Rotundone Chez

Vitis Vinifera

L C V Duras ___________________________

Olivier Geffroy
email: olivier.geffroy@vignevin.com

Thibaut Scholasch

Thierry Dufourcq

Eric Serrano

UNDERSTANDING AND MAPPING ROTUNDONE SPATIAL VARIABILITY IN VITIS VINIFERA L. C.V DURAS COMPREHENSION ET CARTOGRAPHIE DE LA VARIABILITE SPATIALE EN

Keywords: rotundone, peppery aroma, spatial variability, geostatistics, remote and proximal sensing

Recent research works highlighted large variability in rotundone concentration -the main compound responsible for peppery aroma in red wines -within a vineyard block associated with variation in land underlying the vineyard and vine water deficit. In order to further study and determine practical ways to map rotundone spatial variability, a study was carried on a 0.41 ha Duras vineyard from the South West of France under oceanic climatic influence. The smart point method was used and in order to optimize vineyard sampling, variability in vine architecture of the plot was assessed through measurements of trunk circumference (TC). On the basis of the three classes of TC obtained, six smart points made of 50 vines were positioned and followed independently over the vine growing season and winter dormancy for shoot elongation, stem water potential (Ψ stem), mineral uptake (petiole analysis and Dualex® measurements), airborne multispectral video imagery (NDVI), fruit quality, pruning wood weights (Physiocap® and manual weighing) and yields at harvest. At each smart point, rotundone was measured in wines obtained by microvinification techniques (1 L Erlenmeyer). Differences between smart points were very weak from an agronomic standpoint. However, a few plant physiological indicators (i.e. Ψ stem, Dualex® measurements) allowed discriminating vine performances. Despite this very low variability between plant and fruit measured variables (including those related to grape and wine quality but excluding TC), rotundone concentration was on average more than 50% higher in the wines from the high TC area. Thus, our data set suggests that a link exists between plant architecture (TC) and rotundone, and that TC can be used to approach rotundone spatial distribution. Our results also tend to strengthen the hypothesis that rotundone is very sensitive to fine variations of grapevine water status.

Résumé

Des travaux récents ont mis en avant une importante variabilité de concentration en rotundone -le principal composé responsable de l'arôme poivré des vins rouges -au sein d'une même parcelle, en lien avec les caractéristiques du sol et le niveau de contrainte hydrique. Afin d'aller plus loin et d'identifier une méthode pour cartographier de manière simplifiée cette variabilité spatiale, une étude a été lancée sur une parcelle de Duras N de 0,41 ha du Sud-Ouest, sous influence climatique de type océanique. La méthode dite des « smart points » a été mise en oeuvre et afin d'optimiser l'échantillonnage, la variabilité parcellaire dans l'architecture de la vigne a été évaluée grâce à des mesures de circonférence moyenne de tronc (TC). Sur la base des trois classes de TC obtenues, six « smart points » de 50 pieds ont été définis et suivis indépendamment sur la période végétative et le repos hivernal pour l'élongation des rameaux, le potentiel hydrique foliaire de tige, la nutrition minérale (analyses pétiolaires et mesures de Dualex®), l'indice de végétation par imagerie aérienne (NDVI), la qualité des raisins, le poids des bois de taille (Physiocap® et pesage manuel) et les rendements à la récolte. Pour chaque smart point, la rotundone a été mesurée dans des vins élaborés en condition de microvinification (Erlenmeyer d'1 L). Les différences observées entre les smart points sont très faibles d'un point de vue agronomique. Cependant, peu d'indicateurs de la physiologie de la plante (par exemple les mesures de potentiel hydrique foliaire de tige et de Dualex®) ont permis de discriminer le comportement de la vigne. Malgré cette faible variabilité pour la plupart des variables mesurés sur la vigne et les raisins y compris ceux liés à la qualité du raisin et du vin à l'exception du TC, la concentration en rotundone est en moyenne 50% plus élevée dans les vins issus de la zone à fort TC. Nos données suggèrent qu'il existe un lien entre la rotundone et l'architecture de la vigne (TC) et que la circonférence des troncs peut être utilisée afin d'approcher la distribution spatiale de la rotundone. Nos résultats tendent également à renforcer l'hypothèse selon laquelle la rotundone est très sensible à de fines variations de contrainte hydrique.

Mots-Clés: rotundone, arome poivré, variabilité spatiale, géostatistiques, télé et proxi-détections

Introduction

Rotundone is the main compound responsible for peppery aroma in red wines which has been identified in a growing number of cultivars. Recent research works highlighted large variability in rotundone concentration within a vineyard block associated with variation in land underlying the vineyard [START_REF] Scarlett | Within vineyard variation in the 'pepper'compound rotundone is spatially structured and related to variation in the land underlying the vineyard[END_REF]) and vine water deficit [START_REF] Geffroy | Effect of ripeness and viticultural techniques on the rotundone concentration in red wine made from Vitis vinifera L. cv. Duras[END_REF]. The last author identified good correlations (R 2 = 0.76 in 2011 and R 2 = 0.74 in 2012) between rotundone in wine and δ 13 C measured on grape sugar at harvest within the same vineyard. These findings suggest that it may be possible to manipulate rotundone content and the peppery character in wines through selective harvest. In order to further study and determine practical ways to map rotundone spatial variability, a study was carried on a Duras vineyard from the South West of France.

Unlike the approach proposed by [START_REF] Scarlett | Within vineyard variation in the 'pepper'compound rotundone is spatially structured and related to variation in the land underlying the vineyard[END_REF] which was more focused on relationships between land characteristics (soil conductivity, elevation, slope) and rotundone, our experiment proposed to focus on potential relationships involving plant and fruit physiological variables. To reach that goal we implemented an optimized sampling protocol to develop an agronomic approach based on vine physiological behavior.

Material and methods

Experimental vineyard, climatic background and sampling optimization protocol

The experiment was carried out on a 0.41 ha Guyot trained Duras vineyard from the South West of France planted in 1999 with 2.20 m × 1 m vine spacing. The plot that has been followed for 5 years was selected because both high concentration of rotundone and spatial variability in this variable were previously recorded. It is located in an area under oceanic climatic influence with normal annual rainfall and mean air temperature over the 1981-2010 period of 638 mm and 12.9°C, respectively. The vintage of study was characterized by a rainy and warm winter, a normal spring and an extremely rainy summer with 160 mm of rain recorded between mid-July and the end of August. Climatic conditions over the maturation period were dry and hot. The smart point method was used and vineyard sampling was optimized according to the approach proposed by [START_REF] Meyers | Improving vineyard sampling efficiency via dynamic spatially explicit optimisation[END_REF]. Variability in vine architecture of the plot was assessed during winter dormancy through measurements of mean trunk circumference (TC). Twenty-five plots made of four consecutive vines in a row were selected and TC was monitored at each plot at three different heights. According to [START_REF] Tisseyre | Precision viticulture and water status, mapping the predawn water potential to define within-vineyard zones[END_REF], this integrative variable reflects, among other factors, vine response to water and mineral deficits since plantation and thus can be used to determine areas of high water deficit when monitored along with other variables to assess vine nutrition. Data collected were normalized and spatialized through the kriging method (Figure 1A). On the basis of the three classes of TC obtained, six smart points were positioned with A and B corresponding to high, C and D to average and E and F to low TC classes. Each smart point consists of 50 vines selected over five different rows and the six smart points represents 17.5% of the whole block area. They were followed independently over the vine growing season.

Vegetative growth, mineral uptake and water deficit

Several indicators were used in order to reflect the vegetative growth and expression of the vine. Shoot elongation, an early indicator of water deficit [START_REF] Pellegrino | Towards a simple indicator of water stress in grapevine (Vitis vinifera L.) based on the differential sensitivities of vegetative growth components[END_REF]) was monitored weekly on three smart points (A, D and F) between 100 and 500 growing degree days (GDD) on trellised shoots (four shoots per vine and four consecutive vines per smart point). Remotely sensed imagery (airborne multispectral video imagery with a ground resolution of 50 cm) of the plot was obtained at veraison, on 4 August. Number of shoots and pruning wood weights were measured in January 2015 on the whole plot through Physiocap®, a new system to measure the number and the diameter of vine shoots. On each smart point, pruning wood mass (kg/vine) was also measured individually for 30 out of 50 vines. The uptake of mineral elements was accessed through petiole analyses (major and trace minerals) on an average sample of 50 basal leaves (1 per vine) at six key time points between flower cluster with floret separation and harvest, on three smart points (A, D and F). On each smart point, nitrogen status was also characterized on 50 leaves (1 leaf per vine) through measurements of Dualex® (Force A, Orsay, France), a fluorescence based sensor which allows the calculation of a chlorophyll flavonols ratio named NBI® (Nitrogen Balance Index). For each smart point, stem water potentials (Ψ stem) were measured from 10 of the 50 vines according to the protocol defined by [START_REF] Choné | Stem water potential is a sensitive indicator of grapevine water status[END_REF] at 4 time points (closing of the bunch, mid-veraison and harvest).

Fruit composition and rotundone

For each smart point, six grape samples were collected between 10 days before mid-veraison and harvest. Each sample consisted of 200 berries from both sides of the row and several parts of the cluster. After weighting, crushing and centrifugation, potential alcohol, titratable acidity, pH, malic acid, yeast assimilable nitrogen (YAN) and polyphenolic compounds (harvest sample only) were measured. For each smart point, 600 berry samples were also collected at harvest for indirect determination on rotundone in wines prepared by microvinification techniques (1-L Erlenmeyer) according to the protocol described by [START_REF] Geffroy | Effect of ripeness and viticultural techniques on the rotundone concentration in red wine made from Vitis vinifera L. cv. Duras[END_REF]. For each smart point, the 50 sample vines were harvested and yields at harvest (kg/vine) were monitored individually.

Statistical analysis

Statistical analyses were conducted with Xlstat software (Addinsoft, Paris, France). Analytical data with replicated measurements (stem water potentials, NBI, yields at harvest and pruning wood weights) were subjected to an ANOVA treatment. Fisher's least significant difference test was used as a post-hoc comparison of means at P < 0.05. Principal component analysis (PCA) was employed using means for each of the smart point for most of indicators followed, with rotundone as an additional variable.

Results and discussion

Agronomical behavior of the smart points

No differences in budburst date nor in other phenological stages were observed between the smart points. From 350 GDD which corresponds to 5 days before berry set, shoot elongation started to slow down on smart point F (Figure 2A). This deceleration could be linked rather to an early water deficit than to a nitrogen deficiency (Table 1) as higher NBI values were recorded on lower TC smart points (E and F). Both indicators i.e. petiole analysis by reference method and NBI showed lower nitrogen content in the high TC smart point at berry set which suggests a dilution of this element. While Physiocap® measurements showed a relative good homogeneity of the plot for pruning wood biomass (Figure 2B), manual weighing tends to strengthen this hypothesis (Table 1) as higher average values -non-significant statistically -were recorded on smart points A and B. As the vineyard block was characterized by a significant rate of missing plants (15%) with high mortality due to trunk diseases, Physiocap® measurements which occur at a frequency of 1 Hz may have been biased toward the detection of spatial variation reflecting areas with more missing plants than biomass differences between heatlhy vines. The month preceding veraison and far beyond until the end of August, frequent rain events contributed to the development of the interrow grass cover crop. Despite potential signal perturbation due to the presence of this green cover at veraison and the high percentage of missing plants, NDVI spatial pattern (Figure 2C) tends to visually overlaying the TC map. At closing of the bunch and veraison, probably because of the heavy rains, Dualex® measurements were perturbed. Due to bad weather conditions, measurement of stem water potential at veraison was delayed of 25 days and at harvest, leaves were in a bad condition to perform the measurement. Ψ stem especially at bunch closure, allows a perfect discrimination of the smart point according to the classes of TC defined. This observation corroborates previous research works [START_REF] Tisseyre | Precision viticulture and water status, mapping the predawn water potential to define within-vineyard zones[END_REF]. According to the threshold defined by van [START_REF] Van Leeuwen | Vine water status is a key factor in grape ripening and vintage quality for red Bordeaux wine. How can it be assessed for vineyard management purposes[END_REF] water deficit experienced by the vine on the plot was null to weak. Surprisingly at harvest and possibly earlier in the season, nitrogen status from the average TC class as reflected by the NBI values in leaves and YAN in grapes presented higher nitrogen levels which may be the combined effect of a better assimilation in comparison with the low TC smart points in relation with water deficit, and of a lesser dilution in comparison with the high TC smart points. In terms of sugar accumulation over the maturation period (Figure 2B) and polyphenolics in grapes at harvest (Table 1), little differences were noticed between the smart points. The slight decrease observed in the last sample may reflect more a measurement artefact than a real loss in sugar by respiration.

Differences between the smart points were very weak which is consistent with the hypothesis that low water deficit leads to low spatial contrast within the smart points [START_REF] Taylor | Identification and significance of sources of spatial variation in grapevine water status[END_REF]. Indicators used in routine and well known by winegrowers such as yields at harvest, fruit quality, mineral uptake through petiole analysis, pruning wood weights were not enough powerful to discriminate the smart points from an agronomical point of view while much better results were obtained with trunk circumference, Ψ stem, NBI.

Discussion about the average TC class and rotundone concentration in wine

It is always difficult to position an average class and to relate it either to a lower or to a higher group. This is not an unimportant matter as the average TC area of the studied plot represents almost half of the whole surface (Figure 1A). The wide range of indicators followed during our study and their analysis through a PCA treatment (Figure 3) should allow us to discuss it. It appears that smart points C and D have more similarities with smart points E and F and should be merged with the low TC class when considering selective harvest. This is consistent with the rotundone concentration, with average values of 94.5, 60.5 and 61 ng/L found in wines from the high, average and low TC area respectively. The PCA plots show that rotundone is correlated with variables related directly or indirectly to water deficit such as Ψ stem, weight of 200 berries and malic acid level, which is consistent with [START_REF] Geffroy | Effect of ripeness and viticultural techniques on the rotundone concentration in red wine made from Vitis vinifera L. cv. Duras[END_REF].

Conclusion

The present works allowed us to set up a methodology for assessing rotundone spatial distribution through the smart point method, under oceanic climatic influence. Despite very small differences between smart points for most of the measured variables including those related to wine quality, rotundone was on average more than 50% higher in the wines from the high TC area. Our results suggest that a link exists between rotundone in wine and plant architecture (TC) and that this costeffective indicator -the time to perform the 25 measurements on the plot didn't exceed one hour -can be used to approach rotundone spatial distribution. As TC is known to be closely correlated with apparent soil electrical conductivity (EC a ) (Bramley et al. 2011), our results are in agreement with recent research works [START_REF] Scarlett | Within vineyard variation in the 'pepper'compound rotundone is spatially structured and related to variation in the land underlying the vineyard[END_REF]) that show correlations between EC a and rotundone concentration. Our study also tends to strengthen the link existing within a vineyard between small variations in water deficit and rotundone in wine, highlighting the high sensitivity of the compound to fine variation of grapevine water status. In 2015, the same method will be implemented on a Syrah plot and grapes will be harvested selectively thanks to a new commercial harvester.

Acknowledgments

This study was carried out with financial support from FranceAgriMer and Midi-Pyrénées region. We are grateful to Manon Morlet and Olivier Garcia (both: CIVC, Epernay, France) for the Physiocap® measurements. Jérémie Diesel (IFV), Malek Hamouda and Sébastien Payen (Fruition Sciences) are also acknowledged for their technical assistance.

Literature

  CitedBramley R.G.V., Trought, M.C.T., Praat J. P. 2011. Vineyard variability in Marlborough, New Zealand: characterising variation in vineyard performance and options for the implementation of Precision Viticulture. Australian Journal ofGrape and Wine Research, 17,[72][73][74][75][76][77][78]

Figure 1 :

 1 Figure 1: Variation in (A) trunk circumference, (B) pruning wood biomass, and (C) NDVI index in a 0.41 ha Duras vineyard in the South West of France.Figure 1 : Variation (A) de circonférence de tronc, (B) biomasse de bois de taille, et (C) d'index NDVI au sein d'une parcelle de 0,41 ha du Sud-Ouest de la France.

Figure 1 :

 1 Figure 1: Variation in (A) trunk circumference, (B) pruning wood biomass, and (C) NDVI index in a 0.41 ha Duras vineyard in the South West of France.Figure 1 : Variation (A) de circonférence de tronc, (B) biomasse de bois de taille, et (C) d'index NDVI au sein d'une parcelle de 0,41 ha du Sud-Ouest de la France.

Figure 2 :

 2 Figure 2: Differences between the smart points in (A) shoot elongation from 100 to 500 GDD and (B) sugar accumulation between 10 days before mid-veraison and harvest.Figure 2 : Différences observées entre les smart point pour (A) l'élongation des rameaux sur la période 100-500 et (B) l'accumulation en sucre entre 10 jours avant la mi-véraison et la récolte.

Figure 2 :

 2 Figure 2: Differences between the smart points in (A) shoot elongation from 100 to 500 GDD and (B) sugar accumulation between 10 days before mid-veraison and harvest.Figure 2 : Différences observées entre les smart point pour (A) l'élongation des rameaux sur la période 100-500 et (B) l'accumulation en sucre entre 10 jours avant la mi-véraison et la récolte.

Figure 3 :

 3 Figure 3: Principal component analysis of data collected on the six smart points. (A) PC1 vs PC2; (B) PC1 vs PC3.Figure 3 : Analyse en composantes principales des données collectées sur six smart points. (a) PC1 vs PC2; b) PC1 vs PC3.

Figure 3 :

 3 Figure 3: Principal component analysis of data collected on the six smart points. (A) PC1 vs PC2; (B) PC1 vs PC3.Figure 3 : Analyse en composantes principales des données collectées sur six smart points. (a) PC1 vs PC2; b) PC1 vs PC3.

••Understanding••

  Compound responsible for peppery aroma in wines discovered recently in Australia (Wood et al. 2008) • Identified in a growing number of cultivars (Duras, Négrette, Fer, Abouriou, Syrah, Malbec, Pinot Noir, Gamay, Mondeuse, Pineau d'Aunis, Arani, Vespolina, Schioppettino, Grüner Veltliner, Maturana T…) • Perception threshold of 8 ng/l (water) and 16 ng/l (red wine) • 20 mg are enough to "contaminate" an Olympic swimming pool • One of the 16 most impacting aroma compound in wine (Ferreira 2012) • 20% of anosmia • Well perceived by consumers Reminder about rotundone Correlation between rotundone and δ13C at harvest • Good correlation between rotundone and δ13C on a 428 m2 Duras block • Rotundone variability is spatially structured which was also confirmed in Australia (Scarlett et al. 2014) • Possibility to identify regions of water deficit within a vineyard, which would facilitate the identification of zones for selective harvesting Work consists in optimizing the sampling by getting access to vine architecture using precision viticulture (Meyers et al. 2011) • Conducted in 2014, a wet vintage, in collaboration with Fruition Sciences on a 0,4 ha Duras block (PDO Gaillac) under Atlantic climate conditions • Trunk Circumference (TC) which reflects vine response to mineral and water deficit since the plantation (Tisseyre et al. 2005) was measured at 3 levels on 25 plots made of 4 consecutive vines • Data were normalized, spatialized through the krigeage method • On the basis of the 3 classes obtained, 6 'smart points' of 50 vines were positioned => 17,5% of the whole block Remotely sensed imagery (NDVI) • Physiocap® measurements, a sensor to assess wood biomass • Pruning wood weights (PWW) • Petiole analyses and Dualex® measurements (NBI), a fluorescence based sensor at 5 time points • Stem water potentials (Ψstem) + δ13C at 3 time points • Fruit quality (Brix, TA, pH, YAN, phenolics) at 5 time points and grape yields at harvest • Rotundone in wines prepared by microvinification techniques The average TC class (C,D) should be merged with the low TC class (E,F) • Rotundone is correlated with variables related directly or indirectly to water status Results overview Nitrogen uptake and water status • Low water deficit due to rainy summer conditions which leads to little differences between the smartpoints in accordance with previous works (Taylor et al. 2010) • Ψstem (veraison+25) allows a very good discrimination between the 6 smartpoints • Dualex® was powerful to discriminate the smartpoints in the early stages (until berry set +10) • Lower nitrogen content in the high TC smartpoint => dilution a Flower cluster with floret separation Understanding and mapping rotundone spatial variability NDVI index, Physiocap® and PWW measurements • Despite the signal disturbance due to the presence of grass cover (heavy summer rains), NDVI spatial pattern tends to visually overlying the TC map • Physiocap® shows a good homogeneity of wood biomass on the block while manual weighting of PWW shows (non-significant) higher values for the high TC area • As Physiocap® measurements were carried out with a frequency of 1Hz, the map might reflect more area with more missing plants (15% on the block) grape yield and rotundone at harvest (50% ver.+44d) • Little differences were noticed between the smartpoints (except for TA and YAN) • While rotundone concentration was on average 50% higher in the grapes from the high TC area => it is possible under Atlantic climate conditions and during vintage with low water deficit to produce from the same block, 2 kinds of wines differing only by their rotundone concentration (mg/kg)

Table I :

 I Differences between the smart points for several variables followed over the vine growing season and winter dormancy a Pruning wood weights. b Flower cluster with floret separation.

Table I :

 I Differences entre les smart points pour plusieurs variables mesurés au cours du cycle de la vigne et lors du repos hivernal. a Poids des bois de taille. b Boutons floraux séparés.


Vivien Despax
email: vivien.despax@lmpt.univ-tours.fr

A family of centered random walks on weight lattices conditioned to stay in Weyl chambers

. We also conjecture our results remain true in the more general case of a drift in the frontier of the Weyl chamber.

Introduction

Since the seminal work of O'Connell, various links between conditioned random walks and representation theory have been explored. In [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF], the authors generalize O'Connell's result to a wider class of random walks defined from minuscule representations of simple Lie algebras over C. According to the standard root systems classification, the main result of [O] appears in [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF] as related to the vector representation in type A. In general, the first octant should be replaced by the closure of a Weyl chamber associated to the considered root system. Also one has to replace the path transformation used by O'Connell -which is based on the Robinson-Schensted-Knuth correspondenceby another map (the generalized Pitman transform) introduced by Biane, Bougerol and O'Connell in [BBO]. It is defined in terms of Littelmann's path model -or, equivalently, in terms of Kashiwara's crystal basis theory -applied to tensor powers of the minuscule representation used to define the steps of the random walk considered. Concerning these topics in representation theory, we refer the reader to the articles [L1], [L2], [L3] and [K], [KN]. Beside these algebraic aspects, probabilistic tools from the Martin boundary theory are also used in [O] and in [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF]. Mention also the possibility to use the Gessel-Zeilberger reflection principle (a multi-dimensional generalization of the classical geometric argument introduced by André to solve the two candidates ballot problem) to derive a simpler combinatorial proof in the minuscule case, see [C].

Continuing in this vein but in the more general context of the representation theory of Kac-Moody algebras, the same authors proposed in [START_REF] Lecouvey | Conditioned random walks from Kac-Moody root systems[END_REF] a different approach which does not require this time any result from the Martin boundary theory and can also be used to obtain the conditioning of random paths defined from non-minuscule representations (one should then replace the random walks by continuous stochastic processes).

The assumption that the drift of the considered random walk belongs to the (open) Weyl chambers is an essential feature of all these works. This indeed yields a positive probability that it always remains in the closed cone, which makes its conditioning easy to define. Nevertheless, one can observe that the expressions obtained for the transition matrice of the conditioned process still make sense without this assumption. This follows from the fact that the generalized Pitman transform applied to our random walk always provides us with Markov chain whatever its drift.

It is then natural to ask if the Markov chain obtained as the Pitman transform of a random walk with drift outside the (open) Weyl chamber still has a probabilistic interpretation in terms of conditioning. The purpose of this article is to answer this question in the zero drift case. Our approach is the following : one can first consider the law of the centered random walk conditioned by the event "Until instant n, the random walk stays in the closed cone" for any positive integer n and let then the instant n tends to infinity. One can also start from the known laws corresponding to a drift in the (open) Weyl chamber and let the drift tends to zero. The main result of our paper consists in proving that these two notions of conditioning coincide. Our proofs use algebraic properties of the random walks we study together with a recent probabilistic theorem of Denisov and Wachtel on the exit time from cones of centered random walks, see [DW]. We also refer the interested reader to [D] for an elementary study of the simple random walks on the integers only based on the reflection principle and the strong law of large numbers. This approach could give a flavor of our general problem but it seems unfortunately difficult to adapt these arguments in higher dimension.

The paper is organized as follows. Section 2 is a recollection on known results on representation theory and the related random walks we study. We choose to restrict ourselves to the case of random walks defined from minuscule representations because a more general treatment would impose to consider random continuous trajectories (random Littelmann paths in fact) and thus would require a more general version of the deep result of Denisov and Wachtel. We refer to [B], [H], [S], [FH] for basics on root systems, weight lattices and representation theory. We also review some specific facts about tensor multiplicities and the random walks we consider that are important for our purpose. According to [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF] and [START_REF] Lecouvey | Conditioned random walks from Kac-Moody root systems[END_REF], we define particular probability distributions on trajectories in order to permit an algebraic treatment of the conditioning to always stay in the (closed) Weyl chamber. We develop a three-dimensional working example to illustrate our main definitions. The law of these conditioned random walks in the case of a drift in the open cone is then recalled in paragraph 2.4 as stated in [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF]. In Section 3, we turn to the specific case of the zero drift where the probability that our random walk always stays in the (closed) Weyl chamber becomes zero. We introduce the conditioning to stay in the (closed) Weyl chamber until an instant n and then study what happens when n tends to infinity. We then prove our main theorem by using Denisov and Wachtel estimate on the tail distribution of the exit time from this cone. Let us give a first statement of our main result. The reader is referred to Theorem 3.3.5 for a definitive version.

Theorem The transition probabilities corresponding to the conditioning to stay in the (closed) Weyl chamber until an instant n of a centered random walk with steps the weights of a minuscule representation converge when n tends to infinity. The limits so obtained can be recovered by letting the drift tends to zero in the transition probabilities corresponding to the same conditioning for random walks with the same steps but with drifts in the open Weyl chamber : We write W = W (R) for the Weyl group of the root system. Let us choose a base S = (α 1 , . . . , α d ) of R and denote by R + the related finite set of positive roots. The family of fundamental dominant weights Ω = (ω 1 , . . . , ω d ) corresponding to this choice of S is then defined by

2 ω i , α j α j , α j = δ i,j 1 ≤ i, j ≤ d. The Weyl chamber C = C (R) is then the open convex cone of E such that C = d i=1 R >0 ω i .
We write C = ⊕ d i=1 R ≥0 ω i and ∂C = C \ C for, respectively, its closure and its frontier in E. The weight lattice P = P (R) is the subgroup of E generated by {ω 1 , . . . , ω d } :

P = d i=1 Zω i .
It can be identified with an integral sublattice of h * R . The subset of the dominant weights

P + = P + (R) is P + = P ∩ C = d i=1 Z ≥0 ω i .
Although the Lie algebra gl d+1 (C) is not simple, we will consider in type A d the weight lattice of gl d+1 (C) rather than that of the simple Lie algebra sl d+1 (C).

Example 2.1.1 For the Lie algebra g = so 7 (C) and its Cartan subalgebra of diagonal matrices h (the B 3 case), we have :

• Rank : d = 3 • D = 3 • Root system : R = {±ε 1 , ±ε 2 , ±ε 3 , ±ε 1 ± ε 2 , ±ε 1 ± ε 3 , ±ε 2 ± ε 3 }
• Weyl group : W is the semidirect product of S 3 , the symmetric group operating by permutation on the set {ε 1 , ε 2 , ε 3 }, and (Z/2Z) 3 acting on the same set by

ε i → (±1) i ε i . The group W is of order 2 3 3! = 48. • A base : S = (α 1 , α 2 , α 3 ) = (ε 1 -ε 2 , ε 2 -ε 3 , ε 3 ) • Corresponding positive roots : R + = {ε 1 , ε 2 , ε 3 , ε 1 ± ε 2 , ε 1 ± ε 3 , ε 2 ± ε 3 } • Corresponding fundamental weights : Ω = (ω 1 , ω 2 , ω 3 ) = ε 1 , ε 1 + ε 2 , 1 2 (ε 1 + ε 2 + ε 3 ) • Corresponding Weyl chamber : C = x 1 ε 1 + x 2 ε 2 + x 3 ε 3 : (x 1 , x 2 , x 3 ) ∈ R 3 x 1 > x 2 > x 3 > 0 • Corresponding Weight lattice : P = (Zε 1 ⊕ Zε 2 ⊕ Zε 3 ) + Z 1 2 (ε 1 + ε 2 + ε 3 ) • Corresponding set of dominant weights : P + = λ 1 ε 1 + λ 2 ε 2 + λ 3 ε 3 : (λ 1 , λ 2 , λ 3 ) ∈ Z 3 + Z 1 2 , 1 2 , 1 2 λ 1 ≥ λ 2 ≥ λ 3 ≥ 0

Finite-dimensional representations

Consider the settings of the previous paragraph.

Weights, characters, decomposition into irreducibles and tensor multiplicities

Recall that P is identified with an integral sublattice of h * R . Given a finite-dimensional representation V of g and γ in P , we denote by V γ the subspace of V defined by

V γ = {v ∈ V : hv = γ (h) v for any h in h} .
When V γ is nonzero, we say that γ is a weight of V and the set of weights of the representation V is denoted by P (V ). One has the decomposition

V = γ∈P V γ = γ∈P (V ) V γ .
The character of the finite-dimensional representation V is then defined as the Laurent polynomial in

D variables x = (x 1 , . . . , x D ) char V (x) = γ∈P dim V γ x γ = γ∈P (V ) dim V γ x γ where x γ = x γ 1 1 . . . x γ N D if γ = D i=1 γ i ε i . Remark that dim V = char V 1 D
where we denote by 1 D the vector D i=1 ε i . The irreducible finite-dimensional representations of g are labelled by the elements of P + , that is by the dominant weights : to each dominant weight λ corresponds a unique (up to isomorphism) irreducible finite dimensional representation of g denoted by V (λ). One has V (0) = {0}. Given a dominant weight λ and a weight γ, the dimension of the subspace

V (λ) γ = {v ∈ V (λ) : hv = γ (h) v for any h in h} is denoted by K λ,γ .
The finite set of weights of V (λ) is denoted by P (λ) instead of P (V (λ)) and we write s λ for the character of V (λ). We thus have

s λ (x) = char V (λ) (x) = γ∈P K λ,γ x γ = γ∈P (λ) K λ,γ x γ .
The family (s λ ) λ∈P + is a basis of the vector space

C W [P ]. Any finite-dimensional repre- sentation V of g decomposes as a direct sum of irreducible components V (λ) with finite multiplicities m V,λ : V ≃ λ∈P + V (λ) ⊕m V,λ .
We use some standard notations when V is obtained by tensor products of irreducible finite-dimensional representations. For any dominant weights λ, δ, Λ and any nonnegative integer n, the multiplicity of the irreducible component V (Λ) in "the" decomposition into irreducibles of the finite-dimensional representation V (λ) ⊗ V (δ) ⊗n will be denoted by

f n Λ/λ,δ : V (λ) ⊗ V (δ) ⊗n ≃ Λ∈P + V (Λ) ⊕f n Λ/λ,δ . When n = 1, we write m Λ λ,δ instead of f 1 Λ/λ,δ . Remark that m Λ λ,δ = m Λ δ,λ .
In the sequel, the dominant weight δ will be fixed and we will then write λ Λ for two dominant weights λ, Λ if V (Λ) appears with a positive multiplicity in "the" decomposition of V (λ) ⊗ V (δ) into irreducibles :

V (λ) ⊗ V (δ) ≃ Λ∈P + V (Λ) ⊕m Λ λ,δ = λ Λ V (Λ) ⊕m Λ λ,δ . The inequality m Λ λ,δ ≤ K δ,Λ-λ (1)
can be proved, for example, by using Kashiwara's crystal basis theory, see Proposition 5.3 in [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF].

Minuscule weights and representations

If δ is a dominant weight such that its orbit under the action of W contains P (δ), we say that V (δ) and δ are minuscule. Here is the type-classification of the minuscule weights.

Type Minuscule weights

D Decomposition on B A d ω i , i = 1, . . . , d d + 1 ε 1 + . . . + ε i , i = 1, . . . , d B d ω d d 1 2 (ε 1 + . . . + ε d ) C d ω 1 d ε 1 D d ω 1 , ω d-1 , ω d d ε 1 , 1 2 (ε 1 + ε 2 + . . . -ε d ) , 1 2 (ε 1 + ε 2 + . . . + ε d ) E 6 ω 1 , ω 6 8 2 3 (ε 8 -ε 7 -ε 6 ) , 1 3 (ε 8 -ε 7 -ε 6 ) + ε 5 E 7 ω 7 8 ε 6 + 1 2 (ε 8 -ε 7 )
Minuscule weights are fundamental weights and, in the minsucule case, every subspace

V (δ) γ = {v ∈ V (δ) : hv = γ (h) v for any h in h} γ ∈ P
which is nonzero is one-dimensional and so

s δ (x) = γ∈P (δ) x γ .
When δ is minuscule, one can also prove

m Λ λ,δ = K δ,Λ-λ ∈ {0, 1} λ, Λ ∈ P + .
In this case, we have then

λ Λ ⇐⇒ m Λ λ,δ = 1 ⇐⇒ Λ -λ ∈ P (δ) λ, Λ ∈ P + , so V (λ) ⊗ V (δ) = λ Λ V (Λ) λ ∈ P + and this gives s λ (x) s δ (x) = λ Λ s Λ (x) λ ∈ P + . ( 2
)
Example 2.2.1 Consider the settings of Example 2.1.1.

• Minuscule weight :

ω 3 = 1 2 (ε 1 + ε 2 + ε 3 )
• Weights of the minuscule weight :

P (ω 3 ) = 1 2 (±ε 1 ± ε 2 ± ε 3 )
• Character of the minuscule weight :

s ω 3 (x 1 , x 2 , x 3 ) = x ± 1 2 1 x ± 1 2 2 x ± 1 2 3
Given a minuscule weight δ, tensor multiplicities have a combinatorial interpretation that will be important for our purpose. When γ, Γ are in P and when n is a positive integer, we denote by π n (γ, Γ) the set

(π 0 , π 1 , . . . , π n ) ∈ P n+1 : (π 0 , π n ) = (γ, Γ) , π i+1 -π i ∈ P (δ) 0 ≤ i ≤ n -1 ,
that is the set of paths in P with steps in P (δ) of length n starting at γ and ending at Γ. When λ, Λ are two dominant weights, the subset of paths always remaining in P + is denoted by π + n (λ, Λ). Again, the following result can be proved thanks to considerations on Kashiwara crystals or Littelmann paths, see Corollary 5.5 in [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF].

Proposition 2.2.2 We have Card π + n (λ, Λ) = f n Λ/λ,δ λ, Λ ∈ P + , n ≥ 1.

A family of random walks on the weight lattice

Consider the settings of paragraph 2.1 and let us choose once for all a minuscule weight δ. We endow the finite set P (δ) with probability distributions that have the property that the probability of any path in P with steps in P (δ) only depends on its two extremal points and its length. They are constructed as follow. Fix θ = (θ 1 , . . . , θ d ) in R d >0 . The dominant weight δ being minuscule, given a step s in P (δ), there exists w in the Weyl group W such that s = wδ. But, δ being a dominant weight, δ -wδ is a positive root : we can then write δ -s = δ -wδ = d i=1 m i α i where the m i are nonnegative integers. Let us then denote by θ [s] the positive real θ

[s] = θ m 1 1 . . . θ m d d and define Σ = s∈P (δ)
θ [s] .

(3)

The probability p s = p s (θ) of the step s is then defined by

p s = θ [s] Σ .
With this construction, one can remark that if s, S in P (δ) are such that S = s -α i (geometrically, S is the image of s under the orthogonal reflexion of E of hyperplan {α} ⊥ ) for some i in {1, . . . , d}, then we have

p s × θ i = p S .
We can represent this situation by the following oriented and colored graph

s i -→ S .
In order to obtain a formulation of our result in terms of characters, we introduce

x = x (θ) = D i=1 x i ε i , a solution in ⊕ D i=1 R >0 ε i of the system x α i = θ -1 i 1 ≤ i ≤ d. (4)
By using the Log function and elementary linear algebra, observe that this system possesses at least one solution since the family S = (α 1 , . . . , α d ) of R D is of rank d. We then can rewrite the sum Σ Σ = x -δ s δ (x)

and the probability distribution (p s ) s∈P (δ) is now given by

p s = x s s δ (x)
s ∈ P (δ) .

Also notice that when one chooses θ = (1, . . . , 1) = 1 d , one has the obvious solution x = D i=1 ε i = 1 D and this yields the uniform distribution on P (δ) : in this particular case, each step has probability 1/dim V (δ).

For the remainder of this paragraph, we fix a θ in R d >0 and (p s ) s∈P (δ) , the associated probability distribution on P (δ) constructed this way. We also pick x in ⊕ D i=1 R >0 ε i , a solution of System (4). We endow the denumerable set P with the σ-algebra of all subsets. Let ν 0 be a probability measure on P with full support

ν 0 (γ) > 0 γ ∈ P.
Consider (S (0) , X (1) , X (2) , . . .) a sequence of independent random variables defined on a probability space (Ω, T , P) such that 1. S (0) : Ω → P has law ν 0 . For any γ in P , we write P γ instead of P S(0)=γ or P [ . | S (0) = γ] for the conditional probability defined by

P γ [ . ] = P [ . ∩ S (0) = γ] P [S (0) = γ] .
2. X = (X (n) : Ω → P ) n≥1 are identically distributed with law ν = s∈P (δ) p s δ {s} . We denote by

d = d (θ) = E [X (1)] = s∈P (δ) P [X (1) = s] = 1 s δ (x) s∈P (δ)
x s s the common expectation of the random variables X .

We define a sequence of random variables S = (S (n)) n≥1 by setting

S (n) = n i=1 X (i) n ≥ 1.
The sequence of random variables W = (W (n)) n≥0 defined by W (0) = S (0) and

W (n) = S (0) + S (n) n ≥ 1
is then called the random walk on P with steps in P (δ), with initial distribution ν 0 and with law of increments ν. Its drift is the vector d. The sequence W is a (timehomogeneous) Markov chain with initial distribution ν 0 , state space P and transition matrix p = p (θ) given by

p (γ, Γ) = P [W (i + 1) = Γ | W (i) = γ] = ν (Γ -γ) = x Γ-γ s δ (x) if Γ -γ is in P (δ) 0 otherwise γ, Γ ∈ P, i ≥ 0.
With the previous construction, one can use independance to check that for any γ, Γ in P and any positive integer n, we have

P γ [W (n) = Γ] = P n i=1 λ + S (i) = π i = x Γ-γ s δ (x) n (π 0 , π 1 , . . . , π n ) ∈ π n (γ, Γ) . (5)
As desired, this number only depends on the two extremal points and on the length of the considered path. In Example 2.3.1, one can see that the drift d is in the open cone C if and only if the parameter θ is in ]0, 1[ 3 and this vector is zero if and only if θ = 1 3 , that is if and only if (p s ) s∈P (ω 3 ) is the uniform distribution on P (ω 3 ). This is a general fact, but it is not a trivial one, see Lemma 7.2 in [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF] for a proof.

Example 2.3.1 Consider the settings of Example 2.2.1.

• Solution of System (4) : (x 1 , x 2 , x 3 ) = 1 θ 1 θ 2 θ 3 , 1 θ 2 θ 3 , 1 θ 3 • Sum (3) : Σ = 1 + θ 3 + θ 2 θ 3 + θ 1 θ 2 θ 3 + θ 2 θ 2 3 + θ 1 θ 2 θ 2 3 + θ 1 θ 2 2 θ 2 3 + θ 1 θ 2 2 θ 3 3
• Relation between the steps :

-+ + 1 ր 3 ց + + + 3 -→ + + - 2 -→ + -+ -+ - 2 -→ --+ 3 -→ --- ց 3 ր 1 + --
Here, the eight step-vectors 1 2 (±ε 1 ± ε 2 ± ε 3 ) are replaced by the signs of their coordinates in the standard basis B.

• Probability distribution p on P (ω 3 ) :

• p +++ = 1 Σ • p ++-= θ 3 p +++ = θ 3 Σ • p +-+ = θ 2 p ++-= θ 2 θ 3 Σ • p -++ = θ 1 p +-+ = θ 1 θ 2 θ 3 Σ • p +--= θ 3 p +-+ = θ 2 θ 2 3 Σ • p -+-= θ 1 p +--= θ 3 p -++ = θ 1 θ 2 θ 2 3 Σ • p --+ = θ 2 p -+-= θ 1 θ 2 2 θ 2 3 Σ • p ---= θ 3 p --+ = θ 1 θ 2 2 θ 3 3 Σ
• Relations between probabilities :

• p ++-p +-+ = p +--p +++ • p ++-p -++ = p -+-p +++ • p ++-p --+ = p ---p +++ • p +-+ p -++ = p --+ p +++ • p +-+ p -+-= p ---p +++ • p -++ p +--= p ---p +++ • Drift : d = (1 -θ 1 ) θ 2 θ 3 (1 + θ 3 ) Σ ω 1 + (1 -θ 2 ) θ 3 (1 + θ 1 θ 2 θ 3 ) Σ ω 2 + (1 -θ 3 ) 1 + θ 2 θ 3 + θ 1 θ 2 θ 3 + θ 1 θ 2 2 θ 2 3 Σ ω 3
Let P ′ be a denumerable set and p ′ a stochastic matrix on P ′ , that is a map p ′ : P ′ × P ′ → [0, 1] such that y∈P ′ p ′ (x, y) = 1 for any x in P ′ . One can build a measurable space (Ω ′ , T ′ ) such that, for each probability measure ν ′ 0 on (Ω ′ , T ′ ), there exists a probability measure P ν ′ 0 on (Ω ′ , T ′ ) and a sequence of P ′ -valued maps defined on this probability space which is a Markov chain with initial distribution ν ′ 0 , state space P ′ and transition matrix p ′ . For a detailed construction, see [W] for example. We adopt this point of view to define a Markov chain from W that we call "the" random walk conditioned to always stay in the dominant weights.

Conditioning to always stay in the (closed) Weyl chamber when the drift is in the (open) Weyl chamber

Let us consider the event

W ∈ C = n≥0 W (n) ∈ C = n≥0 W (n) ∈ P + .
For any λ in P + , we also denote by λ + S ∈ C the event

λ + S ∈ C = n≥1 λ + S (n) ∈ C = n≥1 λ + S (n) ∈ P + .
The set C being stable under addition, we have the inclusions

S ∈ C ⊂ λ + S ∈ C λ ∈ P + .
Using independence and the previous remark, we have thus

P λ W ∈ C = P λ + S ∈ C ≥ P S ∈ C = P 0 W ∈ C λ ∈ P + (6)
and the total probability formula gives then

P W ∈ C = λ∈P + ν 0 (λ) P λ + S ∈ C . ( 7
)
If the event S ∈ C has a positive probability, so has the event W ∈ C by ( 6) and ( 7). Then, from a probabilistic point of view, to take the positive probability event W ∈ C into account leads to introduce the conditional probability P W∈C defined by

P W∈C [ . ] = P . ∩ W ∈ C P W ∈ C .
Let us observe how the transition probability between two states chosen in P + is modified if one replaces P by P W∈C . One can easily derive the following proposition from (8) with Proposition 3.1.1 -its finite-time equivalent -or prove it directly.

Proposition 2.4.1 Assume the event S ∈ C has a positive probability. We have

P W∈C [W (i + 1) = Λ | W (i) = λ] = p (λ, Λ) P Λ W ∈ C P λ W ∈ C λ, Λ ∈ P + , i ≥ 0.
This proposition leads us to introduce the function

h : P + -→ R λ -→ P λ W ∈ C = P λ + S ∈ C .
Observe that the transitions of our random walk under P W∈C are obtained by a particular transformation of the restriction of the matrix p to the subset P + . In the general setting, given a pair (P ′ , p ′ ) where P ′ is a denumerable set and p ′ is a substochastic matrix on P ′ , that is a map p ′ : P ′ × P ′ → [0, 1] such that y∈P ′ p ′ (x, y) ≤ 1 for any x in P ′ , and a positive function h ′ : P ′ → R >0 , we can define another matrix p ′ h ′ on P ′ by setting

p ′ h ′ (x, y) = p ′ (x, y) h ′ (y) h ′ (x)
x, y ∈ P ′ .

Then the matrix p ′ h ′ is stochastic on P ′ if and only if h ′ satisfies

h ′ (x) = y∈P ′ p ′ (x, y) h ′ (y) x ∈ P ′ .
In this case, we say that h ′ is a positive harmonic function on P ′ for p ′ . As we will see it in the following, in the positive drift case, the function h that we introduced previously is an example of positive harmonic function on P + for the restriction of the matrix p to P + . Then, Proposition 2.4.1 shows that the map p + = p + (θ) defined on P + × P + by

p + (λ, Λ) = p (λ, Λ) h (Λ) h (λ) λ, Λ ∈ P +
is a stochastic matrix on P + . Any Markov chain associated to the pair (P + , p + ) is called "the" random walk with law ν conditioned to always stay in the dominants weights and we denote it by W + .

Proposition 2.4.2 For any drift d, the function h is harmonic on the set P + for the restriction of the matrix p to P + .

Proof. The proof is almost direct. It is not more difficult than in the most elementary case of simple random walks on the integers, see [D].

Under the crucial assumption that the drift of the random walk W is in the open convex cone C, Lecouvey, Lesigne and Peigné achieved to explicit the law of W + in terms of irreducibles characters, see Theorem 7.6 and Corollary 7.7 in [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF].

Theorem 2.4.3 (Lecouvey, Lesigne, Peigné) Assume that the drift d is in the (open) Weyl chamber C (equivalently : the parameter θ is in ]0, 1[ d ). We have

h (λ) = x -λ s λ (x) α∈R + 1 -x -α λ ∈ P + .
The transition matrix p + of the Markov chain W + is given by

p + (λ, Λ) = p (λ, Λ) x -Λ s Λ (x) x -λ s λ (x) = 1 s δ (x) s Λ (x) s λ (x) if λ Λ 0 otherwise λ, Λ ∈ P + .
As we noticed it in the introduction, these expressions still make sense when the parameter θ is 1 d . In the next section, we prove that the equality obtained for this value of the parameter can be still probabilistically interpreted in terms of conditioning for the corresponding centered random walk to always stay in the (closed) Weyl chamber.

3 The zero drift case

Conditioning to stay in the dominant weights until an instant

We introduce (W (≤ n)) n≥0 the increasing sequence of positive probability events defined by

W (≤ n) ∈ C = n i=0 W (i) ∈ C = n i=0 W (i) ∈ P + n ≥ 0 which limit is W ∈ C.
We also introduce for each λ in P + the increasing sequence

(λ + S (≤ n)) n≥1 defined by λ + S (≤ n) ∈ C = n i=1 λ + S (i) ∈ C = n i=1 λ + S (i) ∈ P + n ≥ 1 which limit is λ + S ∈ C.
Observe that we have

P [W (≤ n)] = λ∈P + ν 0 (λ) P λ [W (≤ n)] n ≥ 0, P λ [W (≤ n)] = P [λ + S (≤ n)] λ ∈ P + , n ≥ 1, P λ + S (≤ n) ∈ C ≥ P S (≤ n) ∈ C ≥ P n i=1 X (i) = +1 = p n δ > 0 λ ∈ P + , n ≥ 1.
When the drift is zero, Theorem 3.2.1 shows in particular that the event W ∈ C has probability zero. Then we can use these positive probability approximations of the event W ∈ C in order to build a notion of conditioning of always staying in the dominant weights in the zero drift case. Let us introduce the sequence of conditional probabilities P W(≤n)∈C n≥0 defined by

P W(≤n)∈C [ . ] = P . ∩ W (≤ n) ∈ C P W (≤ n) ∈ C n ≥ 0.
Since the sequence P W (≤ n) ∈ C n≥0 converges to P W ∈ C , one has, under the assumption that P W ∈ C is positive (that is when the drift is in the (open) Weyl chamber C or when θ is in ]0, 1[ d ),

P W(≤n)∈C [E] ---→ n→∞ P W∈C [E] E ∈ T . If we set p + n (λ, Λ, i) = P W(≤n)∈C [W (i + 1) = Λ | W (i) = λ] λ, Λ ∈ P + , n ≥ 0, i ≥ 0,
the previous remark gives in particular

p + n (λ, Λ, i) ---→ n→∞ p + (λ, Λ) λ, Λ ∈ P + , i ≥ 0 (8) whenever the parameter θ is in ]0, 1[ d .
When the drift is zero, that is when the simple random walk considered is the centered one, it is no longer obvious whether the sequences

P W(≤n)∈C [W (i + 1) = Λ | W (i) = λ] n≥0 λ, Λ ∈ P + , i ≥ 0
are convergent, since the denominator P W (≤ n) ∈ C tends to zero when n tends to infinity.

As done in the positive drift case, we start by observing how the transition probability between two states in P + is modified when we replace P by one of the P W(≤n)∈C .

Proposition 3.1.1 Let λ, Λ be in P + . One has

p + n (λ, Λ, i) = p (λ, Λ) P Λ [W(≤n-i-1)] P λ [W(≤n-i)] if i ≤ n -2 p (λ, Λ) if i > n -2 n ≥ 2, i ≥ 0.
In particular, one has for any nonnegative integer i

p + n (λ, Λ, i) = p + n-i (λ, Λ, 0) n ≥ i + 2.
Proof. Once again, the proof is not more difficult than in the most elementary case of simple random walks on the integers, see [D].

Exit time from a cone in the zero drift case

Given a dominant weight λ, we introduce the random variable

τ λ = inf n ≥ 1 : λ + S (n) ∈ C ,
that is the first time the random walk starting at µ exits from the closed cone C. In a more general setting, Denisov and Wachtel found out an asymptotic for the tail distribution of the exit time from a cone of a centered random walk under some conditions on the cone and the moments of the law of increments, see [DW]. As C is a closed convex polyhedral cone (i.e. the convex hull of a finite set of half lines) and as our set of steps P (δ) is finite, the conditions they require are fulfilled in our particular situation.

Theorem 3.2.1 (Denisov,Wachtel) Assume that the drift of the random walk W is zero, that is (p s ) s∈P (δ) is the uniform distribution on the set of steps P (δ) (equivalently : θ is 1 d ). There exists two positive reals κ 1 , κ 2 and a positive function

V : C → R >0 such that P (τ λ > n) ∼ n→∞ κ 1 V (λ) n -κ 2 λ ∈ P + .
Remark 3.2.2 1. From this theorem, the existence of the positive function V and the polynomial rate of convergence to zero are the only facts that we need in the following, see Proposition 3.3.2. For a complete presentation including detailed discussion about hypotheses, constants and the positive function appearing in the conclusion of this theorem, see [DW].

2. In the case of the symmetric random walk on the integers, one can prove this result with elementary tools, see [D].

Main result

Recall that δ is a fixed minuscule weight and P (δ) is the finite set of weights of the minuscule representation V (δ). Let θ be in R d >0 and (p s ) s∈P (δ) the associated probability distribution on P (δ). Remind that x is a solution in ⊕ D i=1 R >0 ε i of System (4). From a probabilistic point of view, it is convenient to introduce a subset of P + which is more adapted to our situation. According to Proposition 2.2.2, the support of the conditioned random walk W + defined in paragraph 2.4 is the subset P +,δ such that

P +,δ = λ ∈ P + : ∃n ≥ 0 f n λ/0,δ > 0 .
Introduce also the sequence of subsets P +,δ n n≥0

defined by

P +,δ n = λ ∈ P +,δ : f n λ/0,δ > 0 n ≥ 0.
Observe that P +,δ 0 = {0} and P +,δ = n≥0 P +,δ n .

In general, we have P +,δ = P + . With this notation, we have

V (δ) ⊗n ≃ λ∈P + V (λ) ⊕f n λ/0,δ = λ∈P +,δ n V (λ) ⊕f n λ/0,δ n ≥ 0.
By restricting the matrix p to P +,δ instead of P + , we have a simple tool to derive our main result, see Lemma 3.3.3.

The following identity can be proved by elementary computations on characters, see Proposition 5.3 in [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF].

f n Λ/λ,δ = κ∈P +,δ n m Λ λ,κ f n κ/0,δ λ, Λ ∈ P + , n ≥ 0. (9)
Lemma 3.3.1 We have

Λ ∈ P +,δ n+1 ⇐⇒ ∃λ ∈ P +,δ n λ Λ Λ ∈ P + , n ≥ 0.
Proof. On the one hand, we have

V (δ) n+1 ≃ Λ∈P + V (Λ) ⊕f n+1 Λ/0,δ
and, in the other hand, we have

V (δ) n+1 ≃ V (δ) ⊗ V (δ) ⊗n ≃ V (δ) ⊗   λ∈P + V (λ) ⊕f n λ/0,δ   .
Using Equality (2), this gives

Λ∈P + f n+1 Λ/0,δ s Λ (x) = s δ (x) λ∈P + f n λ/0,δ s λ (x) = λ∈P + f n λ/0,δ s λ (x) s δ (x) = λ∈P + f n λ/0,δ λ Λ s Λ (x) = Λ∈P + λ Λ f n λ/0,δ s Λ (x) .
We deduce then the equality

f n+1 Λ/0,δ = λ Λ f n λ/0,δ
which gives the expected result. Now we prove that the sequences

p + n (λ, Λ, i) n≥0 λ, Λ ∈ P + , i ≥ 0
are convergent. Mimicking the previous section, Proposition 3.1.1 leads us to introduce the two sequences of positive functions (ψ n = ψ n (δ)) n≥1 and (h n = h n (δ)) n≥1 defined by

ψ n : P +,δ -→ R >0 λ -→ P λ W ∈ C = P λ + S (≤ n) ∈ C h n : P +,δ -→ R >0 λ -→ ψn(λ) ψn(0)
n ≥ 1.

Theorem 3.3.2 Assume that the drift is zero (equivalently, θ = 1 d ). The sequence (h n ) n≥1 pointwise converges on P +,δ to an harmonic function on P +,δ for the restriction of the matrix p to P +,δ .

Proof. Let λ be in P +,δ and n a positive integer such that n ≥ 2. Remark that

h n (λ) = ψ n (λ) ψ n (0) = P [τ λ > n] P [τ 0 > n] .
By Theorem 3.2.1, one has the pointwise convergence of the sequence (h n ) n≥1 on P +,δ . Now we have

ψ n (λ) = λ Λ p (λ, Λ) ψ n-1 (Λ) λ ∈ P +,δ ,
and so

h n (λ) = ψ n-1 (0) ψ n (0) λ Λ p(λ, Λ)h n-1 (Λ) λ ∈ P +,δ .
Theorem 3.2.1 gives also

ψ n-1 (0) ∼ n→∞ ψ n (0)
and the result follows.

The following lemma helps us to identify the pointwise limit given by the previous theorem. This is an easy adaptation of a lemma called "Maximum principle" in [W].

Lemma 3.3.3 Let h 1 , h 2 be two harmonic functions on P +,δ for the restriction of the matrix p to P +,δ . If they coincide at 0 and if h 1 ≤ h 2 , then they coincide on P +,δ .

Proof. For h 1 , h 2 satisfying the assumptions of the proposition, consider the nonnegative function h = h 2 -h 1 . Both functions h 1 and h 2 being harmonic on P +,δ for the restriction of p to P +,δ , so is the function h. As P +,δ 0 = {0}, one has h = 0 on P +,δ 0 . Assume now that h = 0 on P +,δ n for some nonnegative integer n and suppose there exists Λ 0 in P +,δ n+1 such that h (Λ 0 ) > 0. Using Lemma 3.3.1, one can then find a weight λ 0 in P +,δ n such that λ 0 Λ 0 . The function h being harmonic and nonnegative on P +,δ , we obtain

h (λ 0 ) = p (λ 0 , Λ 0 ) h (Λ 0 ) + λ 0 Λ Λ =Λ 0 p (λ 0 , Λ) h (Λ) > 0.
This is a contradiction. So h is zero on P +,δ n+1 . By induction, h is zero on P +,δ . We already have an example of such positive harmonic function : by construction of the probability distribution p and by Equality (2), the function

P +,δ -→ R λ -→ x -λ s λ (x)
has the required property for every choice of θ in R d >0 . All that we have to do is to compare the pointwise limit given by Theorem 3.3.2 to this candidate. Given Λ in P +,δ , using (5) and Proposition 2.2.2, one gets x Λ-λ s δ (x) n = Card π + n (λ, Λ)

x Λ-λ s δ (x) n = f n Λ/λ,δ

x Λ-λ s δ (x) n .

This shows

ψ n (λ) = Λ∈P +,δ n f n Λ/λ,δ
x Λ-λ s δ (x) n . x κ s δ (x) n = x -λ s λ (x) ψ n (0) .

We state now our main result. For the last time, recall that the drift is zero if and only if θ = 1 d , that is if and only if p is the uniform distribution on P (δ) :

p s = 1 dim V (δ)
s ∈ P (δ) .

Theorem 3.3.5 Assume that the drift of the random walk is zero. The sequence (p + n ( . , . , 0)) n≥0 pointwise converges on P +,δ × P +,δ and its limit p + = p + 1 d is given by

p + (λ, Λ) = p 1 d (λ, Λ) s Λ 1 D s λ (1 D ) = 1 dim V (δ) dim V (Λ) dim V (λ) if λ Λ 0 otherwise λ, Λ ∈ P +,δ .
By Theorem 2.4.3, we therefore have the pointwise convergence on P +,δ × P +,δ p + (θ) -----→

θ→1 d θ∈]0,1[ d p + 1 d .
Proof. In the zero drift case, Lemma 3.3.3 and Proposition 3.3.4 show that the pointwise limit on P +,δ of the sequence (h n ) n≥1 is λ → s λ 1 D = dim V (λ) .

Using Proposition 3.1.1, we obtain the expected result.

To conclude, observe that the assumption on the drift is only required in Theorem 3.3.2. Thus, if Theorem 3.2.1 is generalized in the near future to the case of a drift in the boundary of the cone, the generalization of our Theorem 3.3.5 will automatically follow.

Conjecture 3.3.6 Assume the drift of the random walk is in ∂C. The sequence (p + n ( . , . , 0)) n≥0 pointwise converges on P +,δ × P +,δ and its limit p + = p + (θ) is given by p + (λ, Λ) = p (θ) (λ, Λ)

x -Λ s Λ (x) x -λ s λ (x) =

1 s δ (x) s Λ (x) s λ (x)
if λ Λ 0 otherwise λ, Λ ∈ P +,δ .

  Proposition 3.3.4 We haveh n (λ) ≤ x -λ s λ (x) λ ∈ P +,δ , n ≥ 1.Proof. When n is a positive integer such that n ≥ 2, one hasψ n (λ) = P (i) ∈ P +,δ ∩ λ + S (n) = Λ .

  (i) ∈ P δ + ∩ λ + S (n) = Λ = π∈π + n (λ,Λ) π=(π 0 ,π 1 ,...,πn)

Λ∈P

  Now we use Identity (9) to writeψ n (λ) = x -λ s δ (x) n .Now remark that for any κ in P +,δ n , one hasΛ∈P +,δ n K λ,Λ-κ x Λ-κ ≤ γ∈P K λ,γ x γ = s λ (x)and one getsψ n (λ) ≤ x -λ s λ (x)

  Weight lattices associated to root systems . . . . . . . . . . . . . . . . . . 2.2 Finite-dimensional representations . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Weights, characters, decomposition into irreducibles and tensor multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Minuscule weights and representations . . . . . . . . . . . . . . . . 2.3 A family of random walks on the weight lattice . . . . . . . . . . . . . . .

	1 Introduction
	2 Background
	2.1

  Let d be a positive integer and g a simple Lie algebra over C of rank d. Denote by h a Cartan subalgebra of g. The relative root system R is realized as a finite subset of a d-dimensional subspace E of a standard Euclidean space R D whose standard basis and standard inner product are respectively denoted by B = (ε 1 , . . . , ε D ) and . , . . The classification of root systems distinguishes four infinite families A d , B d , C d , D d (corresponding respectively to the classical Lie algebras sl d+1 (d ≥ 1), so 2d+1 (d ≥ 3), sp 2d (d ≥ 2), so 2d (d ≥ 3)) and five exceptional types E 6 , E 7 , E 8 , F 4 , G 2 .

	Law of the random walk with drift in the (open) Weyl chamber conditioned to stay in the (closed) Weyl chamber until instant n	----→ n→∞	Law of the random walk with drift in the (open) Weyl chamber conditioned to always stay in the (closed) Weyl chamber
	↓ d = 0		↓ d = 0
	Law of the random walk with zero drift conditioned to stay in the (closed) Weyl chamber until instant n	----→ n→∞	Law of the random walk with zero drift conditioned to always stay in the (closed) Weyl chamber
	2 Background
	2.1 Weight lattices associated to root systems

Acknowlegments. The author would like to deeply acknowledge O. Durieu, E. Journé, C. Lecouvey, E. Lesigne, M. Peigné, K. Raschel for helpfull discussions and support during the writing of this article.