Robert Troc
email: r.troc@int.pan.wroc.pl

Ryszard Wawryk

Wojciech Miiller

Halina Misiorek

Malgorzata Samsel-Czekala

R Troć

M Samsel-Czekała

Bulk properties of the Kondo-like system UCoGe

Keywords: actinides, superconductors, Kondo effect, thermal properties, transport properties, magnetoresistance, magnetization, electrical transport PACS numbers: 74.25.-q, 74.25.Ha, 75.20.Hr, 75.47.-m

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Introduction

The various UTM ternaries (T = transition metal) with M = Si and Ge were first prepared by arc melting many years ago [1]. Their orthorhombic crystal structure, mainly of the TiNiSi-type (space group: Pnma) with the U atoms arranged in characteristic zigzag chains along the a axis, as well as the magnetic properties were recognized. A rule of thumb was found, stating that together with reducing the delectron occupation the magnetism weakens and magnetic ordering disappears [1,2] Considering only the germanide series, three of them, namely UFeGe [3], UCoGe and URuGe [1] with reduced d-electron occupation have just been found to be paramagnetic down to 4.2 K. The remaining studied compounds in this series, namely [START_REF]FPLO-5.00-18 and 5.10-20 [improved version of the original FPLO[END_REF] within the local (spin)-density approximation (L(S)DA), used in [START_REF] Diviš | [END_REF]8], and the WIEN2k package within L(S)DA [START_REF] Diviš | [END_REF] as well as LSDA+U [7] methods.

Recently a number of papers [10][11][12][13][14][15][16][17][18] have indicated the co-existence of weak ferromagnetic (FM) and SC states in UCoGe below the critical temperature, T S = 0.5 -0.7 K. The p-wave type of superconductivity and an axial SC state with nodes along the easy magnetization direction along the c axis was postulated [10]. Very recently a weak itinerant FM order has been detected in zero-field muon-spin rotation (µSR) measurements [16]. As the authors of Ref. [16] declared, the µSR study provides an unambiguous proof for ferromagnetism present in the whole sample volume of the above germanide confirming the conclusions of earlier papers about the coexistence of ferromagnetism with superconductivity on the microscopic scale. Ferromagnetic quantum critical fluctuations (FM-QCF) and anomalous coexistence of ferromagnetism and superconductivity in UCoGe have been revealed also by Co-NMR and NQR studies on a powder sample [17]. As a source of superconductivity in UCoGe the proximity of the system to the FM-QCP has been suggested. Interestingly, a high-pressure study of a single-crystalline UCoGe has recently revealed [18] that ferromagnetism is smoothly depressed and near the FM critical point superconductivity is enhanced to a maximum (T s =0.60 K) at 1.11 GPa and is still present in the paramagnetic phase at pressures even above 2 GPa. Because of a limitation to only low temperature studies of all cited works [10][11][12][13][14][15][16][17][18] published on the above germanide up to now, we report here a set of new observations of the magnetic, transport and thermal properties in most cases for the first time given in a wide temperature range, measured also on a single-crystalline sample of UCoGe.

Experiment and calculations

The single crystal having a shape of a cylinder of dimensions l = 5.0 mm × φ = 1.5 mm was grown by the Czochralski technique at the University of Silesia. The starting elements (with purity in weight percent) were: U (99.98), Co (99.99), and Ge (99.999). The quality of the single crystal before annealing (called as grown sample)

was not high as indicates its temperature dependence of the resistivity (see below).

The situation was completely changed after annealing the as grown sample at 900 K for 7 days in a high vacuum.

Magnetization and magnetic susceptibility measurements have been performed using a commercial superconducting quantum interference device (SQUID) magnetometer in the temperature region from 1.72 K up to 400 K and in fields up to 5 T. The electrical resistivity measurements were performed using a steady-current four-point method with spot-welded wire contacts on cylindricallyshaped specimen at temperatures between 0.3 K and 300 K and in transverse magnetic fields (j⊥B) up to B = µ 0 H = 8 T. The geometrical factor of the sample was determined with accuracy not lower than 20%. The thermoelectric power in the low temperature range (0.4 -50 K) and at the higher temperatures from 40 K to 360 K has been measured as described in Refs. [19,20], respectively. For thermoelectric power examinations the specimens were cut out to the dimensions of about 2.5×1×1 mm 3 along the cylinder axis, being the [001] direction of the crystal. The temperature difference (∆T) at lowest T was kept of the order 0.3 K along the sample while above 20 K ∆T≅ 2 K. The thermal conduction was probed using the stationary heat flux method in the temperature range 5 -300 K. The difference ∆T along the sample was For the purpose to estimate diffusion term of thermoelectric power, we used the densities of states (DOS) of UCoGe that has been reported and discussed in [8].

They were calculated by employing the fully relativistic version of the FPLO method [START_REF]FPLO-5.00-18 and 5.10-20 [improved version of the original FPLO[END_REF] in which the 4-component Kohn-Sham-Dirac equation, containing implicitly spinorbit coupling up to all orders, is solved self-consistently. The Perdew-Wang parametrization [21] of the exchange-correlation potential in the LDA approach was utilized. In the calculations the experimental values at room temperature of lattice parameters and atomic positions in the unit cell were used from tables 1 and 2 in [8]. The adopted valence basis sets were as follows: the U: 5d5f;6s6p6d;7s7p, Co: 3s3p3d;4s4p and Ge: 3s3p3d;4s4p states. The size of the k-point mesh of 12 x 12 x12 in the Brillouin zone turned out to be sufficient.

Results and discussions

Magnetic properties

The susceptibility data of UCoGe were measured in a field of 0.5 T along the cylindrical axis of the sample and perpendicular to it, which are approximately equivalent to the directions parallel, and perpendicular, respectively, to the crystallographic c axis χ i (i means B||c and B⊥ ⊥ ⊥ ⊥ c). show linear isotherms, also that intersecting the origin of the coordinate system, which in the case of a ferromagnet indicates the T C temperature, as it was shown for the polycrystalline samples in Refs. [10] and [11]. As seen in figure 1(b), the above dependence measured at 1.72 and 5 K in higher values of µ 0 H/M are almost identical to each other, but in lower values of µ 0 H/M both show a strong curvature due to a substantial magnetic correlation effect and any T C can not be detected.

The temperature dependences of the inverse molar susceptibility along the above directions are displayed in figure 2. The obtained χ -1 i (T) curves showing pronounced curvatures in all the temperatures measured can be fitted in the temperature range 100 -400 K to the modified Curie-Weiss (MCW) law (denoted by the dashed lines) with the parameters given in the figure. Below about 100 K the temperature derivatives of the χ -1 i (T) curve become gradually increasing compared to the fitted MCW curves and, thus, this function diverges from the MCW fit

1 2 0 1 ] ) ( 3 [ ) ( - - Θ - + = p B eff A T k N T µ χ χ
where χ 0 , N A , k B . µ eff , and Θ p have usual meaning. From the point of view of the crystal electric field (CEF) theory such a rapid χ -1 i vs. T bending can be easily explained by thermal depopulation of CEF levels with decreasing temperature. In the case of the 5f [24]) a strong curvature of their χ -1 i (T) takes place. It should be mentioned that the obtained high temperature results are in reasonable accordance with the previous data published in Ref. [1]. There is, however, some difference in the effective magnetic moments between our present value µ eff = 1.98(5) µ B (this paper) and that given in

Ref. [12] of 1.6(1) µ B (no curve is shown). A similar curvilinear behavior of χ -1 i (T) has been reported for the solid solutions of the polycrystalline U(Co 1-x Ru x )Ge samples for x = 0.4 -1.0 (see Ref. [11]) measured in a field B = 1 T in the temperature range 2 -300 K. In the upper inset of figure 2 we show the temperature variation of the magnetization σ i (T) taken at 0.5 T while in the lower inset we display a field dependence of the magnetization M i (B) measured only at 1.72 K. As seen, in our case the curvilinear decrease in M a (B) o%served down to 0 K for both orientations of the sample is without hysteresis, i.e., no magnetic remanence exists.

Therefore, it can be related only to the presence of ferromagnetic correlations (FMC) in contrary to having the spontaneous moment observed earlier on both the polycrystalline [10,12] (0.029 µ B ) and single-crystalline [13] As is apparent in figure 3, the quality of our single crystal before annealing was not so high. The transport measurement data show rather a poor residual resistivity ratio RRR of about 2. It seems that this was mainly caused by a small deviation of the c axis of crystallites from the cylinder axis yielding an additional electron scattering on the crystallite separations. It appears, however, that the quality of the same single crystal, but annealed at 900 K for one week, has been distinctly 3). An analogous electrical behavior measured up to 25 K on polycrystalline UCoGe has recently been reported by Huy and de

Visser [25]. In turn, in the right hand inset of figure 3 the temperature derivative of the resistivity is plotted against the temperature. For both samples a wide maximum around T inf is well apparent underlining strongly the fact that at this temperature there is a crossover probably dividing the energy scale for the here-studied compound into low-temperature and higher temperature regions. The aforementioned linearity is better seen on an enlarged picture of the low-temperature curve of ρ(T) presented in figure 4, where also no sign of a distinct anomaly around 3 K is observed, i.e., at the temperature claimed as a Curie point for the polycrystalline and single-crystalline samples studied by other authors (see Refs. [10][11][12][13][14][15][16][17][18]). Recently, such a linear behavior above the corresponding T sc 's has been found in the super clean samples of URu 2 Si 2 [26] and NpPd 5 Al 2 [27]. Also for the URh 0.62 Ru 0.38 Ge solid solution lying on the border of ferromagnetism the non-Fermi liquid (NFL) regime has been reported [28].

Thus, in the latter U(Rh,Ru)Ge system the randomness interactions between localized and itinerant electrons acting near the critical concentration x cr ≈ 0.38 are thought to be an origin of a NFL behavior (see e.g. Ref. [29]), supported also by the lowtemperature C p ~ TlnT dependence [30].

On the contrary, the T 2 Fermi liquid law has been reported in all cases mentioned above [10][11][12][13][14]25]. In Ref. [15] The T 2 dependence is reported below T C and T 5/3 above this temperature up to 10 K. Since all these samples were weakly ferromagnetic at low temperatures, such a behavior was explained, e.g., in Ref. [12] by scattering on magnons with the same exponent. , where ρ is a measured resistivity and ρ 0 is the residual one, plotted against temperature for the annealed sample of UCoGe (∆ρ 1 ) and for reference sample of ThNiGe (∆ρ 2 ). We have not used here as a reference, e.g., ThCoGe because the latter compound exhibits some weak ferromagnetic behavior up to about 200-250 K, as we and the authors of Ref. [31] have found. The crystal structure of these two Th-based ternaries is, however, of the hexagonal AlB 2 -type and, therefore, both these ternaries are not isostructural with the orthorhombic TiNiSi-type of UCoGe, but at present no other possibility exist to use a better reference material.

As a result of using the non-magnetic reference we display in figure 5 the resulting curve for UCoGe (∆ρ 1,2 = ∆ρ 1 -∆ρ 2 ) which distinctly shows the Kondo-like behavior above about 80 K. As is also clear from this figure, as well as from figure 6, where we plotted the above difference in a manner ∆ρ 1,2 vs. logT, the high temperature Kondo-like effect is preceded by a knee in these curves. The maximum as observed above, marked as T coh, is usually associated with the coherence effect in the Kondo-lattice system [32]. The fitting parameters are also shown in this figure.

The measured overall temperature dependence of the electrical resistivity of UCoGe resembles that of other intermetallic uranium ternaries, but especially that of URu 2 Si 2 [34]. In a similar way to the case of URu 2 Si 2 , our ∆ρ 1,2 (T) decreases also rapidly below 50 K down to a residual resistivity at its superconducting temperature.

Previously, the only existing ρ(T) curve, obtained for UCoGe in a wide temperature range 1.5 -300 K, has been reported by Canepa et al. [35]. Despite annealing of their polycrystalline sample (at 1250 K for 3-5 days), the overall resistivity values given by them are almost one order of magnitude larger than those of our annealed sample.

Moreover, the latter authors observed some sharp maximum in their ρ(T) at T max = 46 negative increase in TMR with lowering T is apparent with an appearance of a sharp negative minimum at 8 K which reason is at present unknown. Furthermore, as the inset to this figure clearly shows, the characteristic temperature T inf is marked at about 13 K, i.e. exactly at the same temperature as found from the temperature derivative of the resistivity as the crossover temperature described above.

Next, we show in figure 8 the TMR as a function of a field B (applied up to 8 2 dependencies with the α coefficient varied with temperature.

Thermoelectric power

The measured Seebeck coefficient S(T) of UCoGe is displayed in figure 9. As seen, the main feature of this curve is a pronounced positive maximum at T max = 13 K which coincides well with the temperatures of maxima in the temperature derivatives of both the resistivity (right hand inset of figure 3) or magnetoresistiviy (insert of figure 7(b))

and also with T max of the smaller peak in ∆C p (T) (see below the inset of figure 12(b)).

As we have pointed out in the previous sections of this paper, this temperature is in our system under study a measure of the characteristic crossover temperature, denoted as T inf . In the case of thermoelectric power (TEP) such a maximum in S(T) can often be interpreted as being due to the Kondo lattice effect [36], where T max = T K /3 or for very clean samples as a phonon drag effect [START_REF] Blatt | Physics of Electronic Conduction in Solids[END_REF]. However, the latter should take place of figure 12(a)). On the other hand, the TEP of UCoGe resembles the behavior of superconducting, spin fluctuating UPt 3 [START_REF] Visser | [END_REF]39], where the positive peak at T ~ 8 K is considered as a measure of the spin fluctuation temperature T sf . It should be also mentioned here that a very similar TEP and resistivity behavior exhibits U 3 Au 3 Sn 4 [40], which is considered as a heavy fermion system with γ(0)= 280 mJ K -2 mol -1 U.

The large γ value is predominantly caused by a reduced f-d hybridization.

Furthermore, according to the phenomenological model which considers scattering conduction electrons by the 5f quasi-particle band of a Lorentzian form, used in Ref. [41] at higher temperatures for the TEP results of correlated cerium and uranium systems, S(T) can be described using the however, below the crossover temperature become more itinerant. The latter is also consistent with the linear behavior of S(T) characteristic of normal metals as is observed for UCoGe below about 2.5 K (upper inset of figure 9). According to the equation followed by metals: S(T) = αT, where α = π 2 k B /3|e|T * , in our case α ~ 2.8 µV K -2 which yields T * ~ 100 K, just as the value T K , determined from the Kondo-like peak in the specific heat (see figure 12(b) and the corresponding text). The value α is practically the same for both kinds of samples used in our studies, so that we present in figure 9 only the results obtained on the as grown sample. The difference between

TEPs in these two samples, except for a small difference in their magnitudes, is well seen only below about 0.8 K where the annealed sample starts to go into the SC state,

where TEP drops to 0.

In order to estimate an overall magnitude of the temperature dependence of TEP in UCoGe, we have applied the Mott model for diffusion TEP, S d (T), of a twoband conductor [START_REF] Blatt | Thermoelectric Power of Metals[END_REF] in an analogous manner as it was done for UCoGa 5 in Ref. [START_REF] Troć | [END_REF] using the following formula:

             -          = = = F F E E E E 2 B 2 d dE (E) dN (E) N 1 dE (E) dN (E) N 1 3e T k π ) T ( S pdf pdf s s , (1)
where N s and N pdf denote the sums of contributions to densities of states coming from all conduction electrons, i.e s and p, d, f ones, respectively. We used our calculated, employing the FPLO code [START_REF]FPLO-5.00-18 and 5.10-20 [improved version of the original FPLO[END_REF] in the LDA approach, N s and N pdf densities in paramagnetic UCoGe, reported in [8], and computed their derivatives over energy at E F , being presented in figure 10. As is seen in this figure, the N s contribution in UCoGe is equal to 0 (see triangles), thus also its derivative. Hence, only the second term connected with the N pdf density remains non-zero in the above equation. In this give the resulting shape of our measured TEP depicted in figure 9. Such an approach, e.g., has been proposed for the superconducting cuprates [44] and could also be applied in our case, but it requires at least to perform neutron inelastic scattering measurements on a large single crystal of UCoGe. It turned out, that for several uranium compounds, like UNi 4 B, UPt 2 In, and UCu 3+x Ga 2-x as well as for UNi 2 Al 3 and UPd 2 Al 3 , two 5f-band Mott's Lorentzian model, where both i = N and W cases had to be used together. Thus, this signals the presence of two energy scales in these systems. It is common, however, for all these materials that their TEP increases with an upward curvature, i.e., in an opposite way to the TEP of UCoGe, and with a tendency to saturate in the vicinity of RT. Nevertheless, calculations taking only i = N, yield both the TEP-curve shape and the above parameters (see Table 2 in Ref. [45 ]) close to ours.

Another feature of the here-described S(T) is a change of the sign from positive to negative at about 75 K, which is easy to explain in the case described

Thermal conductivity

Thermal conductivity κ tot (T) data obtained on the annealed sample of UCoGe, taken over a range 5 -300 K, are given in figure 11. This shows that κ tot first increases very steeply with rising temperature, then in the vicinity of 25 K it goes through a wide knee and finally increases monotonically to RT but having a small anomaly of unknown origin around 200 K. At low temperatures up to about 15 K, this dependence can be approximated as a linear one. The total thermal conductivity of a non-magnetically ordered material as UCoGe is usually expressed as a sum of the electronic κ e and lattice κ lat contributions. At first, we have estimated the contribution of the electron thermal conductivity to the total thermal conductivity by using the Wiedemann-Franz (W-F) law: κ e = L 0 T/ρ, where L 0 is the So mmerfeld value 2.45×10 - This dependence goes through a well apparent maximum at T max ~ 25 K where this ratio reaches a value of about 18 and then it monotonically declines with further increasing temperature to reach finally at RT a value being 7 times higher than L 0 itself. Such a large value at higher temperatures is often an obstacle to observe a Kondo-like behavior in the thermal conductivity [46]. This is possible, however, e.g.

for CeCu 2 and YCu 2 , also crystallizing in an orthorhombic system like UCoGe, which however show up L ~ L 0 [47]. On the other hand, the overall shape of the L/L 0 (T) function found for UCoGe is very similar to those, e.g. of UPd 2 Al 3 [48] or CeCu 2 Si 2 [49] and other heavy fermion superconductors. Figure 11 shows that separation of the lattice contribution trough κ lat = κ tot -κ e yielded the temperature dependence κ lat (T) close to that of κ tot (T). Thus, the large Lorenz ratio is due to unusual large ratio κ lat /κ e .

Furthermore, the lattice component is generally assumed to be composed of several other contributions, like phonon, non-magnetic bipolaron and excitonic ones, depending on the kind of heat carriers [START_REF] Smirnov | Handbook on the Physics and Chemistry of Rare Earths[END_REF][START_REF] Berman | Thermal Conductivity in Solids[END_REF]. Then, the phonon conductivity, κ ph may be described by the Debye model given by Callaway's equation [START_REF] Callaway | [END_REF] as described in details in Refs. [53] and [54].

κ ph = GT 3 dx x x T ∫ Θ - / 0 2 1 4 ) 2 / ( sinh τ , (2)
where x = h ω/k B T ( h ω is the phonon energy), G is a constant, Θ D is the Debye temperature, and τ -1 is the sum of inverse relaxation times for various phonon scattering processes [53,54]. In the calculations we have used two parameters, i.e., sound velocity υ ph = 900m/s and Debye temperature Θ D = 250 K. For the other contributions, such as the bipolar (κ bip ) and excitonic (κ exc ) ones, the heat transport is well described in a review given by Smirnov and Oskotski in Ref. [START_REF] Smirnov | Handbook on the Physics and Chemistry of Rare Earths[END_REF].

Figure 11 presents the separation of κ lat (exp.) into κ ph (calc.) and a difference ∆κ lat . It is clearly seen that Calleway's equation well describes the phonon contributions, κ ph (calc.), to κ tot at low temperatures, but above 25 K it diverges from κ lat (exp.) and diminishes with increasing temperature to reach a value at RT as that of κ e one. The remaining part after the subtraction of κ ph (calc.)(T) from κ lat (exp.)(T), i.e., the difference ∆κ lat , we have assumed to be mainly due to κ bip (T) in such a semimetallic compound as UCoGe. This differential function starting from 25 K follows a shape almost that of high temperature κ tot (T). As we have pointed out in the paper [53], the Callaway equation, however, omits several other possible phonons scattering contributions, e.g., the scattering on the moments in the PM state [START_REF] Smirnov | Handbook on the Physics and Chemistry of Rare Earths[END_REF] and hence the difference ∆κ lat may consist (except for bipolarons) also of other different but difficult to be identified contributions. If we have actually to do with bipolarons in the compound under consideration, a question arises about their role in the occurrence of SC in UCoGe. Nevertheless, these values given for UCoGe are not so large compared to those of real heavy-fermion systems (more than 400 mJ K -2 mol -1 ). On the other hand, the Debye temperature of UCoGe is, as expected, somewhat smaller than that for ThNiGe, i.e., 248 against 266 K. The latter value is comparable to 255 K found in Ref. [31] and to 246 K reported in Ref. [55]. Despite different crystal structures of these two compounds, the closeness in their Debye temperatures strongly indicates a similarity in their lattice dynamics and justifies using here ThNiGe as a reference non-magnetic compound. The upper inset illustrates the C p /T vs. T dependences for our two kinds of samples of UCoGe compared to that reported by Huy et al. [13] but in contrast to our results showing the ferromagnetic properties with T C ~ 3 K. It is surprising that our annealed sample, though having no sign of FM order, shows a similar in shape anomaly but being more enhanced as to its magnitude. Certainly both of them do not 21 reveal a λ-type-shape anomaly as expected in the case of the magnetic order. It should be noted that our as grown sample although exhibits the low-temperature linear behavior in ρ(T), it gives a completely smooth trend in the C p /T against T curve which allowed for a simple extrapolation to T = 0 K to get the γ(0) value for UCoGe.

Thermal properties

On the other hand, the revealed large hump existing in our low temperature specific heat measurements in the annealed sample of UCoGe (see the upper inset of figure 12 (a)) makes it difficult to judge about any divergent tendency of the C p /T vs. logT function expected for the NFL state, which is well manifested by the observed linearity in ρ(T) described above. As seen from this inset, all three considered there cases approach the same C p /T values above 5 K. In addition, there is also shown a smooth C ph (T) curve defined as a sum of the Debye and Einstein contributions to C p of ThNiGe, as specified below. This procedure allows to eliminate C el of a given Thcompound and signals that C ph is more complex than usual when one uses only the Debye function. Thus, in figure 12(b) we display the specific heat difference ∆C p = C p -C ph plotted against T. We found the difference ∆C p between these two curves of large numbers, being magnetic and electronic in origin. An expected error in ∆C p is usually a sum of errors made during measurements of both heat capacities. To minimize the experimental uncertainties in an extraction ∆C p , we have used as C ph (T), i.e. a smooth curve mentioned above, which is without C el of ThNiGe (see below).

As Figure 12(b) indicates, as a result of the above procedure one gets a "camel curve", i.e. there are possible two wide anomalies spanned in the temperature range 1.0 -90 K. Their maxima are centered at about 13 and 45 K and their amplitudes are in a rough ratio of about 1:2, respectively. In addition, we present at This is well seen in the inset to this figure when subtracting the overall electronic specific heat C el * = γ * T from ∆C p (T) of UCoGe, but with a gamma value γ * amounting only ~ 1/3 γ(0), as presented in figure 12(b). Of course, the temperature change in a gamma value from γ(0) = 54.0 mJ K -2 molU -1 found at T= 0 K to a constant value γ * = 18.5 mJ K -2 molU -1 , valid above 90 K, is unknown. However the "camel curve" as well as the straight-line C el * = γ * T are well followed by experimental points, which may indicate a good extracting procedure applied and that the latter can not be treated as any Schottky contribution. The occurrence of the 13Kanomaly seems to be associated with the crossover described above in the section treated on the transport properties, while that of 45K-one, extended in a wide temperature range and having an amplitude about 1.3 J K -1 mol -1 , which is close to that expected for S=1/2 case of a Kondo-lattice effect C K [56]. In analogy to the situation found for Ce-based compounds, Rietschel et al. have analyzed the uranium heavy-fermion superconductors: UB 13 , UPt 13, and U 2 Zn 17 in a similar way [57]. The value of the determined above amplitude is several time too small to ascribe it to a Schottky-type anomaly. For uranium compounds, due to usually large splittings of the CEF levels one cannot expect such a well defined anomaly peak (e.g. see Ref. [58]).

Taking into account the fact reported by Desgranges and Schotte [56] that T max = 0.45 Moreover, in figure 13 the C p vs. T curve is drawn separately for ThNiGe which may allow to perform an analysis of the lattice contribution to the total C p (T) dependence just in UCoGe, assuming that for this aim the Th-based compound as a non-magnetic reference may be used. As seen from this figure, only by combining the Debye and Einstein modes together with the electronic heat contribution we could fit fairly well the experimental data, by a smooth curve through the experimental points, mentioned above. Due to this fitting we got a little higher overall γ ht than γ(0).

However, no satisfactory agreement between the fitting results and the experimental data has been obtained (not shown) using only the Debye approach. This is better apparent in figure 14 where we have plotted the C p /T 3 against T dependence for both ternaries considered here. It is well seen there that the low-temperature Einstein modes in the lattice heat do exist undoubtedly (see the appropriate maximum at T max for the Th-based material, while for the U-based one there is only a marked hump centered at the same T max being less apparent due to a large value of the electronic specific heat contribution found for the latter compound). This supports again the fact that the lattice dynamics of both these compounds are similar to each other. From found to require about seven oscillators per f.u. for the Debye phonons and about two oscillators per f.u. for the Einstein low-lying optic phonons. This procedure is taken from Ref. [59] where the case of YbGaGe was analyzed. Of course, as mentioned already above, the fitted electronic contribution is here represented by a high temperature coefficient, γ ht , which is a little bit higher than γ(0) of ThNiGe.

Conclusions

Our results of bulk measurements done on an annealed single crystal show superconductivity but do not reproduce the previously reported coexistence with ferromagnetism. The magnetization studies, however, show the presence of lowtemperature strong ferromagnetic correlations in UCoGe, possibly due to the proximity of this compound into the quantum critical point. The recent paper by Slooten et. al. [18] has shown that near the ferromagnetic critical point p c = 1.4 GPa superconductivity is not only enhanced but also it is still present in the paramagnetic region induced by higher pressures. Thus, this result clearly indicates again the possibility for UCoGe to show up superconductivity without weak FM order.

The most essential to underline here is the fact that the electrical resistivity data exhibit a linear behavior at low temperatures for our non-magnetic sample of UCoGe being followed by the coherence-type knee in ρ(T) at around 50 K and then at much higher temperatures this dependence reveals a single-ion Kondo-like effect. In turn, the transverse magnetoresistivity contains only a negative contribution in its field and temperature dependences in the whole range of the paramagnetic state of UCoGe and by this the presence of the Kondo-like state is confirmed in this germanide at higher temperatures. This state is also reflected by the corresponding high temperature Kondo-peak in the specific heat. The positive extremum in thermoelectric power, the maxima in the temperature derivatives of the resistivity and magnetoresistivity and also a low-temperature anomaly in the specific heat, all these occur almost at the same temperature of around 13 K. Thus, this temperature gives rise to the crossover transition in UCoGe from a more localized behavior of the 5felectron states in the higher temperature region and, hence, the appearance of a Kondo-like effect to a more metallic state in the lower temperature region allowing for the superconductivity. Finally, the measured thermal conductivity points to the fact of a predominant role of the lattice contribution in UCoGe, which may play some special role in both the exhibited superconductivity and thermopower behavior.

Nevertheless, the remarkable large values of the upper critical field B C2 (0) ppovides evidence for spin-triplet superconductivity which persists in the paramagnetic phase of the considered here germanide [18].

  0.1-0.5 K. Particular care was taken to avoid a parasitic heat transfer between the sample and its environment. Specific heat measurements in the temperature range from 0.35 to 180 K were performed in a commercial multi-purpose measurement device (Quantum Design PPMS) with a 2τ-relaxation-type method.

Figure 1 (

 1 a) shows the temperature change of magnetic susceptibilities measured in these two directions for both kinds of samples, i.e., as grown and annealed ones. As seen from this figure, the differences χ i (T) curves obtained for the same sample but after different thermal treatment are negligible in the whole temperature range of measurements (shown are only data determined up to 100 K). It means that these various treatments of the sample do not influence so much its magnetic behavior. Note, however, an existing pronounced anisotropy in the magnetic behavior between these two orientations of a single crystal. No sign of the susceptibility saturation in approaching T= 0 K is apparent. In the inset of this figure, we have presented the field dependence of the magnetization M (in µ B ) in applied fields B||c up to 5 T taken at various temperatures from 1.72 to 25 K. One can see pronounced curvatures of this dependence diminishing with increasing temperature up to 25 K, but without remanence M 0 at each temperature measured. The lack of the field dependence of the susceptibility appears only above this temperature giving rise to a normal paramagnetic behavior.So that the FM correlations in UCoGe start to develop below 20 K, which is well evidenced from the inset of figure1(a). This is a clear sign of an existence of only strong magnetic correlations in this annealed sample over this temperature range. The induced magnetic moment at 1.72 K in a field of 5 T is equal to 0.14 µ B . A very similar magnetization dependence at the same temperature region (not shown) is noted also for B⊥ ⊥ ⊥ ⊥ c but with lower induced moments at 5 T. For purpose of a comparison, we have also plotted in this inset the magnetization curve measured by Huy et al. [9] (solid line) on a single crystal of UCoGe at 2 K oriented along the caxis. In contrast to our results, this curve displays the ferromagnetic behavior due to the presence of a spontaneous magnetization M s with a value of 0.07 µ B . The total (intrinsic and induced) moment at 5 T was found to be about 0.21 µ B . After subtracting the value of their M s one obtains the same value of the induced moment work. In order to better emphasize the behavior of the magnetization M, in figure 1(b) we have plotted the Arrott diagram (i.e. µ 0 H/M vs. M 2 ) at fixed temperatures from 1.72 K up to 25 K. The isotherms that intersect the M 2 axis indicate usually the presence of a spontaneous magnetization. In our case such an intersect is negligible and unusual only at 1.72 K. A typical Arrott diagram should

  improved by reaching the RRR ratio of 19. The resistivity data of this annealed sample are also plotted against temperature in this figure. Nevertheless, despite the difference in the absolute magnitudes of the resistivity, especially at low temperatures, and the lack of the superconductivity down to 0.3 K for the nonannealed sample, the temperature characteristics of the resistivity for both samples have almost identical features, as shown in the insets to figure3. In contrast to the previously published data reported by other authors, both these samples exhibit at low temperatures the linear ρ(T) dependence and, in addition, the inflection point T inf at about 13 K with further linear behavior when the temperature is increased above this the left hand inset of figure

Figure 4 11 difference

 411 also clearly illustrates the SC transition at T SC = 0.65 K. Furthermore, in the next figure5we show the ∆ρ i (ρ -ρ 0 )

12 K

 12 instead of our smooth "knee", which they interpreted as an evidence of nonidentified magnetic transition at this temperature.Additionally, in the inset to figure6we have plotted the temperature dependence of the resistivity for ThNiGe with fitting these experimental results into the modified Bloch-Grüneisen-Mott equation[33] allowing to determine the Debye temperature Θ R D (=139 K) for this compound. Our overall results of the resistivity found for this Ni-based ternary material is by a factor of about 5 lower than those published in the literature[31]. Finally, in figures 7 (a) and 7 (b) the low temperature resistivity ρ(T), measured in fields B= 0 and 8 T as well as transverse magnetoresistivity (TMR), defined as ∆ρ/ρ = [ρ(Β,Τ)-ρ(0,Τ)]/ρ(0,Τ) ×100 (%) are shown, respectively. If for B= 0, as indicated above, this former dependence is linear in temperature, the situation is changed into the Fermi liquid (FL) behavior at 8 T (see the inset of figure 7(a)). So that the FL state has been recovered in applied magnetic fields. In figure 7(b), where we have plotted the ∆ρ/ρ(T) curve below 60 K measured at 8 T, a strong

  Θ D /10, i.e., at about 25 K for Θ D = 248 K being the Debye temperature derived in the low temperature region of the specific heat of UCoGe (see lower inset

2 + T 2 )

 22 following equation: S(T)= A i T/(B i , where i= N or W in the case of a narrow or wide band of the 5f electrons, respectively. From a fitting of this equation to the low temperature S(T) results we can estimate roughly the f-electron level position to the Fermi level, ∆= ε f -E F , and the width Γ of the band in a Lorentzian form, because A N = 2∆/|e| and B N 2 = 3(∆ 2 + Γ 2 )/(π k B ) 2 . As shown in the lower inset of figure 9, the TEP peak in UCoGe can be well fitted by the narrow, single Lorentzian band (SLB) model, but only up to 30 K.Above this temperature, the theoretical curve diverges upward from the experimental one. This deviation from the SLB model observed at higher temperatures may be connected with scattering processes that are beyond this model, like a phonon-drag effect (see below), crystal field interactions, and so on. For the parameters A N and B N given in the inset we find ∆= 0.26 meV and Γ= 2 meV. This might indicate that in UCoGe we have to do with more localized character of the 5f-electrons which,

  obtained the linear dependence of diffusion TEP as follows: S d (T)= -14.7×10 -2 T (µV K -1 ), which is plotted in figure 9 (dashed straight line). It is seen that the calculated S d (T) yields a negative linear temperature variation reflecting the general trend of the experimental TEP at higher temperatures.It is interesting to add that in the case of the strong scattering of phonons by electrons, which has been noted in connection with the thermal conductivity of UCoGe considered below, the TEP comprises mainly diffusion S d and a phonon drag S ph contributions. If one takes the former one, S d (T), being negative in the whole temperature region studied, just as that found from the above calculations, then the latter, S ph (T), should have only positive values, and their sum S d (T) + S ph (T), would

  i.e., due to the sum of negative S d (T) and positive S ph (T) contributions. Apart from this description, the positive values of S found in the lower temperature region may indicate that the principal charge carriers are holes. It also means that the lack of charge compensation by the transfer of conduction electrons from uranium to Co and Ge atoms creates holes, and only in the high temperature region the electron-type conductivity starts to dominate. The large value of S of about 20 µV K -1 observed for UCoGe at T max ≡ T inf is attributed to Kondo-lattice type of interactions also reflected by fairly large C p /T values in this region of temperature (see below). At higher temperatures S(T) has a tendency to saturate into a large negative value of -30 µV K -1 .

  figure 12(a) the temperature dependences of the specific heat C p measured up to 180 K for both UCoGe (non-annealed) and ThNiGe; the latter is taken as a non-magnetic reference of the phonon contribution to the total C p , as it has been done in the case of the transport properties. At lower temperatures, the specific heat over temperature which can be fitted by the equation: C p /T = γ(0) + BT2 , where the first and second terms are electronic and phonon contributions, respectively, is shown in the lower inset. The linear response of the above function is apparent for both studied compounds up to 7 K. The values of the electronic specific heat coefficient γ(0) and the Debye temperature Θ D obtained for these two compounds are displayed in this inset. Our γ(0) values being of 54 and 7 mJ K -2 mol -1 are close to the literature values of 57[13] and 5[55] or 7.6 mJ K -2 mol -1[31], respectively. The enhancement of the γ(0) value by the existence of the 5f-electrons is here almost 8 times.

  presumably bar errors estimated as ±5 and ±2 %, respectively. Similar error values have been used by Rietschel et al.[57] in their f-derived ∆C p for a number of magnetically non-ordered uranium and rare-earth compounds as UBe 13 , UPt 3 , CeCu 2 Si 2 and CeCu 6 showing superconductivity at very low temperatures, where they considered the superposition of Kondo and Schottky peaks in these compounds, having also to do with subtracting from each other the large numbers.

  where T K is a Kondo temperature, one finds T K for UCoGe being equal to about 100 K. This value is just equal to about 2 T coh . In addition, the temperature variation of the magnetic entropy, S m , is shown in the inset of figure12(b). At 100 K S m is equal to about 0.3Rln2.

  rough fitting (solid line) we have obtained the approximate weighted equation, displayed in this figure, where C D (C E ) represents the specific heats due to the Debye (Einstein) phonons (involving the total nine expected modes, i.e., three acoustic and six optic modes) and with the electronic term C el also added. The corresponding characteristic temperatures Θ D and Θ E are also given in this figure. It is thus clear from the above formula that for the best possible fitting, the lattice heat is

Figure 1 .

 1 Figure captions

Figure 2 .

 2 Figure 2. (Color online) The inverse magnetic susceptibility χ -1 of the UCoGe crystal measured along two directions (indicated) as a function of temperature. The insets show the magnetization M against both temperature T (upper one) and applied magnetic fields B at 1.72 K (lower one) for both crystallographic directions The latter was measured in increasing (open symbols) and decreasing (solid symbols) fields.

Figure 3 .

 3 Figure 3. (Color online) Electrical resistivity ρ(T) curves for as grown and annealed samples of UCoGe. Left hand inset shows the low temperature behavior of ρ(T), and right hand one presents the temperature derivative dρ(T)/dT of the resistivity.

Figure 4 .

 4 Figure 4. (Color online) The low temperature resistivity on the expanded temperature scale.

Figure 5 .

 5 Figure 5. (Color online) The determination of the differential resistivity ∆ρ 1,2 (= ∆ρ 1 -∆ρ 2 ) against temperature for the annealed sample of UCoGe after subtracting the phonon reference data based on ThNiGe.

Figure 6 .

 6 Figure 6. (Color online) The differential resistivity ∆ρ 1,2 (= ∆ρ 1 -∆ρ 2 ) as in figure5vs. temperature on a logarithmic scale. The inset displays a fitting of the experimental resistivity taken for ThNiGe to the Bloch-Grüneisen-Mott equation[33].

Figure 7 .

 7 Figure 7. (Color online) The low-temperature dependences of the resistivity in B=0 and 8 T (a) and their transverse magnetoresistivity ∆ρ/ρ 0 (Τ) at B= 8 T (b) for the annealed sample of UCoGe. The insets illustrate the Fermi-liquid behavior of the resistivity measured at 8 T (a) and the temperature derivative of the magnetoresistivity, d(∆ρ/ρ 0 (Τ))/dT, at B= 8 T (b). T inf (=13 K) is the characteristic temperature of the crossover (see text). The solid squares indicate the values of TMR taken from figure 8 at 8 T.

Figure 8 .

 8 Figure 8. (Color online) Transverse magnetoresistivity against applied magnetic fields ∆ρ/ρ 0 (B) up to 8 T measured at various temperatures 4.2 -40 K. The dashed curves depict the square B 2 field dependencies. In the inset the coefficient α is plotted against T.

Figure 9 .

 9 Figure 9. (Color online) Measured (symbols) and calculated (dashed line) TEP for UCoGe. The upper inset shows the low temperature S(T) dependences for two samples of UCoGe: as grown and annealed (open and solid circles, respectively) and, additionally, the first temperature derivative of S(T) for the latter sample (squares). One can see from this inset an almost linear variation of S(T)= αT below about 2.5 K. The lower inset displays a fitting of the phenomenological modified Mott equation to the low temperature variation of S(T).

Figure 10 .

 10 Figure 10. (Color online) The calculated two parts of densities of states DOS in paramagnetic UCoGe, i.e., N s (E) and N pdf (E), defined in the text. The insert shows the energy derivative of N pdf (E) drawn in the enlarged scale around E F .

Figure 11 .

 11 Figure 11. (Color online) Thermal conductivity κ tot vs. T (squares). After using the W-F law and making separation of κ tot , we obtained its two main contributions: electronic κ e (dotted curve) and lattice κ lat (exp.) (circles) parts, respectively. The dashed line presents the difference ∆κ lat obtained by extracting from κ lat (exp.) the κ ph (calc.) part (solid line) based on the Calleway model which is discussed in the text. The inset shows the normalized L/L 0 ratio vs. T.

Figure 13 .

 13 Figure 13. (Color online) Total specific heat C p measured for ThNiGe (solid circles). A deconvolution of the total C p into the Debye (C D ), Einstein (C E ), and electronic (C el ) specific heat contributions is also presented as C fit (solid line). The two former contributions define the smooth C ph (T) curve, shown in figure 12(a). The fitting parameters shown in the figure are described in the text.

29 Figure 14 .

 2914 Figure 14. (Color online) The C p /T 3 vs. T plot presents all the specific heat contributions into C p of ThNiGe, described in figure13and additionally such a total plot for UCoGe.

  curve) at temperatures above 15 K, but below this temperature it becomes an S-shaped one in the range of 15 -10 K to become, finally, almost concave in fields at 4.2 K as is expected for a ferromagnet. At the latter temperature, ∆ρ/ρ(B) achieves the large negative overall value up to 8 T, where it attains about -25 %, however it still increases negatively up to -28 % reaching its maximum at about 8 K. The character of field and temperature dependencies of the as grown sample of UCoGe was analogous, but the maximum value of TMR was found by a factor of 2.8 lower (-10%). The increase in the absolute value of TMR with lowering the

	behavior (convex
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	T) taken at different temperatures (4.2 -40 K). For all temperatures measured we
	have obtained the negative dependences of TMR, which character, however, changes
	with decreasing temperature. It appears that this is typical of the Kondo-lattice-like
	13

temperature, as is presented in figure

8

, indicates that down to almost T inf an expected Kondo-lattice dependence takes place. A change in the observed overall slopes gives rises to the crossover around 13 K mentioned above. The dashed lines illustrate theα(µ 0 H)

  The steep decrease of the resistivity below 50 K, owing to the coherence effect, is here connected with an almost linear drop of κ tot (T) at the same temperature region. A calculation of the Wiedemann-Franz-Lorenz number L= κ tot ρ/T using the measured thermal conductivity gives abnormal high values for L(T) with a clear evidence that a large lattice term κ lat exists in the total thermal conductivity of UCoGe, several times larger than that of the electronic κ e one. The temperature dependence of the normalized Lorenz number, L/L 0 , is given in the inset of figure11.

	electrical resistivity (see figure 3) being far from that exhibited by typical metallic
	systems.
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	8 WΩ K -2 . As seen from this figure, κ e is essentially a linear function of temperature
	but having only about 15% value of κ tot at RT. Such a small magnitude of the
	electronic part of the thermal conductivity is caused by large overall values of the
	18

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgments

We are gratefud to B. Coqblin for his interest in this paper and E. Talik for synthesizing the single crystal. Technical assistance of G. Badurski and R. Gorzelniak with some experiments as well as a fitting of the thermal conductivity by J. Mucha are acknowledged.


Keywords:

STATISTICS OF GEOMETRIC RANDOM SIMPLICIAL COMPLEXES E. FERRAZ AND A. VERGNE

Abstract. Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the Cech complex associated to the coverage of each point. We compute explicitly the variance of number of k-simplices as well as the variance of the Euler's characteristic. The solution strategy used to compute the second moment can be used to compute analytically the third moment and allows to stablish a conjecture for the nth moment. We apply concentration inequalities on the results of homology and the moments of the Euler's characteristics to find bounds for the for the coverage probability

Motivation

As technology goes on [START_REF] Kahn | Mobile networking for smart dust[END_REF][START_REF] Lewis | Wireless Sensor Networks[END_REF][START_REF] Pottie | Wireless integrated network sensors[END_REF], one can expect a wide expansion of the so-called sensor networks. Such networks represent the next evolutionary step in building, utilities, industrial, home, agriculture, defense and many other contexts [START_REF] Chong | Sensor networks: Evolution, opportunities, and challenges[END_REF].

These networks are built upon a multitude of small and cheap sensors which are devices with limited transmission capabilities and power. Each sensor monitors a region around itself by measuring some environmental quantities (e.g., temperature, humidity), detecting intrusion, etc, and broadcasts its collected informations to other sensors or to a central node. The question of whether information can be shared among the whole network often is of crucial importance.

Many researches have recently been dedicated to this problem considering a variety of situations. It is possible to categorize three main scenarios: those where it is possible to choose the position of each sensor, those where sensors are arbitrarily deployed in the target region with the control of a central station and those where the sensor locations are random in a decentralized system.

The problem of the first scenario is that, in many cases, placing the sensors is impossible or implies a high cost. Sometimes this impossibility comes from the fact that the cost of placing each sensor is too large and sometimes the network has an inherent random behavior (like in the ad hoc case, where users move). In addition, this policy cannot take into account the configuration of the network in the case of failure of some sensor.

The drawback of the second scenario is a higher unity cost of sensors, since each one has to communicate with the central station. Besides, the central station itself increases the cost of the whole system. Moreover, if sensors are supposed to know their positions, an absolute positioning system has to be included in each sensor, making their hardware even more complex and then more expensive.

It is thus important to investigate the third scenario: randomly located sensors, no central station. Actually, if we can predict some characteristics of the topology of a random network, the number of sensors (or, as well, the power supply of them) can be a priori determined such that a given network may operate with high a) b) c)

Figure 1. a) Sensors and their coverage; b) simplicial complex representation when sensors are monitoring the region; c) simplicial complex representation when sensors are communicating amoung them.

probability. For instance, we can choose the mean number of sensors such that, if they are randomly deployed, there is more than 99% of probability the target region to be completely covered. Usually, sensors are deployed in the plane or in the ambient space, thus mathematically speaking, one has to deal with configurations in R 2 , R 3 or a mainfold. The recent works of Ghrist and his collaborators [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF][START_REF] Silva | Coordinate-free coverage in sensor networks with controlled boundaries via homology[END_REF] show how, in any dimension, algebraic topology can be used to compute the coverage of a given configuration of sensors. Trying to pursue their work for random settings, [START_REF] Decreusefond | On the poisson random geometric graph: the one dimensional case[END_REF] has completely solved this problem in one dimension, without using, however, the sofisticated tools of algebraic topology. Due the fact that we cannot order the points in R d , it is not possible to find as much results as found for the one-dimensional case, but the results in this work holds for any dimension d.

The principal idea of the problem is that each sensor can control some environmental information (such as temperature, pression, presence of an intruder, etc) around them. The homology of the coverage of this network, as shown in [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF], can be represented by a simplicial complex. A simplicial complex is a generalization of a graph, so, while we represent a graph with points and edges, a simplicial complex can be represented by points, edges, filled triangles, filled tetraedrons and so on. Almost all the work considers the radius of monitoring ǫ but a different interpretation can be done if the sensors are communicating amoung them. In this case, we suppose that sensors have a power suply allowing them to transmit theirs ID's and, at the same time, sensors have receivers which can identify the transmitted ID's of other sensors above a threshold power. The sensors, knowing mutually the ID's of the close neighbors, are considered connected, creating an information network. The problem remains analogous as the previous one, except that we substitute the coverage radius ǫ by a communication one of ǫ/2. We can see examples of simplicial complexes representations given by sensors communicating amoung them or monitoring a region in Fig. 1.

In this work we consider that sensors are points of a Poisson point processus. This assumption reflects the fact that, due the lack of control of the sensors positioning, only a random fraction of the availables sensors will actually lie in the target region or some sensors may shut down by running out of energy, moreover, the position of each sensor, a priori, does not interact with the positions other sensors. Instead of using the Euclidean norm, we use the maximum norm along this paper. We consider this for three reasons: this norm represents a superior and an inferior limits for the euclidean norm (we can inscribe and circunscribe a circle with two squares); due to the random interactions with the environment (causing shadowing and fading), even the euclidean norm cannot capture with precision the real behavior of this kind of sensor networks, so we choose the norm that allows us to simplify the calculations; using the maximum norm, the Cech complex become equal to the Rips-Vietoris complexes, which allows us to apply directly some results of the algebraic topology. Finally, we assume that sensors lie over d-torus. We justify this choice for three reasons: it avoids the border effects, it helps to determine weather or not a sensor network in the d-box is completely covered and if ǫ is small compared to a, the calculations for all parameters in the d-torus are a good approximation for the [0, a] d box. The coverage of a point and the region where points can lie in are illustrated in Fig. 1, representing the case where a point is deployed over a plan. One of the main results of this paper is the explicit expression for the variance of the number of k-simplices, N k , the covariance between N k and N l and the variance of the Eulers's characteristic, χ, in such complex, which allows us to apply concentrations inequalities. For d ≥ 2, χ is expressed by a power serie and if d = 1, it is possible to find its closed-form expression. A complex closed-form expression for the third order central moment of N k is explicitely calculated and using the same strategy solution of this case, it is possible to find an expression that allows the computation of the moments of any order of the number of simplices.

The paper is organized in the following way: Section 2 presents the preliminaires of the topological tools used in the paper as well as the tools used from the Malliavin calculus; the calculations of the second order moments as well an application of the concentration inequlities are presented in the Section 3; in section 4, we present generalization of the procedure to find the second order moments allowing the computation of the n-th moment of the number of k-simplices.

Algebraic Topology

For further reading on topology, see [START_REF] Hatcher | Algebraic Topology[END_REF][START_REF] Armstrong | Basic Topology[END_REF][START_REF] Munkres | Elements of Algebraic Topology[END_REF]. Graphs can be generalized to more generic topological objects known as simplicial complexes. While graphs model binary relations, simplicial complexes represent higher order relations. Given a set of points V , a k-simplex is an unordered subset {v

0 , v 1 , • • • , v k } where v i ∈ V and v i = v j for all i = j. The faces of the k-simplex {v 0 , v 1 , • • • , v k } are defined as all the (k -1)-simplices of the form {v 0 , • • • , v j-1 , v j+1 , • • • , v k } with 0 ≤ j ≤ k.
A simplicial complex is a collection of simplices which is closed with respect to the inclusion of faces, i.e., if {v 0 , v 1 , • • • , v k } is a k-simplex then all its faces are in the set of (k -1)-simplices.

Given U = (U v , v ∈ T) a collection of open sets, the Cech complex of U denoted by C(U), is the abstract simplicial complex whose k-simplices correspond to (k + 1)-tuples of distinct elements of U that have non empty intersection, so

{v 0 , v 1 , • • • , v k } is a k-simplex if and only if k i=0 U v k = ∅.
One can define an orientation for a simplicial complex by defining an order on vertices. A change in the orientation corresponds to a change in the sign of the coefficient as

[v 0 , • • • , v i , • • • , v j , • • • , v k ] = -[ v 0 , • • • , v j , • • • , v i , • • • , v k ].
For each integer k, C k (X) is the vector space spanned by the set of oriented ksimplices of X. The boundary map ∂ k is defined to be the linear transformations

∂ k : C k → C k-1 which acts on basis elements [v 0 , • • • , v k ] via ∂ k [v 0 , • • • , v k ] = k i=0 (-1) k [v 0 , • • • , v i-1 , v i+1 , • • • , v k ].
Examples of such operations are given in Fig. 3. This map gives rise to a chain complex: a sequence of vector spaces and linear transformations

v0 v1 v2 v0 - v2 + [v0, v1] + [v1, v2] ∂ -→ [v2] -[v0] v0 v1 v2 v0 v1 v2 [v0, v1, v2] ∂ -→ [v1, v2] -[v0, v2] +[v0, v1] v0 v1 v2 v3 Filled Empty v0 v1 v2 v3 [v0, v1, v2, v3] ∂ -→ +[v1, v2, v3] -[v0, v2, v3] +[v0, v1, v3] -[v0, v1, v2]
• • • ∂ k+2 -→ C k+1 (X) ∂ k+1 -→ C k (X) ∂ k -→ C k-1 (X) • • • ∂2 -→ C 1 (X) ∂1 -→ C 0 (X).
A standard result then asserts that for any integer k,

∂ k • ∂ k+1 = 0. If one defines Z k = ker ∂ k and B k = im∂ k+1 , this induces that B k ⊂ Z k . 0 0 0 C k Z k B k C k+1 C k-1 ∂ -→ ∂ -→ Figure 4. A chain complex showing the sets C k , Z k and B k .
The k-dimensional homology of X, denoted H k (X) is the quotient vector space,

H k (X) = Z k (X) B k (X) •
and the k-th Betti number of X is its dimension:

β k = dim H k = dim Z k -dim B k .
Let s k be the number of k-simplices in a simplicial complex X. The well known topological invariant named Euler characteristic for X, denoted by χ(X), is an integer defined by:

χ(X) = ∞ i=0 (-1) i β i .
A well known theorem states that this is also given by:

χ(X) = ∞ i=0 (-1) i s i .
The simplicial complexes we consider are of a special type. They can be considered as a generalization of geometric random graphs.

Definition 1. Given ω a finite set of points on the torus. For ǫ > 0, we define

U ǫ (ω) = {B d∞ (v, ǫ), v ∈ ω} and C ǫ (ω) = C(U ǫ (ω)), where B d∞ (x, r) = {y ∈ T a d , x -y ∞ < r}. Theorem 1. Suppose ǫ < a/4. Then C ǫ (ω) has the same homotopy type as U ǫ (ω).
In particular they have the same Betti numbers.

Proof. This will follow from the so-called nerve lemma of Leray, as stated in [START_REF] Rotman | An Introduction to Algebraic Topology[END_REF]Theorem 7.26] or [START_REF] Björner | Topological methods[END_REF]Theorem 10.7]. One only needs to check that any non-empty intersection of sets B d∞ (v, ǫ) is contractible.

Consider such a non-empty intersection, and let x be a point contained in it. Then, since ǫ < a/4, the ball B d∞ (x, 4ǫ) can be identified with a cube in the Euclidean space. Then each B d∞ (v, ǫ) containing x is contained in B d∞ (x, 4ǫ), hence also becomes a cube with this identification, hence convex. Then the intersection of these convex sets is convex, hence contractible. Definition 2. Let ω be a finite set of points in T a d . For any ǫ > 0, the Rips-Vietoris complex of ω, R ǫ (ω), is the abstract complex whose k-simplices correspond to unordered (k + 1)-tuples of points in ω which are pairwise within distance less than ǫ of each other.

Lemma 2. For the torus

T d a equipped with the product distance d ∞ , R ǫ (ω) has the homotopy type of the Cech complex C 2ǫ (ω)
The proof is given in [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF] in a slightly different context, but it is easy to check that it works here as well. It must be pointed out that Cech and Rips-Vietoris simplicial complexes can be defined similarly for any distance on T d a but it is only for the product distance that the homotopy type of both complexes coincides.

By Lemma 2, k points are forming a (k -1)-simplex whenever they are twoby-two closer than 2ǫ from each other. We define along the paper h

(v 1 , • • • , v k ) as h(v 1 , • • • , v k ) = h k (v 1 , • • • , v k ) = 1≤i<j≤k 1 [ vi-vj <2ǫ] ,
which determines if a set of k distinct ordered points generates a (k -1)-simplex.

Proposition 3. Let ω ∈ T d
a be a set of points, generating the simplicial complex

C ǫ (ω). Then, if i > d, β i (ω) = 0.
Proof. By Theorem 1, C ǫ (ω) has the same homology as U ǫ (ω). But U ǫ (ω) is an open manifold of dimension d, so its Betti numbers β i (ω) vanish for i > d, see for example [START_REF] Greenberg | Algebraic Topology: a First Course[END_REF]Theorem 22.24].

Proposition 4. Let ω ∈ T d
a be a set of points, generating the simplicial complex C ǫ (ω). There are only two possible values for the d-th Betti number of C ǫ (ω):

i) β d = 0, or ii) β d = 1.
If the second holds, then we also have χ(C ǫ (ω)) = 0.

Proof. By Theorem 1, C ǫ (ω) has the same homology as U ǫ (ω). Now, U ǫ (ω) is an open submanifold of the torus, so there are only two possibilities:

i) U ǫ (ω) is a strict open submanifold, hence non-compact ii) U ǫ (ω) = T d a .
In the first case, β d (ω) = 0 by [START_REF] Greenberg | Algebraic Topology: a First Course[END_REF]Corollary 22.25]. In the second case C ǫ (ω) has same homology as the torus, hence β d (ω) = 1 and χ(ω) = 0.

2.1. Application to sensor networks. We now interpret the topological properties of simplicial complexes in terms of connectivity and coverage. In terms of coverage in a network, a 0-simplex represents a single sensor and the existence of a k-simplex means that the (k + 1) points of this simplex are covering the convex hull containing those points. We can see in Figure 1, examples of some simplices and their interpretation in terms of sensor networks.

In a very intuitive fashion, the number of k-simplices itself shows some tendency in the network: if in two networks with identical number of sensors, one of them has more 1-simplices than the other, this first one has a tendency to be more connected; by the same reason, if a network has more 2-simplices than another one, the region on the first case tends to be more strongly covered.

In a more sophisticated way, Theorem 1 formalizes that, in order to determine coverage of sensors, it suffices interpret them as Cech complexes. Unfortunately, a moment of thought shows that constructing the Cech complex cannot be done by pairwise only communications between sensors. Thus, the only complex that can be computed this way is the Rips-Vietoris complex.

An interpretation to Euler characteristic is given by Proposition 4, where we see that χ = 0 is a necessary condition to have a complete coverage of the torus, and β d = 1 is a necessary and sufficient condition. This could in turn translate into conditions for coverage in [0, a] d when considered as embedded in Euclidean space (i.e., not as a torus), but then one needs to be careful about border effects. For example, one can say that

β d = 1 is a sufficient condition for coverage of [ǫ, a -ǫ] d .

Stochastic Model

Poisson point process.

To characterize the randomness of the system, we consider that the set of points is represented by a Poisson point process ω with intensity λ in a Polish space Y . The space of configurations on Y , is the set of locally finite simple point measures (cf [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]):

Ω Y = ω = n k=0 δ(x k ) : (x k ) k=n k=0 ⊂ Y, n ∈ N ∪ {∞} ,
where δ(x) denotes the Dirac measure at x ∈ Y . Simple measure means that ω({x}) ≤ 1 for any x ∈ Y . Locally finite means that ω(K) < ∞ for any compact K

Sensor network coverage Cech complex representation

S 1 v 1 S 1 S 2 v 1 v 2 S 1 S 2 S 3 v 1 v 2 v 3 S 1 S 2 S 2 v 1 v 2 v 3 S1 S2 S3 S4 ǫ ǫ ǫ ǫ v 1 v 2 v 3 v 4
Table 1. Topological representation of the coverage of a sensor network. Each node v represents a sensor. From top to bottom, the highest order simplex is a vertex, an edge, a triangle, three edges, a tetrahedron. of Y . It is often convenient to identify an element ω of Ω Y with the set corresponding to its support, i.e.,

n k=0 δ(x k ) is identified with the unordered set {x 1 , • • • , x n }. For A ∈ B(Y ), we have δ(x k )(A) = 1 [x k ∈A] , so ω(A) = x k ∈ω 1 [x k ∈A] = A dω(x),
counts the number of atoms in A. The configuration space Ω Y is endowed with the vague topology and its associated σ-algebra denoted by F Y . Since ω is a Poisson point process of intensity λ: i) For any A, ω(A) is a random variable of parameter λS(A), i.e.,

P(ω(A) = k) = e -λS(A) (λS(A)) k k! • ii) For A ′ ∈ B(Y )
, for any disjoints A, A ′ , the random variables ω(A) and ω(A ′ ) are independent.

Define ∆ n = {(x 1 , • • • , x n ) ∈ Y n | x i = x j , ∀i = j}. Let f (x 1 , • • • , x n
) be a measurable function and let F (ω) be a random variable given by

F (ω) = xi∈ω∩A,1≤i≤n xi =xj if i =j f (x 1 , • • • , x n ) = A∩∆n f (x 1 , • • • , x n ) dω(x 1 ) • • • dω(x n ),
A well known property of the Poisson point processes [START_REF] Daley | An introduction to the theory of point processes[END_REF] states that

E λ [F (ω)] = A f (x 1 , • • • , x n ) dλ(x 1 ) • • • dλ(x n ). A real function f : Y n → R is called symmetric if f (x σ(1) , • • • , x σ(n) ) = f (x 1 , • • • , x n )
for all permutations σ of S n .The space of symmetric square integrable random variables is denoted by

L 2 (λ) •n . For f ∈ L 2 (λ) •n , the multiple Poisson stochastic integral I n (f n ) is then defined as I n (f n )(ω) = ∆ n f n (x 1 , • • • , x n )( dω(x 1 ) -dλ(x 1 )) • • • ( dω(x n ) -dλ(x n )). If f n ∈ L 2 (λ) •n and g m ∈ L 2 (λ) •m , the isometry formula (1) E λ [I n (f n )I m (g m )] = n!1 [m=n] f n , g m L 2 (λ) •n
holds true (see [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]). Furthermore, we have:

Theorem 5. Every random variable F ∈ L 2 (Ω Y , P) admits a (unique) Wiener-Poisson decomposition of the type

F = E λ [F ] + ∞ n=1 I n (f n ),
where the series converges in L 2 (P) and, for each n ≥ 1, the kernel f n is an element of L 2 (λ) •n . Moreover, we have the isometry

(2) F -E λ [F ] 2 L 2 (λ) •n = ∞ n=1 n! f n 2 L 2 (R+) •n .
For

f n ∈ L 2 (λ) •n and g m ∈ L 2 (λ) •m , we define f n ⊗ l k g m , 0 ≤ l ≤ k, to be the function: (3) (y l+1 , • • • , y n , x k+1 , • • • , x m ) -→ Y l f n (y 1 , • • • , y n )g m (y 1 , • • • , y k , x k+1 , • • • , x m ) dλ(y 1 ) . . . dλ(y l ).
We denote by

f n • l k g m the symmetrization in n + m -k -l variables of f n ⊗ l k g m , 0 ≤ l ≤ k.
This leads us to the next proposition, shown in [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]:

Proposition 6. For f n ∈ L 2 (λ) •n and g m ∈ L 2 (λ) •m , we have I n (f n )I m (g m ) = 2(n∧m) s=0 I n+m-s (h n,m,s ), where h n,m,s = s≤2i≤2(s∧n∧m) i! n i m i i s -i f n • s-i i g m belongs to L 2 (λ) •n+m-s , 0 ≤ s ≤ 2(m ∧ n).
In what follows, given f ∈ L 2 (λ) •q (q ≥ 2) and t ∈ Y , we denote by f ( * , t) the function on Y q-1 given by (

x 1 , • • • , x q-1 ) -→ f (x 1 , • • • , x q-1 , t).
Definition 3. Let Dom D be the the set of random variables F ∈ L 2 (P ) admitting a chaotic decomposition such that

∞ n=1 qq! f n 2 < ∞.
Let D be defined by

D : Dom D → L 2 (Ω Y × Y, P × λ) F = E λ [F ] + n≥1 I n (f n ) -→ D t F = n≥1 nI n-1 (f n ( * , t)).
It is known, cf. [START_REF] Ito | Generalized poisson functionals[END_REF], that we also have

D t F (ω) = F (ω ∪ {t}) -F (ω), dP × dt a.e..
Definition 4. The Ornstein-Uhlenbeck operator L is given by

LF = - ∞ n=1 nI n (f n ),
whenever F ∈ Dom L, given by those F ∈ L 2 P such that their chaotic expansion verifies

∞ n=1 q 2 q! f n 2 < ∞.
Note that E λ [LF ] = 0, by definition and (1).

Definition 5. For F ∈ L 2 (P) such that E λ [F ] = 0, we may define L -1 by

L -1 F = - ∞ n=1 1 n I n (f n ).
Combining Stein's method and Malliavin calculus yields the following theorem, see [START_REF] Peccati | Stein's method and normal approximation of poisson functionals[END_REF]:

Theorem 7. Let F ∈ Dom D be such that E λ [F ] = 0 and Var(F ) = 1. Then, d W (F, N (0, 1)) ≤ E λ 1 - Y [D t F × D t L -1 F ] dλ(t) + Y E λ |D t F | 2 D t L -1 F dλ(t).
Another result from the Malliavin calculus used in this work is the following one, quoted from [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]:

Theorem 8. Let F ∈ Dom D be such that DF ≤ K, a.s., for some K ≥ 0 and DF L ∞ (Ω,L 2 (Y )) < ∞. Then (4) P(F -E λ [F ] ≥ x) ≤ exp - x 2K log 1 + xK DF L ∞ (Ω,L 2 (Y ))
.

Proposition 9. Let X a compact subset of R d and consider the map τ : X → Y as x i = ky i for x i ∈ X, y i ∈ Y and k a positive real constant. Denote by τ * ω the image measure of ω by τ , i.e., τ * :

Ω X → Ω Y maps ω = ∞ i=1 δ(x i ) to τ * ∞ i=1 δ(kx i ).
The application τ * : Ω are the same for any k if λ τ = λ/k d and ǫ τ = kǫ.

Proof. A slightly changing on Propositions 6.1.7 and 6.1.8 of [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF] is enough to show that τ * maps the Poisson measure λ on Ω X to the Poisson measure λ τ = λ/k d on Ω Y . Then, it suffices to realize that for x i ∈ X and for y i ∈ Y :

h(x 1 , • • • , x k ) = 1≤i<j≤k 1 [ xi-xj <2ǫ] = 1≤i<j≤k 1 [ kxi-kxj <2kǫ] , hence h(y 1 , • • • , y k ) = 1≤i<j≤k 1 [ yi-yj <2ǫτ ] ,
which concludes the proof.

Second order moments

4.1. Number of k-simplices. We use all the definitions of the previous section.. The number of (k -1)-simplices can be counted by the expression:

N k = v1,...,v k ∈ω vi =vj if i =j h(v 1 , • • • , v k ) k! = ∆ k h(v 1 , • • • , v k ) k! dω(x 1 ) • • • dω(x k ) (5)
Lemma 10. We can rewrite N k as

N k = 1 k! k i=0 k i λ k-i I i (T d a ) i h(x 1 , . . . , x k ) dx 1 . . . dx k-i .
Proof. We have that

∆ k h(x 1 , • • • , x k )( dω(x 1 ) -λ dx 1 ) . . . ( dω(x i ) -λ dx i )λ dx i+1 . . . λ dx k = i j=0 (-1) j i j ∆ k h(x 1 , • • • , x k ) dω(x 1 ) . . . dω(x j )λ dx j+1 . . . λ dx k .
Thus, after some algebrism with the binomial factors, we have

1 k! k i=0 k i i j=0 (-1) j i j ∆ k h(x 1 , • • • , x k ) dω(x 1 ) . . . dω(x j )λ dx j+1 . . . λ dx k = k! ∆ k h(x 1 , • • • , x k ) dω(x 1 ) . . . dω(x k ) = N k ,
concluding the proof. Definition 6. Let C 1 and C 2 be two simplices with common vertices. For L ∈ P({1, 2}), let us denote m L the number of vertices belonging exactly to the list L of simplices.

Then M = m 12 + m 1 + m 2 is the total number of vertices and J 2 represents the integral on these two simplices:

J 2 (m 12 , m 1 , m 2 ) = ∆ m 12 +m 1 ∆ m 12 +m 2 h m12+m1 h m12+m2 dx 1 . . . dx M . with x 1 , • • • , x M being the M vertices. -1 0 1 x 2 x m12 . . . 1 x m12+1
x m12+m1 . . .

J 2 (m 12 , m 1 , m 2 ) = m 12 + m 1 + m 2 + 2m 1 m 2 m 12 + 1 . ( 6
)
Proof. Let us split the integration domain of J 2 in two domains S 1 and S 2 corresponding to the cases:

(1) All the vertices are connected with each other, thus there is only one simplex. The integral on S 1 is simply the number of points in the simplex:

M = m 12 + m 1 + m 2 .
(2) There are at least two vertices at distance d > 1, which leads to two simplices. By symetry we can choose to order the m L vertices for each L ∈ P({1, 2}) from lowest to greatest or the opposite and choose which simplex is on which side of the axis. Thus we have the integral on S 2 equal to 2m 12 !m 1 !m 2 !A, with A an integral whose calculation is detailed below. We choose to enumerate the vertices of the simplexes such that:

• x 1 , • • • , x m12 are the m 12 common vertices.

• x m12+1 , • • • , x m12+m1 are the m 1 vertices of only C 1 . • x m12+m1+1 , • • • , x M are the m 2 vertices of only C 2 .
Without loss of generality we can choose the origin to be x 1 . The vertices are now order as described in Fig. 5:

0 ≤ x m12 ≤ x m12-1 ≤ • • • ≤ x 2 ≤ 1, -1 ≤ x 2 -1 ≤ x m12+1 ≤ x m12+2 ≤ • • • ≤ x m12+m1 ≤ x m12+1 + 1, x m12+1 ≤ x m12+m1+1 -1 ≤ x M ≤ x M-1 ≤ • • • ≤ x m12+m1+1 ≤ 1, Let us denote J a (f )(x) = x a f (u) du then we write the composition J (2) a (f )(x) =
x a u a f (v) dv du. We also denote m = m 12 + 1 and n = m 12 + m 1 + 1, then we have:

A = 1 0 J (m12-2) 0 (1)(x 2 ) 0 x2-1 -J (m1-1) xm+1 (1)(x m ) 1 xm+1 J (m2-1) xn-1 (1)(x n ) dx n dx m dx 2 .
We easily find that:

J (m12-2) 0 (1)(x 2 ) = x m12-2 2 (m 12 -2)! , -J (m1-1) xm+1 (1)(x m ) = 1 (m 1 -1)! , J (m2-1) xn-1 (1)(x n ) = 1 (m 2 -1)! .
Thus we have:

A = 1 (m 12 -2)!(m 1 -1)!(m 2 -1)! 1 0 x m12-2 2 0 x2-1 -x m dx m dx 2 = 1 (m 1 -1)!(m 2 -1)!(m 12 + 1)! ,
concluding the proof.

Theorem 12. Let ǫ ≤ a/6. Then, the covariance between the number of (k -1)simplices, N k , and the number of (l -1)-simplices, N l , for l ≤ k is given by

(7) Cov λ [N k , N l ] = l-1 i=0 1 i!(k -l + i)!(l -i)! (λ(2ǫ) d ) k+i a 2ǫ d k + i + 2 i(k -l + i) l -i + 1 d . Proof. We want to evaluate E λ [(N k -E λ [N k ])(N l -E λ [N l ])]
. By Lemma 10, this can be written as

E λ 1 k! k i=1 k i λ k-i I i f k i 1 l! l i=1 l i λ l-i I i f l i ,
where

f n j = (T d a ) j h(v 1 , • • • , v n ) dv 1 . . . dv n-j .
Using the isometry formula, given by Eq. ( 1), we have

Cov λ [N k , N l ] = 1 k!l! l i=1 k i l i λ k+l-2i E λ I i f k i I i f l i = 1 k!l! l i=1 k i l i λ k+l-2i i! f k i f l i L 2 (λ) •i = l-1 i=0 1 i!(k -l + i)!(l -i)! λ k-l+2i f k l-i f l l-i L 2 (λ) •(l-i) . (8)
Hence, we are reduced to compute

f k j f l j L 2 (λ) •(j) = (T d a ) j (T d a ) l-j h(v 1 , • • • , v l ) dv j+1 . . . dv l (T d a ) k-j h(v 1 , • • • , v k ) dv j+1 . . . dv k λ dv 1 . . . λ dv j .
Since a > ǫ/6,the integration region is convex (see Fig. 6) and we have

f k j f l j L 2 (λ) •(j) = [0,a] d λ dv 1 ([0,a] d ) k-1 h(0, v 2 , • • • , v k ) × h(0, v 2 , • • • , v j , v ′ 1 , • • • , v ′ l-j ) dv ′ l-j . . . dv ′ 1 dv k . . . dv j+1 λ dv j . . . λ dv 2 . Moreover, if v i = (u i,1 , • • • , u i,d ) and v ′ i = (u ′ i,1 , • • • , u ′ i,d
) and we proceed to the following substitutions:

u i,1 = 2ǫx i if 2 ≤ i ≤ j, u i,1 = 2ǫy k-j if j + 1 ≤ i ≤ k, u ′ i,1 = 2ǫz i if 1 ≤ i ≤ l -j, v 0 v 1 v 2 v 0 v 1 v 2 a) b)
Figure 6. a) Maximum cover in T a and ǫ = a/6. The red region shows the cover of a point v 0 , the blue region is the cover of v 1 and the greer region is the cover of v 2 . b) Maximum cover in the same conditions of a) when ǫ = a/5. In this case, we the three covers intersect each other pairwise, but there is no intersection of the three covers.

This results in a Jacobian (2ǫ) k+l-2i-1 and we recognize the integral to be exactely J 2 (j, kj, lj) as defined in Definition 6. Thus, we have:

f k j f l j L 2 (λ) •(j) = λ i a d (2ǫ) k+l-2i-1 (J 2 (j, k -j, l -j)) d .
Finally, using Eq. ( 6) and Eq. ( 8) gives the result.

Remark. We remark that the possibility of writing Var(N k ) as Eq. ( 7) is due the fact that we use the maximum norm. This simplifies the calculations since we can treat each component individually. However, considering the Euclidean norm it is still possible to find analytically a closed-form expression for Var(N k ), but its calculation involves nasty integrals and a generic term cannot be found. When we consider the Rips-Vietoris complex in T 2 a , the variance of the number of 1-simplices and 2-simplices are given by:

V λ [N 2 ] = a 2ǫ 2 π 2 (4λǫ 2 ) 2 + π 2 (4λǫ 2 ) 3 ,
and

V λ [N 3 ] = a 2ǫ 2 (4λǫ) 3 π 6 π - 3 √ 3 4 + (4λǫ 2 ) 4 π π 2 2 - 5 12 - π √ 3 2 +(4λǫ 2 ) 5 π 2 4 π - 3 √ 3 4 2   • 4.2.
Euler's characteristic. Since we have an expression for the variance of the number of k-simplices, it is possible to calculate one for the Euler characteristic.

Theorem 13. Let ǫ ≤ a/6. Then, the variance of the Euler characteristic in a d torus is:

V λ [χ] = a 2ǫ d ∞ n=1 c d n (λ(2ǫ) d ) n ,
where

c d n = n j=⌈(n+1)/2⌉   2 j i=n-j+1 (-1) i+j (n -j)!(n -i)!(i + j -n)! n + 2(n -i)(n -j) 1 + i + j -n d - 1 (n -j)! 2 (2j -n)! n + 2(n -j) 2 1 + 2j -n d .
Proof. The variance of χ is given by:

V λ [χ] = E λ (χ -E λ [χ]) 2 = E λ   ∞ k=1 (-1) k N k - ∞ k=1 (-1) k E λ [N k ] 2   = E λ   ∞ k=1 (-1) k (N k -E λ [N k ] 2   = E λ   ∞ i=1 ∞ j=1 (-1) i+j (N i -E λ [N i ])(N j -E λ [N j ])   .
We remark that N i ≤

N i 1 i! , so there is a constant c such that E λ   ∞ i=1 ∞ j=1 |(N i -E λ [N i ])(N j -E λ [N j ])|   ≤ ∞ i=1 ∞ j=1 N i 1 i! N j 1 j! ≤ cE λ [e n1 ] 2 < ∞.
Thus the alternating serie converges absolutely allowing us to exchange the mean with the sums and we can write

V λ [χ] = ∞ i=1 (-1) i ∞ j=1 (-1) j Cov λ [N i , N j ] .
The result follows by Eq. ( 7) and some tedious but straightforward algebra.

Lemma 14. Let n be a positive integer, then

n j=1 n j j -1 n -j -1 - j -1 n -j = (-1) n .
Proof. We first simplify the expression:

n j=1 n j j -1 n -j -1 - j -1 n -j = n j=1 2n -3j j n j j n -j ,
Then, applying hypergeometric functions, we solve the sum:

n j=1 2n -3j j n j j n -j = (-1) n .
Theorem 15. In one dimension, the expression of the variance of the Euler characteristic is:

V λ [χ] = a λe -2λǫ -4λ 2 ǫe -4λǫ .
Proof. If d = 1, according to Theorem 13:

V λ [χ] = a 2ǫ ∞ n=1 c 1 n (2λǫ) n , (9)
and we define

α n = n j=⌈ n+1 2 ⌉   2 j i=n-j+1 (-1) i+j n (n -j)!(n -i)!(i + j -n)! - n (n -j)! 2 (2j -n)!   . and β n = c 1 n -α. It is well known that 2j-n i=0 (-1) i j i = (-1) 2j-n-1 j -1 2j -n ,
using Stiffel's relation, we obtain:

α n = (-1) n n n! n j=⌈ n+1 2 ⌉ n j 2 2j-n i=0 (-1) i j i + 2(-1) n n j = 1 (n -1)! n j=⌈ n+1 2 ⌉ 2 n j j -1 n -j -1 - n j j n -j -2(-1) n n j = 1 (n -1)! n j=⌈ n+1 2 ⌉ n j j -1 n -j - j -1 n -j -1 -2(-1) n n j . (10)
The identity n j = n n-j allows us to write that

n j=⌈(n+1)/2⌉ (-2(-1) n ) n j = n j=0 n j = 2 n , n odd, n j=⌈(n+1)/2⌉ (-2(-1) n ) n j = n n/2 + n j=0 - n j = -2 n + n n/2
, n even.

Since j-1 n-j = 0 for j < n+1

2

, we have

n j=⌈ n+1 2 ⌉ n j j -1 n -j - j -1 n -j -1 = n j=1 n j j -1 n -j - j -1 n -j -1 for n odd and n j=⌈ n+1 2 ⌉ n j j -1 n -j - j -1 n -j -1 = - n n/2 + n j=1 n j j -1 n -j - j -1 n -j -1 .
for n even. According to Lemma 14, we get:

n j=⌈(n+1)/2⌉ n j j -1 n -j -1 - j -1 n -j = -1, n odd, n j=⌈(n+1)/2⌉ n j j -1 n -j -1 - j -1 n -j = 1 - n n/2 , n even.
Then, we substitute these two last expressions in Eq. ( 10) to obtain

α n = (-1) n (1 -2 n )1 [n≥1] (n -1)! ,
and thus

∞ i=0 α n x n = -xe -x + 2xe -2x .
Proceeding along the same line, β n is given by

β n = n j=⌈ n+1 2 ⌉   2 j i=n-j+1 (-1) i+j 2(n -i)(n -j) (n -j)!(n -i)!(i + j -n + 1)! - 2(n -j) 2 (n -j)! 2 (2j -n + 1)! = (-1) n (-2 + 2 n )1 [n≥1] (n -1)! - 21 [i≥2] (i -2)! ,
and again we can simplify the power serie

∞ i=0 β n x n : ∞ i=0 β n x n = 2xe -x -2(x + x 2 )e -2x .
Then, substituting α n and β n in Eq. ( 9) yields the result.

Indeed, Corollary 15 suggests the possibility of finding a simple expression for variance of the Euler's characteristic in higher dimensions. Applying it for d = 2, we have a sum in a squared term given by:

n + 2(n -i)(n -j) 1 + i + j -n 2 = n 2 + 4n(n -i)(n -j) 1 + i + j -n + 2(n -i)(n -j) 1 + i + j -n 2 •
With respect to the case in one dimension, no extra knowledge is needed to simplify the first two terms, and, if x = λ(2ǫ) 2 , they are given respectively by:

a 2ǫ 2 (-x + x 2 )e -x + 2(x -2x 2 )e -2x
and a 2ǫ 2 4(xx 2 )e -x + 4(-x + 2x 2 )e -2x .

Unfortunately, we are not able to find a way to express the third term without an infinite serie. This holds for any dimension: we can always find a closed-form for the terms depending on n d and n d-1 , but we cannot go any further.

Theorem 16. We have Dχ ≤ 2 and Dχ L ∞ (Ω,L 2 (T d a )) < ∞ and

P(χ -χ ≥ x) ≤ exp - x 4 log 1 + 2x V λ [χ] .
Proof. In two dimensions, the Euler characteristic is:

χ = β 0 -β 1 + β 2 .
Therefore we can bound Dχ by the variation of β 0β 1 added to the variation of β 2 when we add a vertex to a simplicial complex.

If we add a vertex on the torus, either the vertex is isolated or not. In the first case, it forms a new connected component incrementing β 0 by 1, and the number of holes that is β 1 is the same. Otherwise, as there is no new connected component, β 0 is the same, but the new vertex can at most fill a hole incrementing β 1 by 1. Therefore, the variation of β 0β 1 is at most 1. Now, let us look at the variation of β 2 when we add a vertex to a simplicial complex. According to Proposition 3 is at most 1, showing that Dχ ≤ 2. Then, we use Eq. ( 4) to complete the proof.

N th order moments

For this section, without loss of generality, using Proposition 9, we can choose k = 1/2ǫ, so λ τ = λ(2ǫ) d , ǫ τ = 1/2 and ak = a/2ǫ.

We are interested in the central moment, so we introduce the following notation for the centralized number of (k -1)-simplices: Ñk = N k -Nk .

Finally, let us denote that i j = 0 as soon as i ≤ 0 or j ≤ 0 or ij ≤ 0 for i and j integers. Definition 7. We extend the Definition 6 used in the second order moments calculations.

Let C 1 , C 2 and C 3 be three simplices with common vertices. For L ∈ P({1, 2, 3}), let us denote m L the number of vertices belonging exactly to the list L of simplices.

Then M = m 123 + m 12 + m 13 + m 23 + m 1 + m 2 + m 3 is the total number of vertices and J 3 represents the integral on these three simplices:

J 3 = ∆ p 1 ∆ p 2 ∆ p 3 h p1 h p2 h p3 dx 1 . . . dx M .
with p i being the number of vertices of simplex

C i for i = 1, • • • , 3, for instance p 1 = m 123 + m 12 + m 13 + m 1 , and x 1 , • • • , x M being the M vertices.
Definition 8. We denote J 3 (i, j, s, t) the integral defined above such that

• m 123 = 2t -i -j + s ∨ 0 • m 12 = i + j -s -t ∨ 0 • m 13 = i -t ∨ 0 • m 23 = j -t ∨ 0 • m 1 = k -i ∨ 0 • m 2 = k -j ∨ 0 • m 3 = k -s ∨ 0.
Theorem 17. The third moment of the number of (k -1)-simplices is given by:

E λ Ñk 3 = i,j,s,t λ 3k-i-j t! k i k j k s i t j t t i + j -s -t J 3 (i, j, s, t), with s ≥ |i -j|.
Proof. The chaos decomposition of the number of (k -1)-simplices is as shown in lemma 10:

Ñk = I 1 (f 1 ) + • • • + I k (f k ) = k i=1 I i (f i ), with f i (x 1 , • • • , x i ) = k i h(x 1 , • • • , x k )λ k-i dx k . . . dx i+1 , And
I i (f i ) = f i ( dω(x 1 ) -dλ(x 1 )) . . . ( dω(x i ) -dλ(x i )).
Then, we define g i,j,i+j-s = i+j-s∧i∧j t=⌈ i+j-s 2 ⌉ t! i t j t t i+j-s-t f i • u-t t f j and using the chaos expansion (cf Proposition 6):

Ñk 3 = (I 1 (f 1 ) + • • • + I k (f k )) 3 =   k i=1 k j=1 I i (f i )I j (f j )   (I 1 (f 1 ) + • • • + I k (f k )) = k i,j=1 i+j s=|i-j| I s (g i,j,i+j-s )(I 1 (f 1 ) + • • • + I k (f k )) = k i,j,l=1 i+j s=|i-j| I s (g i,j,i+j-s )I l (f l ).
When taking the expectation of Ñk , we use the isometry formula in Eq. ( 1). Denoting u = i + js, we obtain:

E λ Ñk 3 = E λ   k i,j=1 i+j∧k s=|i-j|∨1 I s (g i,j,u )I s (f s )   = k i,j=1 i+j∧k s=|i-j|∨1 g i,j,u f s λ s dx 1 . . . dx s = k i,j=1 i+j∧k s=|i-j|∨1 u∧i∧j t=⌈ u 2 ⌉ λ s t! i t j t t u -t (f i • u-t t f j )f s dx 1 . . . dx s .
Then we recognize the integral defined in Definition 8:

E λ Ñk 3 = k i,j,=1 i+j∧k s=|i-j|∨1 u∧i∧j t=⌈ u 2 ⌉ λ 3k-i-j t! k i k j k s i t j t t u -t J 3 (i, j, s, t).
Finally, relaxing the boundaries on the sums conclude the proof. Then M = L∈P({1,••• ,n}) m L is the total number of vertices and J n represents the integral on these n simplices:

J n = ∆ p 1 • • • ∆ pn h p1 . . . h pn dx 1 . . . dx M .
with p i being the number of vertices of simplex

C i for i = 1, • • • , n, and x 1 , • • • , x M being the M vertices.
Theorem 18. The expression of the nth power of the number of (k -1)-simplices is given by:

(11) Ñ n k = i1,••• ,in s1,•••sn-2 t1,•••tn-2   n-2 j=1 t j ! m j,1 t j m j,2 t j t j u j -t j   I a (• j∈A f ij )I b (• j∈ Āf ij ). With for j ∈ {1, • • • , n -2}: • 1 ≤ i 1 , • • • , i n ≤ k, • s j ≥ |m j,1 -m j,2 |, • m j,1 = i 2j-1 if 1 ≤ j ≤ ⌊ n 2 ⌋ and s 2(j-⌊ n 2 ⌋)-1 otherwise, • m j,2 = i 2j if 1 ≤ j ≤ ⌊ n 2 ⌋ and s 2(j-⌊ n 2 ⌋) otherwise, • u j = m j,1 + m j,2 -s j , • A ⊂ {1, • • • , n}, • If n is even, then a = s n-3 and b = s n-2 , • If n is odd, then a = s n-2 and b = i n .
Proof. The decomposition of the centralized number of (k -1)-simplices is:

Ñk = I 1 (f 1 ) + • • • + I k (f k ) = k i=1 I i (f i ).
Now, we raise Ñk to the nth power:

Ñk n = k i=1 I i (f i ) n .
First, we consider the case where n is even, we can group the factors by 2:

Ñ n k = k i1=1 I i1 (f i1 ) k i2=1 I i2 (f i2 ) . . .   k in-1=1 I in-1 (f in-1 ) k in=1 I in (f in )   .
We then use the chaos expansion of Proposition 6:

I i (f i )I j (f j ) = 2(i∧j) s=0 I i+j-s   s≤2t≤2(s∧i∧j) t! i t j t t s -t f i • s-t t f j   = i+j s=|i-j| I s   i+j-s≤2t≤2(i+j-s)∧i∧j) t! i t j t t i + j -s -t f i • i+j-s-t t f j   .
Let us denote g s = t! i t j t t i+j-s-t f i • i+j-s-t t f j , so we can re-write, relaxing the boundaries on the sums:

I i (f i )I j (f j ) = s≥|i-j| t I s (g s ).
Thus, we have:

Ñ n k = k i1,i2=1 s1≥|i1-i2| t1 I s1 (g s1 ) • • • k in-1,in=1 s n/2 ≥|in-1-in| t n/2 I s n/2 (g s n/2 ).
We go on grouping terms by 2 until we only have a product of 2 chaos left: First we made n/2 chaos expansions, leading to n/2 sums with indexes s

j , j = 1, • • • , n/2.
To reduce the number of chaos to 2, we have to make other chaos expansions. For j ≥ n 2 + 1, the sum indexed by s j represents the expansion of the chaos indexed s 2(j-n 2 )-1 and s 2(j-n 2 )-1 . We have 2 chaos remaining when j = 2(j -n 2 ) + 2, i.e. when j = n -2.

Moreover, there are as much sums indexed with t j as with s j , that is n -2. Thus we can write: • m j,1 = i 2j-1 if 1 ≤ j ≤ n 2 and s 2(j-n 2 )-1 otherwise,

Ñ n k = k i1,••• ,in=1 s1,•••sn-2 t1,•••tn-2
• m j,2 = i 2j if 1 ≤ j ≤ n 2 and s 2(j-n 2 ) otherwise. Then, denoting u j = m j,1 + m j,2s j and A the subset of {1, • • • , n} such that if j ∈ A then the chaos i j is expanded in the chaos s n-3 , we have: The notation • j∈A f ij represents the product defined in Eq. ( 3) of the functions f ij for j ∈ A, but whom variables depend on all the i 1 , • • • , i n , s 1 , • • • , s n-2 , and

t 1 , • • • , t n-2 .
Now, if n is odd, we consider n -1 which is even, therefore we have: • m j,1 = i 2j-1 if 1 ≤ j ≤ n-1 2 and s 2(j-n-1 2 )-1 otherwise, • m j,2 = i 2j if 1 ≤ j ≤ n-1 2 and s 2(j-n-1 2 ) otherwise. And with u j = m j,1 + m j,2s j , Theorem 19. The expression of the nth moment of the number of (k -1)-simplices is given by:

Ñ n k = k i1,••• ,in-1=1
E λ Ñk n = i1,••• ,in s1,••• ,sn-3 t1,••• ,tn-2 λ nk+c   n j=1 λ -ij k i j     n-2 j=1 t j ! m j,1 t j m j,2 t j t j u j -t j   J n (i 1 , • • • , i n , s 1 , • • • , s n-3 , t 1 , • • • , t n-2 ).
With for j ∈ {1, • • • , n -2}: Proof. The expression of the nth power of the number of (k -1)-simplices is given in Eq. ( 11):

• if j ≤ n -3, s j ≥ |m j,1 -m j,2 |, • m j,1 = i 2j-1 if 1 ≤ j ≤ ⌊ n
Ñ n k = k i1,••• ,in=1 s1,••• ,sn-2 t1,••• ,tn-2   n-2 j=1 t j ! m j,1 t j m j,2 t j t j u j -t j   I a (• j∈A f ij )I b (• j∈ Āf ij ).
If n is even, we have:

Ñ n k = k i1,••• ,in=1 s1,••• ,sn-2 t1,••• ,tn-2   n-2 j=1 t j ! m j,1 t j m j,2 t j t j u j -t j   I sn-3 (• j∈A f ij )I sn-2 (• j∈ Āf ij ).
So let us focus on the only part of the equation which is likely to change when we take the expected value, that we will denote:

K = sn-3 sn-2 I sn-3 (• j∈A f ij )I sn-2 (• j∈ Āf ij ).
We then use the property of Eq. ( 1) and recognize the integral from Definition 9: Then for n odd we directly write:

E λ [K] = sn-3   n j=1 λ k-ij k i j   λ sn-3 J n (i 1 , • • • , i n , s 1 , • • • , s n-3 , t 1 , • • • , t n-2 ) = sn-3 λ nk+sn-3   n j=1 λ -ij k i j   J n (i 1 , • • • , i n , s 1 , • • • , s n-3 , t 1 , • • • , t n-2 ),
K ′ = in sn-2 I in (• j∈I f ij )I sn-2 (• j∈ Ī f ij ), E λ [K ′ ] = in λ nk+in   n j=1 λ -ij k i j   J n (i 1 , • • • , i n , s 1 , • • • , s n-3 , t 1 , • • • , t n-2 ),
with i n ∈ {|m n-2,1m n-2,2 | ∨ 1, k}.

The binomials with the i j allow us to relax the boundaries on the sums on i j , concluding the proof.

2 aFigure 2 .

 22 Figure 2. Ilustration of the coverage of a point and the region where points can lie, in the 2 dimensional case

Figure 3 .

 3 Figure 3. Examples of boundary maps. From left to right. An application over 1-simplices. Over a 2-simplex. Over a 3-simplex, turning a filled tetrahedron to an empty one.

1 Figure 5 .

 15 Figure 5. Example of relative positions of the points

Definition 9 .

 9 Let C 1 , • • • , C n be n simplices with common vertices. For L ∈ P({1, • • • , n}), let us denote m L the number of vertices belonging exactly to the list L of simplices.

I sn- 3

 3 (φ sn-3 )I sn-2 (φ sn-2 ), With s j ≥ |m j,1m j,2 | for j ∈ {1, • • • , n -2} if we denote:

I

  sn-3 (φ sn-3 )I sn-2 (φ sn-2 ) = I sn-3 (• j∈A f ij )I sn-2 (• j∈ Āf ij ).

  s1,••• ,sn-3 t1,••• ,tn-3I sn-4 (φ sn-4 )I sn-3 (φ sn-3 ) k in=1 I in (f in ) = k i1,••• ,in=1 s1,••• ,sn-2 t1,••• ,tn-2 I sn-2 (φ sn-2 )I in (f in ), with s j ≥ |m j,1m j,2 | for j ∈ {1, • • • , n -2}using the same notations for n -1 instead of n:

I sn- 2 (

 2 I sn-2 (• j∈{1,••• ,n-1} f ij ),concluding the proof.

2 ⌋

 2 and s 2(j-⌊ n 2 ⌋)-1 otherwise,• m j,2 = i 2j if 1 ≤ j ≤ ⌊ n 2 ⌋ and s 2(j-⌊ n 2 ⌋) otherwise, • m j,3 = s j if 1 ≤ j ≤ n -3 and s n-3 otherwise, • u j = m j,1 + m j,2m j,3 , • If n is even, then c = s n-3 and s n-3 ≥ |m n-2,1m n-2,2 | ∨ |m n-3,1m n-3,2 |, • If n is odd, then c = i n and i n ≥ |m n-2,1m n-2,2 |.

with s n- 3 ≥

 3 |m n-2,1m n-2,2 | ∨ |m n-3,1m n-3,2 |.