I -existence of a Large Deviation Principle

Olivier Faugeras a , James Maclaurin a a Inria Sophia-Antipolis Méditerranée

Abstract

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. Unlike previous works which made the biologically unplausible assumption that the weights were i.i.d. random variables, we assume that they are correlated. We introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of neurons. The result is that the image law through the empirical measure satisfies a large deviation principle with a good rate function. We provide an analytical expression of this rate function.

Résumé

Description asymptotique de réseaux de neurones stochastiques. Iexistence d'un principe de grandes déviation Étant donné un réseau complètement connecté de neurones dans lequel les poids synaptiques sont des variables aléatoires gaussiennes corrélées, nous caractérisons la loi asymptotique de ce réseau lorsque le nombre de neurones tend vers l'infini. Tous les travaux précédents faisaient l'hypothèse, irréaliste du point de vue de la biologie, de poids indépendants. Nous introduisons la mesure empirique sur l'espace des trajectoires solutions des équations du réseau de neurones de taille finie et la loi moyennée (par rapport aux poids synaptiques) des trajectoires de ces solutions. Le résultat est que la loi image de cette loi par la mesure empirique satisfait un principe de grandes déviations avec une bonne fonction de taux dont nous donnons une expression analytique.

Email address: firstname.name@inria.fr (Olivier Faugeras).

Preprint submitted to the Académie des sciences

Version française abrégée

Nous considérons le problème de décrire la dynamique asymptotique d'un ensemble de 2n + 1 neurones lorsque ce nombre tend vers l'infini. Ce problème est motivé par un désir de parcimonie, par celui de rendre compte de l'apparition de phénomènes émergents, ainsi que par celui de comprendre les effets de taille finie. Nous considérons donc un réseau de 2n+1 neurones interconnectés dont la dynamique commune (en temps discret) obéit aux équations stochastiques [START_REF] Cessac | From neuron to neural networks dynamics[END_REF]. Dans celles-ci apparaissent les poids synaptiques ou coefficients de couplage notés J n ij qui sont des variables aléatoires gaussiennes corrélées. Pour répondre à la question posée nous considérons la loi notée Q Vn de la solution à (2) moyennée par rapport aux poids synaptiques ou plus précisément l'image Π n de cette loi par la mesure empirique [START_REF] Cessac | Increase in complexity in random neural networks[END_REF]. Nous montrons que cette loi satisfait un principe de grande déviations avec une bonne fonction de taux H dont nous donnons une expression analytique dans la définition 3.1 et les équations (9) et (12). Ce travail généralise au cas des poids synaptiques corrélés celui d'auteurs comme Sompolinsky [START_REF] Sompolinsky | Chaos in Random Neural Networks[END_REF] et Moynot et Samuelides [START_REF] Moynot | Large deviations and mean-field theory for asymmetric random recurrent neural networks[END_REF] qui ont considéré le cas de poids synaptiques indépendants. Dans ce cas, plus simple d'un point de vue mathématique, mais beaucoup moins plausible d'un point de vue biologique, on observe le phénomène de propagation du chaos. Nous montrons dans un second article [START_REF] Faugeras | Asymptotic description of stochastic neural networks. ii -characterization of the limit law[END_REF] que la bonne fonction de taux a un minimum unique que nous caractérisons complètement. La propagation du chaos n'a pas lieu mais la représentation est parcimonieuse dans un sens défini dans [START_REF] Faugeras | Asymptotic description of stochastic neural networks. ii -characterization of the limit law[END_REF].

Introduction

Neural networks

Our goal is to study the asymptotic behaviour and large deviations of a network of interacting neurons when the number of neurons becomes infinite.

Sompolinsky also succesfully explored this particular topic [START_REF] Sompolinsky | Chaos in Random Neural Networks[END_REF] for fully connected networks of neurons. In his study of the continuous time dynamics of networks of rate neurons, Sompolinsky and his colleagues assumed that the synaptic weights in neuroscience, were random variables i.i.d. with zero mean Gaussian laws. The main result obtained by Sompolinsky and his colleagues (using the local chaos hypothesis) under the previous hypotheses is that the averaged law of the neurons dynamics is chaotic in the sense that the averaged law of a finite number of neurons converges to a product measure as the system gets very large.

The next efforts in the direction of understanding the averaged law of neurons are those of Cessac, Moynot and Samuelides [START_REF] Cessac | Increase in complexity in random neural networks[END_REF][START_REF] Moynot | Etude mathématique de la dynamique des réseaux neuronaux aléatoires récurrents[END_REF][START_REF] Moynot | Large deviations and mean-field theory for asymmetric random recurrent neural networks[END_REF][START_REF] Cessac | From neuron to neural networks dynamics[END_REF][START_REF] Samuelides | Random recurrent neural networks[END_REF]. From the technical viewpoint, the study of the collective dynamics is done in discrete time. Moynot and Samuelides obtained a large deviation principle and were able to describe in detail the limit averaged law that had been obtained by Cessac using the local chaos hypothesis and to prove rigorously the propagation of chaos property.

One of the next challenges is to incorporate in the network model the fact that the synaptic weights are not independent and in effect often highly correlated.

The problem we solve in this paper is the following. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. Like in [START_REF] Moynot | Etude mathématique de la dynamique des réseaux neuronaux aléatoires récurrents[END_REF][START_REF] Moynot | Large deviations and mean-field theory for asymmetric random recurrent neural networks[END_REF] we study a discrete time dynamics but unlike these authors we cope with more complex intrinsic dynamics of the neurons.

Mathematical framework

For some topological space Ω equipped with its Borelian σ-algebra B(Ω), we denote the set of all probability measures by M(Ω). We equip M (Ω) with the topology of weak convergence. For some positive integer n > 0, we let

V n = {j ∈ ) : |j| ≤ n}. Let T = | T +1
, for some positive integer T . We equip T with the Euclidean topology, T ) with the cylindrical topology, and denote the Borelian σ-algebra generated by this topology by B(T ) ). For some µ ∈ M(T ) ) governing a process (X j ) j∈) , we let µ Vn ∈ M(T Vn ) denote the marginal governing (X j ) j∈Vn . For some j ∈ ), let the shift operator S j : T ) → T ) be S(ω) k = ω j+k . We let M S (T ) ) be the set of all stationary probability measures µ on (T ) , B(T ) )) such that for all j ∈ ) d , µ • (S j ) -1 = µ. Let p n :

T Vn → T ) be such that p n (ω) k = ω k mod Vn . Here, and throughout the paper, we take k mod V n to be the element l ∈ V n such that l = k mod (2n + 1). Define the process-level empirical measure μn :

T Vn → M S T ) as μn (ω) = 1 2n + 1 k∈Vn δ S k pn(ω) . (1)
Let (Y j ) be a stationary Process on T such that the Y j s are independent. Each Y j is governed by a law P , and we write the governing law in M S (T ) ) as P ) . It is clear that the governing law over V n may be written as P ⊗Vn (that is the product measure of P , indexed over V n ).

The equation describing the time variation of the membrane potential U j of the jth neuron writes

U j t = γU j t-1 + i∈Vn J n ji f (U i t-1 ) + θ j + B j t-1 , j ∈ V n t = 1, . . . , T. (2) f : | →]0, 1[ is a monotonically increasing Lipschitz continuous bijection. γ
is in [0, 1) and determines the time scale of the intrinsic dynamics of the neurons. The B j t s are i.i.d. Gaussian random variables distributed as N 1 (0, σ 2 )1 . They represent the fluctuations of the neurons' membrane potentials. The θ j s are i.i.d. as N 1 ( θ, θ 2 ). The are independent of the B i t s and represent the current injected in the neurons. The U j 0 s are assumed to be independent random variables with law µ I .

The J n ij s are the synaptic weights. J n ij represents the strength with which the 'presynaptic' neuron j influences the 'postsynaptic' neuron i. They arise from a stationary Gaussian random field specified by its mean and covariance function

E[J n ij ] = J 2n + 1 cov(J n ij J n kl ) = 1 2n + 1 Λ ((k -i) mod V n , (l -j) mod V n ) ,
Λ is positive definite, let Λ be the corresponding (positive) Fourier transform. We make the technical assumption that the series (Λ(i, j)) i,j∈) is absolutely convergent to Λ asum > 0 and convergent to Λ sum > 0.

We note J n the (2n + 1) × (2n + 1) matrix of the synaptic weights, J n = (J n ij ) i,j∈Vn .

The process (Y j ) defined by

Y j t = γY j t-1 + θ + B j t-1 , j ∈ V n , t = 1, • • • T, Y j 0 = U j 0
is stationary and independent. The law of each Y j is easily found to be given by

P = (N T (0 T , σ 2 Id T ) ⊗ µ I ) • Ψ,
where Ψ : T → T is the following affine bijection. Writing v = Ψ(u), we define

     v 0 = Ψ 0 (u) = u 0 v s = Ψ s (u) = u s -γu s-1 -θ s = 1, • • • , T. (3)
We extend Ψ to a mapping T ) → T ) componentwise.

The application Ψ defined in (3) plays a central role in the sequel we introduce the following definition.

Définition 1.1 For each measure µ ∈ M(T Vn ) or M S (T ) ) we define µ to be µ • Ψ -1 .

We next introduce the following definitions.

Définition 1.2 Let E 2 be the subset of M s (T ) ) defined by

E 2 = {µ ∈ M S (T ) ) | E µ 1,T [ v 0 2 ] < ∞}.
We define the process-level entropy to be, for µ ∈ M S (T ) )

I (3) (µ, P ) ) = lim n→∞ 1 (2n + 1) I (2) µ Vn , P ⊗Vn . If µ / ∈ E 2 , then I (3) (µ, P ) ) = ∞.
For further discussion of this rate function, and a proof that I (3) is well-defined, see [START_REF] Deuschel | Microcanonical distributions for lattice gases[END_REF].

We note Q Vn (J n ) the element of M(T Vn ) which is the law of the solution to (2) conditioned on J n . We let Q Vn = E J [Q Vn (J n )] be the law averaged with respect to the weights. The reason for this is that we want to study the empirical measure μn on path space. There is no reason for this to be a simple problem since for a fixed interaction J n , the variables (U -n , • • • , U n ) are not exchangeable. So we first study the law of μn averaged over the interactions.

Finally we introduce the image laws in terms of which the principal results of this paper are formulated. Définition 1.3 Let Π n and R n in M(M S (T ) )) be the image laws of Q Vn and P ⊗Vn through the function μn : T Vn → M S (T ) ) defined by (1):

Π n = Q Vn • μn R n = P ⊗Vn • μn

The good rate function

We obtain an LDP for the process with correlations (Π n ) via the (simpler) process without correlations (R n ). To do this we obtain an expression for the Radon-Nikodym derivative of Π n with respect to R n . This is done in propositions 2.4 and 2.5. In equation ( 13) there appear certain Gaussian random variables defined from the right handside of the equations of the neuronal dynamics [START_REF] Cessac | From neuron to neural networks dynamics[END_REF]. Applying the Gaussian calculus to this expression we obtain equation ( 14) which expresses the Radon-Nikodym derivative as a function (depending on n) of the empirical measure [START_REF] Cessac | Increase in complexity in random neural networks[END_REF]. Using the fact that this function is measurable we obtain equation (15). This equation is essential in a) finding the expression for the rate function H of definition 3.1, b) proving the lower-bound for Π n on the open sets, c) proving that the sequence (Π n ) is exponentially tight.

The key idea is to associate to every stationary measure µ a certain stationary Gaussian process G µ , or equivalently a certain Gaussian measure defined by its mean c µ and its covariance operator K µ .

Given µ in M S (T ) ) we define a stationary Gaussian process G µ , i.e. a measure Q µ ∈ M S (T ) 1,T ). For all i the mean of G µ,i t is given by c µ t , where

c µ t = J T ) f (u i t-1 )dµ(u), t = 1, • • • , T , i ∈ ), (4)
The covariance between the Gaussian vectors G µ,i and G µ,i+k is defined to be

K µ,k = θ 2 δ k 1 T † 1 T + ∞ l=-∞ Λ(k, l)M µ,l , (5)
where

M µ,k st = T ) f (u 0 s-1 )f (u k t-1 )dµ(u), (6)
The above integrals are well-defined because of the definition of f and that the series in ( 5) is convergent since the series (Λ(k, l)) k, l∈) is absolutely convergent and the elements of M µ,l are bounded by 1 for all l ∈ ).

These definitions imply the existence of a Hermitian-valued spectral representation for the sequence M µ,k (resp. K µ,k ) noted M µ (resp. Kµ ) which satisfies

Kµ (θ) = θ 2 1 T t 1 T + 1 2π π -π Λ(θ, -ϕ) M µ (dϕ).
We also use the partial sums, noted K µ,k

[n] , k ∈ V n , in [START_REF] Faugeras | Large deviations of a spatially-ergodic neural network with learning[END_REF], to define another sequence A µ,k

[n] which is used to define in the limit n → ∞

õ (θ) = Kµ (θ)(σ 2 Id T + Kµ (θ)) -1 . (7)
We next define a functional Γ

[n] = Γ [n],1 + Γ [n],
2 , which we use to characterise the Radon-Nikodym derivative of Π n with respect to R n . Let µ ∈ M S (T ) ) and Γ

[n],1 (µ) = - 1 2(2n + 1) log det Id (2n+1)T + 1 σ 2 K µ [n] , (8)
where K µ

[n] the ((2n + 1)T × (2n + 1)T ) covariance matrix of the law Q µ,Vn Because of previous remarks the above expression has a sense. Taking the limit when N → ∞ does not pose any problem and we can define Γ 1 (µ) = lim n→∞ Γ [n],1 (µ). The following lemma whose proof is again straightforward indicates that this is well-defined. Lemma 2.1 When N goes to infinity the limit of (8) is given by

Γ 1 (µ) = - 1 4π π -π log det Id T + 1 σ 2 Kµ (θ) dθ (9)
Nikodym derivative which can be expressed as a function of the functional Γ.

Proposition 2.4 The Radon-Nikodym derivative of Q Vn with respect to P ⊗Vn is given by the following expression.

dQ Vn dP ⊗Vn (u) = E   exp   1 σ 2   j∈Vn Ψ 1,T (u j ), G j - 1 2 G j 2       , (13)
the expectation being taken against the N T -dimensional Gaussian processes (G i ), i ∈ V n given by

G i t = j∈Vn J N ij f (u j t-1 ), t = 1, • • • , T,
and the function Ψ being defined by [START_REF] Deuschel | Microcanonical distributions for lattice gases[END_REF].

Using standard Gaussian calculus we obtain the following proposition. We state the following theorem.

Theorem 3.1 Π n is governed by a large deviation principle with a good rate function H.

The proof is too long to be reproduced here, see [START_REF] Faugeras | Large deviations of a spatially-ergodic neural network with learning[END_REF]. We only give the general strategy. First we prove the lower bound on the open sets. For the upper bound on the closed sets, we simply avoid it by a) proving that (Π n ) is exponentially tight which allows us to b) restrict the proof of the upper bound to compact sets. The proof of b) is long and technical. It is built upon ideas found in [START_REF] Guionnet | Dynamique de Langevin d'un verre de spins[END_REF].

Note that we have found an analytical form for H through equations ( 9) and (12)

Proposition 2 . 5 +∞ if I ( 3 )

 253 The Radon-Nikodym derivatives write asdQ Vn dP ⊗Vn (u) = exp(N Γ [n] (μ n (u)),(14)dΠ n dR n (µ) = exp((2n + 1)Γ [n] (µ)). (15)Here µ ∈ M S (T ) ), Γ [n] (µ) = Γ [n],1 (µ) + Γ [n],2(µ) and the expressions for Γ [n],1 and Γ [n],2 have been defined in equations (8) and (10).3 The large deviation principleWe define the function H : (T ) ) → [0, +∞) as follows.Définition 3.1 Let H be the function M + 1,S (T ) ) → | ∪ {+∞} defined by (µ, P ) ) = ∞ I (3) (µ, P ) ) -Γ(µ) otherwise,where Γ = Γ 1 + Γ 2 .

We note N p (m, Σ) the law of the p-dimensional Gaussian variable with mean m and covariance matrix Σ.

for all µ ∈ M + 1,S (T ) ).

It also follows easily from previous remarks that Proposition 2.1 Γ [n],1 and Γ 1 are bounded below and continuous on M S (T ) ).

The definition of Γ [n],2 (µ) is slightly more technical but follows directly from proposition 2.4. For µ ∈ M S (T ) ) let

where

The ":" symbol indicates the double contraction on the indexes.

It is shown in [START_REF] Faugeras | Large deviations of a spatially-ergodic neural network with learning[END_REF] that φ n (µ, v) defined by ( 11) is a continuous function of µ which satisfies

By a standard argument we otain the following proposition.

µ). We may conclude from propositions 2.1 and 2.3 that Γ [n] is lower-semicontinuous hence measurable.

From these definitions it is relatively easy, and proved in [START_REF] Faugeras | Large deviations of a spatially-ergodic neural network with learning[END_REF], to show that the measure Q Vn is absolutely continuous with respect to P ⊗Vn with a Radon-


Olivier Faugeras

James Maclaurin

Asymptotic description of stochastic neural networks. II -Characterization of the limit law

Introduction

In [START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF] we started our asymptotic analysis of very large networks of neurons with correlated synaptic weights. We showed that the image Π n of the averaged law Q Vn through the empirical measure satisfied a large deviation principle with good rate function H. In the same article we provided an analytical expression of this rate function in terms of the spectral representation of certain Gaussian processes. In the next section we recall some definitions given in [START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF].

Mathematical framework

For some topological space Ω equipped with its Borelian σ-algebra B(Ω), we denote the set of all probability measures by M(Ω). We equip M(Ω) with the topology of weak convergence. For some positive integer n > 0, we let V n = {j ∈ ) : |j| ≤ n}, and |V n | = 2n + 1. Let T = | T +1 , for some positive integer T . We equip T with the Euclidean topology, T ) with the cylindrical topology, and denote the Borelian σ-algebra generated by this topology by B(T ) ). For some µ ∈ M(T ) ) governing a process (X j ) j∈) , we let µ Vn ∈ M(T Vn ) denote the marginal governing (X j ) j∈Vn . For some j ∈ ), let the shift operator S j : T ) → T ) be S(ω) k = ω j+k . We let M S (T ) ) be the set of all stationary probability measures µ on (T ) , B(T ) )) such that for all j ∈ ), µ • (S j ) -1 = µ.

Let p n : T Vn → T ) be such that p n (ω) k = ω k mod Vn . Here, and throughout the paper, we take k mod V n to be the element l ∈ V n such that l = k mod |V n |. Define the process-level empirical measure μn :

T Vn → M S T ) as μn (ω) = 1 |V n | k∈Vn δ S k pn(ω) . (1)
The equation describing the time variation of the membrane potential U j of the jth neuron writes

U j t = γU j t-1 + i∈Vn J n ji f (U i t-1 ) + θ j + B j t-1 , , U j 0 = u j 0 , j ∈ V n t = 1, . . . , T (2)
f : | →]0, 1[ is a monotonically increasing Lipschitz continuous bijection. γ is in [0, 1) and determines the time scale of the intrinsic dynamics of the neurons. The B j t s are i.i.d. Gaussian random variables distributed as N 1 (0, σ 2 ) 1 . They represent the fluctuations of the neurons' membrane potentials. The θ j s are i.i.d. as N 1 ( θ, θ 2 ). The are independent of the B i t s and represent the current injected in the neurons. The u j 0 s are i.i.d. random variables each governed by law µ I .

The J n ij s are the synaptic weights. J n ij represents the strength with which the 'presynaptic' neuron j influences the 'postsynaptic' neuron i. They arise from a stationary Gaussian random field specified by its mean and covariance function Λ, see [START_REF] Faugeras | Asymptotic description of neural networks with correlated synaptic weights[END_REF][START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF]. We note J n the |V n | × |V n | matrix of the synaptic weights, J n = (J n ij ) i,j∈Vn .

where Ψ : T → T is the following affine bijection. Writing v = Ψ(u), we define

     v 0 = Ψ 0 (u) = u 0 v s = Ψ s (u) = u s -γu s-1 -θ s = 1, • • • , T. (3)
We extend Ψ to a mapping T ) → T ) componentwise and introduce the following notation. Définition 2.1. For each measure µ ∈ M(T Vn ) or M S (T ) ) we define µ to be µ • Ψ -1 .

We note Q Vn (J n ) the element of M(T Vn ) which is the law of the solution to (2) conditioned on J n . We let

Q Vn = E J [Q Vn (J n )]
be the law averaged with respect to the weights.

Finally we introduce the image law in terms of which the principal results of this paper are formulated.

Définition 2.2. Let Π n ∈ M(M S (T )
)) be the image law of Q Vn through the function μn : T Vn → M S (T ) ) defined by (1):

Π n = Q Vn • μ-1 n 3 Characterization

of the unique minimum of the rate function

In [START_REF] Faugeras | Asymptotic description of neural networks with correlated synaptic weights[END_REF], to each measure ν ∈ M(T ) ) we associate the measure, noted

Q ν of M(T ) ) defined by Q ν = µ ) I ⊗ Q ν 1,T where Q ν 1,T is a Gaussian measure on T ) 1,T
with spectral density Kµ (θ). We also define the rate function H ν , which is a linear approximation of the functional Γ defined in [START_REF] Faugeras | Asymptotic description of neural networks with correlated synaptic weights[END_REF]. We prove the following lemma in [START_REF] Faugeras | Asymptotic description of neural networks with correlated synaptic weights[END_REF].

Lemma 3.1. For µ, ν ∈ M S (T ) ), H ν (µ) = 0 if and only if µ = Q ν .
As stated in the following proposition, proved in [START_REF] Faugeras | Asymptotic description of neural networks with correlated synaptic weights[END_REF], there exists a unique minimum µ e of the rate function. We provide explicit equations for µ e which would facilitate its numerical simulation. Proposition 3.1. There is a unique distribution µ e ∈ M S (T ) ) which minimises H. This distribution satisfies H(µ e ) = 0 which is equivalent to µ e = Q µe .

Proof. By the previous lemma, it suffices to prove that there is a unique µ e such that Q µe = µ e .

(4) We define the mapping L :

M + 1,S (T ) ) → M + 1,S (T ) ) by µ → L(µ) = Q µ .
By the definition of Q µ we have Q µ 0 = µ ) I , which is independent of µ.

It may be inferred from the definitions in Section 2 of [START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF] that the marginal of L(µ) = Q µ over F 0,t only depends upon the marginal of µ over F 0,t-1 , t ≥ 1. This follows from the fact that Q µ 1,t (which determines Q µ 0,t ) is completely determined by the means {c µ s ; s = 1, . . . , t} and covariances {K µ,j uv ; j ∈ ), u, v ∈

[1, t]}. In turn, it may be observed from equations (4-5) in [START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF] that these variables are determined by µ 0,t-1 . Thus for any µ, ν ∈ M + 1,S (T ) ) and t ∈

[1, T ], if µ 0,t-1 = ν 0,t-1 , then L(µ) 0,t = L(ν) 0,t .
It follows from repeated application of the above identity that for any ν satisfying

ν 0 = µ ) I , L T (ν) 0,T = L(L T (ν)) 0,T . (5)
Defining

µ e = L T (ν), (6)
it follows from (5) that µ e satisfies (4).

Conversely if µ = L(µ) for some µ, then we have that µ = L 2 (ν) for any ν such that ν 0,T -2 = µ 0,T -2 . Continuing this reasoning, we find that µ = L T (ν) for any ν such that

ν 0 = µ 0 . But because Q µ 0 = µ ) I , since Q µ = µ, we have µ 0 = µ )
I . But we have just seen that any µ satisfying µ = L T (ν), where ν 0 = µ ) I , is uniquely defined by (6), which means that µ = µ e .

We may use the proof of proposition 3.1 to characterize the unique measure µ e such that µ e = Q µe in terms of its image µ e . This characterization allows one to directly numerically calculate µ e . We characterize µ e recursively (in time), by providing a method of determining µ e 0,t in terms of µ e 0,t-1 . However we must firstly outline explicitly the bijective correspondence between µ e 0.t and µ e 0,t , as follows. For v ∈ T , we write

Ψ -1 (v) = (Ψ -1 (v) 0 , . . . , Ψ -1 (v) T ). The coordinate Ψ -1 (v) t is the affine function of v s , s = 0 • • • t obtained from equations (3) Ψ -1 (v) t = t i=0 γ i v t-i + θ γ t -1 γ -1 ., t = 0, • • • , T.
Let K µe,l (t-1,s-1) be the (t -1) × (s -1) submatrix of K µe,l composed of the rows from times 1 to (t -1) and the columns from times 1 to (s -1), and c µe (t-1) = † (c µe 1 , . . . , c µe t-1 ).

Let the measures µ e 1 0,t ∈ M(T 0,t ) and µ e (0,l) t,s ∈ M(T 0,t × T 0,s ) be given by

µ e 1 0,t (dv) = µ I (dv 0 ) ⊗ N t c µe (t) , σ 2 Id t + K µe,0 (t,t) dv 1 • • • dv t . µ e (0,l) (t,s) (dv 0 dv l ) = µ I (dv 0 0 ) ⊗ µ I (dv l 0 ) ⊗ N t+s ((c µe (t) , c µe (s) ), σ 2 Id t+s + K µe,(0,l) (t,s) )dv 0 1 • • • dv 0 t dv l 1 • • • dv l s ,
where

K µe,(0,l) (t,s) =    K µe,0 (t,t) K µe,l (t,s) † K µe,l (t,s) K µe,0 (s,s)    .
The inductive method for calculating µ e is outlined in the theorem below. Theorem 3.2. We may characterise µ e inductively as follows. Initially µ e 0 = µ ) I . Given that we have a complete characterisation of µ e (0,l) (0,t-1) , µ e 1 0,t-1 : l ∈ ) ,we may characterise µ e (0,l) (0,t) , µ e 1 0,t : l ∈ ) according to the following identities.

For s ∈ [1, t], c µe s = J | t f Ψ -1 (v) s-1 µ e 1
0,s-1 (dv). ( 7)

For 1 ≤ r, s ≤ t, K µe,k rs = θ 2 δ k 1 T † 1 T + ∞ l=-∞ Λ(k, l)M µ,l rs .
Here, for p = max(r -1, s -1),

M µe,0 rs = | p+1 f (Ψ -1 (v) r-1 ) × f (Ψ -1 (v) s-1 ) µ e 1 0,p (dv), ( 8
)
and for l = 0

M µe,l rs = | r ×| s f (Ψ -1 (v 0 ) r-1 ) × f (Ψ -1 (v l ) s-1 ) µ e (0,l) (r-1,s-1) (dv 0 dv l ). ( 9
)
Remark 1. From a numerical point of view the t-dimensional integral in (7) and the max(r, s)-dimensional integral in (8) can be reduced by a change of variable to at most two dimensions. Similarly the r + s-dimensional integral in (9) can be reduced to at most four dimensions. Remark 2. If we make the biologically reaslistic assumption that the synaptic weights are not correlated beyond a certain correlation distance d ≥ 0, Λ(k, l) = 0 if k or l does not belong to V d it is seen that the matrixes K µe are 0 as soon as k / ∈ V d : the asymptotic description of the network of neurons is sparse.

Convergence results

We use the Large Deviation Principle proved in [START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF][START_REF] Faugeras | Asymptotic description of neural networks with correlated synaptic weights[END_REF] to establish convergence results for the measures Π n , Q Vn and Q Vn (J n ).

Theorem 4.1. Π n converges weakly to δ µe , i.e., for all

Φ ∈ C b (M S (T ) )), lim n→∞ T Vn Φ(μ n (u)) Q Vn (du) = Φ(µ e ).
Similarly,

lim n→∞ T Vn Φ(μ n (u)) Q Vn (J n )(du) = Φ(µ e ) J almost surely
Proof. The proof of the first result follows directly from the existence of an LDP for the measure Π n , see theorem 3.1 in [START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF], and is a straightforward adaptation of the one in [START_REF] Moynot | Etude mathématique de la dynamique des réseaux neuronaux aléatoires récurrents[END_REF]Theorem 2.5.1]. The proof of the second result uses the same method, making use of theorem 4.2 below.

We can in fact obtain the following quenched convergence analogue of the usual lower bound inequality in the definition of a Large Deviation Principle. Theorem 4.2. For each closed set F of M S (T ) ) and for almost all J

lim n→∞ 1 |V n | log Q Vn (J n )(μ n ∈ F ) ≤ -inf µ∈F H(µ).
Proof. The proof is a combination of Tchebyshev's inequality and the Borel-Cantelli lemma and is an adaptation of the one in [3, Theorem 2.5.4, Corollary 2.5.6].

We define QVn (

J n ) = 1 |Vn| j∈Vn Q Vn (J n ) • S -j
, where we recall the shift operator S. Clearly QVn (J n ) is in M S (T Vn ). We now prove the following ergodic-type theorem. We may represent the ambient probability space by W, where ω ∈ W is such that ω = (J ij , B j t , u j 0 ), where i, j ∈ ) and 0 ≤ t ≤ T -1, recall (2). We denote the probability measure governing ω by P. Let u (n) (ω) ∈ T Vn be defined by [START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF]. As an aside, we may then understand Q Vn (J n ) to be the conditional law of P on u (n) (ω), for given J n . Proof. Our proof is an adaptation of [START_REF] Moynot | Etude mathématique de la dynamique des réseaux neuronaux aléatoires récurrents[END_REF]. We may suppose without loss of generality that T Vm h(u)dµ Vm e (u) = 0. For p > 1 let

F p = µ ∈ M S (T ) )| T Vm h(u) µ Vm (du) ≥ 1 p .
Since µ e / ∈ F p , but it is the unique zero of H, it follows that inf Fp H = m > 0. Thus by theorem 3.1 in [START_REF]Asymptotic description of stochastic neural networks. i -existence of a large deviation principle[END_REF] there exists an n 0 , such that for all n > n 0 , Q Vn (μ n ∈ F p ) ≤ exp (-m|V n |) .

However P ω|μ n (u (n) (ω)) ∈ F p = Q Vn (u|μ n (u) ∈ F p ) . Thus ∞ n=0 P ω|μ n (u (n) (ω)) ∈ F p < ∞.
We may thus conclude from the Borel-Cantelli Lemma that P almost surely, for every ω ∈ W, there exists n p such that for all n ≥ n p , 1 |V n | j∈Vn h π Vm S j u (n) (ω) ≤ 1 p .

This yields (10) because p is arbitrary. The convergence of μn (u (n) (ω)) is a direct consequence of (10), since this means that each of the V th m marginals converge.

Corollary 4. 3 .

 3 Fix m and let n > m. For almost every J and all h ∈ C b (T Vm ),lim n→∞ T Vm h(u) QVn,Vm (J n )(du) = T Vm h(u) µ Vm e (du). lim n→∞ T Vm h(u) Q Vn,Vm (du) = T Vm h(u) µ Vm e (du).That is, the V th m marginals QVn,Vm (J N ) and Q Vn,Vm converge weakly to µ Vm e as n → ∞. Proof. It is sufficient to apply theorem 4.1 in the case where Φ in C b (M S (T ) )) is defined by Φ(µ) = T Vm h dµ Vm and to use the fact that Q Vn , QVn (J) ∈ M S (T Vn ).

Theorem 4. 4 .

 4 Fix m > 0 and let h ∈ C b (T Vm ). For u (n) (ω) ∈ T Vn (where n > m) P almost surely,lim n→∞ 1 |V n | j∈Vn h π Vm (S j u (n) (ω)) = T Vm h(u)dµ Vm e (u),(10)Hence μn (u (n) (ω)) converges P-almost-surely to µ e .

We note N p (m, Σ) the law of the p-dimensional Gaussian variable with mean m and covariance matrix Σ.


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Charlotte Lecluze, Loïs Rigouste, Emmanuel Giguet, Nadine Lucas