Nicolas Bossard

Jérémy Jacob

Claude Le Milbeau

Elisabeth Lallier-Vergès

Valery T Terwilliger

Rachel Boscardin

Valery J Terwilliger

Variation in δD values of a single, species-specific molecular biomarker: a study of miliacin throughout a field of broomcorn millet (Panicum

Keywords: Compound-specific isotope analysis, hydrogen isotopes, spatial variability, miliacin, millet, pentacyclic triterpenes

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Compound-specific stable hydrogen isotope analyses are increasingly receiving attention for their potential to record paleoclimatic conditions. The rationale for using this technique is that the hydrogen isotopic composition (δD) of individual molecules records information on the climatic conditions at the time of their biosynthesis. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF], [START_REF] Sauer | Compound-specific D/H ratios of lipid biomarkers from sediments as a proxy for environmental and climatic conditions[END_REF], [START_REF] Huang | Hydrogen isotope ratios of palmitic acid in lacustrine sediments record late Quaternary climate variations[END_REF], [START_REF] Gleixner | Compound-specific hydrogen isotope ratios of biomarkers: tracing climatic changes in the past[END_REF] Once released in the environment, molecules synthesised by a large array of producers in the catchment can then be transported to lake sediments where, if conditions remain favourable, they can be preserved for million years. [START_REF] Yang | Preservation of lipid hydrogen isotope ratios in Miocene lacustrine sediments and plant fossils at Clarkia, northern Idaho, USA[END_REF] The δD of molecules that accumulated through time in sediments can thus provide integrative inferences about past climate changes. Nonetheless, the accuracy of these inferences requires understanding the quantitative relationships between compound-specific δD values and the controlling environmental factors.

An effective means of quantifying relationships between δD of individual molecules and environmental factors is by calibration over natural or controlled environmental gradients.

Calibration studies have demonstrated that δD values of lipids from higher plants are primarily controlled by the δD values of the sources of the hydrogen used for their synthesis, normally meteoric waters, which, in turn, are climatically controlled. [START_REF] Dansgaard | Stable isotopes in precipitation[END_REF] Nonetheless, other factors can also affect the δD values of lipid compounds from higher plants. Unlike submerged, aquatic photosynthesizing organisms; the hydrogen in water that land plants use to synthesize organic compounds will have been exposed to fractionations from evaporation in the soil and from transpiration. Although the precise effects of evaporation and transpiration on the δD values of the source water for lipid synthesis in higher plants is presently a contentious issue, their associated fractionations may lead to a 30‰ enrichment in deuterium in higher plant lipids over those of phytoplankton. [START_REF] Sachse | Hydrogen isotope ratios of recent lacustrine sedimentary nalkanes record modern climate variability[END_REF], [START_REF] Feakins | Controls on the D/H ratios of plant leaf waxes in an arid ecosystem[END_REF], [START_REF] Mcinerney | Hydrogen isotope ratios of leaf wax n-alkanes in grasses are insensitive to transpiration[END_REF] Soil properties affect the vertical distribution of the soil water δD values and thus the δD values of water absorbed by plants depending on their rooting system. [START_REF] Barnes | Isotopic exchange in soil water[END_REF] The type of vegetation (trees/herbs for example) not only controls the depth of root penetration but also the intensity of soil water evaporation and leaf water transpiration. [START_REF] Hou | Hydrogen isotopic variability in leaf waxes among terrestrial and aquatic plants around Blood Pond, Massachusetts (USA)[END_REF] Depending on biosynthetic (isoprenoids/acetogenic lipids) and possibly photosynthetic pathways (C 3 /C 4 /CAM), hydrogen will be subjected to different fractionation during enzymatic processes that, in turn, will lead to varying δD of biochemicals. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF], [START_REF] Sauer | Compound-specific D/H ratios of lipid biomarkers from sediments as a proxy for environmental and climatic conditions[END_REF] , [START_REF] Chikaraishi | Hydrogen and carbon isotopic fractionations of lipid biosynthesis among terrestrial (C3, C4 and CAM) and aquatic plants[END_REF], [START_REF] Smith | Influence of physiology and climate on δD of leaf wax n-alkanes from C3 and C4 grasses[END_REF] Finally, water use efficiency as well as potential variations of water source δD with time may also have impacts on the δD of higher plant lipids. [START_REF] Hou | Evidence for water use efficiency as an important factor in determining the δD values of tree leaf waxes[END_REF], [START_REF] Sessions | Seasonal changes in lipid D/H fractionation by Spartina alterniflora[END_REF] The aforementioned potential sources of variability could lead to a considerable range of δD values contributing to the pool of plant-derived lipids preserved in lake sediments. Part of this variability could be due to environmental differences (soil properties for example) among the source areas of plants. Another source of variability could arise from differences in δD values of the same lipid from different plant types within the same source area. The few studies of possible plant sources to lake sediments within a catchment suggest that the differences in δD values of nalkyl lipids can be large among different plant types. [START_REF] Hou | Hydrogen isotopic variability in leaf waxes among terrestrial and aquatic plants around Blood Pond, Massachusetts (USA)[END_REF], [START_REF] Krull | Compound-specific δ 13 C and δ 2 H analyses of plant and soil organic matter: A preliminary assessment of the effects of vegetation change on ecosystem hydrology[END_REF], [START_REF] Liu | Compound specific D/H ratios and molecular distributions of higher plant leaf waxes as novel paleoenvironmental indicators in the Chinese Loess Plateau[END_REF], [START_REF] Liu | Multiple controls for the variability of hydrogen isotopic compositions in higher plant n-alkanes from modern ecosystems[END_REF] The potential of sediments to accumulate n-alkyl lipids with a large spectrum of δD values raises the question of the confidence level of paleoclimatic records based on compound-specific δD.

A way to discriminate between environmental and biological factors affecting the δD of plant lipids and to avoid the bias induced by plant types is to analyze molecules that are specific to certain plant taxa. Most of the δD work on land plant derived lipids has been restricted to n-acids, n-alkanes and sterols that are common across most taxa. We have detected, in tropical lake sediments, pentacyclic triterpene methyl ethers (PTME) derived mainly from Gramineae. [START_REF] Jacob | Pentacyclic triterpene methyl ethers in recent lacustrine sediments (Lagoa do Caçó, Brazil)[END_REF] One of these PTMEs, miliacin (olean-18-en-3β-ol methyl ether; Figure 1), was also detected in the sediments of Lake le Bourget where, at least within the French Alps, it has a unique source plant: millet (Panicum miliaceum). [START_REF] Jacob | Millet cultivation history in the French Alps as evidenced by a sedimentary molecule[END_REF], [START_REF] Jacob | Carbon isotope evidence for sedimentary miliacin as a tracer of Panicum miliaceum (broomcorn millet) in the sediments of Lake le Bourget (French Alps)[END_REF] The objective of the research reported herein is to examine the variability in δD values of miliacin extracted from millet seeds collected randomly in a field, which constitutes and intermediate scale between a single plant and an entire lake catchment. We address the specific question of variability in δD with environmental change by restricting the study to a single compound from a single species. This allows us to exclusively examine the impacts of local environmental parameters. To the best of our knowledge, this is the first paper to evaluate environmental variation in δD of a land plant derived lipid that, if extracted from sediments, would not be confounded by variation in plant type. Our findings will be readily applicable to the use of miliacin δD in sediments for paleoclimate studies but are also thought to be extendable to other specific compounds.

Site, materials and methods

Sample collection

Samples of P. miliaceum (var. Sunrise) were collected during August 2009 at harvest time from a field in Mézières-lez-Cléry, near Orléans, France (47°50'15", 1°48'20"E and ca. 100m above sea level). Mean δD of precipitation was -41.1 ± 7.2‰ during the 1996-2005 time period at the closest Global Network for Isotopes in Precipitation (GNIP) monitoring station (Orléans-La-Source). This field was chosen considering the variability in soil properties as illustrated by different shades of grey on aerial picture (Figure 2) that indicate contents of organic matter. The heights of plants of P. miliaceum growing in the field were measured from the first node up to the top of the main panicle. Plant heights range from ca. 30 cm up to 120 cm, with higher plants developed on more organic soils, as determined by field measurements. 26 stands representative (numbered from 183 to 210) of the various contexts (heights of the millet plants and soil color) were selected in the field. From 1 to 5 plants (3 in most cases for a total of 61) were taken at each stand at less than 30cm apart. The whole plants were sampled and dried in an oven at 35°C for 48 h.

The weight of the main panicle was determined after drying.

Miliacin extraction and purification

For each plant sample, twenty seeds of the main panicle were ground to powder and weighed. These twenty seeds provide a sufficient amount of miliacin for determination of δD values. We analyzed two sets of twenty seeds from a single panicle of each plant and we observed no significant differences either in millet seed weight, miliacin concentration or miliacin δD values between sets. Lipids were ultrasonically extracted three times with organic solvents (DCM:isopropanol 2:1), and the lipids extracts were then combined. The total extract was then separated into neutral, acidic and polar fractions by ion exchange column on aminopropyl-bonded silica. The neutral lipid fraction was eluted with DCM:isopropanol 2:1 and then dried under nitrogen. Miliacin (Figure 1) was purified from the neutral fraction by flash chromatography on activated silica by using the following solvents of increasing polarity. After the elution of aliphatic and aromatic hydrocarbons, miliacin was collected in fractions eluted with 2mL hexane:toluene (1:1) and 2mL hexane:ethyl acetate (19:1). Miliacin-containing fractions were then combined, dried under nitrogen and stored at -4°C until analysis by Gas Chromatography-Mass Spectrometry (GC-MS) and Gas Chromatography-isotope ratio Mass Spectrometry (GC-irMS). 5α-cholestane was added prior to analysis by GC-MS for quantification purposes.

GC-MS analyses

Miliacin contents and purity were assessed by GC-MS on a TRACE gas chromatograph coupled to a Polaris GCQ mass spectrometer (ThermoFinnigan, Bremen) according to previously described methods. [START_REF] Jacob | Pentacyclic triterpene methyl ethers in recent lacustrine sediments (Lagoa do Caçó, Brazil)[END_REF] The gas chromatograph was fitted with an Rtx-5 MS capillary column (5m column guard, 30 m, 0.25 mm i.d., 0.25 μm film thickness). The gas chromatograph operating conditions were: temperature held at 40°C for 1 min, then increased from 40 to 300°C at 20°C.min - 1 , with final isothermal hold at 300°C over 30min. The sample was dissolved in toluene and injected splitless in a 2µL volume, with the injector temperature set at 280°C. The carrier gas was Helium and the flow was set at 1.4mL.min -1 . The mass spectrometer was operated in the electron ionisation (EI) mode at 70eV ionization energy and scanned from 50 to 650 Da.

GC-irMS analyses

Miliacin δD values were determined by using a Trace gas chromatograph equipped with a Triplus autosampler coupled to a DeltaV Advantage isotope ratio mass spectrometer through a GC-Isolink pyrolysis interface and a ConFlo IV dilution system (ThermoScientific, Bremen).

Chromatographic conditions were the same as those used in GC-MS, except that the GC column (J&W DB5, 30m, 0.25 mm i.d., 0.25 μm film thickness) was slightly different. 1µL of miliacin dissolved in toluene was co-injected with an internal standard constituted by 1µL of a mixture of nalkanes (n-C 16 to n-C 30 ) with δD values ranging from -46.3 to -242.6‰ determined offline (Arndt Schimmelmann, Indiana University, Bloomington, USA). Due to high miliacin purity, no significant coelution of other compounds with n-alkanes was observed (Figure 3). Data were then normalized to the V-SMOW isotopic scale by using the δD of the n-C 25 and n-C 27 alkanes as reference. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF] All miliacin δD values are reported in ‰ units relative to the V-SMOW scale.

The 13 remaining n-alkanes were used for accuracy assessment, with the exception of the n-C 29 alkane that co-eluted with 5α-cholestane (Figure 3). Each sample was randomly (i.e. not one after the other and not in logical order in the sequence of injection) injected at least 3 times. When necessary, supplementary injections were performed and aberrant values where not taken into account by using a Dixon test (p<0.05). The mean precision of miliacin δD values was 5.9‰ (from 0.3 to 12.0‰), i.e. within common precision ranges reported for natural samples. [START_REF] Sessions | Seasonal changes in lipid D/H fractionation by Spartina alterniflora[END_REF], [START_REF] Chikaraishi | δ 13 C and δD identification of sources of lipid biomarkers in sediments of Lake Haruna (Japan)[END_REF], [START_REF] Mugler | Effect of lake evaporation on δD values of n-alkanes: A comparison of Nam Co (Tibetan Plateau) and Holzmaar (Germany)[END_REF] The nalkanes mixture was analysed every nine injections, as external standard. The overall precision for the n-alkane standard (injected with and without sample) is around 3‰ (1σ, n=361), with no notable difference between internal and external standards, which suggests no significant coelution. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF] Measured n-alkanes δD values are in good agreement with those measured offline, except for a significant drift at elevated δD values. The best match between offline and online δD values for n-alkanes is between -240 and -70‰. The expected miliacin δD values (-80 to -150‰) fall within this range. The H3 + factor was determined daily prior to analysis and ranged from 3.7 to 3.8 in the course of analysis.

Results

Miliacin δD values means and standard deviations

Miliacin δD values range from -144.5‰ to -98.1‰; i.e. with a difference of almost 50‰ (Table 1, Fig. 4a). Standard deviations of the repeated analysis of each sample range from 0.3 to 12‰ with a mean standard deviation of 5.9‰. Μiliacin δD values of individual plants were normally distributed (Figure 4b) about an average of -121.1‰ (± 9.1 sd) (Kolmogorov-Smirnov, p<0.05). When δD values average of each plant are again averaged by stand, they range from -132.8 ‰ to -102.8‰ (Figure 4c); i.e. a difference almost reaching 30‰. In this case, standard deviation of δD values in each stand ranges from 2.1 to 19.3‰, with a mean standard deviation of 6.9‰. Μiliacin δD values per stand show a normal distribution (Figure 4d) with an average of -120.7‰ (± 6.5 sd) (Kolmogorov-Smirnov, p<0.05).

Comparison of miliacin δD values with other parameters

To examine whether miliacin δD values varied systematically with environmental condition, we tested the values for correlation with the following environmentally influenced parameters: standard deviation of miliacin δD values, plant height, panicle weight, 20 seeds weight and miliacin concentration. Plant heights range from 20 to 120cm, panicle weight from 0.1 to 8.8g, 20 seeds weight from 25 to 160mg and miliacin concentrations from 0.08 to 4.8μg.g -1 seed (Table 1). No linear relation could be established between miliacin δD values and other parameters (r²<0.15).

Discussion

δD of pentacyclic triterpenes

Very few studies have reported δD values for pentacyclic triterpenes, probably because these compounds are secondary metabolites and constitute minor components in biological and geological systems, when compared to n-alkyl lipids. Average miliacin δD values (-121‰ V-SMOW) are in agreement with some of the few other values that have been reported for pentacyclic triterpenes. δD values of -139 and -171‰ have been reported for lupenol, -142‰ for lupenone, -144‰ for an unidentified pentacyclic triterpenone and -151 and -165‰ for two unidentified pentacyclic triterpenols, all extracted from Spartina alterniflora. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF], [START_REF] Sessions | Seasonal changes in lipid D/H fractionation by Spartina alterniflora[END_REF] For taraxerol, α-amyrin and an unidentified pentacyclic triterpene extracted from the surface sediments of a pond in Massachusetts, δD values were around -170/-180‰. [START_REF] Sauer | Compound-specific D/H ratios of lipid biomarkers from sediments as a proxy for environmental and climatic conditions[END_REF] Various pentacyclic triterpenes extracted from Daucus carota were highly depleted, with -252‰ for δand β-amyrins, and values ranging from -226 to -239‰ for unidentified pentacyclic triterpenols. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF] In contrast to n-alkyl lipids (acetogenic lipids), which are produced from acetyl-CoA in the plastid, pentacyclic triterpenes are produced in the cytosol, also from acetyl-CoA, via the mevalonic pathway. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF], [START_REF] Lichtenthaler | The 1-deoxy-D-xylulose-5-phosphate pathway of isoprenoid biosynthesis in plants[END_REF] This explains a systematic 50 to 100‰ offset between n-alkyl lipids and sterols or triterpenols, directly from plants or from soils and sediments. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF], [START_REF] Chikaraishi | Hydrogen and carbon isotopic fractionations of lipid biosynthesis among terrestrial (C3, C4 and CAM) and aquatic plants[END_REF], [START_REF] Chikaraishi | δ 13 C and δD identification of sources of lipid biomarkers in sediments of Lake Haruna (Japan)[END_REF], [START_REF] Chikaraishi | Carbon and hydrogen isotopic fractionation during lipid biosynthesis in a higher plant (Cryptomeria japonica)[END_REF], [START_REF] Chikaraishi | Fractionation of hydrogen isotopes during phytol biosynthesis[END_REF], [START_REF] Huang | A 62,000 year record of environmental and climatic change inferred from carbon and hydrogen isotope stratigraphy of higher plant biomarkers at Lake Tulane, Florida[END_REF], [START_REF] Chikaraishi | Carbon and hydrogen isotop variation of plant biomarkers in a plant soil-system[END_REF] The origin of this difference is still unclear and could be related either to the source of hydrogen (directly from H 2 O or via NADPH), distinct pools of NADPH within the cell or it could results from kinetic isotope effects associated to hydrogenation. [START_REF] Sessions | Fractionation of hydrogen isotopes in lipid biosynthesis[END_REF], [START_REF] Chikaraishi | Carbon and hydrogen isotopic fractionation during lipid biosynthesis in a higher plant (Cryptomeria japonica)[END_REF], [START_REF] Chikaraishi | Fractionation of hydrogen isotopes during phytol biosynthesis[END_REF] Differences in δD among pentacyclic triterpenes with various structures (such as lupane, oleanane, ursane, taraxerane) could arise from fractionation intervening during the ultimate steps of their biosynthesis that involves the cyclisation of 2,3-oxidosqualene and subsequent rearrangements through methyl group and double bond migrations, and ring expansion/contraction. [START_REF] Xu | On the origins of triterpenoid skeletal diversity[END_REF] In fact, very little is known on the fractionation of hydrogen isotopes associated to the synthesis of pentacyclic triterpenes. The δD of various pentacyclic triterpenes produced by a single plant could therefore provide key information on hydrogen isotopic effects associated to their synthesis, as it was remarkably as was performed for phytol. [START_REF] Chikaraishi | Fractionation of hydrogen isotopes during phytol biosynthesis[END_REF]

Compound-specific δD variations over a small area

The almost 50‰ range in miliacin δD values within the area of a single field is within the range of compound-specific δD variations of less plant specific compounds in larger-scale paleoenvironmental studies (see section 4.4.). The importance of the 50‰ range can be put into perspective by comparing it with the ranges of compound-specific δD values recorded in the few other studies of plants over small areas, either under natural or controlled conditions.

Studies in natural systems of homogeneous climate (such as our study site) reveal large differences in δD values of plant lipids. In a study of 48 land and aquatic plant species grouped by life form growing in the catchment of a small pond, δD values of n-C 30 fatty acids varied by 100 ‰ between all species, 70 ‰ among tree species, and 40 ‰ among grasses. [START_REF] Hou | Hydrogen isotopic variability in leaf waxes among terrestrial and aquatic plants around Blood Pond, Massachusetts (USA)[END_REF] Within trees developed in the same area, the difference in n-alkane δD values can reach 30‰ for deciduous trees (Betula spp. and Quercus spp.) and up to 35‰ for evergreen trees (Pinus strobus and Tsuga canadensis)

. [START_REF] Hou | Evidence for water use efficiency as an important factor in determining the δD values of tree leaf waxes[END_REF] Similarly, a difference of ca. 60‰ was noted between δD values of n-alkanes produced by trees growing in woody grassland (Atalaya sp.) and trees growing in woodland (Acacia sp.) at few tens of meters of distance.. [START_REF] Krull | Compound-specific δ 13 C and δ 2 H analyses of plant and soil organic matter: A preliminary assessment of the effects of vegetation change on ecosystem hydrology[END_REF] When grasses are considered, the differences in δD values of nalkanes extracted from different species growing in individual sites of the US Great Plains was as large as 50‰. [START_REF] Smith | Influence of physiology and climate on δD of leaf wax n-alkanes from C3 and C4 grasses[END_REF] Thus, the range of miliacin δD values we observed falls within the ranges observed for n-alkyl lipids extracted from various plants in natural conditions over a small area.

In controlled environment chambers, environmental variations that could affect the δD of biochemicals are reduced, when compared to natural systems. The range of δD values is thus expected to be narrower. Nevertheless, δD values of n-alkyl lipids from plants have been found to differ by as much as 90‰ between tree and grass species growing in a controlled environment chamber. [START_REF] Liu | Multiple controls for the variability of hydrogen isotopic compositions in higher plant n-alkanes from modern ecosystems[END_REF] δD values of n-alkanes and n-alkanoic acids each differed by a maximum of 60‰ between several grass species. [START_REF] Smith | Influence of physiology and climate on δD of leaf wax n-alkanes from C3 and C4 grasses[END_REF], [START_REF] Hou | Can sedimentary leaf waxes record D/H ratios of continental precipitation? Field, model, and experimental assessments[END_REF] When δD values of n-alkanes were compared among species of a single photosynthetic pathway (C 3 or C 4 ), however, this difference was reduced to 30‰. [START_REF] Smith | Influence of physiology and climate on δD of leaf wax n-alkanes from C3 and C4 grasses[END_REF] Within tree species, a maximum of 30‰ difference in δD values of a single alkanoic acid was found among individuals of a single oak species. [START_REF] Hou | Can sedimentary leaf waxes record D/H ratios of continental precipitation? Field, model, and experimental assessments[END_REF] None of these studies (either under natural or controlled conditions) had more than four replicates per species, however, thereby limiting conclusions on intraspecies variability in δD values. These large ranges in compound-specific δD values reported in both controlled and natural conditions have been attributed to several major causes that are, in most cases, interconnected.

Studies indicate that by taking up isotopically distinct water and through different transpiration dynamics, the δD values of the water used for lipid biosynthesis may vary systematically between plant life forms (e.g. trees, shrubs and grass). The δD values of soil water can decrease with depth due to a decreasing loss of D-depleted water as vapour during evaporation.

Differences in depths of rooting systems among plant life forms can thus lead to uptake of waters with different δD values,. [START_REF] Feakins | Controls on the D/H ratios of plant leaf waxes in an arid ecosystem[END_REF], [START_REF] Smith | Influence of physiology and climate on δD of leaf wax n-alkanes from C3 and C4 grasses[END_REF], [START_REF] Krull | Compound-specific δ 13 C and δ 2 H analyses of plant and soil organic matter: A preliminary assessment of the effects of vegetation change on ecosystem hydrology[END_REF], [START_REF] Sachse | Significant seasonal variation in the hydrogen isotopic composition of leaf-wax lipids for two deciduous tree ecosystems (Fagus sylvatica and Acer pseudoplatanus)[END_REF] Before its use in lipid biosynthesis, the water is exposed to further D-enrichment during transpiration and the magnitude of that enrichment may also vary by life form. The two fractionations; evaporation of soil water and transpiration may minimize or maximise differences between plant life forms in δD values of water used in lipid biosynthesis. For example, grasses tend to take up most of their water at the surface where it is most D-enriched whereas deeper soils are a more important source of water for trees. . Tree leaves may have higher transpiration rates than grasses because their leaves are exposed to direct sunlight and wind exposure, and do not benefit from canopy effect and direct soil moisture. [START_REF] Hou | Hydrogen isotopic variability in leaf waxes among terrestrial and aquatic plants around Blood Pond, Massachusetts (USA)[END_REF] The D-enrichment of water taken up by grasses relative to water taken up by trees may thus be offset by lower water losses via transpiration in grasses than in trees.

Differences in leaf waxes δD values in distinct plants species also arise from differences in plant physiology and photosynthetic pathways. The impact of leaf morphology and architecture is still debated. [START_REF] Smith | Influence of physiology and climate on δD of leaf wax n-alkanes from C3 and C4 grasses[END_REF] Differential fractionation of hydrogen isotopes between C 3 , C 4 and CAM plants has also been proposed but there is still no consensus on whether this arises from isotope effects related to metabolisms or to physiological and morphological differences. [START_REF] Feakins | Controls on the D/H ratios of plant leaf waxes in an arid ecosystem[END_REF], [START_REF] Smith | Influence of physiology and climate on δD of leaf wax n-alkanes from C3 and C4 grasses[END_REF], [START_REF] Liu | Multiple controls for the variability of hydrogen isotopic compositions in higher plant n-alkanes from modern ecosystems[END_REF], [START_REF] Chikaraishi | Carbon and hydrogen isotopic fractionation during lipid biosynthesis in a higher plant (Cryptomeria japonica)[END_REF], [START_REF] Chikaraishi | Fractionation of hydrogen isotopes during phytol biosynthesis[END_REF], [START_REF] Bi | Molecular and carbon and hydrogen isotopic composition of n-alkanes in plant leaf waxes[END_REF] The last parameter to be taken into account for explaining the variability of biochemical δD values relates to the strategies developed by plants for their water use. [START_REF] Hou | Hydrogen isotopic variability in leaf waxes among terrestrial and aquatic plants around Blood Pond, Massachusetts (USA)[END_REF], [START_REF] Smith | Influence of physiology and climate on δD of leaf wax n-alkanes from C3 and C4 grasses[END_REF], [START_REF] Sachse | δD values of individual n-alkanes from terrestrial plants along a climatic gradient -Implications for the sedimentary biomarker record[END_REF] The negative correlation between δ 13 C and δD values of plant lipids suggests that variable water use efficiency influences interspecies differences in leaf wax δD values. [START_REF] Feakins | Controls on the D/H ratios of plant leaf waxes in an arid ecosystem[END_REF], [START_REF] Hou | Evidence for water use efficiency as an important factor in determining the δD values of tree leaf waxes[END_REF], [START_REF] Liu | Multiple controls for the variability of hydrogen isotopic compositions in higher plant n-alkanes from modern ecosystems[END_REF] None of these explanations can be invoked in our case that concerns a single biochemical produced by a single species developed under relatively homogeneous conditions.

Origin of variability in miliacin δD values

Our data bring new information about the spatial variability of compound-specific δD values. By focusing our study on a single compound specific to one plant, we reduce interspecies offsets but also reduce potential variability related to micro-environmental conditions, biosynthetic and photosynthetic pathways and physiological factors (including leaf structure and rooting system). Despite these precautions, a large range of miliacin δD values (50‰) is found in millet seeds collected from plants developed on a ca. 1000 m² area. However, this range is not necessarily attributable to differences in environmental conditions.

The normal distribution of miliacin δD values per plant implies a single dominant controlling parameter. The lack of correlation of miliacin δD values with the standard deviation of these values and miliacin concentrations excludes any analytical bias. No correlation could be found between miliacin δD values and parameters related to plant yields (plant height, panicle weight and weight of 20 seeds). Because plant heights were visually correlated with soil properties in the field, this suggests that there is no relationship between miliacin δD values and soil properties. This is confirmed when intra-and inter-stand variability is compared. If soil properties were a major factor, one could expect larger differences between stands (at decimetric scale) than within a single stand, which shows similar soil properties at decimetric scale. The similarity of intra-stand (mean of standard deviation of miliacin δD values per stand; 6.9) and inter-stand (standard deviation of mean δD values per stand; 6.5) miliacin δD values variability thus allows us excluding decimetric-scale differences in soil properties as a dominant control parameter.

Conversely, the Gaussian distribution of miliacin δD values suggests that precipitation δD and mean hydrological regime that prevail at the field scale constitute the dominant control parameters. [START_REF] Hou | Can sedimentary leaf waxes record D/H ratios of continental precipitation? Field, model, and experimental assessments[END_REF] Because our study was focused on a single plant, several potential parameters (plant form, photosynthetic and biosynthetic pathways, plant physiology) that have been proposed to explain compound-specific δD differences in spatial studies (see 4.2.) can be discarded. Large-scale differences in soil properties cannot either be invoked. Although this remains to be tested, the dispersion of miliacin δD values could result from micro-scale heterogeneities in soil properties and/or in relative humidity. For example, the millet field was heterogeneously but densely covered by the weed Cynodon dactylon (Bermuda grass) at harvest time. The competition for water uptake between these two plants could partially explain local differences in miliacin δD values.

Alternatively, these differences could also be related to intrinsic (phenotypic and genotypic) variability within P. miliaceum. [START_REF] Zhang | Stable isotope studies of crop carbon and water relations: A Review[END_REF]

Consequences on the climatic significance of compound-specific δD

The maximum difference of miliacin δD values between plants analysed from the Mézièreslez-Cléry millet field is 50‰. This range must be compared to known ranges of compound-specific δD values found in both actual and ancient natural systems in order to estimate the confidence level of paleoclimatic interpretations based on the variations of δD values of lipids extracted from sediments archives. Throughout a latitudinal transect (from 70°N to 40°N) in Europe, leaf wax C 27 n-alkane extracted from various deciduous trees had δD values ranging from ca. -140 to -210‰ (i.e. a 70‰ range) whereas sedimentary leaf wax C 27 n-alkane from the same transect showed δD values from -130 to -220‰ (i.e. a 90‰ range). [START_REF] Sachse | Hydrogen isotope ratios of recent lacustrine sedimentary nalkanes record modern climate variability[END_REF], [START_REF] Sachse | δD values of individual n-alkanes from terrestrial plants along a climatic gradient -Implications for the sedimentary biomarker record[END_REF] However, it is not clear whether these ranges of δD values purely reflect a climatic gradient or also attests to vegetation changes through latitude because the n-alkanes studied are common to a wide range of higher plant taxa. In this study, a wide variety of deciduous trees were analysed, with potentially large interspecies differences in compound-specific δD values (see section 4.2.). Nevertheless, such calibration studies permit inferences about shifts in compound-specific δD values with environmental and climatic variations.

In sedimentary archives, a 80‰ shift in n-alkanes or n-acids δD were interpreted as the transition between a savannah and tropical forest due to different hydrological conditions. [START_REF] Schefuss | Climatic controls on central African hydrology during the past 20,000 years[END_REF], [START_REF] Jacob | Paleohydrological changes during the last deglaciation in Northern Brazil[END_REF], [START_REF] Tierney | Northern Hemisphere controls on tropical Southeast African climate during the past 60,000 years[END_REF] Thus, up to 80 ‰ range of compound-specific δD values in recent and ancient systems is indicative of highly contrasted environmental conditions. These ranges are in the same order as that found for miliacin δD values, at a single location. The overlap between the range of local variability under the same environmental regime and the range of drastic environmental changes could, if only the range is considered, induce serious doubt on the confidence level to which the δD of sedimentary lipids can be interpreted in paleoclimatology.

The number of samples of a single species analysed in this study permits a more thorough examination of the significance of the 50‰ miliacin range in δD values. Miliacin δD values follow a normal distribution (average = -121.1‰ of all plants and -120.7‰ of plants averaged by stand; Figure 4). In the meantime, the range of δD values diminishes from ca. 50‰ (all plants) to 30‰ (average by stands). 90% of miliacin δD values per plants are comprised between -112 and -130‰ V-SMOW, i.e. a 20‰ range (Figure 4b). This range is evidently reduced when miliacin δD values are averaged by stand (from -114 to -126‰; i.e. a range of 12‰; Figure 4d). Consequently, the probability for obtaining miliacin δD values that are representative of the whole millet field from remotely sampling a single plant appears rather high.

Sedimentary lipids constitute an integrative signal of all lipids produced in the catchment. If variation in δD values of species specific molecular biomarkers was high within a small, climatically homogeneous area, the resolution of a climatic interpretation from an entire catchment would be low. Our findings suggest that, although noteworthy variation in miliacin δD values under such conditions is possible, variation is more commonly small. Research efforts now need to concentrate on how well the original lipid δD value is archived in sediments. Following the incorporation of their source plants into sediments, the δD values of lipids may be affected by (i) selective degradation during diagenesis; (ii) temporal offsets due to differences in transportation times of plant sources to sediments; and (iii) uneven representation of environmental heterogeneities within the catchment. Significant D depletion in soil relative to overlying plant nalkyl lipids but little difference in δD values of isoprenoids in soil and overlying plants suggested larger diagenetic effects in the former than the latter category of compounds. [START_REF] Chikaraishi | Carbon and hydrogen isotop variation of plant biomarkers in a plant soil-system[END_REF] D depletion of surficial sedimentary lipid compounds relative to surrounding plants has been attributed to offsets between the time lipids were produced and their accumulation in sediments. [START_REF] Sachse | δD values of individual n-alkanes from terrestrial plants along a climatic gradient -Implications for the sedimentary biomarker record[END_REF] If we exclude these temporal and diagenetic offsets, the δD value of sedimentary miliacin would record the weighted mean miliacin δD value of all millet biomass produced in the catchment; i.e. the variability is restrained to the mean value, with no distortion. This hypothesis should be tested by examining the propagation of variability from the source to the sedimentary archive, i.e. the representativeness of sedimentary lipid δD compared to catchment-scale lipid δD. [START_REF] Zhang | Stable isotope studies of crop carbon and water relations: A Review[END_REF]

Conclusion

This study contributes to constraining the climatic inferences that can be made from δD analyses of molecular biomarkers preserved in lake sediments that may have originated under a variety of environmental conditions throughout a catchment. The analysis of the spatial distribution of δD values measured on a single molecule extracted from a single plant species grown in a field with heterogeneous soil properties shows differences as large as 50‰. This large difference is not necessarily related to environmental conditions. Despite the large range between maximum and minimum values, however, most miliacin δD values were tightly clustered in a normal distribution about the mean and thus varied little within the study area. Similarly, the δD values of miliacin in lake sediments may average miliacin δD values over their source areas.

Figure 1 :

 1 Figure 1: Structure of miliacin (olean-18-en-3β-ol methyl ether).

Figure 2 :

 2 Figure 2: Location of P. miliaceum field, heterogeneity of soil properties on aerial picture (grey scale) and location of samples.

Figure 3 :

 3 Figure 3: Chromatogram obtained by GC-irMS illustrating the distribution of miliacin and of a series of n-alkane standards (nC 16 -nC 30 ) with δD values measured offline (acquired from Arndt Schimmelmann, Indiana University, Bloomington, USA) used for normalising miliacin δD values. * indicates the 5α-cholestane, used as quantitation standard.

Figure 4 :

 4 Figure 4: a-Distribution of δD values of miliacin extracted from the seeds of 61 plants collected in the Mézières-lez-Cléry field sorted by increasing plant height, which was considered pertinent due to the visual correlation between plant height and soil properties in the field. Maximum and minimum miliacin δD values and the corresponding range (-98‰ to -144‰ V-SMOW; range of ~50‰ V-SMOW) are reported.; b-Frequency histogram of miliacin δD values of all plants with calculated mean (μ) and standard deviation (σ); c-δD values and standard deviation of miliacin δD values averaged per stand, sorted by increasing plant height. Maximum and minimum of averaged miliacin δD values (from -105 to -135‰ V-SMOW; range of 30‰) are reported; d-Frequency histogram of miliacin δD values averaged per stand with calculated mean (μ) and standard deviation (σ).

  Figure 1

Table captions Table 1 :

 captions1 Inventory of millet plant samples by stand together with plant height, panicle weight, weight of 20 seeds, miliacin concentration in seeds and miliacin δD values and standard deviations of miliacin extracted from seeds. Average and standard deviation of miliacin δD values by stand are calculated as the arithmetic average and standard deviation of miliacin δD values of plants developed on the same stand. n represents the number of replicates per sample in GC-irMS.

	Sample n° Stand	Plant height (cm)	Panicle weight (g)	Weight of 20 seeds (mg)	Miliacin concentration (ng.g -1 seeds)	Μiliacin δD value (‰V-SMOW)	Standard deviation	n
	183-33 183-35	183	33 35	0.35 0.4	74 59	240 139	-112.8 -136.0	-124.4	7.7 1.7	9 16.5 3
	184-65		65	2.04	84	4775	-110.3		6.5		3
	184-70	184	70	1.51	141	1292	-122.3	-116.4	6.6		3
	184-73		73	3.11	136	1813	-116.5		6.5

Acknowledgments

This work was undertaken within the PalHydroMil project, supported by Agence Nationale de la Recherche grant (ANR JCJC, 2011-2013) and NSF project n° BCS0551668. Mr Sévin and Mr Beaulieu are acknowledged for giving access to the millet field. N. Bossard beneficiates from a PhD grant provided by the Ministère de la Recherche et de l'Enseignement Supérieur (Action Thématique Prioritaire, 2009-2012). V.J. Terwilliger is hosted at ISTO under the auspices of Le Studium® (Institute for Advanced Studies). The authors wish to thank J.R. Disnar and S. Gogo for useful discussions and reviewers who greatly improved a previous version of the manuscript.


Sohier Julien
email: juso-hier@gmail.com

The scaling limits of a heavy tailed renewal Markov process

Keywords: heavy tailed Markov renewals processes, scaling limits, fluctuation theory for random walks, regenerative sets. Mathematics subject classification (2000): 60F77, 60K15, 60K20, 60K05, 82B27

In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the α-stable regenerative set. We then apply these results to the strip wetting model which is a random walk S constrained above a wall and rewarded or penalized when it hits the strip [0, ∞) × [0, a] where a is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.

1 Introduction and main results.

The Markov renewal setup.

This work mainly focuses on the scaling limits of a Markov renewal process with heavy tailed renewal times. By a Markov renewal process, we mean a point process τ on N whose increments τ 1 , . . . , τ nτ n-1 , . . . are not necessarily i.i.d., but governed by a Markov chain J with measurable state space (E, µ). Throughout this paper, we will assume that (E, µ) is a compact Polish space equipped with its standard Borel σ-field B(E).

The Markov dependence of J on τ may be described in various equivalent ways, one of them is the following:

1. First consider a kernel of the form K x,dy (n) with x, y ∈ E, n ∈ Z + , that is a function K : E × B(E) × Z + → [0, 1] such that for fixed n ∈ Z + , K •,• (n) verifies the following:

• for each x ∈ E, K x,• (n) is a σ-finite measure on E.

• for every F ∈ B(E), K •,F (n) is a Borel function.

We will always assume that for every n ∈ N and x ∈ E, K x,dy (n) has a density with respect to µ(•) that we denote by k x,y (n), implying in particular that k(x, y) := 1 µ(dy

) n≥1 K x,dy (n) = n≥1 k x,y (n) (1)
is well defined.

2. Then sample J as a Markov chain on E starting from some initial point J 0 = x 0 ∈ E and with transition kernel

P x 0 [J n+1 ∈ dy|J n = x] := k(x, y)µ(dy). (2)
3. Finally sample the increments (τ iτ i-1 ) i≥1 as a sequence of independent, but not identically distributed random variables according to the conditional law:

P x 0 [τ n+1 -τ n = m|{J i } i≥0 ] = k Jn,J n+1 (m) k(J n , J n+1 )
, n, m ≥ 1.

(3) Markov renewal processes have been introduced independently by Lévy [L 56] and Smith [START_REF] Smith | Regenerative stochastic processes[END_REF], and their basic properties have been studied by Pyke [START_REF] Pyke | Markov renewal processes with finitely many states[END_REF], Cinlar [C ¸in68] and Pyke and Schauffele [START_REF] Pyke | Limit theorems for Markov renewal processes[END_REF] among others. A modern reference is Asmussen [START_REF] Asmussen | Applied probability and queues[END_REF][VII,4], which describes some applications related to these processes, in particular in the field of queuing theory. More recently, they have been widely used as a convenient tool to describe phenomena arising in models issued from statistical mechanics, and more particularly in models related to pinning models, such as the periodic wetting model ( [START_REF] Caravenna | Infinite volume limits of polymer chains with periodic charges[END_REF] and the monograph [START_REF] Giacomin | Random polymer models[END_REF][Chapter 3]) or the 1 + 1 dimensional field with Laplacian interaction ([CD08] and [START_REF] Caravenna | Scaling limits of (1 + 1)-dimensional pinning models with Laplacian interaction[END_REF]).

We will show our results in the case where the kernel K satisfies the following assumptions.

Assumption 1.1. We make the following assumptions on the kernel K:

• the transition function of the Markov chain J is absolutely continuous with respect to µ, and its density k(•, •) is continuous on the whole square E 2 .

• There exist α ∈ (0, 1), a strictly positive continuous bounded function Φ(•) on E 2 and L(•) a slowly varying function such that the equivalence

k x,y (n) ∼ Φ(x, y) L(n) n 1+α (4)
holds uniformly for (x, y) ∈ E 2 ; moreover, we assume that the Markovian renewal is non defective, that is for every x ∈ E:

y∈E n≥1 k x,y (n)µ(dy) = 1. (5)
A few remarks about these assumptions are in order:

• We will use the following notation which is valid for every measurable A ∈ σ(τ ): 1 µ(dy) P[A, J 1 ∈ dy] =:

P[A, J 1 = y]. (6)
The fact that k is continuous implies that this definition makes sense everywhere, and not only almost everywhere.

• The strict positivity of the function Φ implies the strict positivity of the transition kernel of J on the whole E 2 . In particular, J is a regular Markov chain (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF][VIII,7]). Regular Markov chains with strictly positive density are arguably the simplest example of Markov chains with continuous state space which still satisfy the basic ergodic theorem. More precisely, it is well known that every strictly positive regular kernel on a closed interval is ergodic, and also that the ergodicity of such a kernel is equivalent to the existence of a strictly positive stationary distribution which we will denote by Π (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF][VIII,7, Theorems 1 and 2]).

A consequence of equation ( 4) is the fact that the Markov renewal process τ has steps distribution with infinite mean. Making use of the Markov renewal theorem (see [START_REF] Greenwood | Empirical estimators for semimarkov processes[END_REF] for this result in the case where J is Harris recurrent, which is the case in our regular setup), this implies in particular that, as n → ∞, the Green function associated to τ verifies:

P[n ∈ τ ] → 0. ( 7
)
We point out that even for ordinary renewal processes, the exact rate of decay of the Green function in the case of infinite mean has been a longstanding problem which has been solved in great generality only recently by Doney (see [START_REF] Doney | One-sided local large deviation and renewal theorems in the case of infinite mean[END_REF]). A consequence of Doney's results is the fact that, given a true renewal τ with interarrival times given by

P[τ 1 = n] = L(n) n 1+α (8)
where L(•) is some slowly varying function and α ∈ (0, 1), the following equivalence holds:

P[n ∈ τ ] ∼ α sin(απ) π 1 L(n)n 1-α . ( 9
)
The extension of this deep result to the framework of Markov renewal theory has been achieved in the case of finite state modulating chain in [START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1 + 1)-dimension[END_REF], but it turns out that their techniques do not extend to the continuous state case we are considering here.

As a matter of fact, our first result deals with the integrated behavior of the Markov renewal function. More specifically, for (x, y, n) ∈ E 2 × N, let us denote by U(n, x, dy) the Markov renewal mass distribution function defined by U(n, x, dy) :=

n i=0 k≥1 P x [τ k = i, J k ∈ dy] =: n i=0 u(i, x, dy
). The purpose of the section 2 of the present work is to give asymptotics on U for large n, generalizing to our setup the well known Tauberian theorems which hold for the classical renewal. We show the following: Theorem 1.2. As n → ∞, for every x ∈ E, the following weak convergence holds :

U(n, x, dy)L(n) n α → α Γ(1 + α)Γ(1 -α) Π[dy] E Π (2) [Φ/k] ( 10
)
where Π (2) is the invariant measure associated to the Markov chain (J i , J i+1 ) i≥0 .

By this we mean that for every continuous bounded function f on E and every fixed x ∈ E, as n → ∞, the following convergence holds :

L(n) n α y∈E U(n, x, dy)f (y) → α Γ(1 + α)Γ(1 -α) y∈E f (y)Π[dy] E Π (2) [Φ/k] (11) Note that dΠ (2) [u, v] = Π(du)k(u, v)µ(dv)
where Π(•) is the invariant measure associated to the Markov chain J, and thus in particular one has the equality

E Π (2) [Φ/k] = E 2 Φ(u, v)Π(du)µ(dv). (12)
It turns out that Theorem 1.2 is sufficient to resolve the large scale behavior of the set τ , which we describe by considering the class C ∞ of closed subsets of [0, ∞) endowed with the Mathéron topology. This topology can be defined as follows: for

t ≥ 0, F ∈ C ∞ , we set d t (F ) := inf(F ∩ (t, ∞)). The function t → d t (F ) is right continuous and F may be recovered from d • (F ) as F = {t ∈ R + , d t -(F ) = 0}.
The space of càdlàg functions may be endowed with the standard Skorohod topology, and this topology gives the Mathéron topology via the previous identification. Under this topology, C ∞ is metrizable, separable and compact, in particular it is a Polish space.

Finally, we introduce the classical notion of α stable regenerative set; recall that a subordinator is a non decreasing Lévy process. It is said to be α-stable (α ∈ (0, 1)) if its Lévy-Khintchine exponent φ(λ) is equal to λ α . We consider A α , the α stable regenerative set, which is defined as being the closure of the range of the α stable subordinator. We stress that A α is a highly non trivial random element from C ∞ , which coincides for α = 1/2 with the zero level set of a standard brownian motion.

If we consider τ as a subset of N, the set τ (N ) := τ ∩[0,N ] N may be viewed as a (random) element of C ∞ . If the Markov renewal process (τ, J) satisfies Assumption 1.1, we show that the following result holds, it is actually the first main result of this paper.

Theorem 1.3. The sequence of rescaled sets (τ (N ) ) N converges in law towards the set

A α ∩ [0, 1].
A proof of such a result in the classical renewal framework can be found in [START_REF] Giacomin | Random polymer models[END_REF][Theorem 2.7] by making use of the pointwise convergence of the Lévy exponent of a Poisson process whose range is equal to τ (N ) towards the one of the α stable-regenerative set, which directly implies the result (see [START_REF] Fitzsimmons | Intersections and limits of regenerative sets[END_REF]); the same idea is not available in our setup since the increments of τ are no longer i.i.d. Another idea which does not work here has been used in [START_REF] Caravenna | Infinite volume limits of polymer chains with periodic charges[END_REF] to prove a very similar result in the context of weakly inhomogeneous wetting models; the authors use in an essential way the finiteness of the state space of the governing Markov chain J, and the local behavior of the Green function given in equation (9).

Application to the strip wetting model.

Definition of the model.

The main motivation for proving Theorem 1.2 has been provided by a model which originates from statistical mechanics and which we describe now.

We consider (S n ) n≥0 a random walk such that S 0 := 0 and S n := n i=1 X i where the X i 's are i.i.d. and X 1 has a density h(•) with respect to the Lebesgue measure. We denote by P the law of S, and by P x the law of the same process starting from x. We will assume that h(•) is continuous and bounded on R, that h(•) is positive in a neighborhood of the origin, that E[X] = 0 and that E[X 2 ] =: σ 2 ∈ (0, ∞).

The fact that h is continuous and positive in the neighborhood of the origin entails that

n 0 := inf n∈N {(P[S n > a], P[-S n > a]) ∈ (0, 1) 2 } < ∞. (13)
In the sequel, we will assume that n 0 = 1 (and thus that (P[-S 1 > a], P[S 1 > a]) ∈ (0, 1) 2 ). We stress that our work could be easily extended to the generic n 0 ≥ 2 case, althought this should lead to some specific technical difficulties (for example one should extend Theorem 1.3 to the case where the transition function of J may vanish on E).

For N a positive integer, we consider the event C N := {S 1 ≥ 0, . . . , S N ≥ 0}. We define the probability law P N,a,β on R N by

dP N,a,β dP := 1 Z N,a,β exp β N k=1 1 S k ∈[0,a] 1 C N ( 14
)
where β ∈ R and Z N,a,β is the normalisation constant usually called the partition function of the system. For technical reasons, for x ∈ [0, a], we will sometimes need to consider the quantity

Z N,a,β (x) := E x exp β N k=1 1 S k ∈[0,a] 1 C N . (15)
Note that Z N,a,β = Z N,a,β (0). P N,a,β is a (1 + 1)-dimensional model for a linear chain of length N which is attracted or repelled to a defect strip of width a. By (1 + 1)-dimensional, we mean that the configurations of the linear chain are described by the trajectories (i, S i ) i≤N of the walk, so that we are dealing with directed models. The strength of this interaction with the strip is tuned by the parameter β. Regarding the terminology, note that the use of the term wetting has become customary to describe the positivity constraint C N and refers to the interpretation of the field as an effective model for the interface of separation between a liquid above a wall and a gas, see [START_REF] Deuschel | Scaling limits of equilibrium wetting models in (1 + 1)-dimension[END_REF].

The purpose of this part is to investigate the behavior of P N,a,β in the large N limit. In particular, we would like to understand when the reward β is strong enough to pin the chain near the defect strip, a phenomenon that we will call localization, and what are the macroscopic effects of the reward on the system. We point out that this kind of questions have been answered in depth in the case of the standard wetting model, that is formally in the a = 0 case, and that the problem of extending these results to the case where the defects are in a strip has been posed in [START_REF] Giacomin | Random polymer models[END_REF]Chapter 2]. Note that a closely related pinning model in continuous time has been considered and resolved in [START_REF] Cranston | Continuous model for homopolymers[END_REF]; we stress however that their techniques are very peculiar to the continuous time setup.

The free energy.

A standard way to define localization for our model is by looking at the Laplace asymptotic behavior of the partition function Z N,a,β as N → ∞. More precisely, one may define the free energy F (β) by

F (β) := lim N →∞ 1 N log (Z N,a,β ) . ( 16
)
The basic observation is that the free energy is non-negative. In fact, one has:

Z N,a,β ≥ E exp β N k=1 1 S k ∈[0,a] 1 S k >a,k=1,2,...,N ≥ P [S j > a, j = 1 . . . , N] . (17)
Choose some M > a such that P[S 1 ∈ [a, M]] > 0. Integrating over S 1 , one gets:

P[S j > a, j = 1, . . . , N] ≥ [a,M ] h(t)P t [S 1 > a, . . . , S N -1 > a] dt. (18)
A consequence of fluctuation theory estimates for killed random walk (see [Soh][Lemma 3.1] for the discrete case) is that for fixed M, the quantity

N 3/2 P t [S 1 > a, . . . , S N -1 > a] ∈ [c, c ′ ]
for every N and every t ∈ [a, M] where c, c ′ are positive constants. Thus:

Z N,a,β ≥ c N 3/2 [a,M ] h(t)dt. (19)
Therefore F (β) ≥ 0 for every β. Since the lower bound has been obtained by ignoring the contribution of the paths that touch the strip, one is led to the following:

Definition 1.4. The model {P N,a,β } is said to be localized if F (β) > 0.
The relevance of this classical definition is discussed for example in [START_REF] Giacomin | Random polymer models[END_REF][Chapter 1] for closely related models.

It is easy to show that F (•) is a convex function, in particular it is a continuous since F (β) ≤ β ∨ 0 for every β ∈ R. On the other hand, F (•) is increasing since for fixed N, Z N,a,β is increasing in β. Therefore, there exists a critical value β a c ∈ R such that the strip wetting model is localized for β > β a c .

Scaling limits of the system.

A common feature shared by the strip wetting model and the classical homogeneous one is the fact that the measure P N,a,β exhibits a remarkable decoupling between the contact level set I N := {i ≤ N, S i ∈ [0, a]} and the excursions of S between two consecutive contact points (see [START_REF] Deuschel | Scaling limits of equilibrium wetting models in (1 + 1)-dimension[END_REF] for more details in the standard homogeneous pinning case). Conditioning on I N = {t 1 , . . . , t k } and on (S t 1 , . . . , S t k ), the bulk excursions e i = {e i (n)} n := {S t i +n } 0≤n≤t i+1 -t i are independent under P N,a,β and are distributed like the walk (S ′ , P St i ) conditioned on the event

S ′ t i+1 -t i = S t i+1 , S ′ t i +j > a, j ∈ {1, . . . , t i+1 -t i -1} .
It is therefore clear that to extract scaling limits on P N,a,β , one has to combine good control over the law of the contact set I N and suitable asymptotics properties of the excursions.

This decoupling is the basis to the resolution of the large scale limits of the laws {P N,a,β }; in fact, we can show that for β > β a c , the diffusive rescaling limit of the laws {P N,a,β } is simply the null function, whereas in the delocalized phase the limiting law is the brownian motion conditioned to stay positive, that is the brownian meander. We stress that the proofs are quite similar although technically more involved that in the classical homogeneous wetting model, see [START_REF] Sohier | On pinning phenomena and random walk fluctuation theory[END_REF] for more details.

What is left by these considerations is the critical case. For simplicity, in this regime we describe the scaling limits in terms of the limit of the sequence {τ (N ) } N .

We point out that additionnal tightness conditions should be made on the free process S in order to ensure the convergence in law of the entire trajectories towards the reflected brownian motion (see [CGZ] to get some insight into this issue).

It turns out that the scaling limits of the set of contact times with the strip is almost the 1/2-stable regenerative set; more precisely, we compute its density with respect to the law of A 1/2 . Theorem 1.5. Consider the set τ := {j ∈ N, S j ∈ [0, a]} where S j is distributed according to the law P N,a,β a c . One has the convergence in law:

τ (N ) ⇒ B 1/2 ∩ [0, 1] ( 20
)
where B 1/2 is a random closed set whose law is absolutely continuous with respect to the one of A 1/2 with Radon Nykodym derivative given by

π 2 1 -max(A 1/2 ∩ [0, 1]) 1/2 . ( 21
)
We stress that the density term that appears in the above result is pushing the rightmost point of the process away from 1 and closer to the origin.

1.3

Outline of the paper.

The remaining of this paper is divided into three main parts, the first one dealing with the generic Markovian renewal setup, the second one being devoted to the application of the main result of the first part to the strip wetting model and the last one focusing on results about random walks. More specifically, the exposition of the paper will be organized as follows:

1. in section 2, we explicit the asymptotic behavior of the Markov mass renewal function which forms the cornerstone to the proof of our two main results. We then show Theorem 1.3 by using this asymptotic behavior.

2. in section 3, we describe the underlying Markov renewal structure inherent to the strip wetting model.

3. in section 3.1, we describe some results issued from fluctuation theory for random walks which will be the basis of our approach for the proof of Theorem 1.5.

4. in section 3.2, we give a suitable representation for the free energy of the strip wetting model and apply it to the proof of Theorem 1.5.

5. in section 4, we prove the results given in section 3.1, relying on some recent results by Doney [START_REF] Doney | Local behavior of first passage probabilities[END_REF].

2 Asymptotic equivalence of the mass Markov renewal function.

2.1 Notations.

Given two kernels A x,dy (n), B x,dy (n) we define their convolution

(A * B) x,dy (n) := n m=0 E A x,dz (m)B z,dy (n -m) (22)
and the k-fold convolution of the kernel A with itself will be denoted by A * k x,dy where by definition A * 0

x,dy := δ x (dy)1 n=0 . For (x, y) ∈ E 2 , λ > 0, we consider the Laplace transform measure U x,dy (λ) associated to (u(n, x, dy)) n defined by U x,dy (λ) := j≥0 e -λj u(j, x, dy).

(23

)
Our main technical step will be to prove the following proposition:

Proposition 2.1. For every x ∈ E, as λ ց 0, one has the following weak convergence:

Γ(1 -α)λ α L(1/λ)E Π (2) [Φ/k] α U x,dy (λ) → Π[dy]. (24)
To prove Proposition 2.1, we will need some technical results which we summarize in the next section.

2.2

Proof of Proposition 2.1

The Markov renewal equation writes:

u(n, x, dy) = k≥0 K * k x,dy (n) (25) which implies U(n, x, dy) = k≥0 n j=0 K * k x,dy (j). ( 26
)
For conciseness, we will denote by z := (x, y) an element of E 2 . Note that Assumption 1.1 implies in particular that 0

< min E 2 Φ ≤ max E 2 Φ < ∞. For z ∈ E 2 , λ > 0, define φ z (λ) := E x [e -λτ 1 |J 1 = y]. (27)
We are able to control the behavior of φ z (•) close to the origin, and this uniformly on E 2 . Proposition 2.2. As λ ց 0, one has:

sup z:=(x,y)∈E 2 k(x, y) (1 -φ z (λ)) λ α L(1/λ) - Φ(z)Γ(1 -α) α → 0. ( 28
)
To prove Proposition 2.2, we will need the following result:

Lemma 2.3. Given ρ > 0 and a family of atomfree measures

V (n) z on R + indexed by z ∈ E 2 , n ∈ N. Denote by Ψ (n)
z (λ) their Laplace transforms. Assume that they verify that for every u > 0, one has

lim n→∞ sup z∈E 2 V (n) z ([0, u]) -u ρ = 0. ( 29
)
and assume moreover that there exists

d > 0 such that sup (n,z)∈N×E 2 Ψ (n) z (d) < ∞.
Then for every λ > d:

lim n→∞ sup z∈E 2 Ψ (n) z (λ) - Γ(ρ + 1) λ ρ = 0. ( 30
)
Note that in our case, the sequence of measures V (n) will have a density with respect to Lebesgue measure, and thus will be without atoms.

Proof of Lemma 2.3. Let t > 0 be fixed; for any λ > a, uniformly on z ∈ E 2 , as n → ∞, we have:

[0,t] e -λu V (n) z (du) → [0,t]
e -λu ρu ρ-1 du.

Indeed, integrating by part in the left hand side above and making use of the fact that the sequence V (n) is atomfree, one obtains:

[0,t] e -λu V (n) z (du) = e -λt V (n) z (t) + λ [0,t] e -λu V (n) z (u)du (32)
and making use of the uniformity statement of equation (29) (which allows us in particular to use dominated convergence) and reintegrating by part, equation (31) is proved.

On the other hand, still for λ > d, it is clear that

[t,∞) e -λu V (n) z (du) ≤ e -(λ-d)t sup (n,z)∈N×E 2 Ψ (n) z (d) (33)
which can be made arbitrarily small (independently of z, n) as soon as t is large enough. Noting that

[0,∞) e -λu ρu ρ-1 du = Γ(ρ + 1) λ ρ , (34)
we get the result. Now we prove Proposition 2.2.

Proof of Proposition 2.2. For z ∈ E 2 , we define the (infinite mass) measure on

R + U z ([0, n]) := j≤n P x [τ 1 > j, J 1 = y].
The following equivalence holds uniformly on E 2 :

U z ([0, n]) ∼ j≤n L(j)Φ(z) αj α ∼ n 1-α L(n)Φ(z) α(1 -α) . (35)
Indeed, we use the general fact that if u n (z) is a positive sequence depending on

z ∈ E 2 verifying sup z∈S |u n (z)| < ∞ and, as n → ∞, sup z∈E 2 n α u n (z) L(n) -Φ(z) → 0, ( 36
) then sup z∈E 2 1 -α n 1-α L(n) j≤n u j (z) -Φ(z) → 0. ( 37
)
Recalling the standard equivalence

n j=1 L(j) j α ∼ n 1-α 1 -α L(n), (38)
the proof of this convergence is straightforward. We denote by Ψ z (λ) the Laplace transform associated to the measure U z (•). Integrating by part, we then have the equality which is valid for z ∈ E 2 , λ ≥ 0:

1 -φ z (λ) = (1 -e -λ ) n≥0 e -λn P x [τ 1 > n|J 1 = y] = (1 -e -λ ) Ψ z (λ) k(x, y) . (39)
Making use of equation ( 35), we give the asymptotic behavior of Ψ z (•) close to the origin; more precisely, uniformly on z ∈ E 2 , the convergence

Ψ z (λ) ∼ L(1/λ)Φ(z)Γ(1 -α) αλ 1-α (40)
holds as λ ց 0. Clearly, if this is true, making use of equation (39), Proposition 2.2 will be proved. We define a sequence of measures

V (n) z on R + by V (n) z ([0, u]) := Uz([0,nu])
Uz([0,n]) . Equation (35) implies that the hypothesis of Lemma 2.3 is verified by the sequence V (n) z with ρ = 1α with an arbitrary d > 0. It is plain that, for every fixed λ > 0,

Ψ (n) z (λ) = Ψ z (λ/n) U z ([0, n]) . ( 41
)
Making use of Lemma 2.3, as n → ∞, the convergence

Ψ z (λ/n) U z ([0, n]) → Γ(2 -α) λ 1-α (42)
holds uniformly on E 2 . Hence we get:

Ψ z (λ/n) ∼ Φ(z)L(n)Γ(2 -α) α(1 -α)(λ/n) 1-α (43)
and this still holds uniformly on E 2 . Extending this convergence to any sequence which decreases to zero (which is straightforward), making use of standard properties of slowly varying functions and of the Gamma function, one gets the equivalence (40).

To finally prove Proposition 2.1, we will need the following easy lemma.

Lemma 2.4. Let (c n ) n be a positive sequence which converges towards c > 0, u k a positive sequence which converges towards u > 0 and g(•) a function R + → R which is o(λ) as λ ց 0. Then, as λ ց 0, one has:

k≥0 c k e -k(λu k +g(λ)) ∼ c uλ . ( 44
)
Proof of Lemma 2.4. Note that one can ignore the first k 1 terms in the sum appearing in equation ( 44), where k 1 ≥ 0 is an arbitrarily large fixed integer. Let ε ∈ (0, min(u, c)) be fixed. For k 1 large enough and λ small enough (independently from k 1 ), one has :

(c -ε) k≥k 1 e -k(λ(u+ε)) ≤ k≥k 1 c k e -k(λu k +g(λ)) ≤ (c + ε) k≥k 1 e -k(λ(u-ε)) . ( 45
) Therefore c -ε c u u + ε ≤ lim inf λց0 λu c k≥0 c k e -k(λu k +g(λ)) ≤ lim sup λց0 λu c k≥0 c k e -k(λu k +g(λ)) ≤ c + ε c u u -ε (46)
and considering ε arbitrarily close to zero yields the statement. Consider now a function f which is continuous and positive on E, and let x ∈ E be fixed. Making use of the Markov renewal equation and of the fact that conditionally on the J i 's, the (τ iτ i-1 ) i≥1 are independent, we get:

E f (y)U x,dy (λ) = n≥0 E f (y) E n-1 n-1 j=0 φ x j ,x j+1 (λ)P x j [J 1 = x j+1 ] ( 47
)
where x 0 := x and x n := y. For (u, v) ∈ E 2 , we define

g u,v (λ) := 1 -φ u,v (λ) - Γ(1 -α)λ α L(1/λ)Φ(u, v) αk(u, v) . ( 48
)
Recall that Proposition 2.2 asserts that sup

(u,v)∈E 2 |g u,v (λ)| = o(λ α L(1/λ)) as λ ց 0.
For fixed n and fixed z ∈ E 2 , one has:

E n-1 n-1 j=0 φ x j ,x j+1 (λ)P x j [J 1 = x j+1 ] = E n-1 exp n-1 j=0 log 1 -(1 -φ x j ,x j+1 (λ)) P x [J 1 = x 1 , . . . , J n = x n ] (49)
Given ε ∈ (0, 1/2), there exists λ 0 such that as soon as λ ∈ (0, λ 0 ), for every

(u, v) ∈ E 2 , there exists c u,v (= c u,v (λ)) ∈ [1/2 -ε, 1/2 + ε]
such that the last term above is equal to:

E n-1 exp n-1 j=0 (1 -φ x j ,x j+1 (λ)) + c x j ,x j+1 (1 -φ x j ,x j+1 (λ)) 2 P[J 1 = x 1 , . . . , J n = x n ] = E exp n-1 j=0 - Γ(1 -α)λ α L(1/λ) α Φ(J j , J j+1 ) k(J j , J j+1 ) -g J j ,J j+1 (λ) + c J j ,J j+1 (1 -φ J j ,J j+1 (λ)) 2 1 Jn=y . ( 50
)
Making use of the ergodic theorem(which holds in our case, since J is a regular Markov chain (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF][VIII,7 Theorem 1]), we get that

1 n n-1 j=0 Φ(J j , J j+1 ) k(J j , J j+1 ) → E Π (2) Φ k = E Π(dx)Φ(x, y)µ(dy) (51)
P x almost surely. When n becomes large, the generic term appearing in the sum of the right hand side of (47) finally becomes equivalent to

y∈E f (y)P x [J n = y] × exp -n Γ(1 -α)λ α L(1/λ) α E Π (2) [Φ/k] + E Π (2) g J 0 ,J 1 (λ) + c J 0 ,J 1 (1 -φ J 0 ,J 1 (λ)) 2 . ( 52
)
The function λ

→ E Π (2) [g J 0 ,J 1 (λ) + c J 0 ,J 1 (1 -φ J 0 ,J 1 (λ)) 2 ] is then o(λ α L(1/λ)).
On the other hand, making use of the classical ergodic theorem, as n → ∞, the following convergence holds:

y∈E f (y)P x [J n = y] → E f (y)Π[dy]. (53)
Finally, we can apply Lemma 2.4 to prove Proposition 2.1.

Proof of Theorem 1.2.

To prove Theorem 1.2, we will use the following result, which is known as the extended continuity theorem (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF][XIII Theorem 2a]):

Theorem 2.5. Let U n be a sequence of measures defined on R + , and let φ n denote their Laplace transforms. For a fixed d > 0, if the sequence of functions φ n converges pointwise towards a function φ on (d, ∞), then the sequence of measures U n converges weakly towards a measure U with Laplace transform φ.

We consider a continuous function f on E and a fixed x ∈ E. Assume first that f (•) is positive; making use of Proposition 2.1, for t ց 0 and fixed λ > 0, we get the convergence:

E U x,dy (λt)f (y) E U x,dy (t)f (y) → 1 λ α . ( 54
)
Consider now the measure m f (•) on R + which is defined by

m f ([0, n]) := E f (y)U(n, x, dy), (55)
and denote by Ψ f (•) its Laplace transform. It is then quite clear that one may rewrite equation (54) as

Ψ f (λt) Ψ f (t) → 1 λ α (56)
when t ց 0. On the other hand, Ψ f (λt) is the Laplace transform associated to the measure m f (u/t), and thus by making use of the extended continuity theorem, this implies the following convergence (for t ց 0):

m f (u/t) Ψ f (t) → u α Γ(1 + α) . (57)
Finally, considering the convergence (57) with u = 1 along the subsequence t n = 1/n, as n → ∞, one gets:

E f (y)U(n, x, dy) ∼ E U x,dy (n)f (y) Γ(1 + α) (58)
and making use of Proposition 2.1, this is exactly the statement of Theorem 1.2. Making use of the linearity of the equivalence (58) and writing f = f + -(-f -) where f + (respectively f -) is the positive (respectively the negative) part of f , we are done.

2.4

Proof of Theorem 1.3.

We finally prove Theorem 1.3. The compactness of C ∞ implies that every sequence of probability measures on C ∞ is tight, so that in order to check the convergence in law of a sequence (P n ) towards P, one just has to show finite dimensional convergence, namely that for every t 1 < t 2 < . . . < t n , the following weak convergence holds:

P N G -1 t ⇒ P N G -1 t ( 59
)
where

G t : C ∞ → [0, ∞] n is defined by G t (F ) := (d t 1 (F ), . . . , d tn (F )).
Let us recall that for t > 0, the law of d t under P α is given by (see [START_REF] Bertoin | Subordinators: examples and applications[END_REF][Proposition 3.1])

P α [d t ∈ dy] = sin(απ) π t α y(y -t) α 1 y>t dy. ( 60
)
For conciseness, we show (59) in the n = 1 case when J starts from an initial state x ∈ E. Considering an arbitrary y > t, we have:

P N [J 0 = x, d t > y] = k≥1 E P x [τ k ≤ Nt, J k ∈ dv, τ k+1 > Ny] = j≤[N t] E k≥1 P x [τ k = j, J k ∈ dv] P v [τ 1 > Ny -j] = j≤[N t] E u(j, x, dv)P v [τ 1 > Ny -j] . (61)
Using Abel's summation, we get:

P N [J 0 = x, d t > y] = j≤[N t] E (U(j, x, dv) -U(j -1, x, dv)) P v [τ 1 > Ny -j] = E U(Nt, x, dv)P v [τ 1 > N(t -y) -1] - 1≤j≤N t E U(j, x, dv)P v [τ 1 = Ny -j -1] -P x [τ 1 > Ny] (62) Of course, P x [τ 1 > Ny] → 0 as N → ∞.
As for the first term in the right hand side of equation ( 62), we get :

E U(Nt, x, dv)P v [τ 1 > N(t -y) -1] → E αt α Γ(1 + α)Γ(1 -α) Π(dv) E Π (2) [Φ/k] E Φ(v, u)µ(du) α(y -t) α = sin(απ) π t y -t α . ( 63
)
For the second term in the right hand side of equation ( 62), first note that for ε ∈ (0, y), making use of the uniform convergence property for slowly varying functions, one has :

εN ≤j≤N t E U(j, x, dv)P v [τ 1 = Ny -j -1] ∼ E εN ≤j≤N t αj α Π[dv] L(j)Γ(1 + α)Γ(1 -α)E Π (2) [Φ/k] × L(Ny -j -1) E Φ(v, u)µ(du) (Ny -j -1) α+1 ∼ α Γ(1 -α)Γ(1 + α) t ε u α (y -u) α+1 du. ( 64
)
On the other hand, we get easily that lim εց0 lim

N →∞ j≤εN E U(j, x, dv)P v [τ 1 = Ny -j -1] = 0. ( 65
)
Thus we are left with checking the equality:

sin(απ) π ∞ y t α u(u t) α du = sin(απ) π t y -t α - α sin(απ) π t 0 u α (y -u) α+1 du (66)
which is easy using integration by parts; just note that

t 0 u α (y -u) α+1 du = 1 α u y -u α t 0 - t 0 u α-1 (y -u) α du (67)
and finally the equality

t 0 u α-1 (y -u) α du = ∞ y t α u(u -t) α du (68)
is obvious.

3 Application to the strip wetting model 3.1 Asymptotics about the kernel of the free process.

The following transition kernel will be of basic importance for the proof of Theorem 1.5:

F x,dy (n) := P x [S 1 > a, S 2 > a, . . . , S n-1 > a, S n ∈ dy]1 x,y∈[0,a] 2 if n ≥ 2, F x,dy (1) := h(y -x)1 x,y∈[0,a] dy. ( 69
)
For all n ∈ N and x ∈ [0, a], the kernel F x,dy (n) has a density f x,y (n) with respect to the Lebesgue measure restricted to [0, a].

We define the sequence of entry times of S into the strip by τ 0 := 0 and τ n+1 := inf{k > τ n , S k ∈ [0, a]}. We also consider the process (J n ) n≥0 where J n := S τn ; the process τ is then a Markov renewal process whose modulating chain is the Markov chain J, and with associated kernel F •,• (•).

Note that the kernel F is defective, in the sense that [0,a] k≥1 F x,dy (k

) = P x [S 1 ≥ 0] ≤ P[S 1 ≥ -a] < 1 for every x ∈ [0, a].
In particular, for every x ∈ [0, a], the quantity

P x [τ 1 = ∞] := 1 -[0,a] k≥1 F x,dy (k) is strictly positive.
We denote by l N the cardinality of {k ≤ N|S k ∈ [0, a]}, and we define

F x (k) := [0,a]
j>k F x,dy (j). With these notations, we write the joint law of (l N , (τ n ) n≤l N , (J n ) n≤l N ) under P N,a,β under the following form:

P N,a,β [l N = k, τ j = t j , J j ∈ dy i , i = 1, . . . , k] = e βk Z N,a,β F 0,dy 1 (t 1 )F y 1 ,dy 2 (t 2 -t 1 ) . . . F y k-1 ,dy k (t k -t k-1 ) F y k (N -t k ) + P y k [τ 1 = ∞] (70)
where

k ∈ N, 0 < t 1 < . . . < t k ≤ N and (y i ) i=1,...,k ∈ R k .
To describe the asymptotic behavior of the kernel F , we will need some results issued from fluctuation theory for random walks. Let us collect some basic facts about this topic.

For n an integer, we denote by T n the nth ladder epoch; that is T 0 := 0 and, for n ≥ 1, T n := inf{k ≥ T n-1 , S k > S T n-1 }. We also introduce the so-called ascending ladder heights (H n ) n≥0 , which, for k ≥ 1, are given by H k := S T k . Note that the process (T, H) is a bivariate renewal process on (R + ) 2 . In a similar way, one may define the strict descending ladder variables process (T -, H -) as (T - 0 , H - 0 ) := (0, 0) and

T - n := inf{k ≥ T n-1 , S k < S T n-1 } and H - k := -S t - k . ( 71
)
We define the following function:

Φ a (x, y) := P[H - 1 ≥ a -y]P[H 1 ≥ a -x] σ √ 2π 1 x,y∈[0,a] . (72)
In the appendix, we show the following theorem, which will be the cornerstone of our approach: Theorem 3.1. The following equivalence holds uniformly on (x, y) ∈ [0, a] 2 :

n 3/2 f x,y (n) ∼ Φ a (x, y). ( 73
)
It is a well known fact that the continuity of h(•) implies the continuity of the distribution function of H 1 as one can deduce from the identity P[H 1 ∈ I] = k≥1 P[T 1 = k, S k ∈ I] which is valid for every interval I. Moreover, as we assumed n o = 1 in equation ( 13), Φ a is continuous and strictly positive on the whole square [0, a] 2 , as one has the equivalence

P[H 1 > a] > 0 ⇐⇒ P[S 1 > a] > 0 (74)
and of course a similar statement holds for the first descending ladder height process.

3.2 Defining the free energy through an infinite dimensional problem.

In this section, we define the free energy in a way that allows us to make use of the underlying Markov renewal structure inherent to this model. For λ ≥ 0, we introduce the following kernel:

B λ x,dy := ∞ n=1 e -λn F x,dy (n) ( 75
)
and the associated integral operator

(B λ h)(x) := [0,a] B λ x,dy h(y). (76)
We then have the Lemma 3.2. For λ ≥ 0, B λ x,dy is a compact operator on the Hilbert space L 2 ([0, a]).

Proof. Of course B λ x,dy has a density with respect to the Lebesgue measure restricted to [0, a] that we denote by b λ (x, y). To show Lemma 3.2, it is sufficient to show that B λ

x,dy is actually Hilbert-Schmidt, that is that

b λ (x, y) 2 1 x,y∈[0,a] dxdy < ∞. ( 77
)
Note that Theorem 3.1 entails in particular that there exists c > 0 such that, for all n ∈ N and x, y ∈ [0, a], f x,y (n) ≤ c n 3/2 . Making use of this inequality, it is then straightforward to show (77). Lemma 3.2 enables us to introduce δ a (λ), the spectral radius of the operator B λ . It is easy to check that δ a (λ) ∈ (0, ∞) for λ ≥ 0; δ a (λ) is an isolated and simple eigenvalue of B λ

x,dy (see Theorem 1 in [START_REF] Zerner | Quelques propriétés spectrales des opérateurs positifs[END_REF]). The function δ a (•) is non-increasing, continuous on [0, ∞) and analytic on (0, ∞) because the operator B λ

x,dy has these properties. The analyticity and the fact that δ a (•) is not constant (as δ a (λ) → 0 as λ → ∞) force δ a (•) to be strictly decreasing.

We denote by (δ a ) -1 (•) its inverse function, defined on (0, δ a (0)]. We now define β a c and F a (β) by:

β a c := -log(δ a (0)), F a (β) := (δ a ) -1 (exp(-β)) if β ≥ β a c and 0 otherwise. ( 78
)
Note that this definition entails in particular the analyticity of F (•) on R \ {β a c }. We discuss below the relevance of this definition in accordance with the definition given in equation ( 16).

The fact that b F a (β) (x, y) > 0 for every (x, y) ∈ [0, a] implies the uniqueness (up to a multiplication by a positive constant) and the positivity almost everywhere of the so-called right and left Perron Frobenius eigenfunctions of B F a (β) x,dy ; more precisely, Theorem 1 in [START_REF] Zerner | Quelques propriétés spectrales des opérateurs positifs[END_REF] implies that there exist two functions v β (•) and

w β (•) in L 2 ([0, a]) such that v β (x) > 0 and w β (x) > 0 for almost every x ∈ [0, a],
and moreover:

y∈[0,a] B F a (β) x,dy v β (y) = 1 e β ∧ 1 e β a c v β (x) (79) x∈[0,a] w β (x)B F a (β) x,dy dx = 1 e β ∧ 1 e β a c w β (y)dy. (80)
Spelling out these equalities, we get that

v β (x) = 1 1 e β ∧ 1 e β a c n≥0 e -F a (ε)n y∈[0,a] f x,y (n)v β (y)dy, (81)
which implies in particular the fact that v β (•) is positive and continuous on the whole [0, a] and not only almost everywhere. Similarly, the function w β (•) is everywhere positive and continuous. These considerations lead us to define the new kernel

K β x,dy (n) := e β F x,dy (n)e -F a (β)n v β (y) v β (x) . (82)
It is then straightforward to check that

y∈R n∈N K β x,dy (n) = e β v β (x) y∈R n∈N F x,dy (n)e -F a (β)n v β (y)1 y∈[0,a] = e β v β (x) y∈R B F a (β) x,dy v β (y) = 1 ∧ e β e β a c . (83)
In particular, when β = β a c , K

β a c
•,• is a non defective Markov renewal kernel, which satisfies Assumption 1.1. Indeed, when n → ∞, it is a consequence of Theorem 3.1 that the following convergence holds uniformly on [0, a] 2 :

n 3/2 K β a c x,dy (n) dy ∼ e β a c v β (y) v β (x) Φ a (x, y), (84)
which is equation (4) with α = 1/2 and L(•) a trivial slowly varying function. Let us define Φa (x, y) := e β a c v β (y) v β (x) Φ a (x, y). Note also that the strict positivity of Φa on the square [0, a] 2 implies that the condition about the strict positivity of the transition kernel of J is also satisfied.

It is not hard to see that for every β ∈ R, the measure Π β (dx) := v β (x)w β (x)dx is invariant for the Markov chain J; moreover, both v β (•) and w β (•) being defined up to a multiplicative constant, one can use this degree of freedom to make Π β a probability measure on [0, a].

Let us point out that it is not clear a priori that the quantity we define in (78) actually coincides with the definition given in equation ( 16). We will not delve into this issue in this paper; however, we stress that this identification relies on applying the Markov renewal theorem to the kernel K β in the localized phase (thus when the Markov renewal associated to K β is recurrent positive), and on direct (polynomial) asymptotics on the partition functions in the delocalized one (for more details, see the thesis [Soh10][Chapter 2]). For all relevant purposes, we will use the definition given in equation (78) as a definition for the free energy.

Proof of Theorem 1.5.

We will need the asymptotic behavior of Z N,a,β a c (x), which is given in the next lemma: Lemma 3.3. As N → ∞ and for x ∈ [0, a], the following equivalence holds:

Z N,a,β a c (x) ∼ N 1/2 v β a c (x)(1 -e -β a c ) πe β a c [0,a] 2 v β a c (t)w β a c (s)Φ a (s, t)dsdt [0,a]
w β a c (s)ds

∼ N 1/2 v β a c (x)C a,Φ . (85)
Proof of Lemma 3.3. For (x, y, n) ∈ [0, a] 2 × N, we denote by κ the Markov renewal mass function associated to the kernel K

β a c •,• , that is κ(n, x, dy) := k≥1 (K β a c ) * k
x,dy (n). Summing over the last contact point of the process with the strip before time N, we have:

Z N,a,β a c (x) = v β a c (x) [0,a] N n=0 κ(n, x, dy) 1 v β a c (y) P y [τ 1 > N -n] + v β a c (x) [0,a] N n=0 κ(n, x, dy) 1 v β a c (y) P y [τ 1 = ∞] = v β a c (x) [0,a] N n=0 κ(n, x, dy) j>N -n [0,a] K β a c y,du (j) v β a c (u) + v β a c (x) [0,a] N n=0 κ(n, x, dy) 1 v β a c (y) P y [τ 1 = ∞]. (86)
Taking into account the fact that v β a c (•) is continuous and positive on [0, a], and that K s,dt (n), we make use of Proposition 1.2 in the case where E = [0, a] is equipped with the Lebesgue measure to get the equivalence:

[0,a] N n=0 κ(n, x, dy) P y [τ 1 = ∞] v β a c (y) ∼ N 1/2 π [0,a] Π β a c (dy) E Π (2) [ Φβ a c /k β a c ] P y [τ 1 = ∞] v β a c (y) . ( 87
)
Then we recall that Π β a c (du) = v β a c (u)w β a c (u)du and we make use of the equality

[0,a] w β a c (s) 1 - j≥1 [0,a] F s,dt (j) ds = 1 -e -β a c [0,a] w β a c (t)dt. ( 88
)
Finally making use of equality (12) we get Lemma 3.3.

We will also need the following deterministic lemma:

Lemma 3.4. Let v n , u n and w n be positive sequences such that v n ∼ w n ∼ √ n and u n ∼ n -3/2 as n → ∞. For any (s, t) ∈ (0, 1) 2 such that s < t, the following convergence holds as N → ∞:

1 v N sN i=1 w i N j=tN (u j-i -u j-i-1 )v N -j → - 3 2 (u,v)∈[0,s]×[t,1] u(1 -v) (v -u) 5/2 dudv. ( 89
)
Proof of Lemma 3.4. Let ε ∈ (0, 1) and l ≥ 2 be fixed. We write the left hand of the convergence appearing in equation (89) as

1 v N sN i=1 w i l-1 m=0 V N,i,m (90)
where for i ∈ [1, sN], m ∈ [0, l -1], we defined

V N,i,m := tN +(1-t)N (m+1)/l n=tN +(1-t)N m/l (u n-i -u n-i+1 )v N -n . ( 91
) Defining (⋆) := tN +(1-t)N (m+1)/l n=tN +(1-t)N m/l (u n-i -u n-i+1 ) N(1 -t) 1 - m l = u N (t+ (1-t)m l -i N )+N( 1-t l ) -u N (t+ (1-t)m l -i N ) N(1 -t) 1 - m l . (92)
for N large enough, the following inequalities hold for every i ∈ [1, sN], m ∈ [0, l-1]:

(1 -ε)(⋆) ≤ V N,i,m ≤ (⋆)(1 + ε). (93)
We get that, as N → ∞, for every i ∈ [1, sN], m ∈ [0, l -1], one has:

t + (1 -t)m l - i N 3/2 N 3/2 u N (t+ (1-t)m l -i N )+N( 1-t l ) -u N (t+ (1-t)m l -i N ) =   1 + 1-t l t + (1-t)m l -i N   -3/2 -1 + o(1), (94)
so that as N → ∞, the following equivalence holds:

N × (⋆) ∼      1 + 1-t l t + (1-t)m l -i N   -3/2 -1    (1 -t)(1 -m/l) t + (1-t)m l -i N 3/2 (95)
As a consequence, as N → ∞, for every l ≥ 2, we get the following inequalities:

(1ε)

u∈[0,s] √ u l-1 m=0      1 + 1-t l t + (1-t)m l -u   -3/2 -1    (1 -t)(1 -m/l) t + (1-t)m l -u 3/2 du ≤ lim inf N →∞ 1 v N sN i=1 w i l-1 m=0 V N,i,m ≤ lim sup N →∞ 1 v N sN i=1 w i l-1 m=0 V N,i,m ≤ (1 + ε) u∈[0,s] √ u l-1 m=0      1 + 1-t l t + (1-t)m l -u   -3/2 -1    (1 -t)(1 -m/l) t + (1-t)m l -u 3/2 du. (96)
For l → ∞, making use of Riemann's sums and of the dominated convergence theorem, the integral appearing in the last term of the above equation becomes equivalent to

- 3 2 u∈[0,s] v∈[0,1] √ u (1 -t) 3/2 √ 1 -v (t + (1 -t)v -u) 5/2 dudv, (97)
from which we deduce the result. Now we go to the proof of Theorem 1.5.

Proof of Theorem 1.5. As in the proof of Theorem 1.3, we just have to show finite dimensional convergence. We denote by P B 1/2 the law of the limiting random set appearing in Theorem 1.3 and by P 1/2 the law of the regenerative set with index 1/2. For F ∈ F , we also define g t (F ) = sup(F ∩ [0, t)).

Let us first compute the limiting quantity we are looking for. For 0 < s < t < 1, we write:

P B 1/2 (d s > t) = P B 1/2 (g t < s, g 1 = g s ) + P B 1/2 (g t < s, g 1 > t) = P B 1/2 (g 1 ≤ s) + π 2 u∈[0,s] v∈[t-u,1-u] w∈[u+v,1] P 1/2 (g t ∈ du, d t -g t ∈ dv) √ 1 -wP 1/2 (g 1 ∈ dw|d t = u + v). (98)
We first make use of the explicit law of g 1 (see [START_REF] Bertoin | Subordinators: examples and applications[END_REF][Proposition 3.1]), and we deduce the following equalities:

P B 1/2 (d s > 1) = π 2 [0,s] √ 1 -uP 1/2 (g 1 ∈ du) = π 2 [0,s] √ 1 -u 1 π 1 u(1 -u) du = √ s. (99)
Then we note that for every x ∈ (t, 1) and w ∈ (x, 1), the following holds:

P 1/2 (g 1 ∈ dw|d t = x) = P 1/2 (x + g 1-x ∈ dw) = 1 π dw (w -x)(1 -w) . (100)
On the other hand, the following equality is well known (see [START_REF] Borodin | Handbook of Brownian motion-facts and formulae[END_REF]):

P 1/2 (g t ∈ du, d t -g t ∈ dv) = dudv 2π √ uv 3 . (101)
Finally one has:

P B 1/2 (d s > t) = √ s + 1 2π (u,v)∈[0,s]×[t,1] 1 -v u(v -u) 3 dudv. (102)
In the same spirit as for the proof of Theorem 1.3, we then establish the convergence of P N,a,β a c (d N s > t) towards P B 1/2 (d s > t) as N → ∞, thus proving Theorem 1.5. A computation similar to the one we made for the proof of Lemma 3.3 leads to the following convergence (for s ∈ (0, 1)):

P N,a,β a c (d N s > 1) → √ s. (103)
Thus we are left with computing the limit of the second term in the right hand side of the equality below:

P N,a,β a c (d N s > t) = P N,a,β a c (d N s ≥ 1) + P N,a,β a c (d N s ∈ (t, 1)). (104)
We write:

P N,a,β a c (d N s ∈ (t, 1)) = v β a c (0) Z N,a,β a c sN n=0 x∈[0,a] κ(n, 0, dx) × y∈[0,a] N m=tN K β a c x,dy (m -n) v β a c (y) Z N -m,a,β a c (y). (105)
Once again, we do not have access to the local behavior of the quantities κ(n, 0, x) for n large, thus we use integration by part as in the proof of Theorem 1.3 and we have:

P N,a,β a c (d N s ∈ (t, 1)) = v β a c (0) Z N,a,β a c x∈[0,a] sN j=0 κ(j, 0, dx) N m=tN y∈[0,a] K β a c x,dy (m -sN) v β a c (y) Z N -m,a,β a c (y) + v β a c (0) Z N,a,β a c sN -1 n=0 (x,y)∈[0,a] 2 n j=0 κ(j, 0, dx) N m=tN K β a c x,dy (m -n) -K β a c x,dy (m -n -1) v β a c (y) Z N -m,a,β a c (y) + v β a c (0) Z N,a,β a c (x,y)∈[0,a] 2 N m=tN K β a c 0,dy (m) v β a c (y) Z N -m,a,β a c (y) =: (⋆) + (⋆⋆) + (⋆ ⋆ ⋆). (106)
Making use of Lemma 3.3, it is straightforward to check that (⋆ ⋆ ⋆) is O(N -1/2 ) as N → ∞. As N → ∞, the following holds:

(⋆) ∼ 1 N 1/2 C a,Φ x∈[0,a] √ sN π Π β a c (dx) E Π (2) [ Φβ a c /k β a c ] N m=tN y∈[0,a] √ N -m C a,Φ Φβ a c (x, y) (m -sN) 3/2 dy → √ s π 1 t √ 1 -u (u -s) 3/2 du. (107)
Making use of Lemmas 3.3, 3.4 and of the asymptotic behavior of K x,dy (n)(which holds uniformly on [0, a] 2 ), we easily get the equivalence:

(⋆⋆) ∼ - 3 2π (u,v)∈[0,s]×[t,1] u(1 -v) (v -u) 5/2 dudv. (108)
Putting everything together, we finally have the following convergence:

P N,a,β a c (d N s ∈ (t, 1)) → √ s π 1 t √ 1 -u (u -s) 3/2 du - 3 2π (u,v)∈[0,s]×[t,1] u(1 -v) (v -u) 5/2 dudv.
(109) On the other hand, integrating by part, we get the equalities:

(u,v)∈[0,s]×[t,1] 1 -v u(v -u) 3 = v∈[t,1] 2 u(1 -v) (v -u) 3/2 dv s u=0 - 3 2 (u,v)∈[0,s]×[t,1] 2 u(1 -v) (v -u) 5/2 dudv = 2 √ s 1 t √ 1 -v (v -s) 3/2 dv - 3 2 (u,v)∈[0,s]×[t,1] 2 u(1 -v) (v -u) 5/2 dudv. (110)
With the help of the above, comparing the equality (102) and the convergence appearing in equation (109), we are done.

Appendix

The aim of this section is to prove Theorem 3.1.

Let us consider the renewal function U(•) associated to the ascending ladder heights process:

U(x) := ∞ k=0 P[H k ≤ x] = E[N x ] = x 0 ∞ m=0 u(m, y)dy (111)
where N x is the cardinality of {k ≥ 0, H k ≤ x} and u(m, y) := 1 dy P[∃k ≥ 0, T k = m, H k ∈ dy] is the renewal mass function associated to (T, H). It follows in particular from this definition that U(•) is a subadditive increasing function, and in our context it is also continuous. Note also that U(0) = 1. We denote by V (x) the analogous quantity for the process H -, and by v(m, y) the renewal mass function associated to the descending renewal (T -, H -) .

For u > 0, we define τ u := inf{k ≥ 1, S k ≥ u}. As we did for the Markov renewal kernel, for any k ≥ 1 and for any B ∈ σ(S 1 , . . . , S k-1 ), we introduce the useful notation:

1 dx P[B, S k ∈ dx] =: P[B, S k = x] (112)
Our proof is based on the following result, which is a specialization of results due to Doney [START_REF] Doney | Local behavior of first passage probabilities[END_REF][Proposition 18]:

Theorem 4.1. For x, y ∈ R + , let us define two sequences x n := x(σ 2 n) -1/2 and y n := y(σ 2 n) -1/2 . For any fixed ∆ 0 > 0, the following estimate holds uniformly for ∆ ∈ (0, ∆ 0 ): Uniformly as x n ∨ y n → 0,

P[S n ∈ (x -y -∆, x -y], τ x > n] ∼ U(x) y+∆ y V (w)dw σ √ 2πn 3/2 . ( 113
)
Moreover, there exists a constant c > 0 such that the left hand side of equation ( 113) is dominated by a multiple of the right hand side.

Proof of Theorem 3.1. We denote by S the random walk whose transitions are given by P x [ S1 ∈ dy] := h(xy)dy (114)

and for positive u, by τu the quantity inf{k ≥ 1, Sk > u}.

Let ε ∈ (0, 1) be fixed. Integrating over the first entry point of S into (a, ∞) and on the last location of S in (a, ∞) before hitting the strip, for x, y ∈ [0, a] 2 , one gets the equalities:

where we defined the subsets of (a, ∞) 2 :

D n 1 := (u, v) ∈ (a, ∞) 2 , u ∨ v ≤ ε √ n D n 2 := (u, v) ∈ (a, ∞) 2 , u ∨ v ≥ ε √ n . ( 116
)
We approximate the integral I n 1 by below and by above by breaking the range of integration over v into subintervals of length η > 0, then we use the estimate given in equation (113) and finally let η ց 0 to get the convergence: as n → ∞.

We recall that as the density h(•) is bounded, it is a consequence of Stone's local limit theorem that there exists a constant c > 0 such that:

sup n∈N sup x∈R √ nP[S n = x] ≤ c. (123)
Relying on this domination, the convergence (122) is easy. In fact, for fixed ε > 0, as soon as n is large enough (c > 0 is a constant which may vary from line to line and which is independent from x and y):

u ∨ v ε √ n 2 h(u -x)h(y -v)dudv ≤ c ε 2 ε √ n-a u 2 h(u)du ≤ c ε 2 ε √ n/2 u 2 h(u)du (124)
and recalling that X ∈ L 2 , the last inequality is sufficient to conclude.

  is non defective (and thus that [0,a]N n=0 κ(n, x, dy) j>N -n [0,a] K β a c y,du (j) = P x [S 1 > 0] ≤ 1), the first term in the right hand side of equation (86) is bounded as N becomes large. As for the second term, denoting by k β a c (s, t)

  f x,y (n) = P x [S 1 > a, . . . , S n-1 > a, S n = y] = (a,∞) 2 h(ux)P u [S 1 > a, . . . , S n-3 > a, S n-2 = v] h(yv)dudv = (a,∞) 2 h(ux)P [S 1 > au, . . . , S n-3 > au, S n-2 = vu] h(yv)dudv = (a,∞) 2 h(ux)P τu-a > n -2, Sn-2 = uv h(yv)dudv =

0 V 0 P

 00 x)V (ua)du (a,∞) U(va)h(yv)dv = 0. x)V (ua)du = ∞ (u)h(u + ax)du = ∞ 0 h(u + ax)du + ∞ 0 y∈[0,u] m≥1 v(m, w)h(u + ax)dudw = P[H 1 ≥ ax, T 1 = 1] + y∈[0,∞) u∈[y,∞) h(u + ax)du m≥1 v(m, y)dy. (118)We use of the well known duality lemma (see[START_REF] Feller | An introduction to probability theory and its applications[END_REF][chapter XII]), which asserts that the following equality holds:v(m, y) = P[S 1 ≤ 0, . . . , S m ≤ 0, -S m = y]. (119)Making use of the Markov property, we then get the equalities:(a,∞) h(ux)V (ua)du = P[H 1 ≥ ax, T 1 = 1] + m≥1 ∞ [S 1 ≤ 0, . . . , S m ≤ 0, S m = -y]P[S 1 ≥ y + ax] = P[H 1 ≥ ax, T 1 = 1] + m≥1 P[H 1 ≥ ax, T 1 = m] = P[H 1 ≥ ax].(120)A very similar computation yields the equality(a,∞) U(va)h(yv)dv = P[H - 1 ≥ ay].(121)To complete the proof of Theorem 3.1, we are left with showing that n 3/2 sup (x,y)∈[0,a]

  sup (x,y)∈[0,a] 2 n 3/2 I n 2 ≤ n 3/2 sup (x,y)∈[0,a] 2 D n 2 h(ux)P Sn-2 = uv h(yv)dudv ≤ cn sup (x,y)∈[0,a] 2 D n 2

Acknowledgement:

The author is very grateful to Francesco Caravenna for constant support during this work.