Antoine Gaboriau

Catherine Dominguès

Analyse du discours citoyen : transitions démocratique et écologique

Keywords: transitions démocratique et écologique. Transitions en tension -Controverses et tensions autour des transitions écologiques, GER "Communication, environnement, sciences

AXE I. Transitions et controverses : discours et interactions

Permettre aux citoyens d'être « véritablement des protagonistes de la fabrique de la ville » selon la maire de Barcelone 1 , nourrir le travail des citoyen•ne•s tiré•e•s au sort dans le cas de la plateforme externe de la Convention citoyenne pour le climat 2 , « réinventer notre modèle de société en étant force de proposition sur les sujets estimés prioritaires » pour les organisateur•trice•s de la consultation du « Jour d'Après » 3 … Autant d'objectifs politiques projetés sur les plateformes numériques de participation, qui se sont multipliées ces dernières années, devenant l'une des manifestations d'une « transition démocratique » appelée de leurs voeux par de nombreux acteurs politiques, associatifs ou individuels.

Ces plateformes de participation paraissent ainsi être symptomatiques du mouvement de fond de numérisation de la participation citoyenne depuis plusieurs années [START_REF] L'édition | de l'enquête du think-tank Décider Ensemble illustre ce point : 87 % des institutions publiques répondantes disposent d'une plateforme numérique de consultation, soit le plus haut taux d'équipement relatif à la participation citoyenne, tous dispositifs confondus[END_REF] . Se pencher sur ces outils, c'est donc chercher à prendre la mesure de la matérialité de la transformation de la représentation politique que certains acteurs de ce mouvement de numérisation de la participation souhaitent atteindre par le biais d'outils numériques, et son appropriation par les citoyen•ne•s qui participent effectivement sur ces plateformes.

A l'intersection de la science politique et l'analyse du discours, plusieurs travaux de recherche se sont penchés sur l'analyse des discours politiques ; ils traitent pour une grande partie d'entre eux du discours des représentants politiques (Mayaffre 2020a, b, c, 2021 par exemple) ou leur traitement par les réseaux sociaux (Smyrnaios & Ratinaud, 2013). L'intérêt spécifique de cette communication est, à notre sens, d'ouvrir un objet peu traité par ces deux disciplines, à savoir l'étude du discours déployé par les citoyen•ne•s dans le cadre de la participation citoyenne. Environ trois mille contributions y ont été déposées, par des participant•e•s évidemment concerné•e•s par le sujet, donc nous serons en mesure de tester l'écart de représentativité avec le reste de la population française. Ceci constitue un corpus particulièrement pertinent pour l'étude du discours par les contributeur•trice•s. Cette communication se divise donc en deux parties. La première cherchera à répondre à une question précise : à partir de la plateforme de la CCC, quelle prise en charge du discours est réalisée par les contributeur•trice•s ? Nous chercherons notamment à distinguer ce qui relève de l'opinion assumée personnellement de ce qui est attribué à d'autres.

La deuxième partie de la communication s'intéressera à la manière dont les contributions des participant•e•s à la plateforme de la CCC s'articulent avec le travail des citoyen•ne•s tiré•e•s au sort en décrivant les modalités de l'argumentation : la prise en charge du discours par le locuteur (je, moi vs nous, on, il.elle.s), l'utilisation de connecteurs logiques (donc, pourtant, cependant, …), de verbes modaux (falloir, devoir, pouvoir, …), du mode conditionnel). La communication s'attachera ensuite à comparer les points de description précédents avec ceux décrivant le corpus de contributions du « Jour d'Après » (où les participant•e•s s'adressent principalement à des député•e•s). Nous nous poserons ainsi deux questions : d'abord, nous chercherons à savoir comment les citoyen•ne•s participant à la plateforme s'adressent-ils aux citoyen•ne•s tiré•e•s au sort au sein de l'« innovation démocratique ». Nous tenterons ensuite de vérifier s'il existe une différence de discours avec une initiative animée exclusivement par les représentant•e•s élu•e•s animateur•trice•s du « Jour d'Après ».

Les deux parties de cette communication doivent ainsi nous permettre d'explorer un champ encore peu étudié de la transition écologique : la manière dont les citoyen•ne•s intervenant dans le débat public par le biais de plateformes numériques traitent du sujet et s'adressent aux commanditaires. Nous espérons que nos résultats nourriront les débats relatifs à la participation citoyenne numérique en science politique (quel rapport des participant•e•s à la représentation et aux « innovations démocratiques » ?), en science de l'information et de la communication (le cadrage préalable de l'outil influe-t-il sur les registres argumentatifs déployés ?) et en analyse du discours (comment caractériser les discours produits par les citoyen•ne•s participant à ces consultations ?).

Bibliographie

L

  'objectif de cette communication est de croiser l'exploration d'une « transition démocratique » avec la thématique de la transition écologique. Plusieurs plateformes ont été dédiées à des consultations citoyennes autour de ce sujet, visant des objectifs différents selon le ou la commanditaire à la manoeuvre. Notre objet d'étude sera la plateforme externe de la Convention citoyenne pour le climat (CCC), ouverte de fin 2019 à début 2020. Cette plateforme visait à recueillir les contributions des citoyen•ne•s extérieur•e•s à la Convention afin de nourrir, informer et enrichir les débats des 150 citoyen•ne•s tiré•e•s au sort.

   Journées d'études Sciences Po Paris : « Quels outils d'analyse pour les Gilets jaunes », 16 et 17 janvier 2020, et en particulier les deux références suivantes : -Bellet A., Keller M., Vauquier N. (2020) Grand débat et IA : quelle transparence pour le traitement des données ? Une tentative de rétro-ingénierie de la synthèse. -Loubère L., Marchand P. (2020). Les thématiques du Grand Débat et du Vrai Débat : approche textométrique.  Marchand, P., Ratinaud, P. et Figeac, J. La campagne présidentielle de 2017 sur les réseaux socionumériques. Toulouse, France, 2018.  Ploux S., Genay M., Leu Ploux-Chillès. Les mots du grand débat national : quelques résultats et réflexions préliminaires. HAL:hal-02441477, 2020.  Smyrnaios, N. et Ratinaud, P., « Comment articuler analyse des réseaux et des discours sur Twitter », tic&société [En ligne], Vol. 7, N° 2 | 2ème semestre 2013, mis en ligne le 26 juin 2014, consulté le 30 septembre 2021. Biographies Antoine Gaboriau est doctorant au CESPRA (EHESS), en contrat Cifre avec l'entreprise Open Source Politics. Ses travaux portent au sein de l'entreprise sur la linguistique appliquée aux corpus de contributions recueillis sur les plateformes de participation numériques, et dans sa thèse sur les conséquences de l'implémentation de ces plateformes dans les administrations municipales, d'un point de vue de sociologie des organisations. Catherine Dominguès est chercheure au LaSTIG (IGN), ingénieure des travaux publics de l'État et docteure en informatique linguistique. Son domaine de recherche croise le traitement automatique des langues et la géomatique, et s'inscrit dans la thématique de l'aide à la représentation cartographique adaptée au contexte.


Laurent Maïmoun
email: l-maimoun@chu-montpellier.fr

Sandrine Alonso

Krishna Kunal Mahadea

Vincent Boudousq

Thibault Mura

Denis Mariano-Goulart

Krishna Kunal Mahadea

Denis Mariano-Goulart1

Cross-Calibration Study of The Stratos And Hologic QDR 4500 A Dual-Energy X-Ray Absorptiometers to Assess Bone Mineral Density And Body Composition

Keywords: Cross-calibration, Dual X-ray absorptiometry (DXA), Stratos, QDR4500A, Areal bone mineral density (aBMD), Body composition, Body mass index

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Introduction

Dual-energy X-ray absorptiometry (DXA) is the reference technique for the evaluation of areal bone mineral density (aBMD). It is noninvasive and characterized by low radiation and high precision and reproducibility, making it useful to monitor changes in aBMD over time and to determine treatment efficacy (1). However, a difference in aBMD calibrations between manufacturers has been well documented (234). Typically, it has been reported that a measurement of spine aBMD on a system from Hologic is lower than on the GE Healthcare Lunar system (23). Moreover, aBMD variations have also been observed for different scanners within the same manufacturer's product lines (56). The official guidelines of the International Society for Clinical Densitometry (ISCD) (7) therefore recommend that patient follow-up be performed with the same device. However, in situations where an old DXA is replaced by a newer one, or to meet the needs of a multicenter study, the ISCD recommends a cross-calibration study between devices to ensure consistent results (7).

In addition to aBMD, there is growing interest in evaluating body composition with DXA, which is today the gold standard in clinic for determining WB, regional lean tissue mass (LTM) and fat mass (FM) (8). Indeed, DXA use for patients with various conditions was advocated in the last ISCD recommendations, published in 2013 9 . These included patients living with HIV and taking antiretroviral agents, obese patients undergoing bariatric surgery (1011), patients with other conditions that induce significant weight loss, and patients with weakness or poor physical functioning. For example, when LTM is evaluated with DXA in weak or poorly functioning patients, it contributes to the sarcopenia diagnosis by adding the concomitant evaluation of muscle quantity or mass to the measurements of muscle strength by handgrip strength and physical performance by gait speed (12). Thus, low LTM can be defined by different cut-offs using appendicular lean mass evaluation 12, 13, 14, which requires the capacity of the device to perform segmental analysis - only possible with DXA technology. As for aBMD, the ISCD recommends a cross-calibration study between devices to ensure consistent results for body composition analysis (7).

To date, no data have been published on the cross-calibration between the Stratos (DMS®) and the QDR 4500A (Hologic Inc.®). To our knowledge, Stratos was only cross-calibrated with Discovery A, another device from Hologic®, but only lumbar spine and hip regions were analyzed, with no information on body composition(15).

The purpose of this study was to perform an in vivo calibration of total and regional body scans between two DXA devices: Stratos (DMS-APELEM; pencil-beam technology) and QDR 4500A (Hologic, Inc.; fan-beam technology), with a focus on aBMD and body composition.

Section snippets

Subjects

Fifty-five adults were recruited via an intra-hospital email invitation or from among patients referred to the Department of Nuclear Medicine, Lapeyronie Hospital (Centre Hospitalier Universitaire de Montpellier, CHRU Montpellier, France) for evaluation of aBMD and body composition. All participants were Caucasian and, to be eligible, they had to be over 18 years old, able to lie supine for 10-15 minutes, and have body weight <160 kg (table weight limit restriction of the QDR 4500A system). Participants were excluded from the study if they had a history of lumbar spine or proximal femur fracture, hip replacement or osteosynthesis and cementoplasty at the lumbar spine, or were pregnant or breast feeding. In accordance with the ISCD recommendations (7), these participants were representative of our patients and represented the full spectrum of scans performed in our Department.

Study procedures

Ethics approval for the study was obtained from the Ethics Review Committee of Nîmes, France (Commission de Protection des Personnes, Sud Méditerranée III on 9 July 2015: reference 2015-06.02 bis), and permission for the clinical trials was granted by the French Medicine and Health Care Products Regulatory Agency (ID RCB: 2015-A00596-43, on 21 September 2015). All study participants were volunteers and gave specific signed consent before the scans. All activities performed in this study were in accordance with the Declaration of Helsinki.

Standing height was measured with a stadiumeter to the nearest 0.1cm and recorded as the mean of two consecutive measurements. Body weight was measured with a calibrated scale with a precision of 0.1kg. Body mass index (BMI) was calculated as weight (kg) divided by the square of height (m).

Dual-energy X-ray absorptiometry measurements

As the two systems were located at the same site, each participant was scanned consecutively on the QDR 4500A (Hologic, Inc.) and immediately after on the Stratos (DMS, Mauguio, France). They wore the same light clothing with all metal and plastic artifacts removed. The QDR 4500A was used as the reference device. The lumbar spine (L1-L4) in the posterioranterior projection, the left proximal femur, the radius and the whole body were scanned on both DXA scanners. This was done after following the procedures described in each densitometer's operating manual. For aBMD analysis, the regions of interest (ROIs) were defined for the lumbar spine, femoral neck, total hip, radius and whole body, while for body composition [i.e., LTM and FM], the ROIs were defined for legs, arms and whole body. Moreover, appendicular lean tissue mass (ALM) was defined as the sum of LTM of the arms and legs ,as described by Heymsfield et al.(13).

Specificity of the DXA

-Stratos employs pencil-beam DXA technology. The generator operates on a 90-kV power supply and a 0.1-to 2-mA current depending on the type of scan and the patient's morphology. The X-ray source has a continuous spectrum with two energy peaks (35 and 65 keV) filtered by a k-edge filter (samarium). A pencil-shaped beam is obtained when the raw X-rays pass through by a circular collimator. Photon counting is done with a scintillation crystal (LaCl 3 ) coupled to a photomultiplier. The X-ray tube is placed beneath the patient's table and the detector is lodged in the arm. Scattered rays are stopped by a brasscollimator placed in front of the detector. The X-ray source and the detector move. Simultaneously along the x and y axes and the maximum scan area is 200 x 65cm 2 . The scanned image is a projection of the object with no magnifying effects and independent of the position of the object along the source/detector axis. Software version V5.2.1.2 was used. Other details and dosimetry were previously published (15). The least significant change (LSC) for Stratos was previously reported, with values of 3.39% for lumbar spine (L1-L4), 5.84% for femoral neck, and 2.38% for total hip (15).

-QDR 4500A employs fan-beam DXA technology. Briefly, the X-ray source has a continuous spectrum with two energy peaks (100 and 140 keV).The detector system is composed of a 216-multichannel detector consisting of CdWO4 scintillators coupled to silicon diodes. Software version 12.6 was used. Other details and dosimetry were previously published (16).

Procedure for analysis of femoral regions

Shared procedure: For the two DXAs, the left hip that is scanned is internally rotated using a hip positioner. Hip is internally rotated because it will open up the femoral neck area, putting the femoral shaft in line with the scanner bed and minimizing the lesser trochanter. Specific procedures: For QDR-4500A, the bottom of the global ROI should be 10 lines below the bottom of the lesser trochanter, 5 lines above the top of the head of the femoral neck, 5 lines laterally to the head of the 5 lines above the top of the head of the femoral neck, 5 lines laterally to the head of the femoral neck, and 5 lines laterally to the great trochanter. When the global ROI is defined, the software automatically places the different ROIs (i.e., head of the femur, greater trochanter and bottom of the lesser trochanter) and the femoral midline. By default, the width of the femoral neck box is a rectangular region of 1.5 * 5 cm and is placed tangent to the crest of the greater trochanter. For Stratos, the bottom of the global ROI is automatically positioned 6 cm below the insertion between the femoral neck axis and the lower limit of the femoral neck ROI (yellow line). The red line defines the femoral neck's axis and cuts the femoral neck in half. The femoral neck box in blue is positioned tangent to the crest of the greater trochanter. Last, the green line should be placed from the inflection point of the trochanter edge up to the boundary of the trochanter. Fig. 1 presents the left proximal femur images of both bone densitometers, including their respective ROIs.

Statistical analysis

Characteristics are expressed as mean ± standard deviation (SD) and frequency with percentage. Correlations between the QDR 4500A and Stratos measurements were calculated using Pearson correlation coefficients (with 95% CI). Inter-scanner differences were evaluated by calculating the systematic difference between the QDR 4500A and Stratos measurements (systematic bias) and tested against the 0 value using a paired Student's t-test. Agreement was analyzed using a Bland-Altman plot.

Cross-calibration equations were estimated using linear regression. Systematic bias and agreement between the Stratos and QDR-4500A measurements converted using the crosscalibration equations were assessed using paired Student's t-tests and Bland-Altman plots.

The BMI effect and effect modification on the cross-calibration model were evaluated by adding an effect and an interaction term with a "Severe-Obesity" variable to the linear regression model. This dichotomization was chosen because it best fit the data for the interaction effects under consideration. All the analyses were performed using SAS software version 9.4 (SAS Institute Inc., Cary, NC).

Results

The characteristics of the participants are described in Table 1. Fifty-five individuals (46 women: 84%) with a mean age of 41 ± 13.0 years (range: 20 to 64) and a mean BMI of 31.9 ± 10 kg/m² (range: 12.2 to 49.5) were included in this study. The repartition of participants according to BMI class is presented in Table 2.

Cross-calibration

Results of the comparisons of mean bone mass parameters and body composition parameters measured by the Stratos and QDR 4500A are given in Table 3. Values of aBMD on the two DXAs were highly correlated (p<0.001 for all) with a coefficient of correlation (r) ranging from 0.89 to 0.99 (r=0.89 for whole body, r=0.92 for radius, r=0.95 for femoral neck, r=0.96 for total hip, and r=0.99 for L1-L4 aBMD). For whole body composition, r ranged from 0.96 for ALM to 1.0 for FM (all p<0.001). All these correlations are presented in Table3 and Figs. 2 and3.

However, although the two measures were highly correlated, the paired t-test indicated a statistically significant bias between the Stratos and QDR 4500A for most of the measurements. Compared to QDR 4500A, Stratos under estimated whole body aBMD with a difference of 0.116 g/cm 2 , corresponding to a mean relative change of 10.4%. Specifically, the mean femoral neck and total hip aBMD were, respectively, a mean 0.07 g/cm 2 and 0.04 g/cm 2 lower with QDR 4500A than Stratos, corresponding to mean relative changes of 9% and 4.5. No significant bias was demonstrated for mean aBMD at L1-L4 and radius between the two DXAs .Compared to QDR 4500A ,Stratos over estimated FM (kg and %) and underestimated LTM (p< 0.001 for the three parameters) .The systematic differences for bone parameters and body composition parameters are depicted in Figs. 4a and5a with Bland-Altman plots (Stratos versus QDR 4500A).

As the subjects included in this study presented a large dispersion of BMI (range 12.2 to 49.5 kg/m 2 ), the concordance of measurements between QDR 4500A and Stratos was analyzed according to BMI class (see Supplementary Table1). For most parameters, the bias between the two measurements obtained by the two DXAs was not influenced by BMI. Only limited significant interactions of BMI with whole body aBMD, whole body BMC and whole body FM (kg and %) were found (see Supplementary Table1 and Supplementary Fig. 1).

Due to the observed differences between the two densitometers, linear cross-calibration equations were derived for bone mass parameters and body composition parameters. These equations are presented in Table4. The agreement between the Stratos measurements and the QDR 4500A measurements after conversion using the cross-calibration equations is depicted in Figs. 4b and 5b with Bland-Altman plots. Specific cross-calibration equations for parameters influenced by BMI are presented for the two classes of BMI (>/=35kg/m 2 or >35kg/m 2 ) in Table 5.

Discussion

For a group of participants presenting a large range of BMI values, in vivo cross-calibration demonstrated good linear agreement between the Stratos and the QDR 4500A for both bone and body composition parameters. However, the regression slope in most of the ROIs was significantly different from unity, demonstrating the need to cross-calibrate these two DXAs from different manufacturers. aBMD Although all the aBMD measurements were significantly correlated between the two densitometers, the aBMD values with the Stratos were significantly lower for whole body and higher for femoral neck and total hip compared to QDR 4500A. For the latter two skeletal sites, mean 0.07 g/cm² and 0.04 g/cm² differences were, respectively, observed, which amounted to percentage variations of 9% and 4.5%. In cases of differences exceeding 1%, the ISCD recommends (7) the development of cross-calibration equations to ensure comparability between the two densitometers; for example, to ensure correct patient management when a currently commercialized Stratos replaces an older QDR 4500A, or to ensure the consistency of measurements when a multicenter study is being conducted. Various technological reasons have been proposed to explain the aBMD offsets ,including differences in the manufacturers'calibration processes, computing algorithms, and detection technologies (17)(18)(19). In the current study, the type of beam (i.e., fanorpencil) may also be implicated. However, in a recent meta-analysis based on 14 studies, Yoon et al. (20) concluded that here was no significant difference in aBMD at the hip region between the two modes of DXA scanners, although pencil-beam underestimated aBMD in whole body compared to fan-beam, as was found in our study. It is never the less interesting to note that only studies comparing DXAs from the same manufacturer (i.e., Prodigy vs iDXA from Lunar or QDR 2000W vs QDR 4500A from Hologic) had been compiled (20). This methodology may have introduced a selective bias because an intra-device cross-calibration may have been performed by the manufacturers to limit aBMD differences within their range of densitometers. Other studies comparing pencil-beam (DPX-NT) and fan-beam (QDR 4500C) DXAs from two different manufacturers reported that, although the two DXAs were highly correlated, pencil-beam overestimated aBMD at the femur (0.082 g/cm 2 ) (21). Our results at the femoral region confirmed these findings.

Body composition

DXA analysis of total and regional body composition has progressively become important in the diagnosis and follow-up of patients with a range of pathologies or clinical condition s(9), including patients with HIV, obese patients undergoing bariatric surgery (1011), and patients with muscle weakness or poor physical function (1213) , (30). However, as DXA findings for body composition differ across the scanning systems produced by different manufacturers (3), the development of cross-calibration equations based on in vivo analysis, instead of in vitro phantoms that may be misleading (31), is required and recommended by the ISCD (9).

As expected, although both DXAs showed excellent reliability for whole body LTM and FM (r=0.98 to 1), there were significant differences between the Stratos and QDR 4500A for body composition evaluation. The whole body LTM determined by Stratos was lower (bias of 4.2 kg corresponding to a mean relative change of 7.8%)

Conclusion

To summarize, the study showed good linear agreement between a currently commercialized pencil-beam Stratos and an older fan-beam QDR 4500A. Using cross-calibration equations generated from a population presenting a large range of BMI, an accurate transformation of the Stratos values was obtained, removing all significant differences between the two DXAs. These results can now be used to assess aBMD or LTM and FM in multicenter studies or when a change from QDR 4500A to Stratos is necessary.

Acknowledgments

The authors would like to thank all the participants.


archive ouverte pluridisciplinaire HAL, est destinée au dépôt

The nadir viewing mode records the solar radiation that has been scattered and reflected by the planet and its atmosphere. From these measurements, we can deduce the amounts of several atmospheric species such as dust, ice clouds [1,2] and ozone [3] but also the surface albedo.

Retrieval:

We have developed an improved retrieval algorithm using a more recent and upto-date parametrisation of the atmosphere and surface characteristics than in previous SPICAM works [4]. The purpose is to deduce simultaneously the following parameters: the ozone column, the dust optical depth (OD), the cloud OD and the surface reflectance. In order to keep independent parameters, the retrieval was limited to 3 parameters. The choice of the non retrieved parameter is made between the cloud OD and the surface albedo for each measurement and depends on the cloud presence.

Cloud detection:

We have therefore developed a cloud detection algorithm based on the principle that clouds appear very bright in the UV compared to the "dark" regolith surface and the absorbing airborn dust. More than 4 Martian years (MY27-30) of SPICAM data have been analysed with our method and were used to produce an ice cloud climatology (cf fig 1). We will present the results obtained for ice clouds and illustrate their spatial and seasonal distributions.

Dust properties' influence:

Climatologies of dust and ozone have also been obtained and will be presented (cf. fig 1). The important influence on the retrieval of using different dust properties was introduced in [4]. We have reprocessed the retrieval on the 4 MYs using other phase functions. We will show the impact of these changes on the retrieved quantities. The influences of a different dust altitude profile was also tested (i.e. a comparison between the use of a priori from MCD v5.0 and v5.2) and will be presented, showing that it improves the retrieval.

  Ice cloud, dust and ozone nadir retrieval using SPICAM/UV and influence of dust properties on the retrieved quantities Y Willame, A C Vandaele, C Depiesse, Franck Lefèvre, Franck Montmessin To cite this version: Y Willame, A C Vandaele, C Depiesse, Franck Lefèvre, Franck Montmessin. Ice cloud, dust and ozone nadir retrieval using SPICAM/UV and influence of dust properties on the retrieved quantities. Scientific Workshop: "From Mars Express to ExoMars", Feb 2018, Madrid, Spain. insu-04396051

Figure 1 :

 1 Figure 1: Ice cloud OD (top), dust OD (middle) and ozone column (bottom) climatologies derived from SPICAM/UV measurements between MY27 and 30 [4]. Acknowledgements: The research program was supported by the Belgian Federal Science Policy Office and the European Space Agency (ESA -PRODEX Program -C 90323, 90113, C 4000107727). The research was performed as part of the "Interuniversity Attraction Poles" program financed by the Belgian government (Planet TOPERS) and a BRAIN research grant BR/143/A2/SCOOP. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under the grant agreement no. 607177 CrossDrive and from the European Union's Horizon 2020 Programme (H2020-Compet-08-2014) under grant agreement UPWARDS-633127.

UPWARDS Ice cloud, dust and ozone nadir retrieval using SPICAM/UV and influence of dust properties on the retrieved quantities Y

  . Willame 1 , A.C. Vandaele 1 , C. Depiesse 1 , F. Lefèvre 2 and F. Montmessin 2 .

	Scientific Workshop: "From Mars Express to ExoMars"	27-28 February 2018, ESAC Madrid, Spain
	1 IASB-BIRA, Belgium, 2 LATMOS, France
	Introduction: The SPICAM instrument on
	board Mars-Express (MEX) provides valuable
	data to investigate the Martian atmosphere
	since 2004. The UV channel, a spectrometer
	operating in the spectral interval between 110-
	320 nm, has been in activity during 11 years.

Workshop organized by ESA-ESAC & IAA-CSIC, with the support of the ESAC Faculty and the EU H2020 project

L'


Ez-Zoubir Amri
email: ez-zoubir.amri@univ-cotedazur.fr

Frédéric Carrière
email: carriere@imm.cnrs.fr

Preprint of DOI: 10.1016/j.biochi.2023.11.002

Biochimie 215 (2023) 1-3.Editorial for the Special issue "Lipids -from Membrane Dynamics to Signaling"

Lipids play music at the cellular membrane: from membrane dynamics to signaling via lipid mediators, vesicles and lipid droplets

The front cover for this special issue is a metaphoric representation of lipid science that takes place at the cellular membrane. By their molecular diversity, lipids can self-organize into membranes from which many biological functions can emerge in a dynamic way, including the production of lipid droplets (LDs) for the storage and subsequent delivery of energy or for their later conversion into milk fat globules (MFGs), the production of lipoproteins (such as LDLs) or the release of extracellular vesicles (exosomes, etc). Last but not least, membrane lipids are used by multiple enzymes for the production of potent lipid mediators (LMs) that are involved in a myriad of physiological and pathophysiological conditions. Lipids are therefore essential for writing the musical score of life. To illustrate the ongoing research in this continuously evolving area, this special issue assembles a collection of original articles and reviews of some of the lipid science that was presented at the 17th GERLI Lipidomics meeting, entitled "LIPIDS: from membrane dynamics to signaling", which was held in Saint-Jean-Cap-Ferrat, France, on November 6-9, 2022. The meeting gathered researchers from various areas, who discussed cutting-edge research on lipid membrane components and lipid remodeling, mechanisms linking lipid membrane dynamics to lipid mediator signaling, up to lipid functions in physiology, pathophysiology and applications in nutrition and health, as well as in diagnosis and therapy of lipid-driven diseases.

One well-known but intriguing LD is the milk fat globule (MFG). It has the particularity to be exclusively secreted from mammary gland epithelial cells. After migration to the apical pole of the cell, the "LD-MFG" docks to the plasma membrane and exits the cell covered by a double lipid layer by a mechanism which may be analogous to pinocytosis but in the reverse direction. The resulting milk fat globule membrane (MFGM) is thus composed of three lipid layers in which various proteins originating from the original LD and the plasma membrane are embedded. The review by Tzirkel-Hancock et al [START_REF] Tzirkel-Hancock | Milk fat globule size: Unraveling the intricate relationship between metabolism, homeostasis, and stress signaling[END_REF] describes this mechanism and the various factors including cellular stress that influence the size and composition of the MFG, and how this can exert an adaptive benefit to the lactating mother or the infant. Some lipases are active on both triglycerides and polar lipids, and they are important enzymes for the release of fatty acids and remodeling of cellular membranes [START_REF] Kuge | PLRP2 selectively localizes synaptic membrane proteins via acyl-chain remodeling of phospholipids[END_REF]. Among them, pancreatic lipase related-protein 2 (PLRP2) displays both phospholipase A1 and galactolipase activities, the latter being associated with the digestion of plant photosynthetic membranes [START_REF] Sahaka | The digestion of galactolipids and its ubiquitous function in Nature for the uptake of the essential alpha-linolenic acid[END_REF]. Kergomard et al. [4] used mixed Langmuir monolayers containing galactolipids, phospholipids and phytosterols as plant membrane model systems to investigate how the lipid composition and phase heterogeneity within the membrane can impact the enzyme adsorption and activity.

Surface tensiometry, ellipsometry and atomic force microscopy after Langmuir-Blodgett transfer are used to reveal the phase heterogeneity and monitor enzyme kinetics.

The review by Valentine et al. [START_REF] Valentine | Lysophospholipid acyltransferases orchestrate the compositional diversity of phospholipids[END_REF] describes the importance of lysophospholipid acyltransferases (LPLATs) in the remodeling of membrane phospholipids. This review reports a concise update on the 14 LPLAT members, which are conserved in mice and humans. It The review by Takeshi Harayama [START_REF] Harayama | Metabolic bias: Lipid structures as determinants of their metabolic fates[END_REF] highlights the importance of various cellular phospholipids in regards to their chemical structures and association to different metabolic pathways. The review focused on a phenomenon named for the first time by Takeshi Harayama: "structure-guided metabolic bias", which is well described and discussed in terms of biological relevance.

Lipid biosensors are useful tools to visualize and quantify specific lipid species in live cells.

The article by Eisenreichová et al [START_REF] Eisenreichova | Effects of biophysical membrane properties on recognition of phosphatidylserine, or phosphatidylinositol 4-phosphate by lipid biosensors LactC2, or P4M[END_REF] focuses on the impact of membrane properties, such as fluidity and membrane charge, on the sensitivity of two widely used lipid biosensors, LactC2 and P4M, to their target lipids, PS or PI4P, respectively. Using dual-color fluorescence crosscorrelation spectroscopy, they investigated the binding properties of these biosensors to liposomes of various lipid compositions with high sensitivity. They found that LactC2 has limited efficacy in detecting low levels of PS in simple lipid mixtures consisting of POPC/POPS. The presence of other anionic lipids in these membranes significantly improves the detection of PS. Comparable experiments for P4M revealed that this PI4P probe is less sensitive to the membrane environment. The presented results raise concerns about the quantification of sensed lipids based on biosensor abundance and highlight the importance of understanding the behavior and limitations of lipid biosensors in the context of the lipid membrane environment.

Ether-lipids (ELs) are specific lipids bearing a characteristic sn-1 ether bond. Alkyl-and alkenyl-ELs differ by the ether or vinyl-ether nature of this bond. The review by Papin et al. [START_REF] Papin | Ether-lipids and cellular signaling: A differential role of alkyl-and alkenyl-ether-lipids?[END_REF] summarizes the roles and mechanisms of action of ELs in cellular signaling, trying to discriminate between alkyl-and alkenyl-ELs. Alkenyl-ELs, which are also referred as plasmalogens, have attracted most of the scientific interest as they are the predominant EL species in eukaryotic cells, while much less is known about alkyl-ELs. ELs have been implicated in various signaling pathways, with amounts altered in pathologies such as cancers or neurodegenerative and cardiovascular diseases. However, it remains unknown whether both alkyl-and alkenyl-ELs play the same roles in these processes. The review also focuses on the involvement of EL-mediated alterations of cellular signaling in diseases and discusses the potential interest of ELs in therapy.

Leukotriene B4 (LTB4) is a lipid mediator generated from arachidonic acid in response to various stimuli. This important lipid exerts its biological activities by binding to two LTB4 receptors: BLT1 and BLT2 which are high and low affinity receptors, respectively. The review by Nakamura et al. [START_REF] Nakamura | Therapeutic target of leukotriene B(4) receptors, BLT1 and BLT2: Insights from basic research[END_REF] focuses on the biosynthesis and biological roles of LTB4 through its two receptors. They describe the effects of receptor deficiencies in pathophysiological conditions, up to the potential of LTB4 receptors as therapeutic targets. Moreover, the current information on the structure and post-translational modifications of BLT1 and BLT2 is discussed. In various diseases, the pathophysiological importance of LTB4 and their receptors have been clarified.

For example, disruption of the BLT1 gene or the use of receptor antagonists reduced disease activity in rheumatoid arthritis and bronchial asthma. In contrast, BLT2 deficiency facilitated several diseases in small intestine and skin. The data suggest that BLT1 blockers and BLT2 agonists could be useful in therapy. Drugs targeting each receptor are being developed by pharmaceutical companies.

Non-Alcoholic Fatty Liver Disease (NAFLD) is classified as a pandemic disease, with no effective treatment available so far. The review by Teixeira et al. [START_REF] Teixeira | Impaired hepatic lipid metabolism and biomarkers in fatty liver disease[END_REF] evaluates our current understanding on the onset, progression, and prevention of NAFLD through the analysis of plasma and liver lipidome. Several biomarkers are discussed among which lipid droplets are highlighted as dynamic organelles that represent potential targets for therapeutic intervention.

Epidermal lipids play important roles in skin homeostasis and diseases. Psoriasis is an inflammatory disease characterized by keratinocyte hyperproliferation and Th17 immune responses. Yamamoto et al. previously reported that ethanolamine-type lysoplasmalogen (P-LPE), preferentially produced by the group IIF secreted PLA2 (sPLA2-IIF/PLA2G2F) that is expressed in the suprabasal epidermis, promotes epidermal hyperplasia in psoriatic inflammation [START_REF] Yamamoto | The role of group IIF-secreted phospholipase A2 in epidermal homeostasis and hyperplasia[END_REF]. In this original article [START_REF] Hakoi | Lysophospholipase D from Thermocrispum limits psoriatic inflammation by hydrolyzing epidermal lysoplasmalogen produced by group IIF secreted phospholipase A(2)[END_REF], Yamamoto et al. further show that the forcible degradation of epidermal P-LPE by topical application of recombinant lysophospholipase D (LyPls-PLD) from Thermocrispum, a lysoplasmalogen-specific hydrolase, attenuated epidermal hyperplasia and inflammation in imiquimod-induced and K5.Stat3C transgenic mouse models of psoriasis. In humans, P-LPE levels are elevated in the tape-stripped stratum corneum of patients with psoriasis. Moreover, in primary cultured human epidermal keratinocytes, the aberrant cell proliferation and activation induced by psoriatic cytokines were sPLA2-IIF/P-LPE-dependent and suppressed by addition of LyPls-PLD. The findings confirm the importance of the sPLA2-IIF/P-LPE axis in psoriasis, and may lead to the development of new therapeutic and diagnostic applications.

Bacteria are rich in lipids with a high diversity, belonging to several lipid families and characterized by different stereospecific isomeric structures. The review by Fong-Fu Hsu [START_REF] Hsu | Multiple stage linear ion-trap mass spectrometry toward characterization of native bacterial lipids-a critical review[END_REF] describes an updated method to elucidate the structure of complex bacterial lipids. Modern mass spectrometry plays an important role, thanks to the continuous improvement of sensitivity and speed of analysis, gaining more complete structural information. This review focuses on the description and use of linear ion trap multiple-stage mass spectrometry with high resolution instruments leading to a more complete structural identification of native bacterial lipids. Several examples are discussed for complex and rare lipids, including those consisting of stereoisomers.

The review by Gaudu et al. [START_REF] Gaudu | Dissolved metal ions and mineral-liposome hybrid systems: Underlying interactions, synthesis, and characterization[END_REF] explores the interaction of minerals with liposomes.

Mineral-liposome hybrid systems currently present interesting applications in the medical field for diagnosis and targeted therapeutics. They can be used for targeting, controling and improving drug delivery by liposomes [START_REF] Leo | Liposomes containing nanoparticles: preparation and applications[END_REF]. Iron-oxide nanoparticles, such as magnetite (Fe3O4)-embedded liposomes, are used as contrast agents in magnetic resonance imaging (MRI), as well as in targeting of tumors [START_REF] Liu | Liposomes embedded with PEGylated iron oxide nanoparticles enable ferroptosis and combination therapy in cancer[END_REF]. They enable payload release near the tumors in response to specific stimuli such as electromagnetic heating, magnetic field or photothermal activation [START_REF] Theodosiou | Iron oxide nanoflowers encapsulated in thermosensitive fluorescent liposomes for hyperthermia treatment of lung adenocarcinoma[END_REF]. Besides such medical applications, they can be used as models for the emergence of life on earth. Indeed, minerals may be key elements used as inorganic catalysts in the formation and energetic sustaining of the first cell-like biological systems [START_REF] Dalai | Mineral-Lipid Interactions in the Origins of Life[END_REF].

To conclude, this special issue assembles several reviews and original articles that each exemplify the multiple roles of lipids, playing a ''symphonic concerto'' at the cell membrane. This concerto transitions from membrane dynamics to lipid signaling, including phospholipid remodeling and the role of lipids as building blocks, their role in the formation of lipid droplets, milk fat globules or other vesicles, their conversion into potent lipid mediators by multiple signaling pathways, and their utilization in nanolipid biotechnology for improved medicine.

Nature may have orchestrated lipids at the cellular membrane to bring together the essential elements of life, like a musician can write the key musical notes for a lovely song.

  highlights the novel nomenclature of LPLATs (1-14) putting together the AGPAT (1acylglycerol-3-phosphate O-acyltransferase) and MBOAT (membrane bound Oacyltransferase) families and reviewing their pathophysiological roles.

Acknowledgements

We are grateful to all researchers who took the time to write the reviews and original articles presented in this special issue. We acknowledge the important role played by our colleagues from the scientific board of GERLI (Groupe d'Etude et de Recherche en Lipidomique) who helped us to identify the topics of interest and potential contributions for the 17th GERLI Lipidomics meeting. Their support is highly appreciated and allowed us to invite internationally-renowned scientists who contributed to the meeting and to this special issue. We would like to thank the Editor of Biochimie, Dr Claude Forest, and the Editorial Office Manager Malika Hassini for handling the review process and assembling this special issue. A special thanks to Franck Aguila (IPMC, Sophia Antipolis, Valbonne, France) for creating the beautiful front cover of the special issue (see the legend on the back of the front cover).

Gérard Lambeau

Institut de Pharmacologie Moléculaire et Cellulaire (IPMC) UMR 7275, Centre National de la Recherche Scientifique -Université Côte d'Azur


A Trokhimovskiy

A Fedorova

Franck Montmessin

Oleg I Korablev

Svetlana Guslyakova

Franck Lefèvre

Jean-Loup L Bertaux

F Lefevre

From Spicam IR to Acs Nir; Dayside Nadir Observations in the Near Infrared

Introduction:

The SPICAM instrument onboard the Mars Express orbiter, operating from January 2004 until present, consists of two spectrometers: UV and near-IR (1000-1700 nm). The spectral range of SPICAM IR allows simultaneous retrievals of the column-integrated abundance of water vapor abundance in the 1.38 µm band and the 1.27 µm O2(a1 Δg) emission rate from the nadir observations [START_REF] Montmessin | SPICAM on Mars Express: A 10 year in-depth survey of the Martian atmosphere[END_REF]. No evidence of instrument degradation or properties drift is noted till now. From mid 2018 ACS NIR instrument onboard ExoMars 2016 mission will extend this remarkable multi-annual nadir monitoring survey adding new possibilities [START_REF] Korablev | The Atmospheric Chemistry Suite (ACS) of Three Spectrometers for the[END_REF].

The infrared channel of SPICAM has provided a nearly continuous survey of the columnintegrated abundance of water vapor since 2004 using the water vapor 1.38μm absorption band. This dataset spanning Martian Years 27-34 enables the investigation of both the seasonal and the inter-annual variations of atmospheric water. At the same time an indirect measurement of the ozone column may be accomplished via the O2(a1g) dayglow detected with the IR channel. As the O2(a1 Δg) dayglow originates from ozone photolysis, it reflects the distribution of ozone that is one of the most chemically reactive species of the Martian atmosphere. The excited O2 molecule can subsequently emit a photon at 1.27μm or be quenched by CO2. The latter process dominates at altitudes below 20 km and therefore dayglow only reflects the ozone abundance above 20 km. The comparison with simultaneous water vapor observations has showed that the revealed O2(a1 Δg) dayglow variations truly depend on the water vapor variations, and their anticorrelation prediction is clearly confirmed [3]. Seasonal patterns of H2O and O2(a1Δg) dayglow are rather stable from year to year with minor interannual variations. Dedicated radiative transfer calculations indicate that water vapor retrieval based on the optically thin 1.38μm absorption band in nadir is mostly sensitive to the water present in the first kilometers of the atmosphere above the surface. By that the Emission Phase Function (EPF) observations of water vapor band allow accessing poorly known H2O vertical distribution at lower altitudes for different seasons and locations. This is not yet applicable for the ExoMars mission pointing strategy, but can be used to reanalyze existing SPICAM dataset and plan future Mars Express observations. Preliminary results will be presented. The upcoming ACS NIR experiment follows the lineage of the SPICAM-IR instrument to characterize the Martian climatic processes and aeronomy phenomena [START_REF] Korablev | The Atmospheric Chemistry Suite (ACS) of Three Spectrometers for the[END_REF]. ACS NIR with its tenfold better spectral resolution will significantly enhance the fidelity of atmospheric measurements. The unique characteristics of the TGO orbit will allow a detailed sampling of the diurnal cycle of the main climatic properties of the Martian atmosphere. As designed, the TGO orbit during the Science Phase will produce a complete daily cycle every two months, yielding a representative atmospheric state every hour of the day. Another advantage of TGO orbit is the high freqiuency of occultation events, up to 24 per day. Measured vertical profiles of H2O in solar occultation mode down to the altitudes of 10 km will be used to specify water distribution during nadir data processing. Furthermore, unlike SPICAM data treatment, where climatology and aerosol load were derived from outer sources and models, the ACS NIR data processing will be supported by dust, ice and temperature profiles measurements in parallel by TIRVIM channel. By this ACS science products will be very self consistent. No other spacecraft in the past was able to perform this kind of survey and TGO will open a new era of investigations, constraining Martian climate variability from diurnal through seasonal up to interannual timescales.

Workshop organized by ESA-ESAC & IAA-CSIC, with the support of the ESAC Faculty and the EU H2020 project UPWARDS

Acknowledgements: Mars Express is a space mission of ESA. Science operations of SPICAM are funded by CNES and Roscosmos. We thank our collaborators at LATMOS/France for the support of science operations. AT, AF, and OK acknowledge RFBR grant 16-52-16011. JLB acknowledges grant of Ministry for Education and Science of Russian Federation #14.W03.31.0017.


Mohamed Slim Kammoun

Mylène Maïda

Small cycle structure for words in conjugation invariant random permutations

 from the product of permutations to any non-trivial word in the permutations and also as an extension of the results of [Nica, 1994] from uniform permutations to general conjugation invariant random permutations. In particular, we get optimal assumptions in the case of the commutator of two such random permutations.

Introduction and statement of the results

The object of this paper is to study the behavior of the cycles up to a fixed size for a word in several independent conjugation invariant random permutations.

We start by recalling that the asymptotics of the short cycle structure of one permutation chosen uniformly in the symmetric group is well known and Poisson. If we denote by S n the symmetric group of size n ∈ N * , and by # ℓ σ the number of cycles of length ℓ of the permutation σ, we have : Theorem 1.1. [Arratia et al., 2000, Theorem 3.1] For ρ n following the uniform distribution on S n , for any

d ′ ≥ 1, (# 1 ρ n , # 2 ρ n , . . . , # d ′ ρ n ) (d) ---→ n→∞ (ξ 1 , . . . , ξ d ′ ), where (d)
---→ n→∞ is the convergence in distribution and (ξ 1 , . . . , ξ d ′ ) are independent variables, where the distribution of ξ ℓ is Poisson with parameter 1 ℓ . Moreover, the joint moments converge and in particular, E (# 1 ρ n ) ---→ n→∞ 1. * Université de Toulouse -Institut de Mathématiques de Toulouse, France and Université de Lyon, ENSL, CNRS, France. Email: slim.kammoun@ens-lyon.fr † Univ. Lille, CNRS, UMR 8524 -Laboratoire Paul Painlevé, F-59000 Lille, France and IRL CRM-CNRS, International Reserach Lab, Montréal, Canada. Email: mylene.maida@univ-lille.fr.

1

It is also known that the asymptotic short cycle structure for words in independent uniformly distributed permutations is Poisson. More precisely, let k > 1 be fixed and F k := x 1 , . . . , x k be the free group on k generators {x 1 , . . . , x k }. In the sequel, the index k will always refer to the number of generators. An element w of F k can be written w = x εr kr . . . x ε 1 k 1 , with ε j = ±1 and if k j+1 = k j then ε j+1 = ε j . Such a form is unique, we call it in the sequel the canonical form of the word w.

For every group G, w = x εr kr . . . x ε 1 k 1 induces a word map from the Cartesian product G k into G by substitution :

w : G k → G (g 1 , . . . , g k ) → g εr kr . . . g ε 1 k 1 . We consider the case when G = S n , the symmetric group of size n ∈ N * , equipped with its composition law. When g 1 , . . . , g k is a k-tuple of random permutations, their image by the word map w is again a random permutation and one can study the distribution of its cycle structure. The more natural case to consider is independent uniformly distributed random permutations. It has been investigated by Nica, and more recently by Puder and coauthors. They have shown that the limiting distribution depends only on the algebraic structure of the word w. For any word Ω, and d ∈ N * , Ω d denotes the composition of d copies of the word Ω. We say that the word w is a power if there exists a word Ω = 1 and d ≥ 2 such that w = Ω d . Note that any word w = 1 can be written w = Ω d where Ω is not a power and d ≥ 1. We have the following : Proposition 1.2. [START_REF] Hanany | [END_REF]Puder, 2020, Theorem 1.3] [Nica, 1994, Theorem 1.1] Let w = Ω d = 1 be a word in F k such that d ≥ 1 and Ω is not a power. Let ρ 1,n , . . . , ρ k,n be independent and uniformly distributed on S n . Then the random vector (# 1 w(ρ 1,n , . . . , ρ k,n ), ρ k,n ), . . . , # d ′ w(ρ 1,n , . . . , ρ k,n )) converges in distribution to a limit that depends only on the power d and the maximal length d ′ of the cycles under consideration. Moreover, the joint moments converge also to the corresponding moments of the limiting law. In particular,

E (# 1 w(ρ 1,n , . . . , ρ k,n )) ---→ n→∞ ψ(d) (1)
where ψ(d) := ℓ|d 1 is the number of divisors of the integer d.

Remarkably, the limiting law does not depend on Ω. In particular, the limiting law is the same if we replace the word Ω by the word x 1 in just one letter.

Our goal in this paper will be to explore universality for these beautiful results. Before stating ours, let us give a few motivations to explore words in random permutations. The initial motivations of [Nica, 1994] was related to free probability theory and the notion of asymptotic freeness. Without going too much into the details, one can say that for a k-tuple of random matrices X 1,N , . . . , X k,N with a prescribed joint law, their asymptotic joint distribution in the framework of free probability theory is determined by the limit of E( 1 N Tr w(X 1,N , . . . , X k,N )), for any word w. If the random matrices are independent and their distribution is unitarily invariant, the behavior of the limit is well known and related to the notion of freeness -one says that such random matrices are asymptotically free. It was natural in this context to ask the same question for permutation matrices. This is the point of view of Nica. As the trace of a permutation matrix is related to the number of fixed points, one can see that the question of asymptotic freeness in this case is closely related to the study of the short cycle structure. In the same direction, one can also mention the theory of traffic distribution, developed by Male and coauthors (see e.g. [Male, 2020, Au et al., 2021]), exploring the case when unitary invariance is replaced by permutation invariance and the notion of second order freeness introduced by Mingo and Speicher (see e.g. [Mingo andSpeicher, 2006, Male, 2021]. Talking about links with random matrix theory, [Dubach, 2021] studies the total number of cycles for the commutator of two independent uniform permutations by using a connection with the moments in non-Hermitian random matrices. In a series of papers [START_REF] Puder | Measure preserving words are primitive[END_REF], Hanany and Puder, 2020, Magee and Puder, 2021], Puder and coauthors use the properties of the distribution of some statistics of w(ρ 1,n , . . . , ρ k,n ), as the number of fixed points, short cycles etc. to characterize the algebraic properties of the word w, through what they call the primitivity rank of the word w. Understanding w(ρ 1,n , . . . , ρ k,n ) can then give access to properties of the word map for other groups G.

In the present work, the goal is to extend Proposition 1.2 to a k-tuple (σ 1,n , . . . , σ k,n ) of independent and conjugation invariant random permutations.

Definition 1.3. For any n ≥ 1, let (σ 1,n , . . . , σ k,n ) be random permutations in (S n ) k . We say that the (distribution of the) k-tuple (σ 1,n , . . . , σ k,n ) is conjugation invariant if for any fixed permutations σ 1 , . . . , σ k ∈ S n , (σ -1 1 • σ 1,n • σ 1 , . . . , σ -1 k • σ k,n • σ k )
has the same distribution as (σ 1,n , . . . , σ k,n ). One can give several natural motivations to extend the results known for uniform permutations in this direction. First, if we go back to the question of asymptotic freeness mentioned above, seeking for universality for these questions has always been a strong motivation. For example, the work [Benaych-Georges, 2010] is clearly presented as an extension of [Nica, 1994] with motivations arising from free probability theory and studies some particular conjugation invariant distributions. It is also worth mentioning a connection with random maps : in [START_REF] Budzinski | Universality for random surfaces in unconstrained genus[END_REF] for example, the study of random maps is strongly related to the study of the cycle structure of the product of two permutations, one with prescribed cycle structure (corresponding to the faces of the map), the other being an involution without fixed points (corresponding to the edges), both naturally inheriting conjugation invariant distributions.

Before stating our theorem, let us detail our assumptions on random permutations having few short cycles.

Assumption 1.4. Let S 1 , S 2 be two finite subsets of N * . Let (σ n ) n∈N * be a sequence of random permutations (with for all n ≥ 1, σ n ∈ S n ). We say that the sequence (σ n ) n∈N * has few short cycles with respect to (S 1 , S 2 )

• in the weak sense if ∀i ∈ S 1 , # i σ n √ n P ---→ n→∞ 0, (2)
and ∀i ∈ S 2 , # i σ n n P ---→ n→∞ 0, (3)
where P ---→ n→∞ stands for the convergence in probability,

• and in the strong sense if

∀p ∈ N * , c 1 , . . . , c p ∈ S 1 , lim n→∞ E p i=1 # c i σ n √ n = 0 (4) and ∀i ∈ S 2 , lim n→∞ E # i σ j,n n = 0. (5)
Otherwise stated, we ask for a good probabilistic control on some cycles of length less than the maximum of S 1 ∪ S 2 . This control may be in probability for the weak version or on in mixed moments for the strong version.

In the assumptions of our results, we will need to choose the sets S 1 and S 2 according to the word w we are interested in. We detail now this choice.

For a given word w in F k , we will alternatively use its canonical form -that we have already introduced above : w = x εr kr . . . x ε 1 k 1 , with ε j = ±1 and if k j+1 = k j then ε j+1 = ε j -and its reduced form. Any word w ∈ F k can be written w :

= x β 1 α 1 . . . x β ℓ α ℓ with β i ∈ Z * , ℓ ≥ 1, α i ∈ k , α i+1 = α i ,
where k = {1, . . . , k}. We call this rewriting the reduced form of the word.

We define the following sets of integers : for any w ∈ F k with reduced form

x β 1 α 1 . . . x β ℓ α ℓ and 1 ≤ j ≤ k, let N ′ j (w) := {|β i |, α i = j}, N j (w) := {|d| : ∃β ∈ N ′ j (w), d|β} and N j (w) := max N j (w) ∪ {N ′ j (w) + N ′ j (w)}.
Note that N j (w) ⊂ 2 max( N j (w)) = 2 max(N ′ j (w)) .

For example, for w = , 2, 3, 4, 5, 6, 9, 12}. Our main result is the following extension of Proposition 1.2.

x 1 x 3 2 x -1 1 x 6 2 , N ′ 1 (w) = N 1 (w) = {1}, N 1 (w) = {1, 2}, N ′ 2 (w) = {3, 6}, N 2 (w) = {1, 2, 3, 6}, N 2 (w) = {1
Theorem 1.5. Let w be a word in F k with canonical form w = x εr kr . . . x ε 1 k 1 , where k 1 = k r (there is at least two different letters). We write w = Ω d , with Ω which is not a power and d ≥ 1.

Let σ 1,n , . . . , σ k,n be independent random permutations, with conjugation invariant distributions such that ∀j ∈ k , the sequence (σ j,n ) n≥1 has few short cycles with respect to ( N j (w), N j (w)) in the strong sense.

Then, for any d ′ ∈ N * , (# 1 w(σ 1,n , . . . , σ k,n ), . . . , # d ′ w(σ 1,n , . . . , σ k,n )) converges in distribution to a universal limit that depends only on the power d and the maximal length d ′ of the cycles under consideration. Moreover, the joint moments converge also to those of the limit.

The convergence obtained in Theorem 1.5 is a convergence in distribution and for all the moments. By standard probabilistic arguments, that will be briefly presented at the end of Section 4, the assumptions can be weakened to get the following corollary.

Corollary 1.6. Let w be a word in F k with canonical form w = x εr kr . . . x ε 1 k 1 , where k 1 = k r (there is at least two different letters). We write w = Ω d , with Ω which is not a power and d ≥ 1.

Let σ 1,n , . . . , σ k,n be independent random permutations, with conjugation invariant distributions such that ∀j ∈ k , the sequence (σ j,n ) n≥1 has few short cycles with respect to ( N j (w), N j (w)) in the weak sense.

Then, for any d ′ ∈ N * , the random vector (# 1 w(σ 1,n , . . . , σ k,n ), . . . , # d ′ w(σ 1,n , . . . , σ k,n )) converges in distribution to a universal limit that depends only on the power d and the maximal length d ′ of the cycles under consideration.

Let us first comment on the condition, k 1 = k r . For example, if w = x 1 x 2 x -1 1 , w(σ 1 , σ 2 ) has the same cycle structure as σ 2 so obviously there is no universality.

One can easily check the following cyclic invariance : for any σ

= (σ 1 , . . . , σ k ) ∈ S k n and w ∈ F k , for any ℓ ≥ 1, i ∈ k , # ℓ (σ i w(σ)) = # ℓ (w(σ)σ i ).
Using this remark, one can check that when k 1 = k r , there are 3 cases :

• either w = 1 : in this case there is nothing to prove since w(σ) = I n , that fixes all integers,

• or w = w 1 x α j w -1 1 : in this case, the cycle structure of w(σ) is the same as that of σ α j and there is no universality,

• or w = w 1 w 2 with w 1 and w 2 two words such that the canonical form of w

′ := w 2 w 1 = x ε ′ r ′ k ′ r ′ . . . x ε ′ 1 k ′ 1 satisfies k ′ 1 = k ′ r ′ .
In this case, the cycle structure of w ′ is the same as that of w. In other words, if, after successive cyclic simplifications, the word w still contains at least two different letters, then, by applying this word to independent permutations with conjugation invariant distributions and few short cycles, we get a random permutation with an universal limiting joint distribution for short cycles.

Similar results have already been proved in the case when the permutations are uniformly chosen among permutations with restrictions on cycle lengths in [Benaych-Georges, 2010, Theorem 3.7] (the latter being an extension of [Neagu, 2007], in which only the fixed points of the words in permutations were studied). They can also be seen as an extension of our previous work [START_REF] Kammoun | A product of invariant random permutations has the same small cycle structure as uniform[END_REF] for the product of permutations to any non-trivial word in the permutations. In the case of the product, we could get optimal assumptions, requiring the permutations to have only few fixed points and cycles of size 2. These assumptions were reminiscent of the connectivity assumptions for random maps appearing in [START_REF] Budzinski | Universality for random surfaces in unconstrained genus[END_REF]. In the case when |β i | = 1, for all i ≤ ℓ, we recover the exact same assumptions. For general words, we nevertheless require a more stringent control on short cycles, that may not be optimal. Optimality is discussed in Section 5, it does not hold in full generality but we can show that the assumptions we got for the product are also optimal in the case of the commutator of two permutations.

Note that in this work we focus only on short cycles but in the literature, there is also interest in the limit shape of the Young diagram associated to the cycle structure of a random permutation. We refer to [START_REF] Cipriani | The limit shape of random permutations with polynomially growing cycle weights[END_REF] for further details and exact definitions. This kind of convergence requires also the study of long cycles, which is an interesting question beyond the scope of this study. In the framework of conjugation invariant permutations, short cycles are still very interesting since they gives access to local events i.e. if σ n and ρ n are conjugation invariant random permutations such that

(# 1 σ n , . . . , # ℓ σ n ) d = (# 1 ρ n , . . . , # ℓ ρ n ) for some ℓ < n
2 , then for any 1 ≤ i 1 , i 2 , . . . , i ℓ ≤ n and for any 1 ≤ j 1 , j 2 , . . . , j ℓ ≤ n, the equality

P(σ n (i 1 ) = j 1 , . . . σ n (i ℓ ) = j ℓ ) = P(ρ n (i 1 ) = j 1 , . . . ρ n (i ℓ ) = j ℓ )
holds true. This remark is a direct consequence, for example, of [Hamaker and Rhoades, 2022, Equation (5.68)] (see also [Diaconis, 1989]). Theorem 1.5 can then be seen as a local convergence result.

Acknowledgements: Both authors would like to thank Maxime Février for very useful discussions. S.K. would also like to thank Zachary Hamaker and Camille Male for bibliographical help. M.M. is partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and S.K. is partially supported by a Leverhulme Trust Research Project Grant (RPG-2020-103), Labex CIMI (ANR-11-LABX-0040) and an ERC Project LDRAM (ERC-2019-ADG Project 884584).

Technical lemmas

The technique of proof is inspired by [START_REF] Kammoun | A product of invariant random permutations has the same small cycle structure as uniform[END_REF]. It is based on a graphical representation of the image of several points by the permutations composing the word w. The strategy of the proof will be to identify the class of graphs giving the main contribution in the uniform case and then to show that, under our assumptions, the non vanishing contributions are the same in the conjugation invariant case.

Let us introduce some definitions. Some of them have already been used in [START_REF] Kammoun | A product of invariant random permutations has the same small cycle structure as uniform[END_REF] but we will need colored versions of the graphs used therein.

• We denote by G n ℓ the set of oriented graphs with vertices n having exactly ℓ (oriented) edges and G n the set of oriented graphs with vertices n . We allow here loops but not multiple edges.

We denote also by H n ℓ the set of oriented graphs with vertices n each colored by red or white and having exactly ℓ (oriented) edges and H n the set of oriented graphs with vertices n , each colored by red or white. Given a (possibly colored) graph g we denote by E g the set of its edges.

• Given g ∈ G n and h ∈ H n , we say that h is of type g and we denote it by ḣ = g if E g = E h i.e. g is the non-colored version of h.

• For any σ ∈ S n , we denote by g σ the graph with vertices n and oriented edges

n ℓ=1 {(ℓ, σ(ℓ))} .
• Given g ∈ G n , we denote by

S n,g := {σ ∈ S n , ∀(i, j) ∈ E g , σ(i) = j} .
In other words, S n,g is the set of permutations σ such that g is a subgraph of g σ .

• A vertex i of g is called isolated if E g does not contain any edge of the form (i, j) or (j, i) nor a loop (i, i). Let g ∈ G n , we denote by g the graph obtained from g after removing isolated vertices.

• A connected component of g is called trivial if it is reduced to one isolated vertex.

• Two graphs (resp. colored graphs) are isomorphic if there exists a permutation of their vertices that preserves edges (resp. edges and colors). Let R be the equivalence relation on colored graph such that h 1 Rh 2 if h1 and h2 are two colored graphs that are isomorphic and Ṙ be the equivalence relation on non-colored graph such that g 1 Ṙg 2 if g1 and g2 two graphs that are isomorphic. We denote by

H ℓ := ∪ n≥1 H n ℓ / R and G ℓ := ∪ n≥1 G n ℓ / Ṙ the respective set of equivalence classes of ∪ n≥1 H n ℓ and ∪ n≥1 G n
ℓ and we set H := ℓ≥1 H ℓ and G := ℓ≥1 G ℓ .

We will now introduce the graphs useful for our purpose. We start from an integer m and look at its orbit along the word w(σ 1 , . . . , σ k ). If the reduced form of the word is w = x β ℓ α ℓ . . . x β 1 α 1 , we perform from m a walk of length |β 1 | on the graph g σα 1 , then a walk of length |β 2 | on the graph g σα 2 , etc. This will provide the vertices and edges of the k graphs corresponding to each permutation and each vertex is colored red if it is the entering or exiting point of the walks (see the definitions and the proof of Lemma 2.1 below for more details). In comparison to [START_REF] Kammoun | A product of invariant random permutations has the same small cycle structure as uniform[END_REF], considering graphs colored this way will allow more accurate control to show that the graphs containing loops do not contribute to the limiting distribution.

• Let n ∈ N * and σ = (σ 1 , . . . , σ k ) ∈ (S n ) k . Let m ∈ n be fixed. For a word with canonical form w = x εr kr . . .

x ε 1 k 1 , we define i m,w 0 (σ) := m, i m,w 1 (σ) := σ ε 1 k 1 (m), . . . , i m,w r (σ) := σ εr kr . . . σ ε 1 k 1 (m) = w(σ)(m), for any a ≥ 1 and 0 ≤ b ≤ r -1, i m,w ar+b (σ) := w a (σ)(i m,w b (σ)).
We also define the colors of the vertices as follows : the color of the vertex j with respect to the permutation σ i is given by

c m,w i,j (σ) = red, if      j = m and i = k 1 or j = w(σ)(m) and i = k r or ∃ℓ such that j = i m,w ℓ (σ), i ∈ {k ℓ , k ℓ+1 } and k ℓ = k ℓ+1
, and white otherwise.

For any i ∈ k , G m,w i (σ) ∈ G n is the graph with vertices n and edges Given s ⊂ n , let G s,w i (σ) be the graph such that E G s,w i (σ) = m∈s E G m,w i (σ) and let H s,w i (σ) be the graph such that E H s,w i (σ) = m∈s E H m,w i (σ) and the color of a vertex j is red in H s,w i (σ) if and only if the color of j is red in H m,w i (σ) for some m ∈ s.

E G m,w i (σ) :=   {ℓ:k ℓ =i,ε ℓ =1} i m,w ℓ (σ), i m,w ℓ+1 (σ)     {ℓ:k ℓ =i,ε ℓ =-1} i m,w ℓ+1 (σ), i m,w ℓ (σ)   , H m,w i (σ) ∈ H n is
For example, if w = (x 2 1 x 3 2 ) 2 , σ 1 = (4, 2, 3, 8)(1, 7)(5, 6) and σ 2 = (1, 2, 3, 4, 5)(9, 10), we have

G 1,w 1 (σ) = 1 2 3 4 7 , G 1,w 2 (σ) = 1 2 3 4 5 , H 1,w 1 (σ) = 1 2 3 4 7 , H 1,w 2 (σ) = 1 2 3 4 5 .
One can notice that for any s, i and w, G s,w i (σ) is a subgraph of g σ i . Fro any permutation σ, the graph g σ and all its subgraphs lie in classes of type C γ,γ ′ , that we define now.

• Let ℓ, ℓ ′ ∈ N * , γ := (γ 1 , . . . , γ ℓ ) ∈ (N * ) ℓ and γ ′ := (γ ′ 1 , . . . , γ ′ ℓ ′ ) ∈ (N * ) ℓ ′ .
We denote by C γ,γ ′ the set of classes of graphs g γ,γ ′ defined as follows :

for n ≥ ℓ + ℓ i=1 γ i + ℓ ′ i=1 γ ′ i , g γ,γ ′ ∈ G n has ℓ + ℓ ′ non
trivial connected components, for any j, j ′ ∈ ℓ + ℓ ′ , if j = j ′ then j and j ′ are in two distinct connected components. For j ≤ ℓ ′ , the component containing j is a directed cycle of length γ ′ j (the cycles of length 1 being loops). For ℓ ′ + 1 ≤ j ≤ ℓ + ℓ ′ , the component containing j has γ j edges and γ j + 1 vertices : j is a vertex of incoming degree 0 and outgoing degree 1, that we call the head, one vertex of incoming degree 1 and outgoing degree 0, that we call the tail and γ j -1 vertices of incoming degree 1 and outgoing degree 1. We call such a component straight. For example,

3 5 6 1 2 4 7 ∈ C (1,2),(2) = C (2,1),(2) .
Note that the multiplicity of the entries in the vectors γ and γ ′ is relevant, but their order is not.

• By convention, we will extend the previous definition to the case when ℓ ′ = 0, that is the case when there is no loop nor cycle. In this case, for ℓ ∈ N * , for γ := (γ 1 , . . . , γ ℓ ) ∈ (N * ) ℓ , we denote by T γ the class C γ,∅ of graphs g γ having ℓ non trivial connected components, all of them being straight, the component containing j ≤ ℓ having γ j edges and these components being pairwise distinct.

For example,

3 5 6 1 2 4 ∈ T (2,1) = T (1,2) = C (2,1),∅ .
For a tuple γ := (γ 1 , . . . , γ ℓ ) ∈ (N * ) ℓ , and h ∈ N * , h • γ ∈ (N * ) hℓ is the t-tuple obtained by repeating h times the t-uple γ.

For example T 3•(1,4,5) = T (1,4,5,1,4,5,1,4,5) = T (1,1,1,4,4,4,5,5,5) .

• For any j ∈ k , we define γ i (w) := (|β j |) α j =i , and γ(w) := (γ 1 (w), . . . , γ k (w)).

For example, for w = x 4 1 x -3 2 x 2 3 x 5 2 , γ 1 (w) = (4), γ 2 (w) = (3, 5) and γ 3 (w) = (2).

The structure of the graphs and their colored versions follow the following rules :

Lemma 2.1. Let w be a word in F k with reduced form w = x βq αq . . . x β 1 α 1 , with q ≥ 2 and 1 ≤ i ≤ k. For any s ⊂ n ,

1. A connected component of H s,w i (σ)
either is a cycle or is straight.

2. A connected component of H s,w i (σ) has at least one red vertex.

3. If a connected component of H s,w i (σ) has exactly one red vertex, then it is a cycle of length d dividing β ℓ for some ℓ such that α ℓ = i.

If a connected component of H s,w

i (σ) is straight, then the head and the tail are red.

5. If a connected component of H s,w i (σ) has exactly two red points, then it is either

• a straight component of length |β ℓ | for some ℓ such that α ℓ = i, • a cycle of length |β ℓ | + |β ℓ ′ | for some ℓ, ℓ ′ such that α ℓ = α ℓ ′ = i,
• a cycle of length j < |β ℓ | for some ℓ such that α ℓ = i.

6. Under the condition w(σ)(m) = m, if ℓ is red in H s,w i (σ) , then it is red in H s,w j (σ) for some j = i.

Proof. Let w = x βq αq ...x β 1 α 1 and σ = (σ 1 , . . . , σ k ). For each m, we can describe the sequence i m,w 0 (σ), . . . , i m,w r (σ) as follows : starting from m, we perform a walk of length |β 1 | on the graph g σα 1 , following the oriented edges if β 1 > 0 or going backward if β 1 < 0, then, after |β 1 | steps, we walk the same way on g σα 2 during |β 2 | steps etc. The colored graph H m,w i (σ) is then the part of g σ i explored by this process where each vertex is colored by red if and only it is the entering or exiting point for one of the walks. Given this description, the proof of the lemma is straightforward.

We gather hereafter several technical bounds that we will need in the proofs of our main results. For σ ∈ S n and j ∈ n , c j (σ) is the length of the cycle of the permutation σ containing j.

For a graph g of class T γ , the probability of the event {σ n ∈ S n,g } is controlled by the length of the cycle of a given point, say 1. This is stated in the following lemma that we recall from [START_REF] Kammoun | A product of invariant random permutations has the same small cycle structure as uniform[END_REF]:

Lemma 2.2. Let γ := (γ 1 , . . . , γ ℓ ) ∈ (N * ) ℓ and n ≥ v := ℓ + ℓ i=1 γ i .
For any graph g ∈ G n of class T γ and σ n a random permutation with conjugation invariant distribution, we have

1 - ℓ i=1 P(c 1 (σ n ) ≤ γ i ) - ℓ -1 n -1 ℓ i=1 γ i ≤ P(σ n ∈ S n,g )(n -ℓ)! (n -ℓ -ℓ i=1 γ i )! ≤ 1.
Proof. This is exactly Lemma 2.9 in [START_REF] Kammoun | A product of invariant random permutations has the same small cycle structure as uniform[END_REF]].

We will also need a control on P(σ n ∈ S n,g ) when g is a graph of class C γ,γ ′ , with ℓ ′ > 0. This will require to look at the length of the cycles of ℓ ′ given points, for example 1, . . . , ℓ ′ , provided they lie in pairwise distinct cycles. More precidely, we have:

Lemma 2.3. Let γ := (γ 1 , . . . , γ ℓ ) ∈ (N * ) ℓ , γ ′ := (γ ′ 1 , . . . , γ ′ ℓ ′ ) ∈ (N * ) ℓ ′ and n ≥ v := ℓ+ ℓ i=1 γ i + ℓ ′ i=1 γ ′ i . For ℓ ′ > 0, we define A γ ′ = σ ∈ S n : ∀1 ≤ j ≤ ℓ ′ : c j (σ) = γ ′
j and , ∀1 ≤ j = j ′ ≤ ℓ ′ , j and j ′ are not in the same cycle .

1. For ℓ ′ > 0, if g ∈ G n is a graph of class C γ,γ ′ , for any random permutation σ n with conjugation invariant distribution, we have

P(σ n ∈ S n,g )(n -ℓ -ℓ ′ )! (n -ℓ -ℓ i=1 γ i -ℓ ′ i=1 γ ′ i )! ≤ P(σ n ∈ A γ ′ ).
Moreover,

P(σ n ∈ A γ ′ ) ≤ E ℓ ′ i=1 # γ ′ i σ n (n -ℓ ′ )! n! ℓ ′ i=1 γ ′ i . (6)
2. If ρ n is distributed according to the uniform law then,

P(ρ n ∈ A γ ′ ) 1 - ℓ ℓ ′ i=1 (γ ′ i -1) n -ℓ ′ 1 - ℓ ℓ i=1 γ i n -γ ′ i ≤ P(ρ n ∈ S n,g )(n -ℓ -ℓ ′ )! (n -ℓ -ℓ i=1 γ i -ℓ ′ i=1 γ ′ i )! . Proof. Let n ≥ v := ℓ + ℓ i=1 γ i + ℓ ′ i=1 γ ′ i and g ∈ G n be a graph of class C γ,γ ′ .
By conjugation invariance, one can assume without loss of generality that g = g γ,γ ′ as in the definition of C γ,γ ′ above and that the non-isolated vertices are v . We denote by F := {y = (y i ) ℓ+ℓ ′ +1≤i≤v , y i ∈ {ℓ + ℓ ′ + 1, . . . , n}, pairwise distincts} and for any y ∈ F, we denote by g y the graph isomorphic to g obtained from g by fixing the vertices 1, . . . , ℓ + ℓ ′ and, for any i ∈ {ℓ + ℓ ′ + 1, . . . , v}, replacing i by y i .

It is easy to check that, for y, y ′ ∈ F, if y = y ′ , we have S n,gy S n,g y ′ = ∅. From there, we easily get the first bound of point 1.:

P(σ n ∈ S n,g ) = P σ n ∈ y∈F S n,gy cardF ≤ P(σ n ∈ A γ ′ ) (n-ℓ-ℓ ′ )! (n-v)!
.

For the second inequality of point 1., we write

E ℓ ′ i=1 γ ′ i # γ ′ i σ n = (i 1 ,...,i ℓ ′ )∈ n ℓ ′ E   ℓ ′ j=1 1 {c i j (σn)=γ ′ i }   ≥ (i 1 ,...,i ℓ ′ )∈ n ℓ ′ pairwise distinct E   ℓ ′ j=1 1 {c i j (σn)=γ ′ i }   = n! (n -ℓ ′ )! E   ℓ ′ j=1 1 {c j (σn)=γ ′ i }   ≥ n! (n -ℓ ′ )! P σ n ∈ A γ ′ .
We now go to the bound in the uniform case and define :

B γ,γ ′ := A γ ′   ℓ i=1 γ i j=1 σ ∈ S n , σ j (i + ℓ ′ ) > ℓ + ℓ ′   C γ,γ ′ := A γ ′   ℓ i=1 γ i j=1 σ ∈ S n , σ j (i + ℓ ′ ) > ℓ ′   = A γ ′   ℓ ′ i=1 γ ′ i -1 j=1 σ ∈ S n , σ j (i) > ℓ + ℓ ′   .
By observing that for g = g γ,γ ′ ,

P(σ n ∈ S n,g |σ n ∈ S n \ B γ,γ ′ ) = 0, and
P(σ n ∈ S n,g |σ n ∈ B γ,γ ′ ) = (n -v)! (n -ℓ -ℓ ′ )! ,
we obtain that

P(σ n ∈ S n,g ) = P(σ n ∈ B γ,γ ′ ) (n -v)! (n -ℓ -ℓ ′ )! .
Moreover, by definition, B γ,γ ′ ⊂ C γ,γ ′ ⊂ A γ ′ and then

P(σ n ∈ B γ,γ ′ ) = P(σ n ∈ B γ,γ ′ |σ n ∈ C γ,γ ′ )P(σ n ∈ C γ,γ ′ |σ n ∈ A γ ′ )P(σ n ∈ A γ ′ ).
For any conjugation invariant permutation,

P(σ n ∈ C γ,γ ′ |σ n ∈ A γ ′ ) ≥ 1 - ℓ ′ i=1 γ ′ i -1 j=1 ℓ p=1 P σ j n (i) = p + ℓ ′ |σ n ∈ A γ ′ = 1 - ℓ ′ i=1 ℓ(γ ′ i -1) n -ℓ ′ .
When ρ n is a uniform random permutation, then conditionally to

{ρ n ∈ C γ ′ ,γ }, the restriction of ρ n to n \ {σ j (i), 1 ≤ i ≤ ℓ, 1 ≤ j ≤ γ ′ i } is uniform. Therefore, P(ρ n ∈ B γ,γ ′ |ρ n ∈ C γ,γ ′ ) ≥ 1 - ℓ ℓ i=1 γ i n -γ ′ i .
This concludes the proof of the lower bound in the uniform case.

Asymptotics of fixed points

For readability reasons, before proving the general statements, we will prove the convergence of the number of fixed points under weaker assumptions. The statement is the following :

Theorem 3.1. Let w be a word in F k with canonical form w = x εr kr . . . x ε 1 k 1 ,
where k 1 = k r (there is at least two different letters). For j ≤ k, we denote by r j := card{i, k i = j}. We write w = Ω d , with Ω which is not a power and d ≥ 1.

Let σ 1,n , . . . , σ k,n be independent random permutations, with conjugation invariant distributions satisfying the following assumptions:

∀j ∈ k , ∀p ∈ N * , c 1 , . . . , c p ∈ N j (w) such that p i=1 c i ≤ r j , lim n→∞ E p i=1 # c i σ j,n √ n = 0 (7) and ∀j ∈ k , ∀i ∈ N j (w), lim n→∞ E # i σ j,n n = 0. ( 8
)
Then we have that

E (# 1 w(σ 1,n , . . . , σ k,n )) ---→ n→∞ ψ(d),
where we recall that ψ(d) is the numbers of divisors of the integer d.

In particular, in the case when d = 1, Theorem 3.1 reads :

Corollary 3.2. Let Ω ∈ F k with canonical form Ω = x εr kr . . . x ε 1 k 1 = 1 with k 1 = k r ,
and which is not a power.

Under Assumptions (7) and (8), we have that

E (# 1 Ω(σ 1,n , . . . , σ k,n )) ---→ n→∞ 1.
We start with the identification of the graphs that are contributing to the limit in the uniform case :

Proposition 3.3. Let w = Ω d be a word such that Ω is not a power. We write the reduced form of Ω as x β ℓ α ℓ . . . x β 1 α 1 , with α ℓ = α 1 . Assume that ρ 1,n , . . . , ρ k,n are independent and uniformly distributed on S n . Then, for any ĝ1 , ĝ2 , . . . ĝk ∈ G,

lim n→∞ nP ∀i ∈ k , G 1,w i (ρ 1,n , . . . , ρ k,n ) = ĝi , c 1 (w(ρ 1,n , . . . , ρ k,n )) = 1 = h|d k i=1 ½ {ĝ i =T h•γ i (Ω) } ,
where the sum runs over all divisors h of d and ∀i ∈ k , γ i (Ω) = (|β j |) α j =i .

Proof. We first recall that for any random permutation τ n ,

E(# 1 (τ n )) = E n i=1 ½ {c i (τn)=1} = n i=1 E(½ {c i (τn)=1} ) = n i=1 P(c i (τ n ) = 1)
and if τ n is conjugation invariant then P(c i (τ n ) = 1) = P(c 1 (τ n ) = 1) and

E(# 1 (τ n )) = nP(c 1 (τ n ) = 1).
According to Proposition 1.2, for ρ 1,n , . . . , ρ k,n independent and uniformly distributed,

lim n→∞ E(# 1 w(ρ 1,n , . . . , ρ k,n )) = ψ(d).
This is equivalent to the following:

lim n→∞ (ĝ i ) 1≤i≤k nP ∀i ∈ k , G 1,w i (ρ 1,n , . . . , ρ k,n ) = ĝi , c 1 (w(ρ 1,n , . . . , ρ k,n )) = 1 = ψ(d).
Our candidates for being the main contributions are the k-tuples of classes (T h•γ 1 (Ω) , . . . , T h•γ k (Ω) ) with γ i (Ω) = (|β j |) α j =i and h|d. As there are ψ(d) such k-tuples, one only needs to show that

lim n→∞ nP ∀i ∈ k , G 1,w i (ρ 1,n , . . . , ρ k,n ) = T h•γ i (Ω) , c 1 (w(ρ 1,n , . . . , ρ k,n )) = 1 ≥ 1.
We denote by t := ℓ i=1 |β i | the total length of the word Ω and r = td the length of w. Let h|d and, for n large enough, let j 1 , j 2 , . . . , j ht-1 be ht -1 indices in {2, . . . , n} which are two by two distinct.

Let σ := (σ 1 , . . . , σ k ) be such that

i 1,w 0 (σ) = 1, i 1,w 1 (σ) = j 1 , , . . . , i 1,w ht-1 (σ) = j ht-1 , i 1,w ht (σ) = 1. By construction, ∀1 ≤ i ≤ k, G 1,w i (σ) = T h•γ i (Ω) .
To simplify the notations, for any i ∈ k , we denote by g i := G 1,w i (σ). By Lemma 2.2 and independence, we have the following lower bound :

P (∀i ∈ k , ρ i,n ∈ S n,g i ) ≥ k i=1 (n -hℓ i -h ℓ i j=1 γ i j (Ω))! (n -hℓ i )!   1 -h ℓ i j=1 P(c 1 (ρ n,i ) ≤ γ i j (Ω)) - hℓ i -1 n -1 h ℓ i j=1 γ i j (Ω)   ,
where ℓ i is the number of non-trivial connected components of γ i (Ω).

From there, we get that

P ∀i ∈ k , G 1,w i (ρ 1,n , . . . , ρ k,n ) = T h•γ i (Ω) , c 1 (w(ρ 1,n , . . . , ρ k,n )) = 1 ≥ k i=1 (n -hℓ i -h ℓ i j=1 γ i j (Ω))! (n -hℓ i )!   1 - ℓ i j=1 P(c 1 (ρ n,i ) ≤ γ i j (Ω)) - hℓ i -1 n -1 h ℓ i j=1 γ i j (Ω)   (n -1)! (n -ht)! Since, for any i ∈ k , ℓ i j=1 P(c 1 (ρ i,n ) ≤ γ i j (Ω)) + hℓ i -1 n-1 h ℓ i j=1 γ i j (Ω)
→ 0, and we get that

lim inf n→∞ nP ∀i ∈ k , G 1,w i (ρ 1,n , . . . , ρ k,n ) = T h•γ i (Ω) , c 1 (w(ρ 1,n , . . . , ρ k,n )) = 1 ≥ lim inf n→∞ n 1 n h k i=1 ℓ i j=1 γ i j (Ω) n ht-1 ≥ 1,
where the last inequality comes from the fact that k i=1 ℓ i j=1 γ i j (Ω) = t. This concludes the proof.

We are now ready for the proof of Theorem 3.1.

Proof. It is enough to show that, under the assumptions ( 7) and ( 8), for any ĝ1 , ĝ2 , . . . ĝk ∈ G,

lim n→∞ nP ∀i ∈ k , G 1,w i (σ 1,n , . . . , σ k,n ) = ĝi , c 1 (w(σ 1,n , . . . , σ k,n )) = 1 = h|d k i=1 ½ {ĝ i =T h•γ i (Ω) } .
Let g be of class T γ , with γ = (γ 1 , γ 2 , . . . , γ ℓ ). Then, by Lemma 2.2, for any random permutation σ n with conjugation invariant distribution on S n and ρ n uniformly distributed we have

P(σ n ∈ S n,g ) ≤ 1 1 -1 n ℓ i=1 γ i -ℓ-1 n-1 ℓ i=1 γ i P(ρ n ∈ S n,g ). (9)
Therefore, for any ε > 0, there exists n 0 such that for any n ≥ n 0 ,

P(σ n ∈ S n,g ) ≤ (1 + ε)P(ρ n ∈ S n,g ). (10)
Similarly, for any random permutation σ n with conjugation invariant distribution on S n satisfying Assumption (8), we have

P(σ n ∈ S n,g ) ≥ 1 - ℓ i=1 P(c 1 (σ n ) = i) - ℓ -1 n -1 ℓ i=1 γ i P(ρ n ∈ S n,g ) ≥ (1 -ε)P(ρ n ∈ S n,g ). (11)
Therefore, if for any i ∈ k , there exists γ i such that ĝi = T γ i , from Proposition 3.3, we get that

lim n→∞ nP ∀i ∈ k , G 1,w i (σ 1,n , . . . , σ k,n ) = ĝi , c 1 (w(σ 1,n , . . . , σ k,n )) = 1 = h|d k i=1 ½ {ĝ i =T h•γ i (w) } .
We now want to show that graphs containing loops do not contribute. Let g be a fixed (non-colored) graph. If h is a colored graph such that ḣ = g, then ĥ belongs to a finite set of classes of colored graphs, with cardinal independent of n. Therefore, if we fix some classes of colored graphs ( ĥi ) 1≤i≤k such that one of the components is not straight, it is enough to show that

lim n→∞ nP ∀i ∈ k , H 1,w i (σ 1,n , . . . , σ k,n ) = ĥi , c 1 (w(σ 1,n , . . . , σ k,n )) = 1 = 0. (12)
One can assume that there exists (σ 1 , . . . , σ k ) such that c 1 (w(σ 1,n , . . . , σ k,n )) = 1 and ∀i ∈ k , H 1,w i (σ 1 , . . . , σ k ) = ĥi , otherwise ( 12) is trivial. To simplify the notations, we denote in the sequel h i := H 1,w i (σ 1 , . . . , σ k ), for any i ∈ k . Let h i be the same graph as h i after removing one edge from each cycle having at least two red vertices.

In particular, h i and h i have the same set of non-trivial vertices. We denote by C γ i ,γ ′i the class of ˙ h i , ℓ i and ℓ ′ i being the number of non-zero components of γ i and γ ′i respectively. Let V i be the set of non-trivial vertices of h i and r i is the number of red vertices of h i .

A direct consequence of Lemma 2.1 is that

card k i=1 V i ≤ k i=1 (card(V i ) -r i /2) ≤ k i=1 ℓ i j=1 γ i j + ℓ ′ i j=1 γ ′ j i -1/2 =: deg. There is at most O(n deg-1 ) possible k-tuples of graphs (h ′ i , . . . , h ′ k ) such that ∀i ∈ k , h ′ i = h i and there exists (σ 1 , . . . , σ k ) satisfying ∀i ∈ k , H 1,w i (σ 1 , . . . , σ k ) = ĥ′ i and c 1 (w(σ 1 , . . . , σ k )) = 1.
Indeed, one need to choose at most deg -1 non-trivial vertices in 2, n (1 is a non-trivial vertex), and then choose their positions (not all choices are admissible).

We have

P ∀i ∈ k , H 1,w i (σ 1,n , . . . , σ k,n ) = ĥi , c 1 (w(σ 1,n , . . . , σ k,n )) = 1 ≤ O(n deg-1 )P ∀i ∈ k , H 1,w i (σ 1,n , . . . , σ k,n ) = h i ≤ O(n deg-1 )P ∀i ∈ k , σ i,n ∈ S n, ḣi ≤ O(n deg-1 ) k i=1 P σ i,n ∈ S n, ḣi
By Lemma 2.3, we have

P σ i,n ∈ S n, ḣi ≤ P σ i,n ∈ S n, ˙ h i = O n -ℓ i j=1 γ i j - ℓ ′ i j=1 γ ′ j i +ℓ ′ i P(σ 1,n ∈ A γ ′i ). (13)
Moreover, when ℓ ′ i = 0, by Lemma 2.1, for any j ≤ ℓ ′ i , γ ′i j |β ℓ for some ℓ such that α ℓ = i. Consequently, using (6) and Assumption (7),

P(σ i,n ∈ A γ ′i ) ≤ o(n ℓ ′ i 2 )O(n -ℓ ′ i ) = o(n -ℓ ′ i 2 )
Therefore, if there exists i ≤ k such that ℓ ′ i = 0, then (12) holds.

We now consider the case when ∀i ≤ k, ℓ ′ i = 0. It means that every cycle of the h i 's has at least two red vertices. In this case, by Lemma 2.1, we have

card k i=1 V i ≤ k i=1 (card(V i ) -r i /2) ≤ k i=1 ℓ i j=1 γ i j = deg.
The second inequality is strict as soon as one component has at least three red vertices. In this case, we trivially bound P(σ i,n ∈ A γ ′i ) by 1 in (13) to conclude the proof. If all the components have at most two red vertices, then, by Lemma 2.1, the components are straight or a cycle of length |β ℓ | + |β ℓ ′ | for some ℓ and ℓ ′ or a cycle of length j < |β ℓ | for some ℓ such that α ℓ = i, for some 1 ≤ i ≤ k. In this case, one can conclude similarly to the preceding case, using Assumption (7) to conclude.

4 Proof of Theorem 1.5 and Corollary 1.6

Let us first introduce some more notations. Let λ = (λ 1 , . . . , λ m ) be a Young diagram, that is a sequence of nonnegative integers in decreasing order λ 1 ≥ . . . ≥ λ m ≥ 0. It is customary to represent them with λ i empty boxes on row number i. For two Young diagrams λ, µ, we say that µ ⊂ λ if µ can be obtained from λ by removing some rows of λ. We denote by ℓ(λ) the number of non-empty rows of λ and |λ| = ℓ(λ) i=1 λ i the number of boxes of the diagram.

For example, µ = (3, 1) = ⊂ = (3, 3, 1) = λ

For any pair (λ, µ) of Young diagrams such that µ ⊂ λ, for any n ≥ ℓ(λ), an admissible filling of type (λ, µ, n) is a filling of µ such that • all entries are in n and are pairwise distinct,

• the entries in the first column are increasing,

• in every row, the first entry is the smallest entry,

• ∀i ≤ ℓ(λ), i is in a row of µ of length λ i .

Let K λ,µ,n be the number of admissible fillings of type (λ, µ, n). For example,

• K (3,1),(1),n = 0, because 1 should be in a row of µ of length 3.

• K (2,2,2,1),(2,1),n = 0, because 1, 2 and 3 should go in the first row of µ, which has only two boxes.

• K (3,3,1),(3,1),n = 2(n -3) because the only admissible fillings in this case are of the form 1 2 j 3 and

1 j 2 3 for some 4 ≤ j ≤ n.
An important remark is that, for any Young diagram λ, we obviously have λ ⊂ λ and, for any n ≥ |λ|,

K λ,λ,n = (n -ℓ(λ))! (n -|λ|)! ,
because 1, 2, . . . , ℓ(λ) should go in the first position of each row of λ and we then complete freely the |λ|ℓ(λ) remaining boxes with distinct numbers between ℓ(λ) + 1 and n. By a similar argument, for any µ ⊂ λ, there exists C λ,µ such that for any n ≥ |λ|,

K λ,µ,n = C λ,µ (n -ℓ(λ))! (n -|µ|)! = C λ,µ n |µ|-ℓ(λ) (1 + o(1)). (14)
Let us now go to the proof of Theorem 1.5 itself. The scheme is similar to the proof of Theorem 3.1, except we have to follow simultaneously the trajectory of several starting points 1, 2, . . . , m. As we will see, a crucial point is that it is enough to consider the case when they lie in different cycles. We introduce the following event : for a Young diagram λ, j ≤ ℓ(λ), k 1 , . . . , k j distinct integers in ℓ(λ) , we denote by

A λ,Ω k 1 ,...,k j := σ ∈ (S n ) k , ∀i ≤ j, c k i (Ω(σ)) = λ k i
and ∀i = j, k i and k j lie in pairwise distinct cycles of Ω(σ)} .

It is an extension of the event A γ ′ defined in Lemma 2.3, in the sense that if λ is a Young diagram with rows γ ′ 1 , . . . , γ ′ ℓ in decreasing order then A λ,x 1 1,...,ℓ(λ) = A γ ′ .

The first key step is the following lemma :

Lemma 4.1. Let σ n and ρ n be two conjugation invariant random permutations in S n . Then the following two properties are equivalent :

1. For any Young diagram µ, lim n→∞ n ℓ(µ) P σ n ∈ A µ,x 1 1,...,ℓ(µ) = lim n→∞ n ℓ(µ) P ρ n ∈ A µ,x 1 1,...,ℓ(µ) .

For any

P ∈ R[x 1 , x 2 , . . . , x d ′ ], monomial, lim n→∞ E (P (# 1 σ n , . . . , # d ′ σ n )) = lim n→∞ E (P (# 1 ρ n , . . . , # d ′ ρ n )) .
Proof. For any m ∈ N * , p 1 , . . . , p m ∈ N,

a n p 1 ,...,pm := E n i=1 1 {c i (σn)=1} p 1 . . . n i=1 1 {c i (σn)=m} pm = 1 ≤ i 1 1 , . . . , i 1 p 1 i m 1 , . . . , i m pm ≤ n P ∀j ≤ m, ∀s ≤ p j , c i j s (σ n ) = j .
We denote by λ the Young diagram (m, . . . , m, . . . , 1, . . . , 1) where j appears p j times. For any j ≤ m, let us denote by t j := card{i j 1 , . . . , i j p j } ≤ p j the number of distinct indices among i j 1 , . . . , i j p j and π the Young tableau (m, . . . , m, . . . , 1, . . . , 1) where j appears t j times so that ℓ

(π) = t 1 + • • • + t m .. Note that π ⊂ λ.
Then, by conjugation invariance of the distribution, we have,

P ∀j ≤ m, ∀s ≤ p j , c i j s (σ n ) = j = P ∀j ≤ m, ∀s such that 1 + j-1 ℓ=1 t ℓ ≤ s ≤ j ℓ=1 t ℓ , c s (σ n ) = j , so that a n p 1 ,...,pm = π⊂λ n t 1 +...+tm (1 + o(1))   m j=1 P(p j , t j )   P (∀i ≤ t 1 + . . . + t m , c i (σ n ) = π i ) = π⊂λ n ℓ(π) (1 + o(1))   m j=1 P(p j , t j )   P (∀i ≤ ℓ(π), c i (σ n ) = π i ) ,
where P(p, t) is the number of partitions of p with t parts.

We now denote by

B π = {σ ∈ S n , ∀i ≤ ℓ(π), c i (σ) = π i }.
∀σ ∈ B π , we define f π (σ) as follows : let π σ be the Young diagram following the cycle structure of σ restricted to the cycles having at least one element in ℓ(π) . Then f π (σ) is the filling of π σ with the elements of the cycles of σ starting with the smallest element in each cycle and so that the first column is increasing.

For example, for σ = (1, 7, 8)(9, 3, 2)(4, 6)(10, 13, 6)(12, 11) and π = (3, 3, 3, 2), f π (σ) = 1 7 8 2 9 3 4 6 . Note that for n large enough, f π (σ) is necessarily an admissible filling of type (π, π σ , n). We have then,

P(σ n ∈ B π ) = µ⊂π f admissible filling of type (π,µ,n) P(f π (σ n ) = f ).
By conjugation invariance, for any admissible filling of type (π, µ, n),

P(f π (σ n ) = f ) (n -ℓ(µ))! (n -|µ|)! = P(σ n ∈ A µ,x 1 1,...,ℓ(µ) ) so that P (∀i ≤ ℓ(π), c i (σ n ) = π i ) = µ⊂π K π,µ,n K µ,µ,n P σ n ∈ A µ,x 1 1,...,ℓ(µ) . a n p 1 ,...,pm = (1 + o(1))   n ℓ(λ) P σ n ∈ A λ,x 1 1,...,ℓ(λ) + µ⊂λ,λ =µ C λµ n ℓ(λ) K λ,µ,n K µ,µ,n P σ n ∈ A µ,x 1 1,...,ℓ(µ)   = (1 + o(1))   n ℓ(λ) P σ n ∈ A λ,x 1 1,...,ℓ(λ) + µ⊂λ,λ =µ C ′ λ,µ n ℓ(µ) P σ n ∈ A µ,x 1 1,...,ℓ(µ)   ,
where in the last equality, we have used (14). By replacing σ n by ρ n in the previous calculus, the first implication is direct and the other implication can be obtained by iteration on ℓ(λ).

We give now the counterpart of Proposition 3.3.

Proposition 4.2. Let Ω a word which is not a power. Suppose that its reduced form x β ℓ α ℓ . . . x β 1 α 1 , satisfies α ℓ = α 1 . Assume that ρ 1,n , . . . , ρ k,n are independent and uniformly distributed on S n . Then, for any ĝ1 , ĝ2 , . . . ĝk ∈ G, for any Young diagram µ,

lim n→∞ n ℓ(µ) P ∀i ∈ k , G ℓ(µ) ,Ω i (ρ 1,n , . . . , ρ k,n ) = ĝi and (ρ 1,n , . . . , ρ k,n ) ∈ A µ,Ω 1,...,ℓ(µ) = k i=1 ½ {ĝ i =T |µ|•γ i (Ω) } , (15)
where we recall that |µ| = ℓ(µ) j=1 µ j . Consequently, if σ 1,n , . . . , σ k,n are independent, with conjugation invariant distribution satisfying Assumptions (4) and (5), then, for any Young diagram µ, we have that

P (σ 1,n , . . . , σ k,n ) ∈ A µ,Ω 1,...,ℓ(µ) = P (ρ 1,n , . . . , ρ k,n ) ∈ A µ,Ω 1,...,ℓ(µ) (1 + o(1)). (16)
Proof. A direct consequence of Proposition 1.2 and Lemma 4.1 is that for any Ω which is not a power,

P (ρ 1,n , . . . , ρ k,n ) ∈ A µ,Ω 1,...,ℓ(µ) = P ρ 1,n ∈ A µ,x 1 1,...,ℓ(µ) (1 + o(1)).
Since, when n > |µ|, P ρ 1,n ∈ A µ,x 1 1,...,ℓ(µ) = (nℓ(µ))! n! ,

we have, lim n→∞ n ℓ(µ) P (ρ 1,n , . . . , ρ k,n ) ∈ A µ,Ω 1,...,ℓ(µ) = 1.

If r is the total length of the word Ω written under its canonical form and t = r • |µ|ℓ(µ), let j 1 , . . . , j t be t indices in {ℓ(µ) + 1, . . . , n} which are two by two distinct. Let (g 1 , . . . , g k ) be equal to (G ℓ(µ) ,Ω 1 (σ), . . . , G ℓ(µ) ,Ω k (σ)) when i 1,Ω µ 1 0 (σ) = 1, i 1,Ω µ 1 1 = j 1 , , . . . , i 1,Ω µ 1 µ 1 r (σ) = 1, i 2,Ω µ 2 0 (σ) = 2, i 2,Ω µ 2 1 = j µ 1 r , . . . , i 2,Ω µ 1 µ 2 r (σ) = 2, . . . (r•µ i -1) , . . . , i 1,Ω µ ℓ(µ) µ ℓ(µ) r (σ) = ℓ(µ).

For such a k-tuple of graphs, by Lemma 2.3 and independence, we have the following lower bound :

P (∀i ∈ k , ρ i,n ∈ S n,g i ) ≥ k i=1 (n -|µ|ℓ i -|µ| ℓ i j=1 γ i j (Ω))! (n -|µ|ℓ i )!   1 -|µ| ℓ i j=1 P(c 1 (ρ n,i ) ≤ γ i j (Ω)) - |µ|ℓ i -1 n -1 |µ| ℓ i j=1 γ i j (Ω)   ,
where ℓ i is the number of non-trivial connected components of γ i (Ω).

From there, as in proof of proposition 3.3, we get that lim inf n→∞ n ℓ(µ) P ∀i ∈ k , G ℓ(µ) ,Ω i (ρ 1,n , . . . , ρ k,n ) = T |µ|•γ i (Ω) and (ρ 1,n , . . . , ρ k,n ) ∈ A µ,Ω 1,...,ℓ(µ) ≥ 1.

This concludes the proof of (15). Using similar arguments as in the proof of Theorem 3.1, we get that if σ n satisfies Assumptions (4) and ( 5), then (16) holds.

We can now conclude the proof of Theorem 1.5 itself.

Proof. We assume that w = Ω d , with d ≥ 1 and Ω is not a power. Let d ′ ∈ N * be fixed. If σ n := (σ 1,n , . . . , σ k,n ) satisfy (4) and ρ n := (ρ 1,n , . . . , ρ k,n ) are independent and uniformly distributed, we want to show that (16) implies that for any monomial P,

Indeed, it was proved in [Nica, 1994] that the right handside depends only on d and P. The first remark is that, for any monomial P, there exists a polynomial Q, such that for any fixed σ := (σ 1 , . . . , σ k ), P (# 1 w(σ), . . . , # d ′ w(σ)) = Q(# 1 Ω(σ), . . . , # dd ′ Ω(σ)).

Indeed, for any j ∈ N * , # 1 w(σ) j = On the other hand, # 1 w(σ) j = # 1 Ω(σ) dj = r|dj r# r Ω(σ).

Therefore, by induction, for any j ∈ N * , # j w(σ) can be expressed as a linear combination of {# r Ω(σ)} r|dj and it is enough to show Theorem 1.5 in the particular case when d = 1. As a consequence of Lemma 4.1, we have that, if for any Young diagram µ, lim n→∞ n ℓ(µ) P Ω(σ n ) ∈ A µ,x 1 1,...,ℓ(µ) = lim n→∞ n ℓ(µ) P Ω(ρ n ) ∈ A µ,x 1 1,...,ℓ(µ) , then for any P ∈ R[x 1 , x 2 , . . . , x d ′ ], monomial, lim n→∞ E (P (# 1 Ω(σ n ), . . . , # d ′ Ω(σ n ))) = lim n→∞ E (P (# 1 Ω(ρ n ), . . . , # dd ′ Ω(ρ n ))) .

The convergence of joint moments is therefore a direct consequence of ( 16) and the convergence in distribution follows.

We now go to the proof of Corollary 1.6. Assume that (σ 1,n , . . . , σ k,n ) satisfies the assumptions of Corollary 1.6. For every j ≤ k, let τ j,n be a random permutation, independent of (σ 1,n , . . . , σ k,n ), with Ewens n (0) distribution (that is the uniform law on the subset of S n of permutations having exactly one cycle). We now define, for every j ≤ k, σ j,n := σ j,n , if ∀i ∈ N j (w),

# i σ j,n √
n ≤ 1, τ j,n , otherwise.

Then ( σ 1,n , . . . , σ k,n ) satisfies the assumptions of Theorem 1.5 and (# 1 w(σ n ), . . . , # d ′ w(σ n )) and (# 1 w( σ n ), . . . , # d ′ w( σ n )) have asymptotically the same distribution. w Optimality of our conditions

x 1 x 2 Yes x 1 x 2 2 Yes [x 1 , x 2 ] Yes x 1 x 3 2 No x 1 x 2 x 3 No x 1 x 2 x 1 x 2 Yes x 2 1 x -2 2 Yes x 3 x 1 x -1 3 x 2 No

  the graph with vertices n , the color of vertex j being c m,w i,j (σ), and edges E G m,w i (σ) . Finally, let G m,w i (σ) and H m,w i (σ) be respectively the equivalence class of G m,w i (σ) and H m,w i (σ).

  # 1 w(σ n ), . . . , # d ′ w(σ n ))) = lim n→∞ E (P (# 1 w(ρ n ), . . . , # d ′ w(ρ n ))) .

  σ) j (i)=i = n i=1 1 c i (w(σ)))|j = j# j w(σ) + r|j,r =j r# r w(σ).

Discussion about optimality

In this last section, we make a few remarks on the optimality of our conditions (4), ( 5), ( 7) and (8) on short cycles. We hereafter only consider the case when the permutations are independent and have conjugation invariant distributions. In several cases, in particular the commutator, We can claim that these conditions are sharp. In [START_REF] Kammoun | A product of invariant random permutations has the same small cycle structure as uniform[END_REF] we already discussed the case of the product.

Optimality for the commutator

• Assumption (4) is optimal in the sense that if, for some ℓ ≥ 1, we have

Indeed, one can see that if g is the class of the graph with adjacency matrix I ℓ , the event

)} will contribute to the limit, leading to the term ε 2 ℓ in the limit. • Similarly, Assumption ( 5) is optimal in the sense that if

Indeed, as above, if ĝ′ is the class of the graph with adjacency matrix

(σ n , ρ n )) = (ĝ ′ , ĝ′ )} will contribute to the limit.

Non-optimality in the general framework

On the other hand, one can find words for which the conditions are not optimal. We mention hereafter a few examples where, by easy considerations, sometimes using our previous results on the product, one could improve the assumptions on short cycles.

• Take for example the case w = x 1 x 3 2 . Using our conditions in the case of the product, one could give conditions on fixed points and two cycles of σ 3 2 , that is conditions on cycles of length 1, 3, 6 on σ 2 . Our theorem gives conditions on cycles of lengths, 1, 2, 3, 6 therefore being suboptimal.

• More generally, if w = w 1 w 2 where w 1 and w 2 have disjoint supports, one can try to apply the product theorem to get weaker conditions. For example, for w = (x 1 x 2 )x 3 asking that the number of fixed points of x 1 x 2 are less than ε √ n and the number of two-cycles of x 1 x 2 is less that εn can be obtained with only conditions on fixed points of σ 1,n and σ 2,n .

• In a similar spirit, if for some j ≤ k, gcd N ′ j = d = 1 for some j, one can obtain in many cases better conditions by considering x d j as a new subword y in w.

The table below summarizes some cases where we checked whether our conditions are optimal or not.