Wenbo Yu
email: wbyu@bjtu.edu.cn

Wenzhe Jia

Feng Guo

Zhaoyang Ma

Pengcheng Zhang

Christophe Tromas

Véronique Gauthier-Brunet

Paul R C Kent

Weiwei Sun

Sylvain Dubois

The correlation between N deficiency and the mechanical properties of the Ti 2 AlN y MAX phase

Keywords: Nonstoichiometric Ti2AlN, Anisotropic, Microstructure, Mechanical properties

The role of X deficiency on the mechanical properties of MAX phases was studied by synthesizing Ti2AlN by powder metallurgy in stoichiometric and sub/extra-stoichiometric nitrogen compositions. XRD analyses and ab initio calculations indicate that nitrogen vacancies result in a lattice contraction predominantly along the c-axis. The elastic moduli and intrinsic hardness of substoichiometric Ti2AlN0.9 measured from nanoindentation tests are shown to be slightly smaller than that of Ti2AlN. The key mechanical indexes, bulk (B), shear (G) and Young's (E) moduli as well as the hardness variation are calculated in density functional theory, and show different responses depending on the concentration of N vacancies. This joint experimental and theoretical study provides a full understanding of the energetics, chemical bonding, electronic structure, and mechanics of the N deficient MAX phases which would increase the application of nitride ceramics.

Introduction ‡

MAX phases denoted as Mn+1AXn (n=1-3) are a class of layered hexagonal ternary carbides/nitrides, where M is an early transition metal, A is from group IIIA or IVA element, X is C and/or N. MAX phases exhibit a combination of strong covalent M-X bonds interleaved with A layers through weak M-A bonds. [START_REF] Wang | Recent progress in theoretical prediction, preparation, and characterization of layered ternary transition-metal carbides[END_REF][START_REF] Barsoum | The MN+1AXN phases: A new class of solids: Thermodynamically stable nanolaminates[END_REF] In sharp contradiction with binary carbides/nitrides (hard and brittle, nonmachinable, damage intolerant, and susceptible to thermal shock), MAX phases exhibit machinability, excellent electrical and thermal conductivities, thermal shock and oxidation resistance, stiffness at high temperature and self-lubrication capacity, which bridge the gap between metals and ceramics [START_REF] Barsoum | The MN+1AXN phases: A new class of solids: Thermodynamically stable nanolaminates[END_REF][START_REF] Guitton | Effect of microstructure anisotropy on the deformation of MAX polycrystals studied by in-situ compression combined with neutron diffraction[END_REF][START_REF] Yu | Solid solution effects in the Ti 2 Al (CxNy) MAX phases: Synthesis, microstructure, electronic structure and transport properties[END_REF][START_REF] Bei | Compressive Behavior of Ti3AlC2 and Ti3Al0.8Sn0.2C2 MAX Phases at Room Temperature[END_REF][START_REF] Ward | Corrosion performance of Ti3SiC2, Ti3AlC2, Ti2AlC and Cr2AlC MAX phases in simulated primary water conditions[END_REF][START_REF] Radovic | MAX phases: Bridging the gap between metals and ceramics[END_REF][START_REF] Barsoum | MAX Phases: Properties of machinable ternary carbides and nitrides[END_REF] . Ti-based MAX phase compounds are used as ohmic contacts on wide-gap semiconductors to achieve high-quality of electronic devices [START_REF] Buchholt | Ohmic contact properties of magnetron sputtered Ti3SiC2 on n-and p-type 4H-silicon carbide[END_REF][START_REF] Wang | Ohmic contacts on silicon carbide: The first monolayer and its electronic effect[END_REF] , and have shown good tolerance to neutron irradiation [START_REF] Tallman | Effect of neutron irradiation on select MAX phases[END_REF] .

Concerning the crystal structure, MAX phases and MX phases have similar Ti-N and Ti-C chemical bonding characteristics [START_REF] Ern | Electronic band structure of TiC, TiN, and TiO[END_REF] . In binary transition metal carbide/nitrides, mechanical, chemical and physical properties can be modified by controlling the vacancies in X site (TiCx (x=0.5-0.98) and TiNy (y=0.6-1.1) [START_REF] Ern | Electronic band structure of TiC, TiN, and TiO[END_REF][START_REF] Pierson | Handbook of Refractory Carbides & Nitrides: Properties, Characteristics, Processing and Apps[END_REF][START_REF] Sun | Self-diffusion of Ti interstitial based point defects and complexes in TiC[END_REF] ). The introduction of 12.5% of vacancies in TiN has been shown to reduce shear stiffness from 190 GPa to about 150GPa. [START_REF] Jhi | Vacancy hardening and softening in transition metal carbides and nitrides[END_REF] However, very few studies related to the X deficiency exist for ternary carbides/nitrides MAX phases, possibly due to the difficulty of controlling the defect concentration compared with the binary TiC/TiN. [START_REF] Sun | Self-diffusion of Ti interstitial based point defects and complexes in TiC[END_REF] For example, Yu at al. reported that the synthesis of pure Ti2AlC can only be achieved with the 2Ti:1.05Al:0.85C nominal composition by hot isostatic pressing (HIP) at 1400°C/80MPa for 4h. Guo et al. [START_REF] Guo | Effects of TiC and Ti_3AlC_2 Addition on Combustion Synthesis of Ti_3AlC_2 Powders[END_REF] demonstrated that single-phased Ti2AlCx (x<1) can be fabricated by combustion synthesis starting with 2Ti:Al:0.7C powder ‡ Notice: This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

mixture. The main combustion product is Ti3AlC2 if the molar ratio of the reactant mixture is 2Ti:Al:C. Cabioc'h et al. reported that a large nitrogen deficiency in Ti2AlN can be realized [START_REF] Cabioc | Structural investigation of substoichiometry and solid solution effects in Ti2Al(Cx,N1-x)y compounds[END_REF] , which provides with an effective means to investigate the effect of X site vacancies on the microstructure and properties of MAX phases. Substoichiometric effects allow modifying and, in some cases, improving their physical properties. For example, Scabarozi et al. have shown that substoichiometric Ti2AlN0.95 has a higher resistivity than stoichiometric Ti2AlN [START_REF] Lofland | Elastic and electronic properties of select< equation> M< sub> 2</sub> AX</equation> phases[END_REF] which suggests that vacancies are strong electron scatters.

The MAX phases are also of interest because the ability to etch away the A atoms enables the synthesis of two-dimensional MXenes. [START_REF] Barsoum | The MN+1AXN phases: A new class of solids: Thermodynamically stable nanolaminates[END_REF][START_REF] Barsoum | The MAX phases: unique new carbide and nitride materials[END_REF] Notably, nitride MXenes are predicted to exhibit higher capacitance than carbides as the promising capacitive materials [START_REF] Zhan | Computational Screening of MXene Electrodes for Pseudocapacitive Energy Storage[END_REF] , but they have not been widely studied. The control of the N deficiency in MAX phases would thus result in new MXenes.

In the present work, a series of Ti2AlNy MAX phases in a wide N concentration (y=0.85-1.05) are synthesized by pressureless sintering. The cell parameters, octahedral and trigonal distortion parameters are determined both by the Rietveld method and calculated by ab initio calculations as a function of the vacancy content.

Nanoindentation is used to evaluate intrinsic hardness and elastic moduli of Ti2AlN0.9 and Ti2AlN synthesized by hot isostatic pressing. In order to interpret the experiments, theoretical calculations were performed to determine the N vacancy energetics, the impact on the chemical bonding and electronic structure, and on the mechanical properties. The pioneering correlation between the defects on the X site and the atomic-scale understanding would lay fundamental understanding on this nitride MAX.

Experimental details

Powders of Ti (150-250µm, 99.5% purity), Al (45-150µm, 99.5% purity) and AlN (10µm, 98% purity) were used to synthesize stoichiometric and sub/extra-stoichiometric Ti2AlNy. In order to study the effect of N content on Ti2AlNy lattice parameters, the samples were produced, from different nominal compositions (2Ti:1.05Al:1.05N, 2Ti:1.05Al:1.0N, 2Ti:1.05Al:0.95N, 2Ti:1.05Al:0.90N and 2Ti:1.05Al:0.85N), by pressureless sintering at 1400°C for 0.5h. To investigate the vacancy effect on X site, on the intrinsic hardness and indentation modulus values, 2Ti:1.05Al:1.0N and 2Ti:1.05Al:0.90N nominal compositions were used and dense samples were produced by hot isostatic pressing (HIP) at 1400°C/80MPa for 4h. The synthesis procedure is detailed in [21]. The powder mixtures used in this study contain an excess of 5 at.% of Al to compensate the loss of aluminum by preferential out-diffusion during the sintering process.

Samples used for nanoindentation tests were thus grinded using SiC paper and then polished with a diamond suspension. Finally, a chemo-mechanical polishing has been performed using a neutral suspension of alumina nanoparticles. Such a chemo-mechanical polishing allows to produce a very flat surface and to avoid any work hardening due to conventional grinding.

Phase identification was performed by X-Ray Diffraction (XRD) using a Bruker D501 diffractometer with Cu-Kα radiation. XRD data were refined using the MAUD software [START_REF] Yuan | The pressure induced twisted distortion in flexible oxide Tc2O7[END_REF] in order to extract the lattice parameters of the Ti2AlNy compounds.

Microstructures were examined by Scanning Electron Microscopy (SEM, JEOL 5600LV). Wavelength Dispersive x-ray Spectroscopy (WDS, CAMECA SX100) was used to determine the composition of Ti2AlNy resulting from HIP treatment.

Nanoindentation tests have been performed using a Nano Hardness Tester (NHT) apparatus (CSM instrument, Switzerland) equipped with a Berkovich indenter. The indenter shape has been carefully calibrated for true penetration depth as small as 20nm both by direct Atomic Force Microscopy (AFM) observation and by indenting fused silica samples with a Young's modulus of 72GPa. A large indentation network of 150 indents was performed for each sample in order to probe different crystal grain orientations. The load-penetration depth curves are recorded for each nanoindentation test, and the hardness and elastic modulus are determined from the elastic unloading curve using the equivalent indenter method [START_REF] Pharr | Understanding Nanoindentation Unloading Curves[END_REF][START_REF] Woirgard | An alternative method for penetration depth determination in nanoindentation measurements[END_REF] . Different loads (1, 2, 3, 6, 10, 15, 30, 100 mN) corresponding to different penetration depths were used in order to study the hardness variation with penetration depth. SEM observations were carried out to remove from the statistics the hardness and elastic modulus values that have been measured on indents that were not localized on a single MAX phase grain.

Theoretical methods

Our theoretical work used density functional theory (DFT) and the Vienna atomic simulation package (VASP) [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | Ab initio molecular-dynamics simulation of the liquid-metalamorphous-semiconductor transition in germanium[END_REF][START_REF] Kresse | Ab initio molecular dynamics for open-shell transition metals[END_REF] . We used the projector augmented wave method (PAW) and the generalized gradient approximation (GGA)

Perdew-Burke-Ernzerhof functional [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] . The valence electronic configurations were 2s 2 2p 3 for N, 3s 2 3p 1 for Al, and 3p 6 3d 3 4s 1 for Ti. Geometry optimization utilized 520 eV for the plane wave cutoff energy and convergence criteria of 5×10 -6 eV for the total energy and 0.01eV/Å -1 for the absolute maximum force. Γ centered 15×15×1 k-point grids for Brillouin zone (BZ) integration gave convergence to approximately 3 meV/atom. The special quasi-random structure (SQS) method [START_REF] Zunger | Special quasirandom structures[END_REF] was employed to describe N deficient structures in 4×4×1 supercells containing 128 atoms. 5×5×1

k-point grids were used for these cells. Unless otherwise noted, all lattice parameters and atomic positions were fully relaxed self-consistently. The Electron Localization Function (ELF) [START_REF] Jordi | Theoretical evaluation of electron delocalization in aromatic molecules by means of atoms in molecules (AIM) and electron localization function (ELF) topological approaches[END_REF][START_REF] Yuan | The pressure induced twisted distortion in flexible oxide Tc2O7[END_REF] was calculated from the Kohn Sham orbitals.

The stress-strain method was employed to compute elastic constants, cij. Related mechanical properties were calculated on the basis of the these properties (in Voigt approximation [31] ).. The bulk modulus, B and shear modulus, G and Young's modulus, E of polycrystalline aggregates are derived by :

𝐺𝐺 𝑉𝑉 = 1 15 ((2𝑐𝑐 11 + 𝑐𝑐 33 -𝑐𝑐 12 -2𝑐𝑐 13 ) + 1 5 (2𝑐𝑐 44 + 𝑐𝑐 66 )) 𝐵𝐵 𝑉𝑉 = 2 9 �𝑐𝑐 11 + 𝑐𝑐 12 + 2𝑐𝑐 13 + 1 2 𝑐𝑐 33 � 𝐸𝐸 = 9𝐵𝐵𝐺𝐺 3𝐵𝐵 + 𝐺𝐺

Results and discussion

Microstructural characterizations

Fig. 1(a-e) shows the back-scattered microstructural morphologies of Ti2AlNy fabricated by pressureless sintering at 1400°C for 0.5h from 2Ti:1.05Al:yN (y=0.85,0.9,0.95,1.0 and 1.05) nominal compositions. We can distinguish that the Ti2AlN grain size decreased with increasing N content in the starting powders. For all samples, EDS analyses show that the overall Ti/Al ratio is 2:0. (Ti/Al=2) and some Al2O3 (white dots) are formed. XRD results confirm SEM/EDS analyses. Indeed, XRD pattern shown in Fig. 1(f) demonstrates that only Ti2AlNy peaks are detected for N content in the range 0.9-1.0. In addition to Ti2AlNy, titanium aluminides (TixAly) and Ti4AlNy are respectively detected starting with 2Ti:1.05Al:0.85N and 2Ti:1.05Al:1.05N nominal compositions.

Comparing the XRD patterns, the XRD peak corresponding to (006) planes progressively shifts towards higher 2θ values when N content (y) increases, while the (100) peak shift is less significant. In reference to the interplanar spacing calculation in hexagonal system shown in Equation 1,interplanar spacings (d) for ( 006) and (100) planes are 𝑐𝑐 8 � , and 3𝑎𝑎 4 � , respectively.

𝑑𝑑 = 1 4 3 � (ℎ 2 + ℎ𝑘𝑘 + 𝑘𝑘 2 ) 𝑎𝑎 2 + 𝑙𝑙 2 𝑐𝑐 2 (1)
Herein, we find that N vacancy has a stronger effect on the c lattice parameter than on a The lattice parameters (LPs) of Ti2AlNy as a function of N content, measured from Rietveld refinement of the XRD data using MAUD software, and calculated using ab initio calculations, are summarized in Table I and shown in Fig. 2a. Similar to the findings by Cabioch et al. [START_REF] Cabioc | Structural investigation of substoichiometry and solid solution effects in Ti2Al(Cx,N1-x)y compounds[END_REF] , the volumes of the unit cell remain almost constant independent of y, while different variations for a and c LPs are clearly found: both experimentally and theoretically, the lattice contraction of Ti2AlNy is strongly anisotropic, namely, a parameter slightly increases about 0.0062 Å while c parameter decreases about 0.0488 Å as the N content increases from 0.85 to 1.05. From Fig. 2b, the variation of the c/a ratio (about 0.56%) as a function of N content and consistency with the calculations indicates that the N vacancies are primarily randomly distributed.

The calculated trends of the LPs well reproduce the experimental measurements. The MAX crystal unit cell, as shown in Fig. 3a, can be considered as constituted of

[M6X] octahedrons and [M6A] trigonal prisms [START_REF] Hug | Electronic structures of and composition gaps among the ternary carbides Ti2MC[END_REF][START_REF] Hug | X-ray absorption spectroscopy, EELS, and full-potential augmented plane wave study of the electronic structure of Ti2AlC, Ti2AlN, Nb2AlC, and (Ti0.5Nb0.5)2AlC[END_REF] allowing us to describe the "211"

MAX phases [START_REF] Momma | VESTA: A three-dimensional visualization system for electronic and structural analysis[END_REF] . A cubic octahedron is the unit block of the binary MX, but the fourfold axis is absent in ternary MAX that results in a relaxation owing to the lowered symmetry. The parameter Od, defined as the ratio of the distances between two faces not in the basal planes (d1) and two opposite faces contained in the basal planes (d2) [START_REF] Hug | X-ray absorption spectroscopy, EELS, and full-potential augmented plane wave study of the electronic structure of Ti2AlC, Ti2AlN, Nb2AlC, and (Ti0.5Nb0.5)2AlC[END_REF] , can be used to estimate the non-cubic distortion of the octahedron. It is given by:

𝑂𝑂 𝑑𝑑 = 𝑑𝑑 1 𝑑𝑑 2 = √3 2 � 4𝑍𝑍 𝑇𝑇𝑇𝑇 2 ( 𝑐𝑐 𝑎𝑎 ) 2 + 1 12 (2)
The distortion parameter Pd, defined as the ratio of the M-M distance and the M-A distance [START_REF] Hug | Electronic structures of and composition gaps among the ternary carbides Ti2MC[END_REF] , allows estimating the distortion of the trigonal prism. It is given, in either 211 or 312 MAX phases, by:

𝑃𝑃 𝑑𝑑 = 1 � ( 1 4 -𝑍𝑍 𝑇𝑇𝑇𝑇 ) 2 ( 𝑐𝑐 𝑎𝑎 ) 2 + 1 3 (3)
For an ideal packing of hard spheres of equal diameter, a ratio for c/a as of 8� prisms (blue) represented with VESTA [START_REF] Momma | VESTA: A three-dimensional visualization system for electronic and structural analysis[END_REF] . (b) the Octahedron (Od, circles) and trigonal prism (Pd, squares) distortion parameters versus N content in Ti2AlNy.

Hardness and Young's moduli

To investigate the influence of N vacancies on Ti2AlNy mechanical properties, nanoindentation tests were conducted on samples HIPed from 2Ti:1.05Al:0.9N (Fig. 4a) and 2Ti:1.05Al:1.0N (Fig. 4b) nominal compositions. WDS analysis indicates that the chemical composition of the Ti2AlNy end-products is similar to the starting powders composition. The Al2O3 content, determined by image analysis (from ImageJ software) is about 2vol.% starting with 2Ti:1.05Al:1.0N nominal composition and close to 0 starting with 2Ti:1.05Al:0.9N. Here, the Al2O3 content decreases with the decrease of the AlN content in the initial reactant powder mixture. Thus, the presence of alumina is likely resulting from the oxidation of AlN starting powders.

Fig. 4c shows the variation of the intrinsic hardness and elastic modulus as a function of the indenter penetration depth for Ti2AlN0.9 and Ti2AlN1.0 compounds. The standard deviations result from an average over different grain orientations. From Fig. 4c, one can notice that hardness values strongly decrease with increasing penetration depth. Such an Indentation Size Effect (ISE) [35] was also observed in Ti3SnC2 [START_REF] Tromas | Mechanical Properties of Nanolaminate Ti3SnC2 Carbide Determined by Nanohardness Cartography[END_REF] , Ti3AlC2 [START_REF] Bei | Synthesis, Characterization, and Intrinsic Hardness of Layered Nanolaminate Ti3AlC2 and Ti3Al0. 8Sn0. 2C2 Solid Solution[END_REF] and Ti3(Al0.8Sn0.2)C2 [START_REF] Bei | Synthesis, Characterization, and Intrinsic Hardness of Layered Nanolaminate Ti3AlC2 and Ti3Al0. 8Sn0. 2C2 Solid Solution[END_REF] MAX phase materials. From Fig. 4c, Ti2AlNy nanoindentation modulus remains constant whatever is the penetration depth: the elastic modulus values of Ti2AlN0.9 and Ti2AlN are respectively 268±10 GPa and 278±10 GPa. Such a result demonstrates that the indenter shape calibration is correct and, consequently, that the ISE discussed above is not an artefact due to errors in indenter shape calibration. and 80MPa for 4 h from 2Ti:1.05Al:0.90N and 2Ti:1.05Al:1.0N. Hardness (circles) and nanoindentation modulus (squares) of Ti2AlN and Ti2AlN0.9 (c) as a function of the indenter penetration depth. The red dotted line fits the ISE using the function proposed in the Nix and Gao model [START_REF] Nix | Indentation size effects in crystalline materials: a law for strain gradient plasticity[END_REF] .

The variation of the hardness values is well-fitted by the same function as that proposed in the Nix and Gao model [START_REF] Nix | Indentation size effects in crystalline materials: a law for strain gradient plasticity[END_REF] , except for the higher load (i.e.: 100 mN).

From the fit shown in Fig. 4c, one can deduce that the intrinsic hardness values of Ti2AlN0.9 and Ti2AlN are respectively 8.1 and 9.7 GPa. For the 100 mN load, the size of the indents is in the order of the mean grain size. In that case, several grains are involved in the deformation process and grain boundaries play an important role in macroscopic deformation, as reported by Tromas et al. [START_REF] Tromas | Mechanical Properties of Nanolaminate Ti3SnC2 Carbide Determined by Nanohardness Cartography[END_REF] . For such high loads, nanoindentation conditions (loads, depth of the indent, number of grains involved in the deformation,…) are close to microindentation ones and the hardness values are similar to those reported in microindentation experiments. [START_REF] Barsoum | Processing and characterization of Ti2AlC, Ti2AlN and Ti2AlC0.5N0.5[END_REF] Up to now, the effect of vacancies on the mechanical properties of MAX phases has not been investigated. The hardness values of TiCx and TiNy decrease with increasing vacancy content [START_REF] Pierson | Handbook of Refractory Carbides & Nitrides: Properties, Characteristics, Processing and Apps[END_REF][START_REF] Jhi | Vacancy hardening and softening in transition metal carbides and nitrides[END_REF] . In substoichiometric TiNy, the integrated density of states(DOS)

demonstrates that Ti(3d)-N(2p) bonding states are reduced in the presence of vacancies, leading to a weakened bond strength, shear strength and hardness [START_REF] Magnuson | Bonding mechanism in the nitrides Ti2AlN and TiN: an experimental and theoretical investigation[END_REF] . Furthermore, it is predicted that the introduction of 12.5% of vacancies in TiNy significantly reduces its shear stiffness from 190 GPa to less than 150GPa [START_REF] Jhi | Vacancy hardening and softening in transition metal carbides and nitrides[END_REF] , agreeing well with the experimental results [START_REF] Jiang | Elastic constants and hardness of ion-beam-sputtered TiNx films measured by Brillouin scattering and depth-sensing indentation[END_REF] . A similar trend has been measured experimentally in TiCx [START_REF] Pierson | Handbook of Refractory Carbides & Nitrides: Properties, Characteristics, Processing and Apps[END_REF] .

For the ternary nitride Ti2AlN, intuitively, the N vacancies would play a softening role in the mechanical properties.

From Table II, the Cij are reduced due to the weaker M-X bonding, as the concentration of N vacancies is increased. In the hexagonal lattice, the C11 is the in-plane elastic response, which is larger than C33 derived by the out-of-plane distortion, indicating the average bonding along the in-plane is stronger than the out-of-plane. It is indeed true, somehow determined by the layered structure. However, with higher N deficiency, these two constants become similar as the MAX lattice is becomes more isotropic, which is also shown in Fig. 2. When C33 is close to or smaller than C11, it might be feasible to break the M-A-M bonds significantly damaging the M-X-M bonds [START_REF] Khazaei | Trends in electronic structures and structural properties of MAX phases: a first-principles study on M(2)AlC (M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta), M(2)AlN, and hypothetical M(2)AlB phases[END_REF][START_REF] Khazaei | The effect of the interlayer element on the exfoliation of layered Mo2AC (A = Al[END_REF] . This would open a new route to obtaining the derivative 2D MXenes with clean surfaces by means of mechanical exfoliation, instead of the currently chemical etching processes that lead to complicated surface terminal structures and compositions.

Table I . Lattice parameters, Ti-Al and Ti-N distances (R), Wyckoff position of Ti atom (zTi), and the volume of the unit cell (V), and the formation energy of N vacancy (Ef) of Ti2AlNy (y: 0.85-1.05). The relative volume changes in theory are shown in the parenthesis.

While the shear, bulk and Young's moduli are steadily suppressed with the increasing N vacancy concentration, the three moduli show different magnitudes of response of 13.1%, 7.3%, 11.7%, respectively. Experimentally, for pristine Ti2AlN, the Young's modulus was found to be 278±10GPa, which is in a very good agreement with the result obtained by Resonant Ultrasound Spectroscopy (RUS) on a Ti2AlN bulk polycrystalline sample (E~277GPa [START_REF] Radovic | Elastic properties and phonon conductivities of Ti~ 3Al (C~ 0~.~ 5, N~ 0~.~ 5)~ 2 and Ti~ 2Al (C~ 0~.~ 5, N~ 0~.~ 5) solid solutions[END_REF] ) and 270±20 GPa in another independent measurement by nanoindentation [START_REF] Joelsson | 2</sub>< font face='verdana'> Al</font>< font face='verdana'> N</font></equation> thin films[END_REF] . Note that the calculated E of Ti2AlN with varying N vacancies are all falling into the regime between 258.2 and 293.1 GPa, which is in the line with the above measurements. We concluded that the perturbation of mechanical properties by N vacancies are different, and the incompressibility is the least affected.

Ny LP (Å) RTi-Al(Å) RTi-N(Å) zTi V (Å 3 ) Ef (

ab initio calculations

As discussed above, the combination of metallic and ceramic characteristics in MAX phases provide us with opportunities in many applications, where the formation of defects, their dynamics, and the consequences for the mechanical properties are critical. The formation energy for the varying N vacancies Ef was defined as Ef = (Ev+𝑛𝑛𝜇𝜇 𝑁𝑁 -Ep)/16, where Ev denotes the total energy of Ti2AlN phase with N vacancies,

𝜇𝜇 𝑁𝑁 is the chemical potential for a single N atom as of the positive, which is similar to the finding of N vacancy in Ti4AlN3. [46] It is thus demonstrated that the N vacancies are energetical costly. Moreover, the formation energy of N vacancies in the lower densities are higher than for higher ones.

Having the energetics of the N vacancy formation, we continue to elucidate the mechanical properties and the N vacancies induced changes. Here, we employed two widely used hardness models, Hv,Chen [START_REF] Chen | Modeling hardness of polycrystalline materials and bulk metallic glasses[END_REF] and Hv [START_REF] Tian | Microscopic theory of hardness and design of novel superhard crystals[END_REF] to analyze both defect-free and the deficient Ti2AlN. Note that Pugh's modulus ratio (k) defined as defined as G/B, as the indicator of a material's brittleness, is usually taken into account, and the above two models are expressed as:

Hv,Chen=2(k 2 G) 0.585 -3 (4)
Hv,Tian=0.92k 1.137 G 0.708 (5) We find that in the above two models the hardness is ~20 GPa and decreases monotonically with greater N deficiency. The reduction magnitude of Hv,Chen & Hv,Tian are about 16% and 10% from the pristine to the highest N deficient MAX phase. In Eq. It is well known that the bonding in the MAX phases are a combination of covalent, ionic and metallic bonding. To quantify the chemical bonding characters of Ti2AlN, we compare the ELF in both the N defect free and deficient (N0.90) MAX phases in Fig. 5. In We unravel the physical origin behind the electronic properties based on the two series of bonding: Ti(3d)-N(2p) and Ti(3d)-Al(3p). First of all, the primary characteristics of the chemical bonding could be deduced from the vacancy free MAX phase: i) the Fermi energy is dominated by Ti(3d) states, and the valley shaped DOS near the vicinity of Fermi energy is present. ii) The strong covalent bond is on the Ti(3d)-N(2p) hybridization is primarily situated between [-7, -6] & [-6, -4] eV with a pseudo-gap about -5.8 eV as a result of the band splitting. iii) The hybridization of Ti(3d)-Al(3p) states located at -2 and -1.4 eV is weaker intensity and also in higher energy than those of Ti(3d)-N(2p). In general, for N deficient MAX, the Ti(3d)-N(2p) hybridization states are become more broadened in contrast to the pristine MAX, due to the band splitting originating from the symmetry breaking. Moreover, although lacking hybridization around -6 eV, the primary Ti-N hybridization peaks between -5

and -4 eV do not change much with lower N. One might also notice that the a splitting width of 0.6 eV is maintained as Ny drops to 0.88, showing a width of about 0.5 eV. For Ti-Al, a similar tendency is also observed, where the two hybridizations near -2 and -1 eV does not experience an absence. We therefore suggest that MAX phases are a unique class of ceramic able to tolerate N deficiency. The methodology of synthesizing the controllable N concentration of MAX phase, would stimulate the development of nitrides MAX and MXenes in substoichiometry. Our results show that N deficiency in ternary MAX phase leads to a slow reduction of the bond lengths and mechanical properties, particularly the hardness, resulting in smaller changes than the binary TiNy [START_REF] Jhi | Vacancy hardening and softening in transition metal carbides and nitrides[END_REF][START_REF] Jiang | Elastic constants and hardness of ion-beam-sputtered TiNx films measured by Brillouin scattering and depth-sensing indentation[END_REF] .

Conclusions:

In order to reveal the relation between the X deficiency and the mechanical properties of MAX phase for potential structural materials and other applications, Ti2AlN was synthesized with varying N vacancy concentration by pressureless sintering and hot isostatic pressing. XRD and SEM prove that Ti2AlN could be synthesized with up to 15% N sub-stoichiometry. The reported method will promote the development of N deficient MAX and MXenes. XRD and ab initio calculations reveal that the lattice contraction of Ti2AlNy is mainly along the c-axis. The correspondence between the XRD and calculations indicates that the N vacancies are not highly ordered and are randomly distributed.

The elastic modulus and intrinsic hardness of substoichiometric Ti2AlN0.9

(268GPa and 8.1GPa respectively) are shown to be smaller than that of Ti2AlN (278GPa and 9.7GPa respectively). Overall the variation of moduli and hardness with up to 10% N deficiency is small in magnitude compared to the stoichiometric materials both experimentally and theoretically, confirming that the Ti2AlN exhibits a great potential for tolerating N defects. Ab initio calculations demonstrate that both Ti-N and Ti-Al bonding exhibit only a slight weakening by the N deficiency.

  99, very close to 2:1, which indicates the loss of approximately 5 at.% of the aluminum during reactive sintering. Starting with 2Ti:1.05Al:0.85N nominal composition [Fig.1(a)], titanium aluminide (TixAly) is detected in addition to the desired Ti2AlNy MAX phase. Starting with 2Ti:1.05Al:0.9N [Fig.1(b)], 2Ti:1.05Al:0.95N [Fig.1(c)], and 2Ti:1.05Al:1.0N [Fig. 1(d)] nominal compositions, only Ti2AlNy is observed; no impurities are detected. Starting with 2Ti:1.05Al:1.05N nominal composition [Fig.1(e)], SEM observations coupled with EDS analyses confirm that Ti4AlNy (Ti/Al=4), Ti2AlNy

Fig. 1 :

 1 Fig.1: (a-e) Morphologies of Ti2AlNy samples produced by pressureless sintering at 1400°C for 0.5h from 2Ti:1.05Al:yN (y=0.85,0.9,0.95,1.0 and 1.05) nominal compositions; (f) XRD patterns of different mixtures sintered at 1400°C for 0.5h. The (hkl) indexed peaks correspond to the Ti2AlNy MAX phase.

Fig. 2 .

 2 Fig.2. (a) Variation of the a and c -LPs as a function of nitrogen content in the Ti2AlNy compounds from XRD experiments (blue) and ab initio calculations (white). (b) the c/a ratio versus N content. Error bars for lattice parameters being too small to be clearly observable (Uncertainties of 5×10 -4 Å and 5×10 -3 Å was typically obtained for a and c parameters.

2 3

 2 will be satisfied, and for an ideal octahedron (cubic) and trigonal prism, a unity for both Od and Pd should also be reached. From the position of Ti atoms (ZTi), a and c lattice parameters measured above, both Od and Pd were experimentally determined and plotted as a function of the N content in Ti2AlNy in Fig.3b. Obviously, the relative distortion of the prism is similar to the one of the octahedron whatever the N content in Ti2AlNy, and the variations of Od and Pd are very weakly dependent on the N content, and showing the linearly increasing with the N content in both calculations and experiments. The underestimation of Od and vice versa for Pd might be due to the typical under-bonding of PBE functional in DFT. As shown in Table1, the distance of Ti-Al monotonically decreases with the increase of N content, but the variation of Ti-N distance is not evident, indicating the different interactions between TiN and Al (weak) as well as the Ti and N (weak). To clarify how the N deficiency inside of Ti6C octahedra impacts the Ti-Al & Ti-C bond as well as mechanical properties, we carry out the following work.

Fig. 3 :

 3 Fig. 3: (a) The unit cell of the Ti2AlN phase with [Ti6N] octahedrons (red) and [Ti6Al] trigonal

Figure 4 :

 4 Figure 4: Typical back-scattered images of Ti2AlN0.90 (a) and Ti2AlN (b) samples HIPed at 1400°C

1 2N2

 1 gas molecule, Ep stands for the total energy of the pristine Ti2AlN, and the n is the number of N vacancies in the supercell. Here, we consider the formation energy per unit cell by the division over the size of the supercell, 16. The formation energies of N vacancy in Ti2AlN are

( 4 - 5 )

 45 , the exponential of k, G are the key to deduce different slope of the change, but the overview of the moduli and hardness of nitride MAX provides with a clear insight to the role of the N deficiency. Overall, the drop of the hardness by introducing the N deficiency is not showing a dramatic change in strong contrast to the binary nitrides.

Fig. 5 (

 5 Fig.5(a), the predominate Ti-N bond shows a mixture of covalency (the blue bridge feature) and ionicity (the red spheres on N that gain extra electrons from Ti). For the Ti-Al bond, there is no evident covalent bonding but near the Ti surface, a red triangle electronic cloud appears and represents the Ti-Al bonding. This strong localized feature illustrates that the Ti-Al bonding more ionic feature, and neither covalent (showing shared electrons) nor a metallic (the level of localization would be about 0.5). Moreover, in Fig.5(b), the two N vacancies marked by black circles bring about the expansion of Ti-N block because of lack of chemical bonding. We could also observe a slight change within the interlayer spacing where the arrow pointed. It is thus found that the Ti-N block is suppressed in the N-poor regime, and the interlayer interactions are accompanied to be weakened by N vacancies.

Fig. 5 :

 5 Fig. 5 : 2D cut of the electronic localization function of (a) pristine Ti2AlN in the miller indices (-111), (b) 2D cut of ELF of Ti2AlN0.90 the view) with the miller indices of (-110) for capturing the defects (marked by spheres) and the changes in the interlayer spacing is marked by arrow. Note that the colour bar on the left represents the level of localization.

Fig. 6

 6 Fig.6 The total and projected DOS of Ti2AlN and N deficient MAX phases: (a) Ti2AlN, (b) Ti2AlN0.94, (c) Ti2AlN0.91, (d) Ti2AlN0.88. The total DOS (left side) is represented by black curves, and the projected DOS (right side) for Ti (3d), Al (3p), and N (2p) states is shown in blue, red and orange, respectively.

Table II .

 II The calculated elastic constants Cij and the derived shear moduli, G, bulk moduli, B, Young's moduli, E and hardness, H (GPa).

Acknowledgements

The experimental section was financially supported by the Fundamental Research Funds for the Central Universities (M19JB100020). W.S. and P.K. (initial theory) were supported as part of the Fluid Interface Reactions, Structures and Transport (FIRST) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.


domain: math info

text:
François Castella

Bruno Sericola

Random walk in the complete graph : hitting and cover times

Keywords: Markov chain, Random walk, Hitting times, Cover times, Complete graph, Moments, Asymptotic distribution

We consider the moments and the distribution of the hitting and cover times of a random walk in the complete graph. We study both the time needed to reach any subset of states and the time needed to visit all the states of a subset at least once. We obtain recurrence relations for the moments of all orders and we use these relations to analyze the asymptotic behavior of the hitting and cover times distributions when the number of states tends to infinity.

Introduction

In a random walk, the hitting time of a given subset of states is the first time at which the random walk reaches one of its states, while the cover time of a given subset of states is the first time at which the random walk reaches all its states. These random variables are quite important in many areas of distributed systems and networks. Many applications of random walks on graphs are proposed in [START_REF] Aldous | Applications of random walks on finite graphs[END_REF] as the analysis of electrical networks which are detailed in [START_REF] Doyle | Random Walks and Electrical Networks. The carus mathematical monographs[END_REF]. Conversely, electric networks are of great help to obtain expected values about random walks on graphs that would be more difficult to attain without the electric intuition. They have been extensively used to analyze particular random graphs, see for instance [START_REF] Palacios | Cover times for the stars[END_REF], [START_REF] Palacios | Expected hitting and cover times of random walks on some special graphs[END_REF] and the references therein. Visiting all the nodes of such systems allows the user to collect information about the performance or the reliabilty of the system. These visits can be made using random walks which are a particular case of Markov chains, since the transition probabilities from a given node are uniformly distributed among the number of its neighbours, but they can also be made using general transition probability matrices. Obviously the visiting time of all the nodes is a critical performance measure. For instance, in the context of graph theory for social networks, a notion of centrality has been used to assess the relative importance of nodes in a given network topology and a novel form of centrality based on the second order moments of return times in random walks was proposed in [START_REF] Kermarrec | Second order centrality: distributed assessment of nodes importance in complex networks[END_REF].

Several papers have already been written on both the hitting and cover times and most of these papers only consider expected values of these random variables. Very interesting surveys on random walks on graphs are proposed in [START_REF] Lovász | Random walks on graphs: A survey[END_REF] and [START_REF] Aldous | Reversible Markov Chains and Random Walks on Graphs[END_REF], while [START_REF] Motwani | Randomized Algorithms[END_REF] and [START_REF] Xia | Random walks: A review of algorithms and applications[END_REF] additionally contains many applications in both computer science and networks. It is shown in [START_REF] Feige | A tight upper bound on the cover time for random walks on graphs[END_REF] and [START_REF] Feige | A tight lower bound on the cover time for random walks on graphs[END_REF] that the expected cover time in a random walk with n states is at least (1 -o(1))n ln(n) and at most (4/27 + o(1))n 3 . In the case of a regular graph, it is at most 2n 2 , see [START_REF] Feige | A tight upper bound on the cover time for random walks on graphs[END_REF]. In [START_REF] Ikeda | The hitting and cover times of random walks on finite graphs using local degree information[END_REF], the authors first prove that for any transition probability matrix, both the expected hitting time and the expected cover time of the path graph is Ω(n 2 ). They also prove that for some particular Markov chains called degree-biased random walks, the expected hitting time is O(n 2 ) and the expected cover time is O(n 2 ln(n)). The method proposed in this paper is based on a matrix analysis of the Markov chain evolution equations. The advantage of our method is that it allows to deal with higher moments and with the distribution of the hitting and cover times. This method has been used in [START_REF] Sericola | Hitting times on the lollipop graph[END_REF] to analyze the hitting times on the lollipop graph which consists of a complete graph (a clique) and a path graph. One end of the path graph is connected to one vertex of the clique, and its other end is the target vertex for the hitting times. We have proposed a general recursive way to obtain all the moments of the hitting times on the lollipop graph and we have used this recursion to analyze the asymptotic behavior of both the moments and the distribution of the hitting times when the number of nodes tends to infinity.

In this paper, we consider the moments, the distribution and the asymptotic behavior of the hitting and cover times of a random walk in the complete graph, with and without self-loops. The complete graph or full mesh topology is expensive to design but it clearly provides a high degree of reliability since the failures of several edges will not impact communications. It is quite important as for example in the military domain. Our analysis is based on the general results obtained in [START_REF] Sericola | On cover times of Markov chains[END_REF] for hitting and cover times in general Markov chains.

The remainder of the paper is organized as follows. In section 2, we recall the general results of [START_REF] Sericola | On cover times of Markov chains[END_REF] on the analysis of the hitting time and the cover time of a subset of the state space both in terms of distribution and moments. In section 3, we analyze these results when the graph is a complete graph with self-loops. We obtain simple recurrence relations to compute all the moments of both the hitting and cover times. We also give the asymptotic behavior of these moments when the number of states tends to infinity and we deduce the corresponding asymptotic distributions. We consider in section 4 the case of the complete graph without self-loops. Section 5 concludes the paper.

Basic results

Consider a homogeneous discrete-time Markov chain X = {X n , n ∈ N} over a finite state space S. We denote by P the transition probability matrix of X. We suppose that X is irreducible and, for every non empty subset A of S, we define the hitting time T A of subset A as the first instant at which one state of A is reached by the Markov chain X, i.e.

T A = inf{n ≥ 0 | X n ∈ A}.
The irreducibility of X implies that, for every A ⊂ S, T A is finite almost surely. We also define, for every non empty subset A of S, the cover time C A of subset A as the first instant at which all the states of A have been reached by the Markov chain X, i.e.

C A = inf{n ≥ 0 | A ⊆ {X 0 , . . . , X n }}.
The cover time of the Markov chain X, which is the time needed to reach all the states of S, is simply denoted by C and is given by C = C S . The complementary subset of A in S, S \ A, is denoted simply by A c . The partition {A, A c } of S induces a decomposition of P into four submatrices as

P = P A P A,A c P A c ,A P A c ,
where matrix P A (resp. P A c ) contains the transition probabilities between states of A (resp. A c ) and matrix P A,A c (resp. P A c ,A ) contains the transition probabilities from states of A (resp. A c ) to states of A c (resp.

A). We also denote by 1 the column vector with all its entries equal to 1, its dimension being specified by the context. With these notations, we have, for all i ∈ A c , j ∈ A and ≥ 1, see for instance [START_REF] Sericola | Markov Chains. Theory, Algorithms and Applications[END_REF],

P{T A = , X T A = j | X 0 = i} = (P -1 A c P A c ,A ) i,j . (1)
Summing (1) over all j ∈ A, leads to

P{T A = | X 0 = i} = (P -1 A c P A c ,A 1) i . (2)
In the same way, summing (1) over all ≥ 1, yields

P{X T A = j | X 0 = i} = ∞ =1 (P -1 A c P A c ,A ) i,j = (I -P A c ) -1 P A c ,A i,j . (3)
From (1), we also obtain

E(T A 1 {X T A =j} | X 0 = i) = ∞ =1 (P -1 A c P A c ,A ) i,j = (I -P A c ) -2 P A c ,A i,j . (4)
Using Theorem 1 of [START_REF] Sericola | Hitting times on the lollipop graph[END_REF], the moments of T A and T A 1 {X T A =j} are given, for every r ≥ 1, i ∈ A c and j ∈ A, by

E(T r A 1 {X T A =j} | X 0 = i) = r =1 c r, (I -P A c ) -( +1) P A c ,A i,j (5)
and

E(T r A | X 0 = i) = r =1 c r, (I -P A c ) -1 i (6)
where, for every = 1, . . . , r, c r, = (-1) r j=1 (-1) j j j r .

This gives for instance

E(T A | X 0 = i) = (I -P A c ) -1 1 i , E(T 2 A | X 0 = i) = -(I -P A c ) -1 1 i + 2 (I -P A c ) -2 1 i , E(T 3 A | X 0 = i) = (I -P A c ) -1 1 i -6 (I -P A c ) -2 1 i + 6 (I -P A c ) -3 1 i , E(T 4 A | X 0 = i) = -(I -P A c ) -1 1 i + 14 (I -P A c ) -2 1 i -36 (I -P A c ) -3 1 i + 24 (I -P A c ) -4 1 i .
Let us now consider the cover time C A of subset A. First of all, for any event u and for any random variable U , we introduce the notation P i {u} = P{u | X 0 = i} and E i (U ) = E(U | X 0 = i). Note that, for every j ∈ S, we have

C {j} = T {j} .
Moreover, for every ≥ 0 and for every i ∈ A, with |A| ≥ 2, we have, by definition of C A ,

P i {C A = } = P i {C A\{i} = }.
We thus only need to consider the case where the initial state belongs to the subset A c . If α denotes the initial distribution of X, we get

P{C A = } = i∈S α i P i {C A = } = i∈A α i P i {C A\{i} = } + i∈A c α i P i {C A = }.
The distribution of the cover time C of the Markov chain is then given by

P{C = } = i∈S α i P i {C = } = i∈S α i P i {C S\{i} = }.
The following recursive expression of the distribution of the cover time C A of subset A has been proved in [START_REF] Sericola | On cover times of Markov chains[END_REF]. For every A ⊂ S such that |A| ≥ 2, i ∈ A c and ≥ 0, we have

P i {C A = } =          0 if 0 ≤ < |A| -|A|+1
h=1 j∈A

(P h-1 A c P A c ,A ) i,j P j {C A\{j} = -h} if ≥ |A|. (8)
We also obtained in [START_REF] Sericola | On cover times of Markov chains[END_REF] the following recursive expression of the moments of the cover time of subset A starting from state i ∈ A c . For every A ⊂ S such that |A| ≥ 2, r ≥ 1 and i ∈ A c , we have

E i (C r A ) = E i (T r A ) + r m=1 r m j∈A E i (T r-m A 1 {X T A =j} )E j (C m A\{j} ). (9)
In the following sections, we apply these results to the complete graph to obtain the asymptotic behavior of the moments and distribution of both the hitting and cover times.

3 Analysis of the complete graph with self-loops A random walk on the complete graph with self-loops is a discrete-time Markov chain on the state space S = {0, 1, . . . , n -1}, n ≥ 2, with transition probability matrix P given, for every u, v ∈ S, by P u,v = 1/n.

Hitting times on the complete graph with self-loops

Let A be a proper subset of S with k states, that is |A| = k ∈ {1, . . . , n -1}. The symmetry of the complete graph implies that the hitting time distribution of subset A is the same for any subset of S containing k states. We thus simply use the notation T k for T A . Moreover, we suppose without any loss of generality that the initial state always belongs to the complementary subset A c of A. To simplify the notation, using the symmetry again, we will sometimes drop the initial state because the results remain the same for every initial state i ∈ A c .

Using relation (2), we have for every ≥ 0 and i ∈ A c ,

P i {T k > } = ∞ j= +1 (P j-1 A c P A c ,A 1) i = (P A c 1) i . Since |A c | = n -k we have P A c 1 = (1 -k/n)1
which recursively leads to

P A c 1 = 1 - k n 1 and thus P{T k > } = 1 - k n , (10)
which means that T k is geometrically distributed with parameter k/n. Using relation ( 6), we have, for every r ≥ 1,

E(T r k ) = r =1 c r, n k , (11)
where the coefficients c r, have been defined in [START_REF] Ikeda | The hitting and cover times of random walks on finite graphs using local degree information[END_REF]. The first moments of T k are then given, using [START_REF] Ikeda | The hitting and cover times of random walks on finite graphs using local degree information[END_REF], by

E(T k ) = n k , E(T 2 k ) = n(2n -k) k 2 , E(T 3 k ) = n(6n 2 -6kn + k 2 ) k 3 , E(T 4 k ) = n(24n 3 -36kn 2 + 14k 2 n -k 3 ) k 4 .
Since c r,r = r! we obtain, for every fixed value of k,

E(T r k ) ∼ n-→∞ r! n k r
and, from relation [START_REF] Motwani | Randomized Algorithms[END_REF], we get, for every fixed value of k,

lim n-→∞ P{kT k /n > t} = e -t ,
which means that the sequence of random variables kT k /n converges in distribution, when n tends to infinity, to a random variable that is exponentially distributed with rate 1.

Cover times on the complete graph with self-loops

In the same way, if A is a proper subset of S with k states, that is |A| = k ∈ {1, . . . , n-1} then the symmetry of the complete graph implies that the cover time distribution of subset A is the same for any subset of S containing k states. We thus simply use the notation C k for C A . Moreover, we suppose without any loss of generality, as we did for the hitting times, that the initial state always belongs to the complementary subset A c of A. In the same way, to simplify the notation, using the symmetry again, we will sometimes drop the initial state because the results remain the same for every initial state i ∈ A c . We then have for any initial distribution,

E(C) = E i (C S\{i} ) = E i (C n-1 ) = E(C n-1 ).
For every n ≥ 1, we denote by h n the harmonic sum, i.e.

h n = n k=1 1 k
and we set h 0 = 0. The symmetry of the complete graph implies that E i (C r A ) has the same value for every subset A with k states and for every i ∈ A c . We then write

E i (C r A ) = E(C r k ). In the same way, E j (C m A\{j} )
is related to a set with k -1 states and is the same for every j ∈ A, so we write

E j (C m A\{j} ) = E(C m k-1
). Relation (9) can then be written, for k = 1, . . . , n -1 and r ≥ 1, as

E (C r k ) = E (T r k ) + r m=1 r m E(C m k-1 ) j∈A E i (T r-m A 1 {X T A =j} ) = E (T r k ) + r m=1 r m E(C m k-1 )E(T r-m k ). (12)
The following result gives the mean cover time of a subset of states with k states in the complete graph.

Theorem 1 For every k = 1, . . . , n -1, we have

E(C k ) = nh k .
In particular, when k = n -1, we have

E(C) ∼ n-→∞ n ln(n).
Proof. Let A be a subset of S with k states. The expected cover time of subset A is given by relation [START_REF] Palacios | Expected hitting and cover times of random walks on some special graphs[END_REF] with r = 1. This gives, using the previous results on the hitting times

E(C k ) = E(T k ) + E(C k-1 ) = n k + E(C k-1 )
.

with E(C 1 ) = E(T 1 ) = n. This simple recursion leads to

E(C k ) = nh k . By taking k = n -1, we get E(C) = E(C n-1 ) = nh n-1 and thus E(C) ∼ n-→∞ n ln(n),
which completes the proof.

Observe that this well-known result can also be obtained using geometric random variables. Indeed, it is easy to check that

E(C) = E geo n -1 n + E geo n -2 n + • • • + E geo 1 n = nh n-1 .
Nevertheless the relations obtained in the proof of Theorem 1 are important because they are needed to get the next results. In the following theorem we consider the second order moment of the cover time of a subset of states with k states.

Theorem 2 For every k = 1, . . . , n -1, we have

E C 2 k = n 2 h 2 k + n 2 k =1 1 2 -nh k .
In particular, when k = n -1, we have

E(C 2 ) ∼ n-→∞ n 2 ln 2 (n).
Proof. Let A be a subset of S with k states. The second order moment of the cover time of subset A is given by relation ( 12) with r = 2, that is

E C 2 k = E T 2 k + 2E(C k-1 )E(T k ) + E(C 2 k-1 ).
The previous results on the hitting times and Theorem 1 which give

E (T k ) = n/k and E (C k-1 ) = nh k-1 , yield E C 2 k = n(2n -k) k 2 + 2n 2 h k-1 k + E C 2 k-1 , with E C 2 1 = E T 2 1 = n(2n -1
). This leads to

E C 2 k = n k =1 2n - 2 + 2n 2 k =2 h -1 = 2n 2 k =2 h -1 + k =1 1 2 -nh k . ( 13
)
We consider separately the term between parenthesis. Introducing the notation

ψ k = k =2 h -1 and ϕ k = k =1 h ,
we have

ψ k + k =1 1 2 = ϕ k = k =1 1 r=1 1 r = k r=1 1 r k =r 1 = k r=1 1 r (h k -h r-1 ) = h 2 k - k r=2 h r-1 r = h 2 k -ψ k .
It follows that

ψ k = h 2 k 2 - 1 2 k =1 1 2 .
Inserting this relation inside relation [START_REF] Sericola | Markov Chains. Theory, Algorithms and Applications[END_REF] leads to

E C 2 k = n 2 h 2 k + n 2 k =1 1 2 -nh k . 6 By taking k = n -1, we get E(C 2 ) = E C 2 n-1 = n 2 h 2 n-1 + n 2 n-1 =1 1 2 -nh n-1 and E(C 2 ) ∼ n-→∞ n 2 ln 2 (n),
which completes the proof.

We now consider the third order moment of the cover time of a subset of states with k states.

Theorem 3 For every k = 1, . . . , n -1, we have

E C 3 k = 3n 3 k =1 h 2 + 3n 3 k =2 h 2 -1 + 3n 3 h k k =1 1 2 -6n 2 k =1 h + nh k .
In particular, when k = n -1, we have

E(C 3 ) ∼ n-→∞ n 3 ln 3 (n).
Proof. Let A be a subset of S with k states. The third order moment of the cover time of subset A is given by relation ( 12) with r = 3, that is

E C 3 k = E(T 3 k ) + 3E(C k-1 )E(T 2 k ) + 3E(C 2 k-1 )E(T k ) + E(C 3 k-1 ), with E C 3 1 = E(T 3
1 ). This gives

E C 3 k = k =1 E(T 3 ) + 3 k =2 E(C -1 )E(T 2 ) + 3 k =2 E(C 2 -1 )E(T ).
All the terms on the right hand side have already been calculated in the previous section on hitting times and in Theorems 1 and 2. We thus have

E C 3 k = n k =1 6n 2 -6 n + 2 3 + 3n 2 k =2 (2n -)h -1 2 + 3n 3 k =2 h 2 -1 + 3n 3 k =2 1 -1 u=1 1 u 2 -3n 2 k =2 h -1 = 6n 3 k =2 h -1 2 + k =1 1 3 + 3n 3 k =2 h 2 -1 + 3n 3 k =2 1 -1 u=1 1 u 2 -6n 2 k =2 h -1 + k =1 1 2 + nh k = 6n 3 k =1 h 2 + 3n 3 k =2 h 2 -1 + 3n 3 h k k =1 1 2 - k =1 h 2 -6n 2 k =1 h + nh k = 3n 3 k =1 h 2 + 3n 3 k =2 h 2 -1 + 3n 3 h k k =1 1 2 -6n 2 k =1 h + nh k .
By taking k = n -1, we get

E(C 3 ) = E(C 3 n-1 ) = 3n 3 n-1 =1 h 2 + 3n 3 n-1 =2 h 2 -1 + 3n 3 h n-1 n-1 =1 1 2 -6n 2 n-1 =1 h + nh n-1 .
We denote respectively by A n,1 , A n,2 , A n,3 and A n,4 the four first sums of the right hand side of this expression. We then have

A n,1 = n-1 =1 h 2 .
Since h / 2 ∼ -→∞ ln( )/ 2 and the Bertrand series ln( )/ 2 converges, we have

lim n-→∞ A n,1 = A ≈ 2.404.
Concerning the second sum

A n,2 = n-1 =2 h 2 -1 ,
we observe that h 2 / ∼ -→∞ ln 2 ( )/ and the Bertrand series ln 2 ( )/ diverges. Using the fact that the function ln 2 (x)/x is decreasing for x > e 2 , we obtain

n-1 =2 ln 2 ( ) ∼ n-→∞ n 1 ln 2 (x) x dx = ln 3 (n) 3 .
Hence

A n,2 ∼ n-→∞ ln 3 (n) 3 .
The third sum A n,3 verifies

A n,3 = h n-1 n-1 =1 1 2 ∼ n-→∞ π 2 ln(n) 6 .
The last sum A n,4 is the term ϕ n-1 already considered in the proof of Theorem 2. It follows that we have

A n,4 = n-1 =1 h = h 2 n-1 2 + 1 2 n-1 =1 1 2 ∼ n-→∞ ln 2 (n) 2 .
Gathering these four results leads to

E(C 3 ) ∼ n-→∞ n 3 ln 3 (n),
which completes the proof.

More generally, the subsequent Theorem 3 allows for a recursive computation of all the moments of the cover time of a subset of states with k states. It also gives the asymptotoic behavior of all the moments. It is based on relation [START_REF] Palacios | Expected hitting and cover times of random walks on some special graphs[END_REF]. To prove this theorem, we first need the following lemma which gives a recursive expression of the Stirling numbers of the second kind. The Stirling numbers of the second kind can be defined, for r ≥ 1 and = 1, . . . , r, by

S(r, ) = 1 ! j=1 (-1) -j j j r . ( 14
)
These numbers are well-known in combinatorial analysis and when r ≤ we have, see for instance relation (12.17) in section II.12 of [START_REF] Feller | An introduction to probability theory and its applications[END_REF],

S(r, ) = 0 if r < 1 if r = . (15)
The following original lemma on the Stirling numbers turns out to be useful in the subsequent analysis.

Lemma 4 For every r ≥ 2 and = 2, . . . , r, we have

S(r, ) = 1 r-1 j= -1 r j S(j, -1). ( 16
)
Proof. Let r ≥ 2 and = 2, . . . , r. From relation [START_REF] Sericola | On cover times of Markov chains[END_REF] we have

1 r-1 j= -1 r j S(j, -1) = 1 ! r-1 j= -1 r j -1 k=1 (-1) -1-k -1 k k j = 1 ! -1 k=1 (-1) -1-k -1 k r-1 j= -1 r j k j
Observe that this last sum can be written as

r-1 j= -1 r j k j = (k + 1) r -k r - -2 j=0 r j k j .
We thus get

1 r-1 j= -1 r j S(j, -1) = 1 ! -1 k=1 (-1) -1-k -1 k [(k + 1) r -k r ] - 1 ! -1 k=1 (-1) -1-k -1 k -2 j=0 r j k j .
(17) The term containing the two sums can be written as

1 ! -1 k=1 (-1) -1-k -1 k -2 j=0 r j k j = 1 ! -2 j=0 r j -1 k=1 (-1) -1-k -1 k k j = 1 !   -1 k=1 (-1) -1-k -1 k + -2 j=1 r j -1 k=1 (-1) -1-k -1 k k j   .
Since 1 ≤ j ≤ -2, we have from [START_REF] Sericola | Hitting times on the lollipop graph[END_REF],

-1 k=1 (-1) -1-k -1 k k j = 0. It follows that 1 ! -1 k=1 (-1) -1-k -1 k -2 j=0 r j k j = 1 ! -1 k=1 (-1) -1-k -1 k = (-1) ! . Relation (17) thus becomes 1 r-1 j= -1 r j S(j, -1) = 1 ! (-1) -1 + -1 k=1 (-1) -1-k -1 k (k + 1) r - -1 k=1 (-1) -1-k -1 k k r = 1 ! (-1) -1 + k=2 (-1) -k -1 k -1 k r + -1 k=1 (-1) -k -1 k k r = 1 ! (-1) -1 + -1 k=2 (-1) -k k k r + r + (-1) -1 ( -1) = 1 ! (-1) -1 + k=1 (-1) -k k k r -r -(-1) -1 + r + (-1) -1 ( -1) = 1 ! k=1 (-1) -k k k r = S(r, ),
which completes the proof.

We are now able to prove the following theorem.

Theorem 5 For every k = 1, . . . , n -1 and for every r ≥ 1, E(C r k ) is a r degree polynomial in n, i.e.

E(C

r k ) = r =1 a (k) r, n .
Moreover, for every = 1, . . . , r, we have

a (k) r, ∼ k-→∞
(-1) r+ S(r, ) ln (k).

In particular, when k = n -1, we have

E(C r ) ∼ n-→∞ n r ln r (n).
Proof. Let A be a subset of S with k states. The r-th order moment of the cover time of subset A is given by relation [START_REF] Palacios | Expected hitting and cover times of random walks on some special graphs[END_REF], that is

E (C r k ) = E(T r k ) + r m=1 r m E(C m k-1 )E(T r-m k ) = E(T r k ) + r-1 m=1 r m E(C m k-1 )E(T r-m k ) + E(C r k-1 ), with E (C r 1 ) = E(T r 1
). This yields

E (C r k ) = k j=1 E T r j + r-1 m=1 r m k j=2 E(C m j-1 )E(T r-m j ). ( 18
)
We now proceed by recurrence. For k = 1, we have by relation [START_REF] Palacios | Cover times for the stars[END_REF],

E (C r 1 ) = E(T r 1 ) = r =1 c r, n ,
which is a r degree polynomial in n as claimed. Suppose that the result in the theorem is true for indices i = 1, . . . , r -1 and j = 1, . . . k -1, i.e. suppose that

E(C i j ) = i u=1 a (j)
i,u n u , for i = 1, . . . , r -1 and j = 1, . . . k -1.

We then have from relation (18) and the results on hitting times

E (C r k ) = k j=1 E T r j + r-1 i=1 r i k j=2 E(C i j-1 )E(T r-i j ) = k j=1 r =1 c r, n j + r-1 i=1 r i k j=2 i u=1 a (j-1) i,u n u r-i v=1 c r-i,v n j v = r =1   c r, k j=1 1 j   n + r-1 i=1 r i k j=2 i u=1 a (j-1) i,u n u r-i v=1 c r-i,v j v n v ,
where the c r, are given by [START_REF] Ikeda | The hitting and cover times of random walks on finite graphs using local degree information[END_REF]. This formula already establishes that E (C r k ) is a r degree polynomial in n. To be able to discuss the value of the a (k) r, , we need to push the computation a bit further. Observe that the product of the last two parenthesis is the product of two polynomials with different degrees. This product can be expressed as follows

i u=1 a (j-1) i,u n u r-i v=1 c r-i,v j v n v = r =2   ( -1)∧i t=1∨( -r+i) a (j-1) i,t c r-i, -t j -t   n ,
where x ∨ y = max(x, y) and x ∧ y = min(x, y). We thus obtain

E (C r k ) = r =1   c r, k j=1 1 j   n + r-1 i=1 r i k j=2 r =2   ( -1)∧i t=1∨( -r+i) a (j-1) i,t c r-i, -t j -t   n = r =1   c r, k j=1 1 j   n + r =2 r-1 i=1 r i k j=2   ( -1)∧i t=1∨( -r+i) a (j-1) i,t c r-i, -t j -t   n . ( 19
)
We are now in position to analyze the coefficients a

(k)
r, of polynomial E (C r k ) which are given by ( 19). For = 1 the coefficient a

(k) r,1 of n is a (k) r,1 = c r,1 k j=1 1 j = (-1) r+1 h k ,
and the coefficient a

(k) r, of n for = 2, . . . , r is a (k) r, = c r, k j=1 1 j + r-1 i=1 r i k j=2   ( -1)∧i t=1∨( -r+i) a (j-1) i,t c r-i, -t j -t   . (20)
We prove by recurrence that for every r ≥ 1 and = 1, . . . , r -1, we have

lim k-→∞ a (k) r, ln (k) = (-1) r+ S(r, ),
where S(r, ) are the Stirling numbers of the second kind defined by [START_REF] Sericola | On cover times of Markov chains[END_REF]. It is well-known that the Stirling numbers are positive. The result is true for = 1 since a

(k)
r,1 = (-1) r+1 h k , and S(r, 1) = 1. We consider now the case = 2, . . . , r. Suppose that lim j-→∞ a (j) i,t ln t (j) = (-1) i+t S(i, t), for i = 1, . . . , r -1 and t = 1, . . . i.

Exchanging the sums over j and t in relation (20) leads, for = 2, . . . , r, to

a (k) r, = c r, k j=1 1 j + r-1 i=1 r i ( -1)∧i t=1∨( -r+i) c r-i, -t k j=2 a (j-1) i,t j -t .
We decompose the sum over index i in two distinct sums i ≤ -2 and i ≥ -1. We thus get

a (k) r, = c r, k j=1 1 j + -2 i=1 r i i t=1∨( -r+i) c r-i, -t k j=2 a (j-1) i,t j -t + r-1 i= -1 r i -1 t=1∨( -r+i) c r-i, -t k j=2 a (j-1) i,t j -t .
In the second sum over t, we extract the term corresponding to t = -1. We obtain since c r-i,1 = (-1) r-i+1

a (k) r, = c r, k j=1 1 j + -2 i=1 r i i t=1∨( -r+i) c r-i, -t k j=2 a (j-1) i,t j -t + r-1 i= -1 r i -2 t=1∨( -r+i) c r-i, -t k j=2 a (j-1) i,t j -t + (-1) r r-1 i= -1 (-1) i-1 r i k j=2 a (j-1) i, -1 j . (21)
We consider separately the four sums over j. Since ≥ 2, the first sum over j converges when k tends to infinity to the Riemann series ζ( ), i.e.

∞ j=1 1 j = ζ( ) < ∞.
The second and third sums over j can be written as

k j=2 a (j-1) i,t j -t = k j=2 a (j-1) i,t ln t (j) ln t (j) j -t .
The recurrence hypothesis tells us that the ratio a

(j-1) i,t
/ ln t (j) converges to (-1) i+t S(i, t) when j tends to infinity. This implies that this ratio is bounded by constant uniformly in j, i.e. there is a constant β i,t such that for all j we have a

(j-1) i,t ln t (j) ≤ β i,t .
We thus have for all k ≥

-t ≤ β i,t k j=2 ln t (j) j -t . 2, k j=2 a (j-1) i,t j
The series on the right hand side is a converging Bertrand series whenever -t ≥ 2, a condition that is met in the second and third sums since t ≤ -2. It follows that the second and third sums converge, i.e.

∞ j=2 a (j-1) i,t j -t < ∞.
We now consider the fourth sum over j. Since the sequence a (j-1)

i, -1 / ln -1 (j) converges to the non zero value (-1) i+ -1 S(i, -1) when j tends to infinity, it always keeps the same sign as (-1) i+ -1 S(i, -1) after a certain rank.

Observe now that the function f defined by

f (x) = ln -1 (x) x
is non negative and decreasing on the interval [e -1 , +∞). We thus have

k j=2 ln -1 (j) j ∼ k-→∞ k 1 ln -1 (x) x dx = ln (k) .
It follows that a

(j-1) i, -1 j = a (j-1) i, -1 ln -1 (j) ln -1 (j) j ∼ j-→∞ (-1) i+ -1 S(i, -1) ln -1 (j) j .
The series j ln -1 (j)/j being divergent we have

k j=2 a (j-1) i, -1 j ∼ k-→∞ (-1) i+ -1 S(i, -1) k j=2 ln -1 (j) j ∼ k-→∞ (-1) i+ -1 S(i, -1) ln (k) .
Using these four results in relation ( 21) and appliyng Lemma 4, we obtain

lim k-→∞ a (k) r, ln (k) = (-1) r+ r-1 i= -1 r i S(i, -1) = (-1) r+ S(r, ).
By taking k = n -1, we obtain, since S(r, r) = 1,

E(C r ) = E(C r n-1 ) = r =1 a (n-1) r, n ∼ n-→∞ n r ln r (n),
which completes the proof.

We can now determine the limiting behavior of the distribution of the cover time

C = C n-1 .
Theorem 6 For all t ∈ R \ {1}, we have,

lim n-→∞ P C n ln(n) > t = 1 if t > 1 0 if t < 1.
Proof. From the second part of Theorem 5, we have for every r ≥ 0,

lim n-→∞ E C n ln(n) r = 1.
Let Z be the constant random variable defined by Z = 1. For every r ≥ 0, we have E(Z r ) = 1. Thus the power series r≥0 E(Z r )x r /r! has a positive radius (equal to +∞). This implies, from Theorem 30.1 of [START_REF] Ikeda | The hitting and cover times of random walks on finite graphs using local degree information[END_REF], that the random variable Z is uniquely determined by its moments. It follows from Theorem 30.2 of [START_REF] Ikeda | The hitting and cover times of random walks on finite graphs using local degree information[END_REF] that

C n ln(n) L --→ Z = 1, as n -→ ∞, which means that lim n-→∞ P C n ln(n) > t = 1 if t < 1 0 if t > 1,
which completes the proof.

Observe that the previous theorem does not allow us to conclude when t = 1 because the convergence in distribution occurs only at the points t for which the limiting distribution is continuous.

We can solve this problem by considering the result of [START_REF] Sericola | On cover times of Markov chains[END_REF], which tell us that in the complete graph with self-loops we have

P{C > } =          1 if 0 ≤ < n -1 n-2 j=0 (-1) j n -1 j + 1 n -(j + 1) n if ≥ n -1. ( 22
)
Using this result, we obtain the following limiting distribution which completes the result of Theorem 6 at point t = 1.

Theorem 7 For every t ∈ R, we have

lim n-→∞ P C -n ln(n) n > t = 1 -exp -e -t ,
which is known as the standard Gumbel distribution. At point t = 0, we have

lim n-→∞ P C n ln(n) > 1 = 1 -e -1 ≈ 0.63212.
Proof. Taking = n ln(n) + nt in (22), we obtain

P{C > n ln(n) + nt} = n-2 j=0 (-1) j n -1 j + 1 n -(j + 1) n n ln(n)+nt
.

Introducing the notation

u n,j = (-1) j n -1 j + 1 n -(j + 1) n n ln(n)+nt 1 {j≤n-2} ,
we have

u n,j = (-1) j (n -1) • • • (n -(j + 1)) (j + 1)! exp [n ln(n) + nt] ln 1 - j + 1 n 1 {j≤n-2} .
For any fixed j, we have

u n,j ∼ n-→∞ (-1) j n j+1 (j + 1)! e -(j+1)[ln(n)+t] 1 {j≤n-2} = (-1) j e -(j+1)t (j + 1)! 1 {j≤n-2} .
We thus have, for every j ≥ 0, lim n-→∞ u n,j = (-1) j e -(j+1)t (j + 1)! .

Moreover, since ln(1 -x) ≤ -x for all x ∈ [0, 1), we obtain

|u n,j | = (n -1) • • • (n -(j + 1)) (j + 1)! exp [n ln(n) + nt] ln 1 - j + 1 n 1 {j≤n-2} ≤ n j+1 (j + 1)! e -(j+1)[ln(n)+t] = e -(j+1)t (j + 1)!
and the series j≥0 e -(j+1)t /(j + 1)! is convergent with

∞ j=0
e -(j+1)t (j + 1)! = exp e -t -1.

The Lebesgue dominated convergence theorem for series allows us to conclude that lim

n-→∞ ∞ j=0 u n,j = ∞ j=0 (-1) j e -(j+1)t (j + 1)! = 1 -exp -e -t ,
which is the complementary standard Gumbel distribution function. Take t = 0 to complete the proof.

Observe that, since the limit Z = 1 is constant, we also have the convergence in probability of C/(n ln(n)) to Z = 1, that is, for any ε > 0

lim n-→∞ P C n ln(n) -1 > ε = 0.

Analysis of the complete graph without self-loops

A random walk on the complete graph without self-loops is a discrete-time Markov chain on the state space S = {0, 1, . . . , n -1}, n ≥ 2, with transition probability matrix P given, for every u = v, by P u,v = 1/(n -1) and P u,u = 0. All the results of this section are close to those of the complete graph with self-loops.

Hitting times on the complete graph without self-loops

Let A be a proper subset of S with k states, that is |A| = k ∈ {1, . . . , n -1}. To simplify the notation, using again the symmetry, we will drop sometimes the initial state because the results remain the same for every initial state i ∈ A c . Using relation (2), we have for every ≥ 0 and i ∈ A c ,

P i {T k > } = ∞ j= +1 (P j-1 A c P A c ,A 1) i = (P A c 1) i . Since |A c | = n -k we have P A c 1 = (1 -k/(n - 1
))1 which recursively leads to

P A c 1 = 1 - k n -1 1 and thus P{T k > } = 1 - k n -1 , (23)
which means that T k is geometrically distributed with parameter k/(n -1). Using relation ( 6), we have, for every r ≥ 1,

E(T r k ) = r =1 c r, n -1 k , (24)
a relation that is to be compared with relation [START_REF] Palacios | Cover times for the stars[END_REF]. The first moments of T k are then given by replacing n by n -1 in those obtained for the complete graph with self-loops, i.e. by

E(T

k ) = n -1 k , E(T 2 k ) = (n -1)(2(n -1) -k) k 2 , E(T 3 k ) = (n -1)(6(n -1) 2 -6k(n -1) + k 2 ) k 3 .
In the same way, we have, for every fixed value of k,

E(T r k ) ∼ n-→∞ r! n k r and lim n-→∞ P{kT k /n > t} = e -t ,
which means that the sequence of random variables kT k /n converges in distribution, when n tends to infinity, to a random variable exponentially distributed with rate 1.

The results on the cover times are also very close to those on the complete graph with self-loops.

Cover times on the complete graph without self-loops

Here again, the results on the cover times are also very close to those on the complete graph with self-loops. We consider the first moments of the cover times.

Theorem 8 For every k = 1, . . . , n -1, we have

E(C k ) = (n -1)h k .
In particular, when k = n -1, we have

E(C) ∼ n-→∞ n ln(n).
Proof. Let A be a subset of S with k states. The expected cover time of subset A is given by relation [START_REF] Palacios | Expected hitting and cover times of random walks on some special graphs[END_REF] with r = 1. This gives, using the previous results on the hitting times

E(C k ) = E(T k ) + E(C k-1 ) = n -1 k + E(C k-1 ). with E(C 1 ) = E(T 1 ) = n -1. This simple recursion simply leads to E(C k ) = (n -1)h k . By taking k = n -1, we get E(C) = E(C n-1 ) = (n -1)h n-1 and thus E(C) ∼ n-→∞ n ln(n),
which completes the proof.

Observe that this well-known result can also be obtained using geometric random variables. Indeed, it is easy to check that

E(C) = E geo n -1 n -1 + E geo n -2 n -1 + • • • + E geo 1 n -1 = (n -1)h n-1 .
In the following theorem we consider the second order moment of the cover time of a subset of states with k states.

Theorem 9 For every k = 1, . . . , n -1, we have

E C 2 k = (n -1) 2 h 2 k + (n -1) 2 k =1 1 2 -(n -1)h k .
In particular, when k = n -1, we have

E(C 2 ) ∼ n-→∞ n 2 ln 2 (n).
Proof. It sufficies to follow the same steps of the proof of Theorem 2.

We consider now the third order moment of the cover time of a subset of states with k states. In particular, when k = n -1, we have

E(C 3 ) ∼ n-→∞ n 3 ln 3 (n).
Proof. It suffices to follow the same steps of the proof of Theorem 3.

More generally, we have the result of the following theorem which allows a recursive computation of all the moments of the cover time of a subset of states with k states. It also give the asymptotoic behavior of all the moments. It is based on relation [START_REF] Palacios | Expected hitting and cover times of random walks on some special graphs[END_REF]. For every = 1, . . . , r, we have a Proof. It sufficies to consider the polynomial in n -1 instead of n and to follow the same steps of the proof of Theorem 5.

We can now determine the limiting behavior of the distribution of the cover time C = C n-1 .

Theorem 12 For all t ∈ R \ {1}, we have,

lim n-→∞ P C n ln(n) > t = 1 if t > 1 0 if t < 1.
Proof. It is the same proof as the one of Theorem 6.

Observe that the previous theorem does not allow us to conclude when t = 1 because the convergence in distribution occurs only at the points t for which the limiting distribution is continuous.

We can solve this problem by considering the result of [START_REF] Sericola | On cover times of Markov chains[END_REF], which tell us that in the complete graph without self-loops we have

P{C > } =          1 if 0 ≤ < n -1 n-2 j=0 (-1) j n -1 j + 1 n -1 -(j + 1) n -1 if ≥ n -1. ( 25
)
Using this result, we obtain the following limiting distribution which completes the result of Theorem 12 at point t = 1.

Theorem 13 For every t ∈ R, we have Proof. The proof is identical to the one of Theorem 7 by using relation (25).

Observe that, since the limit Z = 1 is constant, we also have the convergence in probability of C/(n ln(n)) to Z = 1, that is, for any ε > 0 lim n-→∞ P C n ln(n) -1 > ε = 0.

Conclusion

We considered in this paper a random walk on the complete graph and we obtained both the limiting distribution and the moments of the hitting and cover times of a subset of states of the complete graph. Further research will be to analyze other particular graphs such as the Barbell graph, the star graph or the Tadpole graph.

Theorem 10 3 k = 3 (n - 1

 10331 For every k = 1, . . . , n -1, we haveE C n -1)h k .

Theorem 11

 11 For every k = 1, . . . , n -1 and for every r ≥ 1, E(C r k ) is a r degree polynomial in n, i.e.

(- 1 )

 1 r+ S(r, ) ln (k).Moreover, we haveE(C r ) ∼ n-→∞ n r ln r (n).

  lim n-→∞ P C -n ln(n) n > t = 1 -exp -e -t ,which is known as the standard Gumbel distribution. At point t = 0, we havelim n-→∞ P C n ln(n) > 1 = 1 -e -1 ≈ 0.63212.


Nkene Mezui
email: esthernkenemezui@gmail.com

Brancheriau Loic

Ikogou Samuel

Moutou Pitti Rostand

Liens entre stabilité dimensionnelle du bois et tempérament vis-à-vis de la lumière de feuillus tropicaux du Gabon

Mots clefs : stabilité dimensionnelle ; tempérament écologique ; feuillus tropicaux ; retraits ; anisotropie

Contexte et objectifs

Le Gabon est un pays d'Afrique centrale qui renferme de vastes zones forestières représentant 80% de son territoire [START_REF] Ikogou | Experimental study of tropical wood under cyclic compressive loading for sustainable constructions[END_REF] avec plus de 600 espèces arborescentes (Arrêté 2004) d'une diversité qui gagnerait à être valorisée. Dans ces forêts, les espèces d'arbres ont différents degrés de tolérance de l'ombre. Il est ainsi possible de les regrouper en différentes classes comme les espèces d'arbres très héliophiles et pionnières, à l'inverse, des espèces tolérantes à l'ombre ou bien encore des tempéraments intermédiaires avec des espèces dites hémitolérantes. Malheureusement, on note une absence importante d'informations scientifiques sur les caractéristiques techniques du bois de bon nombre d'entre elles tel que leur stabilité dimensionnelle qui est une propriété importante pour leur valorisation. La stabilité dimensionnelle renvoie à la faculté pour le bois de conserver ses dimensions lorsqu'il est exposé à des variations de température et/ou d'humidité. Le bois étant un matériau anisotrope, on considère qu'une essence est stable lorsque ses variations dimensionnelles sont très faibles, c'est-à-dire lorsque l'anisotropie qui est le rapport entre le retrait tangentiel et le retrait radial est proche de 1. L'objectif de cette étude est de mettre en relation la stabilité dimensionnelle du bois de 25 espèces de feuillus tropicaux et leur tempérament vis-à-vis de la lumière.

Matériel et méthodes

L'étude porte sur 25 espèces d'arbres feuillus tropicaux du Gabon prélevés sous forme de plateaux au parc à bois de la maison de la technologique du CIRAD à Montpellier. Ceux-ci ont été répartis en trois tempéraments vis-à-vis de la lumière distincts (Tab. 1 ci-dessous). Parmi les 25 espèces, on compte 5 espèces pionnières (P), 10 espèces hémi-tolérantes (HT) et 10 espèces tolérantes à l'ombre (TO). Les espèces pionnières représentes des espèces qui ne peuvent en aucun cas tolérer l'ombre quel que soit le stade de développement [START_REF] Swaine | On the definition of ecological species groups in tropical rain forests[END_REF]. Les espèces hémi-tolérantes et tolérantes peuvent tolérer l'ombre tout ou une partie de leur vie. Les tempéraments des espèces sont tirés de Meunier et al. (2015), de Doucet (2003), de Bénédet et al. (2019)

Plateau marqué Ebauches

Eprouvettes

Les éprouvettes ont par la suite été immergées dans des bocaux en verre contenant de l'eau distillée (chaque espèce dans un bocal). Afin de permettre l'immersion des éprouvettes, un poids est introduit dans chaque bocal. Un trou est réalisé sur les couvercles des bocaux afin de permettre une bonne circulation de l'air. Les bocaux sont ensuite introduits dans un autoclave (Fig. 2) et des cycles de mise sous vide puis à pression atmosphérique sont réalisés pendant 48 heures maximum en fonction de l'espèce. Au terme de ce temps de saturation, les masses et les dimensions radiale et tangentielle des éprouvettes saturées ont été mesurées. Par la suite, des paliers de stabilisation des éprouvettes (palier 1 : 18 % ; palier 2 : 12 % ; palier 3 : 6% et palier 4 : 0 % à 103°C pendant 48 heures) ont été effectués en enceinte climatique tout en mesurant les différentes variations des dimensions des éprouvettes. Par ailleurs, grâce à une analyse de variance à 1 facteur (ANOVA) réalisée sous RStudio, nous avons pu identifier les propriétés dépendantes du tempérament écologique de l'espèce. Le test post-hoc de Tukey a permis de mettre en évidence les paires de groupes écologiques chez lesquels il y avait des différences statistiques.

Résultats

Les liaisons linéaires entre les propriétés étudiées ont été évaluées grâce à une analyse de corrélation sous le logiciel RStudio pour identifier les variables qui évoluent ensemble, celles qui évoluent en sens opposé et celles qui ne dépendent d'aucune autre propriété. La Fig. 4 présente ces liens.

Conclusion et perspectives

Pour conclure, nous pouvons retenir que le tempérament vis-à-vis de la lumière des espèces d'arbres étudiées est reliée à la stabilité dimensionnelle du bois de ces dernières. Les espèces hémi-tolérantes et tolérantes à l'ombre ont une densité de bois et un retrait radial plus élevés,

(i) (ii) (iii) (iv)

  Une fois les plateaux (parallélépipède rectangle de longueur L = 60 cm) prélevés, ceux-ci ont été découpés en barreaux rectangulaires suivant le marquage des éprouvettes (voir Fig.1les trois premières étapes). Ces barreaux ont ensuite été usinés en de petites éprouvettes de section 20 × 20 × 10 mm (R, T, L). Au total, nous avions 100 éprouvettes pour le groupe des pionniers, 124 pour le groupe intermédiaire et 144 pour les tolérants à l'ombre. Les propriétés étudiées sont les retraits totaux radial (RR en %) et tangentiel (RT en %), l'anisotropie des retraits (RT/RR), le point de saturation des fibres (PSF) et la densité ρ (en kg/m 3 ).

Fig. 1 :

 1 Fig. 1 : Préparation des éprouvettes d'essai

Fig. 2 :Fig. 3 :

 23 Fig. 2 : Conditionnement et mesures des dimensions et des masses des éprouvettes Au terme de ces différentes mesures, les propriétés étudiées ont été déterminées suivant les équations suivantes : RT% = RR% = L s -L 0 L s × 100(1)

Fig. 4 :

 4 Fig. 4 : Corrélogramme des propriétés étudiées : les corrélations positives et négatives sont respectivement en bleu et en rouge et l'intensité de la est indiquée par la taille des cercles.

  Globalement on observe que l'anisotropie des retraits (RT/RR) est corrélée négativement à toutes les autres propriétés et surtout au retrait radial (RR). Seul le retrait radial est corrélé positivement à la densité du bois et cette corrélation est forte. Il est en même temps faiblement corrélé au PSF. Le retrait tangentiel est corrélé positivement avec l'anisotropie des retraits (forte corrélation), le PSF et le retrait radial (corrélations faibles). Ces résultats rejoignent les observations de[START_REF] Masseran | Anisotropie de retrait et structure du bois. Recherche de l'influence des caractères morphologiques transverses des fibres[END_REF] etde Messaoudène et al. (2018).La Fig.5ci-dessous présente les différences statistiques entre les groupes de tempéraments obtenus grâce au test post-hoc de Tukey.

Fig. 5 :

 5 Fig. 5 : Distributions des propriétés statistiquement significatives en fonction du tempérament écologique. a et b : lettres indiquant des différences statistiques ou non. Lorsque deux distributions ont la même lettre, cela signifie qu'elles ne sont pas statistiquement différentes. Toutes les autres propriétés à l'exception du PSF se sont avérées être liées au tempérament visà-vis de la lumière de l'espèce. Sur la Fig. 5 ci-dessus, pour toutes les propriétés on observe que les espèces hémi-tolérantes (HT en rouge) et les espèces tolérantes à l'ombre (TO en bleu) sont statistiquement identiques. Par ailleurs, les espèces pionnières (P en vert) ont une densité du bois (Fig. 5(iv)) et un retrait radial (Fig. 5(i)) plus faible que ceux des espèces des deux autres groupes. Par contre, leur retrait tangentiel est plus élevé (Fig. 5(ii)). Par ailleurs, les espèces hémi-tolérantes et tolérantes à l'ombre apparaissent moins instables dimensionnellement que les espèces pionnières pour lesquelles l'anisotropie est très prononcée (Fig. 5(iii)).

laurentii Louis TO Dacryodes igaganga Aubrev. et Pellegr. TO Daniellia soyauxii Rolfe TO Desbordesia glaucescens Van Tiegh. TO Dialium dinklagei Harms TO Dialium pachyphyllum Harms TO Erismadelphus exsul Mildbr. TO Poga oleosa Pierre TO Scytopetalum klaineanum Pierre TO

  et de Prota4u (référence en ligne).

	Espèces	Tempéraments écologiques
	Anthocleista nobilis G.Don	P
	Cleistopholis glauca Pierre	P
	Croton oligandrus Pierre	P
	Discoglypremna caloneura Prain	P
	Fillaeopsis discophora Harms	P
	Baillonella toxisperma Pierre	HT
	Celtis tessmannii Rendle	HT
	Dacryodes buettneri H.J.Lam	HT
	Dacryodes normandii Aubrev. et Pellegr.	HT
	Gilbertiodendron brachystegioides J.Leon.	HT
	Gossweilerodendron balsamiferum Harms	HT
	Klainedoxa gabonensis Pierre	HT
	Parkia bicolor A.Chev	HT
	Sacoglottis gabonensis Urb.	HT
	Scyphocephalium mannii Warb.	HT
	Beilschmiedia sp	TO
	Brachystegia
	Tab. 1 : Espèces d'arbres étudiées et tempéraments écologiques associés

P : pionnier ; HT : hémi-tolérant ; TO : tolérant à l'ombre

èmes journées du GDR 3544 « Sciences du bois » -Limoges, 22-24 novembre 2023