Matheus Vieira

MD Paul Régnier

PhD Anna Maciejewski-Duval

PhD Alexandre Le Joncour

Prof Guillaume Darasse-Jèze¹

Prof Michelle Rosenzwajg¹

Prof David Klatzmann¹

Prof Patrice Cacoub¹

Prof David Saadoun¹
email: david.saadoun@aphp.fr

David Pr

Saadoun

Interferon signature in giant cell arteritis aortitis

Keywords: Giant cell arteritis, Large-vessel vasculitis, Aortitis, Interferon signature, JAK/STAT

Objectives: Molecular mechanisms underlying large-vessel involvement in giant cell arteritis (LV-GCA) are largely unknown. Herein, we explore the critical involvement of pro-inflammatory signaling pathways in both aorta and T cells from patients with LV-GCA.

Methods:

We analyzed transcriptome and interferon gene signature in inflamed aortas from LV-GCA patients and compared them to non-inflammatory control aorta.

Differential transcriptomic analyses of circulating CD4+ and CD8+ T cells were also performed between patients with active GCA (not under any immunosuppressants or corticosteroid doses higher than 10 mg/day by the time of blood collection) and healthy donors. Interferon-alpha serum levels were measured using ultra-sensitive technique (HD-X Simoa Planar Technology) in GCA patients according to disease activity status.

Results: Transcriptomic analyses revealed 1042, 1479 and 2075 significantly dysregulated genes for aortas, CD4+ and CD8+ cells from LV-GCA patients, respectively, as compared to controls. A great enrichment for pathways linked to interferons (type I, II and III), JAK/STAT signaling, cytokines and chemokines was seen across aortas and circulating T cells. A type I interferon signature was identified as significantly upregulated in the aorta of patients with LV-GCA, notably regarding EPSTI1 and IFI44L genes. STAT3 was significantly upregulated in both aorta and T cells and appeared as central in related gene networks from LV-GCA patients.

Interferon-alpha serum levels were higher in patients with active GCA when compared to those in remission (0.024 vs. 0.011pg/mL; p=0.028).

Conclusion:

LV-GCA presents a clear type I interferon signature in aortas, which paves the way for tailored therapeutical targeting.

Introduction

Giant cell arteritis (GCA) is a granulomatous large-vessel vasculitis that affects patients older than 50 years, representing the most common type of primary vasculitis [START_REF] Sharma | Incidence and prevalence of giant cell arteritis and polymyalgia rheumatica: A systematic literature review[END_REF].

Besides the mainly affected external carotid branches causing the polymorphic and typical cephalic manifestations of GCA, the aorta and its main branches may also be inflamed [START_REF] Brack | Disease pattern in cranial and large-vessel giant cell arteritis[END_REF]. The proportion of GCA patients described with large-vessel vasculitis (LV-GCA) has increased over the past few decades, mostly due to a wider use of aorta imaging such as computed tomography angiography (CTA), magnetic resonance angiography (MRA) or positron emission tomography with computed tomography (PET/CT), which are able to detect large-vessel inflammation in up to 83% of patients [START_REF] Blockmans | Repetitive 18F-fluorodeoxyglucose positron emission tomography in giant cell arteritis: a prospective study of 35 patients[END_REF][START_REF] Prieto-González | Large vessel involvement in biopsy-proven giant cell arteritis: prospective study in 40 newly diagnosed patients using CT angiography[END_REF]. Compared to those with cranial phenotype, LV-GCA patients have a more relapsing disease, requiring longer treatment with higher cumulative dose of corticosteroids [START_REF] Muratore | Large-vessel giant cell arteritis: a cohort study[END_REF]. Furthermore, at twice the risk of having aortic aneurysms (vs. general population), up to 30% of GCA patients develop aneurysm or dissection [START_REF] Saadoun | Medium-and Large-Vessel Vasculitis[END_REF], whose inherent risk of rupture represents a major cause of mortality in GCA [START_REF] Koster | Large-vessel giant cell arteritis: diagnosis, monitoring and management[END_REF].

Vascular pathophysiology in GCA derives from studies of inflammatory temporal arteries, but as its positivity is limited in LV-GCA [START_REF] Koster | Large-vessel giant cell arteritis: diagnosis, monitoring and management[END_REF], the pathophysiological process driving the aortitis subset might not be exactly the same as compared to cranial disease.

Moreover, the difficulty in obtaining aortic samples makes the comprehension and management of LV-GCA phenotype even more challenging. New insights into GCA pathogenesis have emerged as increasing attention is being given to type I interferons (e.g., alpha) in LVV [START_REF] Watanabe | Cellular Signaling Pathways in Medium and Large Vessel Vasculitis[END_REF]. This cytokine signals through the Janus kinase-signal transducer of activators of transcription (JAK/STAT) pathway, inducing the expression of interferon-stimulated genes, the so-c d " n f n gn " [START_REF] Barrat | Interferon target-gene expression and epigenomic signatures in health and disease[END_REF]. Type I interferon signatures have been reported in several autoimmune diseases, but barely in vasculitis [START_REF] Barrat | Interferon target-gene expression and epigenomic signatures in health and disease[END_REF][START_REF] Higgs | Patients with systemic lupus erythematosus, myositis, rheumatoid arthritis and scleroderma share activation of a common type I interferon pathway[END_REF][START_REF] Rönnblom | The interferon signature in autoimmune diseases[END_REF]. Recently, type I interferon transcripts were demonstrated to be increased in the vessel wall of an immunodeficient mice transengrafted with inflamed temporal arteries from GCA patients, and inhibiting JAK1 and JAK3 in this chimeric model was effective to disrupt core vasculogenic effector pathways [START_REF] Zhang | Inhibition of JAK-STAT Signaling Suppresses Pathogenic Immune Responses in Medium and Large Vessel Vasculitis[END_REF]. Indeed, as the JAK/STAT pathway also transduces the signals from several other pro-inflammatory cytokines, its blockade became a paradigm in the treatment of many immune-mediated diseases [START_REF] Nash | Points to consider for the treatment of immune-mediated inflammatory diseases with Janus kinase inhibitors: a consensus statement[END_REF], especially those with an associated interferon signature [START_REF] Barrat | Interferon target-gene expression and epigenomic signatures in health and disease[END_REF]. While clinical trials evaluating JAK inhibitors in GCA are currently ongoing, whether signaling through JAK/STAT pathway plays a major role in large-vessel inflammation remains unclear.

Data on molecular pathways that could drive large-vessel lesions in GCA are lacking.

By using transcriptomic analysis of aorta and circulating T cells from LV-GCA patients, we aimed to assess the pathways involved in this phenotype.

Methods

Study population

The study population consisted of 43 patients with active GCA (33 newly diagnosed and 10 relapsing patients), with median age of 72 [interquartile range 65.75-78] years, all fulfilling the American College of Rheumatology criteria [START_REF] Hunder | The American College of Rheumatology 1990 criteria for the classification of giant cell arteritis[END_REF]. LV-GCA was defined as the presence of inflammation of the aorta (aortitis) or its primary branches in large-

Transcriptomic data of aorta

Aorta tissue samples (n CTRL = 9 and n GCA = 10) for this study were obtained in 10µm cuts. For paraffin extraction, aorta slides were immersed in two consecutive reservoirs containing limonene and ethanol, respectively, being rehydrated with glycerol 3%, and then extracted from the slides and transferred into Eppendorf tubes. Samples were then incubated with DNAase-I for 15 minutes at 20℃, being successively washed thereafter.

After centrifugation, supernatant containing RNA was collected and stored at -80℃.

Deparaffinized tissue was warmed until 55℃ and lysed using High Pure FFPET RNA Isolation Kit (Prosigna, Life Science Roche®). Then, it was incubated overnight at the same temperature. After RNA isolation, samples were submitted to full transcriptomic analysis. All samples passed QC tests for RNA quality, hybridization and noise.

Samples were collected in Affymetrix CEL format and were normalized and analyzed using limma R package. Enrichment analysis of different gene sets within samples and groups was assessed using GSVA and limma R packages as previously described [START_REF] Hänzelmann | GSVA: gene set variation analysis for microarray and RNA-seq data[END_REF]. Samples with RNA concentration <20ng/µL were excluded. For quality control, RNA dilution was performed using Agilent RNA 6000 Nano Kit and 1µL of the sample was run on the Nano chip using an Agilent 2100 electrophoresis bioanalyzer. The quality of total RNA was assessed by the profile of the electropherogram and by the RNA integrity number (RIN). All included samples showed a RIN >7.3. For Illumina Beadarrays, cRNA samples were prepared using Illumina TotalPre-96 RNA Amp kit (LifeTechnologies) and hybridized to Illumina Human HT-12 v4 Beadarrays. Raw data were then normalized and analyzed using limma R package. Enrichment analysis of different gene sets within samples and groups was assessed using GSVA and limma R packages [START_REF] Hänzelmann | GSVA: gene set variation analysis for microarray and RNA-seq data[END_REF].

Transcriptomic data of circulating CD4+ and CD8+ T cells

CD3+

Gene network analysis

Statistically upregulated pathways seen in GCA patients with respect to interferons, JAK/STAT pathways, cytokines and chemokines were selected. Genes with LogFC > 0 as compared to HD were extracted and a network linking genes together was created using igraph R package and STRING protein interaction database. This network analysis relies on whether the respective proteins are already known to interact together [START_REF]STRING: functional protein association networks[END_REF]. Pathways' enrichment analyses of transcriptomic data were performed, and gene network graphs were subsequently built for aorta and circulating CD4+ and CD8+ T cells.

Type I interferon gene signature

In order to specifically evaluate type I interferon activity in our transcriptomes, we searched for upregulated gene signatures from several publications regarding distinct autoimmune diseases [START_REF] Higgs | Patients with systemic lupus erythematosus, myositis, rheumatoid arthritis and scleroderma share activation of a common type I interferon pathway[END_REF][START_REF] Cooles | The interferon gene signature is increased in patients with early treatment-naive rheumatoid arthritis and predicts a poorer response to initial therapy[END_REF][START_REF] Raterman | The interferon type I signature towards prediction of non-response to rituximab in rheumatoid arthritis patients[END_REF][START_REF] Sun | Elevated EPSTI1 promote B cell hyperactivation through NF-κB signalling in patients with m y jög n[END_REF][START_REF] Yao | Identifying Key Genes and nc n y En c d y n jög n' ynd m y W g d n C -Expression Network Analysis[END_REF]. In order to quantify the interferon signature in both aorta and circulating lymphocytes, we used a two-step procedure: first with the aforementioned GSVA R package [START_REF] Hänzelmann | GSVA: gene set variation analysis for microarray and RNA-seq data[END_REF], where enrichment scores for each patient and pathway involved in the dataset were computed; and then with the limma R package,

where direct differential analysis of the presented genes was performed to assess for significance.

Interferon-alpha serum levels

Interferon-alpha was measured in the serum of patients with active or in remission GCA, as well as HD. Inactivated (1% Triton X100; Sigma-Aldrich®, Saint-Quentin-Fallavier, France) plasma samples were tested for interferon-alpha using ultra-sensitive

Results

Interferon gene signature in GCA aortitis

First, we performed aortic tissue transcriptome analysis of GCA patients with LV (n = 10) and compared them to non-inflammatory aorta controls (n = 9). We identified 1042 genes that were significantly dysregulated, being 447 and 595 up-and down-regulated in GCA as compared to controls, respectively. Overall, upregulated pathways were related to interferons, JAK/STAT pathway, pro-inflammatory cytokines and chemokines (Figure 1A). Notably, types I, II and III n f n ' pathways appeared upregulated in aortic tissue from GCA patients.

Next, by using both igraph R package and STRING protein interaction database, we constructed a network of the most up-regulated genes in aortic tissue of GCA patients as compared to CTRL (Figure 1B). The most significant genes involved in this network were STAT3, interferon regulatory factor 7 (IRF7) and the protooncogene SRC.

Additional genes involved in type I interferon signaling (i.e., STAT1, IRF1) and in other inflammatory pathways c N κB (i.e., RELA) and tumor necrosis factor alpha (i.e., TRAF6) were also densely connected in the network and relevant in the interferon context.

We then aimed to precisely assess the relevance of type I interferon signaling and

d g n ' n m z d n c m n f d c ng n m f
CTRL and GCA patients. Using both literature [START_REF] Higgs | Patients with systemic lupus erythematosus, myositis, rheumatoid arthritis and scleroderma share activation of a common type I interferon pathway[END_REF][START_REF] Cooles | The interferon gene signature is increased in patients with early treatment-naive rheumatoid arthritis and predicts a poorer response to initial therapy[END_REF][START_REF] Raterman | The interferon type I signature towards prediction of non-response to rituximab in rheumatoid arthritis patients[END_REF][START_REF] Sun | Elevated EPSTI1 promote B cell hyperactivation through NF-κB signalling in patients with m y jög n[END_REF][START_REF] Yao | Identifying Key Genes and nc n y En c d y n jög n' ynd m y W g d n C -Expression Network Analysis[END_REF] and knowledge, we established and manually curated an interferon-specific gene signature and computed a signature score that allows a statistical estimation of enrichment using limma R package. The following type I-specific interferons response genes were selected:

IFI44L, ISG15, MX1, MX2 and EPSTI1. The customized type I interferon signature was significantly enriched in GCA aortas as compared to controls, with EPSTI1 and IFI44L genes being the most involved genes in this result (Figure 2).

Dysregulated molecular pathways in circulating CD4+ and CD8+ T cells in

GCA

The microarray gene analysis of purified CD4+ T cells of active GCA patients (n = 27)

and HD (n = 39) revealed 1479 significantly dysregulated genes between these groups.

Among them, 509 and 970 were up-and down-regulated in GCA versus HD, respectively. The network of the most up-regulated genes in CD4+ T cells highlighted STAT3 and ubiquitin C in GCA (Figure 3A). In addition to inflammatory pathwaysrelated genes seen in aortas (ie, RELA, TRAF6), IL6 was found to be a relevant contributor of the network.

As for CD8+ T cells, microarray gene analysis between active GCA patients (n = [START_REF] Sun | Elevated EPSTI1 promote B cell hyperactivation through NF-κB signalling in patients with m y jög n[END_REF] and HD (n = 34) demonstrated 2075 significantly dysregulated genes, being 921 upregulated and 1154 down-regulated in GCA versus HD. Consistent with that observed for CD4+ T cells, top up-regulated genes network in CD8+ T cells revealed STAT3 and ubiquitin C as central in GCA, and STAT5A was also relevant besides other activated inflammatory pathways-related genes (ie, RELA, TRAF6) (Figure 3B).

As seen in the aorta, among upregulated pathways in CD4+ and CD8+ T cells, those linked to interferons (type I, II and III), JAK/STAT, cytokines and chemokines were outlined (Supplementary Figure 1 and2), regardless of the presence of aortitis (data not shown). Genes composing the unique interferon signature, however, were not significantly dysregulated in circulating lymphocytes.

JAK/STAT pathway dysregulation in aorta and circulating T cells of GCA

As type I interferons signal through JAK/STAT pathway, we were interested in investigating the specific expression of this pathway throughout aortic tissue and circulating T cells. The comparative heatmap of these g n ' expressions between active GCA patients and CTRL (aorta) or HD (T cells) are represented in Figure 4. In the aorta (top panel), STAT1 and STAT2 transcripts were significantly upregulated as compared to controls. Overall, there was a tendency towards upregulation among all STATs in aortic tissue, except for STAT5A (no difference) and STAT5B

(downregulated). This tendency was inversely reproduced in circulating lymphocytes, as most of differentially expressed STAT transcripts tended to be downregulated in CD4+ (middle panel) and CD8+ (bottom panel) T cells of GCA patients. STAT3 was the sole exception for that, being significantly upregulated in both CD4+ and CD8+ T cells of GCA patients. Also, STAT5A appeared upregulated in CD8+ T cells from active patients as compared to HD.

Higher serum interferon levels in active GCA patients

Finally, we aimed to determine whether interferon signature could be accompanied by elevations in the serum level of type I interferon. Interferon-alpha concentration was shown to be significantly higher in patients with active GCA (aGCA, n = 41) as compared to that of patients in remission (rGCA, n = 15) (0.024 vs. 0.011pg/mL; p = 0.028). There was no statistical difference regarding HD (n = 27) (0.017pg/mL) (Figure 5). The increase seen among active patients occurred regardless of the presence of aortitis (data not shown).

Discussion

We performed for the first time a transcriptomic analysis of the aortic tissue of LV-GCA patients, and then compared the implicated pathways in both aorta and peripheral T cells in GCA. Notably, an enriched unique interferon signature was identified in patients with LV-GCA. Signaling through STAT3 appeared key not only in inflamed aortas but also in circulating CD4+ and CD8+ T cells of GCA patients. Interferon-alpha appeared systemically relevant as well, with higher serum levels in active patients, possibly representing a link between vascular and systemic inflammation.

The interferon signature is increasingly shaping the knowledge of immune-mediated diseases [START_REF] Barrat | Interferon target-gene expression and epigenomic signatures in health and disease[END_REF]. Despite the growing recognition of GCA clinical phenotypes, precise pathophysiological mechanisms driving LV-GCA remain widely unexplored due to the limited availability of tissue samples for study [START_REF] Robinette | The Immunopathology of Giant Cell Arteritis Across Disease Spectra[END_REF]. In inflamed temporal arteries, type II interferon (i.e., gamma) have been consistently identified as central in GCA vascular inflammation [START_REF] Deng | Th17 and Th1 Tcell responses in giant cell arteritis[END_REF][START_REF] Corbera-Bellalta | Blocking interferon γ reduces expression of chemokines CXCL9, CXCL10 and CXCL11 and decreases macrophage infiltration in ex vivo cultured arteries from patients with giant cell arteritis[END_REF][START_REF] Terrier | Interleukin-21 modulates Th1 and Th17 responses in giant cell arteritis[END_REF][START_REF] C N Y R | Interleukin 12 and interleukin 23 play key pathogenic roles in inflammatory and proliferative pathways in giant cell arteritis[END_REF], but very little has been reported on type I interferons. Both can induce interferon-stimulated genes by using the common JAK/STAT signaling pathway. Upon activation, interferon-alpha receptor signals via JAK1 and TYK2, and STAT1/2/3, while interferon-gamma receptor does it through JAK1/2, and mostly STAT1 [START_REF] Barrat | Interferon target-gene expression and epigenomic signatures in health and disease[END_REF][START_REF] Nash | Points to consider for the treatment of immune-mediated inflammatory diseases with Janus kinase inhibitors: a consensus statement[END_REF]. Herein, we demonstrate an upregulated type I interferon signature in inflamed aortas of GCA patients, ultimately leading to increased expression of interferon-related genes. Although the specific interferon signature seen in the aorta was not observed in circulating lymphocytes, interferon-related pathways were also upregulated in circulating lymphocytes, albeit to a lesser degree. This fact underscores an existing dissociation between vascular and systemic inflammation in GCA, and it is possible that the source of tissue-level type I interferon is other than these lymphocytes.

We had previously shown a similar signature in the aortitis of Takayasu's arteritis (TAK), the other prototype of large-vessel vasculitis [START_REF] Régnier | T g ng JAK/ TAT y n T k[END_REF]. The implication of type I interferon in GCA vascular inflammation seems relevant as blocking interferon-gamma solely was insufficient to abrogate inflammation in cultured ex vivo temporal arteries from GCA patients [START_REF] Corbera-Bellalta | Blocking interferon γ reduces expression of chemokines CXCL9, CXCL10 and CXCL11 and decreases macrophage infiltration in ex vivo cultured arteries from patients with giant cell arteritis[END_REF]. The potential role of type I interferon in this setting had first been demonstrated in an animal model, where Zhang and colleagues engrafted inflamed temporal arteries into immunodeficient mice and reconstituted them with immune cells from GCA patients, showing increased transcripts of both type I and II interferon in vasculitic arteries [START_REF] Zhang | Inhibition of JAK-STAT Signaling Suppresses Pathogenic Immune Responses in Medium and Large Vessel Vasculitis[END_REF]. Our study supports the idea that a combined action of interferons might also extends to human aortic tissue in GCA, shedding new light on the mechanisms and potential therapeutic targets in LV-GCA.

Understanding vessel wall inflammatory pathways is critical in avoiding permanent vascular damage. Corticosteroids have been for long the cornerstone of GCA therapy, but by not affecting the interferon-gamma-producing Th1 cells implicated in vascular inflammation [START_REF] Deng | Th17 and Th1 Tcell responses in giant cell arteritis[END_REF], temporal artery T cell infiltration persists over time [START_REF] Maleszewski | Clinical and pathological evolution of giant cell arteritis: a prospective study of follow-up temporal artery biopsies in 40 treated patients[END_REF]. Interferongamma isolated blockade in ex vivo temporal arteries decreased STAT1 activation and expression, but not that of STAT3, allowing to hypothesize that IL-6 could also be involved in this process [START_REF] Corbera-Bellalta | Blocking interferon γ reduces expression of chemokines CXCL9, CXCL10 and CXCL11 and decreases macrophage infiltration in ex vivo cultured arteries from patients with giant cell arteritis[END_REF]. Indeed, IL-6 overexpression and signaling via STAT3 has been implicated in the progression of abdominal aortic aneurysms, and blocking STAT3 is able to slow this process [START_REF] Kokje | IL-6: A Janus-like factor in abdominal aortic aneurysm disease[END_REF][START_REF] Yoshida | STAT3 Activation Correlates with Adventitial Neutrophil Infiltration in Human Aortic Dissection[END_REF][START_REF] Xiao | Experimental abdominal aortic aneurysm growth is inhibited by blocking the JAK2/STAT3 pathway[END_REF]. However, the significance of IL-6-STAT3 axis is less clear in large-vessel vasculitis. Zhang et al. had already put this into question by demonstrating distinctly low levels of STAT3-dependent genes in temporal arteries from their chimeric mice models [START_REF] Zhang | Inhibition of JAK-STAT Signaling Suppresses Pathogenic Immune Responses in Medium and Large Vessel Vasculitis[END_REF]. From a clinical perspective, a phase III trial in TAK evaluating the IL-6 blockade with tocilizumab failed to meet its primary endpoints [START_REF] Nakaoka | Efficacy and safety of tocilizumab in patients with refractory Takayasu arteritis: results from a randomised, double-blind, placebo-controlled, phase 3 trial in Japan (the TAKT study)[END_REF], and a non-negligible portion of GCA patients have persistent aortic inflammation under tocilizumab [START_REF] Reichenbach | Magnetic resonance angiography in giant cell arteritis: results of a randomized controlled trial of tocilizumab in giant cell arteritis[END_REF][START_REF] Quinn | Use of 18Ffluorodeoxyglucose positron emission tomography to monitor tocilizumab effect on vascular inflammation in giant cell arteritis[END_REF]. While IL-6 appeared relevant in our CD4+ T cells gene network, but not in the aorta, STAT3 was consistently reproduced in the interaction networks of both aorta and circulating T cells, suggesting that other mediators (e.g., interferon-alpha) could be the ones that signal through STAT3. Furthermore, us and others [START_REF] Zhang | Inhibition of JAK-STAT Signaling Suppresses Pathogenic Immune Responses in Medium and Large Vessel Vasculitis[END_REF] have demonstrated STAT1 and STAT2 as significantly upregulated in GCA vasculitic arteries, pointing to a need for multiple pathway blockade to control largevessel inflammation. The recognition of the JAK/STAT pathway involvement in LVV has highlighted this therapeutical target as of interest [START_REF] Bursi | Contribution of Janus-Kinase/Signal Transduction Activator of Transcription Pathway 21 in the Pathogenesis of Vasculitis: A Possible Treatment Target in the Upcoming Future[END_REF], with in vitro JAK inhibition in both GCA and TAK showing promising results [START_REF] Zhang | Inhibition of JAK-STAT Signaling Suppresses Pathogenic Immune Responses in Medium and Large Vessel Vasculitis[END_REF][START_REF] Régnier | T g ng JAK/ TAT y n T k[END_REF]. The ongoing trials with JAK1/2 inhibitor baricitinib (NCT03026504) and preferential JAK1 inhibitor upadacitinib (NCT03725202) are expected to shed new light on the impact of JAK inhibitors in GCA.

Another major challenge in GCA management is the reliability on currently available biomarkers. Although erythrocyte sedimentation rate and C-reactive protein are widely used, their sensitivity and specificity in detecting disease relapse are quite limited [START_REF] Camellino | Monitoring and long-term management of giant cell arteritis and polymyalgia rheumatica[END_REF].

By using an ultra-sensitive technique of cytokine detection, we have shown interferonalpha serum levels to be significantly higher in patients with active GCA as compared to those in remission. Although there was no statistical difference between GCA patients and healthy donors, the mean levels of serum interferon alpha observed in patients with active GCA tended to be higher than HD. The correlation of interferonalpha with disease activity could be useful for LV-GCA patients, for which imaging modalities are not yet fully validated as disease activity assessment tools [36]. Also, considering the tissue-level interferon overexpression reported by us and others [START_REF] Zhang | Inhibition of JAK-STAT Signaling Suppresses Pathogenic Immune Responses in Medium and Large Vessel Vasculitis[END_REF], we could hypothesize that interferon-alpha represents a common inflammatory link between vascular and systemic inflammation in GCA.

We acknowledge some limitations. The number of available aortic samples may have not given us enough statistical power to clarify some tendencies. However, it was sufficient to show that STAT1 and STAT2 were significantly upregulated in GCA aortitis, and to highlight STAT3 as central in our gene interaction networks. Our study was also not powered enough to uncover differences between GCA clinical phenotypes for T cells. Finally, as we used whole aortic tissue for transcriptomic analyses, we could not identify the cellular source of type I interferon, and this matter remains to be elucidated.

In conclusion, the unique type I interferon signature in GCA patients with aortitis provides new pathophysiological insights into disease mechanisms. The upregulated signaling pathways pave the way for new therapeutical targets.

  T cells were isolated from PBMCs of active GCA patients and HD by negative isolation using Dynabeads® n c d™ m Patients included in transcriptomic analysis did not receive steroid doses higher than 10mg/day nor immunosuppressants by the time of their blood collection. CD4+ cells were isolated by positive selection using Dynabeads® C 4 n k cc d ng m n f c ' n c n C 8+ c were isolated from the previous CD4-fraction by positive selection using Dynabeads® CD8 isolation kit. Then, total RNA from CD4+ or CD8+ cells was extracted using the NucleoSpin® RNA kit (Macherey-Nagel®), cc d ng m n f c ' instructions. Total RNA was quantified by a NanoDrop ND-1000 spectrophotometer.

  kits (Simoa® IFN-α Adv n g K , Q n x®, N nd ) with HD-X Simoa Planar Technology, according to m n f c ' n c n 2.7. Statistics Continuous variables are presented as means ± standard error of mean (SEM) or as medians and interquartile range [IQR]. Categorical variables are presented as counts and proportions (%). Statistical comparisons were performed by using unpaired moderated d n ' -tests for quantitative data. All statistical tests were two-tailed with a significance threshold of 0.05. Statistical significance was evaluated using R statistical software version 4.1 and GraphPad Prism version 8.4.2 for Windows (GraphPad Software, San Diego, CA, USA).

.

  Figure 1. Upregulated pathways and gene network in GCA aortitis.

Figure 2 .

 2 Figure 2. Type I interferon specific gene signature in GCA aortitis.

Figure 3 .

 3 Figure 3. Gene interaction networks of circulating CD4+ and CD8+ T cells in

Figure 4 .

 4 Figure 4. JAK/STAT signaling pathway in aorta and circulating CD4+ and CD8+

Figure 5 .

 5 Figure 5. Interferon-alpha serum levels in GCA patients.

Table 1 . Main characteristics of GCA patients

 1

	Parameters	GCA patients (n = 43)
	Demographic
	Age, median [IQR] years	72 [65.75-78]
	Female gender, n (%)	30 (69.7)
	Origin, n (%)
	-Europe	37 (86.1)
	-North Africa	5 (11.6)
	-America	1 (2.3)
	Newly diagnosed, n (%)	33 (76.7)
	Relapsed, n (%)	10 (23.3)
	-Disease duration, median [IQR]	135.5 [53.25-229.3]
	months
	Clinical and laboratory features
	General signs, n (%)	33 (76.7)
	Headache, n (%)	28 (65.1)
	Polymyalgia rheumatica, n (%)	23 (53.5)
	Aortitis, n (%)	21 (48.8)
	-Aneurysms	9/21 (42.9)
	-Arterial wall thickening alone	7/21 (33.3)
	-Stenosis	3/21 (14.3)
	-Dissection	3/21 (14.3)
	Jaw claudication, n (%)	17 (39.5)
	Visual symptoms, n (%)	6 (13.9)
	Stroke, n (%)	3 (6.9)

Acknowledgments:

We thank all patients and all physicians involved in the care of the patients. We thank QUANTERIX for providing the kits used in the ultra-sensitive detection of interferon alpha.

Funding: None

Ethical approval information: CPP-Île-De-France VI

Author's contributions: MV, PR and DS conceptualized the study. MV, PR, AM-D and ALJ were involved in data collection and analysis. PR, MV and AM-D did the statistical analysis. MV, PR, AM-D and DS verified the underlying data. MV and DS wrote the initial draft of the manuscript. GD-J, MR, DK and PC critically contributed to the article. All authors revised and approved the submitted version. DS had the final responsibility for the decision to submit for publication.

Conflict of interest:

The authors have declared that no conflict of interest exists.


Pierre Andreoletti
email: pierre.andreoletti@univ-orleans.fr

Pierre Debs
email: pierre.debs@univ-orleans.fr

Simple random walk on Z 2 perturbed on the axis (renewal case)

Keywords: 05C81, 60J55, 60K10, 60J10, 60K40 perturbed random walks, renewal theorem, ergodic theorem

) are submitted to a local field. In our case we assume that the particle evolves freely in the cones but when touching the axis a force pushes it back progressively to the origin. The main result proves that this force can be parametrized in such a way that a renewal structure appears in the trajectory of the random walk. This implies the existence of an ergodic result for the parts of the trajectory restricted to the axis.

Introduction

We consider a random walk X = (X n , n ∈ N) on Z 2 starting at (1, 1), which probabilities of transition differ whether the walk is on the axis, denoted by K c := {(x, y) ∈ Z 2 , xy = 0}, or on K which is made up of four cones. More precisely, X is a simple random walk on K, that is p(x, x ± e i ) = 1 /4, for all x ∈ K, where e i is a vector of the canonical basis, whereas on K c \{(0, 0)} the walk is pushed towards the origin: there exists α ≥ 0 such that for all i > 0 p((i, 0), (i + 1, 0)) = p((i, 0), (i, ±1)) = p((0, i), (0, i + 1)) = p((0, i), (±1, i)) = 1 4i α , p((i, 0), (i -1, 0)) = p((0, i), (0, i -1)) = 1 -3 4i α , and symmetrically when i < 0 (see figure 1). Note that the origin is a "special" point as p((0, 0), ±e i ) = 1 /4. The fluctuations of this random walk can be seen as the movement of a particle which is diffusive on the cones but which is submitted to a field on the axis, this field, which source comes from infinity, is decreasing when the particle gets closer to the origin. Depending on the strength of the local field, represented by α, the behavior of this random walk is more or less perturbed comparing to the simple random walk. In this first work we deal with the case for which the applied force toward 0 is strong, that is α > 3. To give an idea on the changes that provokes such a perturbation, we first present two examples of our main result that are related to the local time of the walk: for all subset A of Z 2 , L (A, n) = n i=1 1 Xi∈A is the local time of X in A until time n. Our first result states as follows Theorem 1.1. Assume α > 3, there exists two positive constants c and c such that: Clearly the behavior of the local time of X is very different than the one of the symmetric random walk S. For example, L S ((0, 0), n)/ log n converges in law to an exponential variable with parameter π (see for instance [R 89], [START_REF] Erdős | Some problems concerning the structure of random walk paths[END_REF]). It turns out that these two results are just simple consequences of a more general result presented below.

log n n L ((0, 0), n) P → c , log n n L (K c , n) P → c.
It is essentially an ergodic-like theorem for the part of the trajectory on the axis. Recall that we assume X 0 = (1, 1) and let us introduce the following stopping times: for any i ∈ N * η i := inf{k > ρ i-1 , X k ∈ K c },

ρ i := inf{k > η i , X k ∈ K}, ρ 0 = 0,
that is respectively the i-th exit and entrance times in K. Also for any n let

N n := max{i ≤ n, ρ i ≤ n}, (1)
which is the index of the last excursion to K before the instant n. Moreover, for any x = (x 1 , x 2 ) ∈ Z 2 , let us denote by x := max(|x 1 |, |x 2 |), the maximum norm of x.

Our main result is the following:

Theorem 1.2. Let B i := (X k , η i ≤ k < ρ i ) and B * n := (X k , η Nn ≤ k ≤ n),
the portions of the trajectory of X restricted to the axis. Let f a positive non-decreasing functional (that is for any

i ≤ k, f (x 1 , • • • , x i ) ≤ f (x 1 , • • • , x k ))
such that there exist 0 < δ < 2 and two positive constants C 1 and C 2 such that for any x ∈ K c ,

E x [f (B 0 )] ≤ C 1 x, (2)
Var x [f (B 0 )] ≤ C 2 x 2-δ , (3)
where B 0 := (X k , 0 ≤ k < ρ) and ρ := inf{k > 0, X k ∈ K}. Assume α > 3 then, in probability

lim n→+∞ log n n Nn i=1 f (B i ) + f (B * n ) = lim n→+∞ log n n Nn i=1 f (B i ) = c f , (4)
where c f is a positive constant that is described below.

Let us give some underlying ideas concerning our main result: the trajectory of X can be split in excursions composed of parts on the axis and parts on the cones. For the parts on the axis, we can prove that at each excursion (before the instant n) the walk escapes from K c in a compact neighborhood of (0, 0) with a probability close to one. This escape coordinate is actually driven by a first invariant probability measure.

The second part on K yields that the number of excursions (composed of the part on the cone and the part on the axis) before the instant n is of order n/ log n. This fact comes essentially from the tail of the first time the walk exits a cone when starting from a coordinate of the neighborhood of (0, 0). It turns out that exit coordinates of the cone are also driven by a second invariant probability measure. This finally makes appears a renewal structure for the trajectory of the walk.

Let us now discuss about the constant which appears in Theorem 1.2

c f := π 8 y E y [f (B 0 )]π * (y) x xπ † (x) = π 8 E π * [f (B 0 )] E π † [X 0 ] , (5)
where π * and π † are respectively the unique invariant probability measures of the Markov chains (X ηi , i) and (X ρi , i). Note that by definition

π * (x) = 0 ⇐⇒ x ∈ K c whereas π † (x) = 0 ⇐⇒ x ∈ ∂K := {x ∈ K/ ∃y ∈ K c , x -y = 1}
. We can actually give some details about these two measures, in particular their tail: lim x→+∞ x 3 π * (x) = c 0 > 0 and lim x→+∞ x α+2 π † (x) = c 2 , both constants c 0 and c 2 are partially explicit (see respectively Proposition 2.1 and Lemma 3.2). With this theorem and the fact that families of function f such that for any k < m, f

(x k , • • • , x m ) = m i=k 1 xi∈K c (to obtain the local time on the axis) or f (x k , • • • , x m ) = m i=k 1 xi=(0,0) (
to obtain the local time at the origin) satisfy (2) and (3), we obtain the first theorem. This also leads to the constants c and c from above expression of c f . Note that the increasing hypothesis on f is only used to prove that log n n f (B * n ) converges in probability to zero and is unnecessary to prove the convergence of log n n Nn i=1 f (B i ) to c f . We can think of several extensions and possible generalizations of this result. The first question is what happen when return-force is lower, that is α ≤ 3. Well in this case the renewal structure does not exist any more, and things worsen when α < 2 as there is no longer concentration of (X ρi , i) in the neighborhood of (0, 0) so a totally new approach is needed. Possible generalization concerns first the shape of the return force, we could easily replace function i → i -α by a sequence (a i , i) such that i i 2 a i < +∞, note that this modification should not change the results. However we chose to keep i -α in order to obtain proof easily readable. For the random walk on K, we could also consider more general random walks, this would need improvement of local limit theorems like the ones proved for simple random in Section 5 (Lemmata 5.1 and 5.4). More specifically what is needed, for example, is uniform convergence for large x of (P y (X η = x), y) (where η := inf{k > 0, X k ∈ K}, see Lemma 5.1). Note that P y (X η = x) is studied for several random walks in [START_REF] Raschel | Random walks in the quarter plane absorbed at the boundary : Exact and asymptotic[END_REF], including simple random walk, but some extra work is needed to obtain uniform convergences.

The paper is organized as follows, in the following section we prove an ergodic result for the first mexcursions, in Section 3 we prove that the number of excursions before the instant n is of order n/ log n and finish with the proof of Theorem 1.2. Finally in Section 4 (resp. Section 5) we resume stochastic estimations for the walk when it remains on the axis (resp. on the cones). For the seek of completeness, we had an appendix in Section 6. Note that although our main result needs the assumption α > 3, we allege this hypothesis whenever it is possible as this will be useful for future works (see more specifically Section 4 which only deals with the trajectory of the walk on the axis).

Ergodic result on the axis during the first m two-types excursions.

We call i th two-types excursion the trajectory of X on the time interval ρ i-1 < k ≤ ρ i , the first part concerning the cones and the second one the axis. In this section we prove that for any positive nondecreasing functional f satisfying (2) and (3), its empirical mean along the trajectory of X on K c during the first m excursions converges. Recall that for i ≥ 1,

B i = (X k , η i ≤ k < ρ i ) and B 0 = (X k , 0 ≤ k < ρ).
Proposition 2.1. Assume α > 3, there exists a probability measure π * such that

1 m m i=1 f (B i ) P → x∈K c π * (x)E x [f (B 0 )] = E π * [f (B 0 )], moreover lim x→+∞ x 3 π * (x) = c 0 > 0, with c 0 := 16 π y π * (y)E y [X ρ ].
To obtain this proposition, we prove two Lemmata, the first one tells that

m i=1 f (B i ) can be approxi- mated by m i=1 E Xη i [f (B 0 )],
the second studies the convergence of this last sum. First Lemma states as follows Lemma 2.2. Assume α ≥ 3, there exists 0 < C < +∞, such that

E   1 m 2 m i=1 f (B i ) -E Xη i [f (B 0 )] 2   ≤ C m . (6)
Proof. Strong Markov property leads to

E   m i=1 f (B i ) -E Xη i [f (B 0 )] 2   = m i=1 E Var Xη i [f (B 0 )] .
By hypothesis (3), above quantity is smaller than

C 2 m i=1 E[X 2-δ
ηi ] with 0 < δ < 2 and we conclude using Lemma 5.5 telling that there exists

C > 0 such that E X 2-δ ηi < C for all i ∈ N * .
Second Lemma, which is essentially an ergodic result, writes Lemma 2.3. Assume α > 3, the Markov chain (X ηi , i) is positive recurrent and its invariant probability measure π * satisfies

1 m m i=1 E Xη i [f (B 0 )] P.a.s. → x∈K c π * (x)E x [f (B 0 )] = E π * [f (B 0 )]. (7)
Proof. (X ηi , i) being obviously irreducible, we just have to prove that (0, 1) is positive recurrent. Introduce, for any x ∈ K c , τ x = inf{k ≥ 0, X η k = x}, then for all k > 0:

P (0,1) τ (0,1) > k = P (∀i ≤ k, X ηi = (0, 1)) = y∈,K c \{(0,1)} P (0,1) ∀i ≤ k -2, X ηi = (0, 1), X η k-1 = y (1 -P y (X η1 = (0, 1))) .
According to (23), there exists 0 < C < 1 such that for all y ∈ K c , P y (X ρ = (1, 1)) > C, implying:

P y (X η1 = (0, 1)) ≥ P y (X ρ = (1, 1)) P (1,1) (X η = (0, 1)) ≥ CP (1,1) (X 1 = (0, 1)) = C 4 .
Consequently, with an obvious induction reasoning:

P (0,1) τ (0,1) > k ≤ y∈K c \{(0,1)} P (0,1) ∀i ≤ k -2, X ηi = (0, 1), X η k-1 = y 1 - C 4 = P (0,1) τ (0,1) > k -1 1 - C 4 ≤ 1 - C 4 k .
Thus E (0,1) τ (0,1) = k≥0 P (0,1) τ (0,1) > k < ∞ and (X ηi , i) is positive recurrent. ( 7) is an application of Birkhoff's ergodic Theorem so we only have to check that E π * [f (B 0 )] exists: first note that by condition (2) for any x, E x [f (B 0 )] ≤ C + x, so we only have to check that x xπ * (x) < +∞.

For that, we have to study the asymptotic in x of π * (x). Let y in K c and δ > 0 small enough such that (1 -δ)α > 3, for any x

P y (X η1 = x) = z≤x 1-δ P y (X ρ = z)P z (X η = x) + z>x 1-δ P y (X ρ = z)P z (X η = x). (8)
By (30), there exists a positive constant c + such that for all z ∈ K, P y (X ρ = z) ≤ (1 + c + )z -α , so using Lemma 5.6, the second sum above is bounded by

(1 + c + ) z>x 1-δ z -α P z (X η = x) ≤ (1 + c + )x -(1-δ)α z>x 1-δ P z (X η = x) ≤ 2(1 + c + )x -(1-δ)α = o(x -3 ). ( 9
)
Local limit result (Lemme 5.1) implies that (x 3 /z)P z (X η = x) ∼ 16/π for any large x uniformly in z with z ≤ x 1-δ , so for the first sum in (8) we get for large x

z≤x 1-δ P y (X ρ = z)P z (X η = x) ∼ 16 πx 3 z≤x 1-δ zP y (X ρ = z) = 16 πx 3 E y [X ρ 1 Xρ≤x 1-δ ]. (10)
Then (8), ( 9) and (10) implies that for any y,

lim x→+∞ x 3 P y (X η1 = x) = 16 π E y [X ρ ] ≤ M
where M does not depend on y by Corollary 4.5. As π * (x) = y∈K c π * (y)P y (X η1 = x):

lim x→+∞ x 3 π * (x) = y π * (y) lim x→+∞ x 3 P y (X η1 = x) = 16 π y π * (y)E y [X ρ ] ≤ 16M π . So x xπ * (x) < +∞ is satisfied. Lemmata 2.2 and 2.3 ensure that 1 m m i=1 f (B i ) -E Xη i [f (B 0 )] tends to 0 and 1 m E Xη i [f (B 0 )] to E π * [f (B 0 )]
in probability which gives Proposition 2.1. It yields following Corollary giving the behaviors (after m excursions) of the local time on the axis and at (0, 0).

Corollary 2.4. Assume α > 3, 1 m L (K c , ρ m ) P → x π * (x)E x [ρ] and 1 m L ((0, 0), ρ m ) P → x π * (x)E x [L (0, ρ)]. Proof. As L (K c , ρ m ) = m i=1 (ρ i -η i ), if we take, for any k < m, f (x k , • • • , x m ) = m -k, then L (K c , ρ m ) = m i=1 f (B i ).
So to prove the result for L (K c , ρ m ) we only have to check that conditions (2) and (3) are full-filled for this f . By Lemma 4.8, for any γ ∈ {1, 2} there exists 0 < ε < 1 such that for any x ∈ K c :

E x [f γ (B 0 )] -E x [f (B 0 )] γ = E x [ρ γ ] -E x [ρ] γ ≤ x γ-ε , so both (2) and (3) are satisfied. L ((0, 0), ρ m ) = m i=1 (L ((0, 0), ρ i ) -L ((0, 0), η i )) is treated similarly.

The number of two-types excursions before the instant n

In this section we prove following Proposition, recall that N n is the number of the last entrance in K before n (see (1)).

Proposition 3.1. Assume α > 3, there exists a probability measure π † such that

log m m N m P → π 8 1 x xπ † (x) =: c 1 . (11)
The main idea comes from decomposition of

ρ k = k i=1 ρ i -η i + k i=1 η i -ρ i-1 , then by Corollary 2.4, k i=1 ρ i -η i is of order k whereas we show below that in probablity k i=1 η i -ρ i-1 is of order k log k.
This last fact comes from the tail of η (the single return instant to the axis) together with the fact that for any i ≤ k, with an overwhelming probability X ρi is in the neighborhood of (0, 0) which size is independent of i and of the coordinates of entry of the walk on the axis. We start with following Lemma

Lemma 3.2. Assume α > 3, (Xρ i , i) is positive recurrent. Moreover its invariant probability measure π † satisfies lim x→+∞ x α+2 π † (x) = c 2 with c 2 = 2 π u uπ † (u).
Proof. The proof of the fact that (X ρi , i) is positive recurrent is very similar than for (X ηi , i). Using the irreducibility of this chain, it suffices to prove that (1, 1) is positive recurrent. Denote by τA = inf{k > 0, X ρ k ∈ A}, then, for all k > 0:

P (1,1) τ(1,1) > k = y∈K\{(1,1)} P (1,1) ∀i ≤ k -2, X ρi = (1, 1), X ρ k-1 = y (1 -P y (X ρ1 = (1, 1))) .
Using (23), there exists 0 < C < 1 such that for all z ∈ K c , P z (X ρ = (1, 1)) > C, implying that P y (X ρ1 = (1, 1)) ≥ C z∈K c P y (X η = z) = C. So, with an induction reasoning:

E (1,1) [τ (1,1) ] = k≥0 P (1,1) τ(1,1) > k ≤ k≥0 (1 -C) k < ∞.
Let us study the tail of invariant probability measure π † . First, note that as (X ρi , i) is obviously aperiodic, using Lemma 4.6, for all x ∈ K:

π † (x) = lim n→+∞ P(X ρn = x) ≤ c x α (12)
implying that π † has a first moment and c 1 = 0. Let δ > 0 small enough such that (α -1)(1 -δ) > 2, assume x = (1, x 2 ) with x 2 > 0 (other cases can be treated similarly), let

L x := {z = (0, z 2 ) with z 2 ≥ x 2 } and L c x the relative complement of L x in K c . As π † (x) = y∈K π † (y)P y (X ρ1 = x): π † (x) = y π † (y) z∈Lx P y (X η = z)P z (X ρ = x) + y π † (y) z∈L c x P y (X η = z)P z (X ρ = x) =: S 1 + S 2 . ( 13
)
When z ∈ L c x , by the second point of Corollary 4.4, there exists C > 0 such that P z (X ρ = x) ≤ C /x 2α , so as α > 3

S 2 ≤ C x 2α y∈K π † (y) z∈L c x P y (X η = z) ≤ C x 2α = o(x -2-α ).
We now deal with the first sum in (13) which we decompose as follows

S 1 = y,y≤x 1-δ π † (y) z∈Lx P y (X η = z)P z (X ρ = x) + y,y>x 1-δ π † (y) z∈Lx P y (X η = z)P z (X ρ = x) =: Σ 1 + Σ 2 .
For Σ 2 , by ( 30) and ( 12), as α + (α -1)(1 -δ) > α + 2:

Σ 2 ≤ C + x α y,y>x 1-δ 1 y α ≤ C + x α+(α-1)(1-δ) = o(x -2-α ),
where C + is a positive constant that may grow from line to line. In view of what we want to prove, S 2 and Σ 2 are negligible. For Σ 1 , we use, as for Σ 2 , (30), the first point of Corollary 4.4 and also Lemma 5.1 telling that uniformly in y, with y ≤ x 1-δ and z ∈ L x , P y (X η = z) ∼ 16y/z 3 π, from this we deduce that

Σ 1 ∼ y,y≤x 1-δ π † (y) z∈Lx 4y πz 3 x α ∼ 2 πx α+2 y,y≤x 1-δ yπ † (y).
Finally as π † has a first moment, lim x→+∞ x α+2 Σ 1 = 2 π u uπ † (u), this finishes the proof. Second Lemma below is a law of large number for the time spent by the walk on the cone during the first m excursions.

Lemma 3.3. Assume α > 3, then in probability lim m→+∞ 1 m log m m i=1 (η i -ρ i-1 ) = 8 π x∈K xπ † (x).
Proof. For any 0 < ε < 1, let us decompose

m i=1 (η i -ρ i-1 ) as follows m i=1 (η i -ρ i-1 ) = m i=1 (η i -ρ i-1 )1 ηi-ρi-1≤εm + m i=1 (η i -ρ i-1 )1 ηi-ρi-1>εm =: Σ 1 + Σ 2 . • Let us prove that Σ 2 = o(m log m): let A := { m i=1 1 ηi-ρi-1>εm ≤ (log m) 1/4 }, Markov inequality gives P (A c ) ≤ 1 (log m) 1/4 m i=1 P(η i -ρ i-1 > εm).
For 0 < δ < 1 /2, one can write:

P(η i -ρ i-1 > εm) = x∈K P(X ρi-1 = x)P x (η > εm) = x,x≤m 1/2-δ P(X ρi-1 = x)P x (η > εm) + x,x>m 1/2-δ P(X ρi-1 = x)P x (η > εm).
According to Lemma 5.4, uniformly in x such that x ≤ m 1 /2-δ for m large enough P x (η > εm) ∼ 8x/πεm. Then for m large enough and as α > 3 using Lemma 4.6:

x,x≤m 1/2-δ P(X ρi-1 = x)P x (η > εm) ≤ 8 εm E[X ρi-1 ] ≤ C + εm .
As α > 3, we can chose δ such that (α -1)(1/2 -δ) > 1 and using again Lemma 4.6:

x,x>m 1/2-δ P(X ρi-1 = x)P x (η > εm) ≤ x,x>m 1/2-δ c x α ≤ c m (α-1)(1/2-δ) = o 1 m . Then P(η i -ρ i-1 > εm) ≤ C + /εm, so finally P(A c ) ≤ C + /(log m) 1/4 ε. A similar computation also prove that P(C c ) ≤ C + /(log m) 1/2 with C := { m i=1 1 ηi-ρi-1>m(log m) 1/2 = 0}. Now, notice that on A ∩ C , Σ 2 = m i=1 (η i -ρ i-1 )1 ηi-ρi-1>εm ≤ m(log m) 1/2 m i=1 1 ηi-ρi-1>εm ≤ m(log m) 3/4 ,
which implies P Σ 2 > m(log m) 3/4 ≤ C+ /((log m) 1/4 ε), and thus, in probability Σ 2 = o(m log m).

• For Σ 1 , assume for the moment that

lim m→+∞ P Σ 1 - m i=1 E Xρ i-1 [η1 η≤εm ] > m(log m) 1/2 = 0. ( 14
)
Let us compute:

E Xρ i-1 [η1 η≤εm ] = E Xρ i-1 η1 η≤(log m) 1/2 + E Xρ i-1 η1 (log m) 1/2 <η≤εm , (15)
the first sum is smaller than (log m) 1/2 . For the second one, we decompose :

E Xρ i-1 η1 (log m) 1/2 <η≤εm = (log m) 1/2 <k≤εm P Xρ i-1 (k < η ≤ εm)(1 Xρ i-1 ≤k 1/2-δ + 1 Xρ i-1 >k 1/2-δ ) =: Σ 3 + Σ 4 . (16)
In order to simplify the writing in the sequel, we introduce the following inequality: for (a, b) such that a < α -1 and ( 1 /2 -δ)(α -a -1) -b > 1, using Lemma 4.6:

E[F (a, b)] := E (log m) 1/2 <k≤εm k b X a ρi-1 1 Xρ i-1 >k 1/2-δ ≤ C + (log m) 1/2 <k≤εm k b r>k 1/2-δ r a-α ≤ C + (log m) 1/2 <k≤εm k b+(a-α+1)( 1 /2-δ) ≤ C + (log m) 1 2 (b+1+(a-α+1)( 1 /2-δ)) .
For Σ 3 , we use Lemma 5.4: for any z ≤ k 1/2-δ , P z (η = k) ∼ 8z/πk 2 , so for large m,

Σ 3 = (1 + o(1)) 8 π X ρi-1 (log m) 1/2 <k≤εm 1 k 1 Xρ i-1 ≤k 1/2-δ = 8 π X ρi-1 (1 + o(1))   (log m) 1/2 <k≤εm 1 k - (log m) 1/2 <k≤εm 1 k 1 Xρ i-1 >k 1/2-δ   = 8 π (log m)X ρi-1 (1 + o(1)) - 8 π X ρi-1 (1 + o(1)) (log m) 1/2 <k≤εm 1 k 1 Xρ i-1 >k 1/2-δ . ( 17
)
Finally uniformly in i ≤ m, as Σ 4 ≤ F (0, 0):

log mX ρi-1 (1 + o(1)) -2F (1, -1) ≤ π 8 E Xρ i-1 η1 (log m) 1/2 <η≤εm ≤ log mX ρi-1 (1 + o(1)) + π 8 F (0, 0).
Thanks to our choice of δ, E[F (0, 0)] and E[F (1, -1)] tend to zero, implying (using a Markov inequality) that

lim m→+∞ P m i=1 (log m) 1/2 <k≤εm (X ρi-1 k -1 1 Xρ i-1 >k 1/2-δ + 1 Xρ i-1 >k 1/2-δ ) > m = 0. ( 18
)
One can notice that i = 1 is a special case as X ρ0 = (1, 1) a.s. and that we should have made its own reasoning. However, as the estimates remain true with even simpler calculations, we have decided to not put it. Collecting (15), ( 16) , ( 17) and ( 18) we obtain that, in probability, when m tends to infinity

1 m log m m i=1 E Xρ i-1 [η1 η≤εm ] ∼ 8 πm log m m i=1 (log m)X ρi-1 = 8 πm m i=1 X ρi-1 .
Then using that (X ρi , i) is positive recurrent, the fact that its invariant probability measure (π † (x), x) admits a first moment (see Lemma 3.2) and the Birkhoff ergodic Theorem, we obtain for large m, that in probability

1 m log m m i=1 E Xρ i-1 [η1 η≤εm ] ∼ 8 πm m i=1 X ρi-1 P.a.s. → 8 π x xπ † (x).
We deduce from that, in probability

lim m→+∞ 1 m log m m i=1 E Xρ i-1 [η1 η≤εm ] = 8 π x xπ † (x).
We are left to prove (14), like in Lemma 2.2 strong Markov property yields

E Σ 1 - m i=1 E Xρ i-1 [η1 η≤εm ] 2 ≤ m i=1 E E Xρ i-1 η 2 1 η≤εm . (19)
Again using Lemma 5.4

E Xρ i-1 η 2 1 η≤εm ) ≤ C + k≤εm kP Xρ i-1 (η > k) ≤ C + (log m) 2 + log m≤k≤εm kP Xρ i-1 (η > k)(1 Xρ i-1 ≤k 1/2-δ + 1 Xρ i-1 >k 1/2-δ ) ≤ C + (log m) 2 + C + log m≤k≤εm X ρi-1 1 Xρ i-1 ≤k 1/2-δ + εmF (0, 0).
as by Lemma 4.6 E[X ρi-1 ] ≤ c for any i, in particular, the mean of the sum in the above inequality is bounded by a constant times εm. Thanks to our previous computations, we have that εmE

[F (0, 0)] = o(m). Therefore E E Xρ i-1 η 2 1 η≤εm ≤ C + mε, so the second moment (19) is smaller that C + εm 2 .
Finally using (second moment) Markov inequality in the probability in (14), yields (14).

The proof of Proposition 3.1 then writes: by Corollary 2.4 and Lemma 3.3, in probability

lim k→+∞ ρ k k log k = lim k→+∞ k i=1 η i -ρ i-1 k log k + k i=1 ρ i -η i k log k = lim k→+∞ k i=1 η i -ρ i-1 k log k = 8 π x xπ † (x),
which yields the result by definition of N m .

Proof of Theorem 1.2

By Propositions 3.1, for any ε > 0 with probability converging to one log n n

(1-ε)c 1 n/ log n i=1 f (B i ) ≤ log n n Nn i=1 f (B i ) ≤ log n n (1+ε)c 1 n/ log n i=1 f (B i ) then, as lim t→+∞ (1-ε)c 1 t /t = (1 -ε)c 1 , Proposition
2.1 gives for the lower bound log n n

c 1 (1-ε)n/ log n i=1 f (B i ) P → (1 -ε)c 1 x∈K c π * (x)E x [f (B 0 )] = c f (1 -ε).
A similar result is true for the upper bound, taking the limit when ε → 0 yields

f (B * n ) > n -ε/2 ): using successively Lemma 5.3 implying P(∪ n i=1 {X ηi > n 1/2+2ε }) ≤ C + n -4ε
, and the increasing property of f giving that f (B * n ) ≤ max 1≤i≤n f (B i ):

P log n n f (B * n ) > n -ε/2 ≤ P log n n f (B * n ) > n -ε/2 , ∩ n i=1 {X ηi ≤ n 1/2+2ε } + C + n -4ε ≤ P n i=1 log n n f (B i ) > n -ε/2 , X ηi ≤ n 1/2+2ε + C + n -4ε ≤ n max 1≤i≤n P log n n f (B i ) > n -ε/2 , X ηi ≤ n 1/2+2ε + C + n -4ε . ( 20
)
To deal with this probability we first compare each random variable f (B i ) with

E Xη i [f (B 0 )].
In the one hand by strong Markov property, Tchebychev inequality and finally condition (3)

P |f (B i ) -E Xη i [f (B 0 )]| > n 1-ε log n , X ηi ≤ n 1/2+2ε = E 1 Xη i ≤n 1/2+2ε P Xη i |f (B 0 ) -E Xη i [f (B 0 )]| > n 1-ε log n ≤ (log n) 2 n 2-2ε E 1 Xη i ≤n 1/2+2ε Var Xη i [f (B 0 )] ≤ C 2 (log n) 2 n 2-2ε E 1 Xη i ≤n 1/2+2ε X 2-δ ηi ≤ C 2 (log n) 2 n 1-6ε+ δ /2+2δε ≤ 1 n 1-6ε+δ/2 , ( 21
)
recall indeed that δ introduced in condition (3) is given and ε can be chosen as small as we want so in particular above probability converges to zero as long as δ/2 > 6ε. On the other hand we have to control the sequence (E Xη i [f (B 0 )], i ≤ n): using condition (2) and lemma 5.5 for 1 < β < 2

P E Xη i [f (B 0 )] > n 1-ε log n ≤ P C 2 X ηi > n 1-ε log n ≤ C (log n) β n β(1-ε) E[X β ηi ] ≤ C (log n) β n β(1-ε) . (22)
Taking

β(1 -ε) > 1 ensures that n max i≤n P E Xη i [f (B 0 )] > n 1-ε log n converges to 0. Collecting (20), (21)
and ( 22) implies that lim n→+∞ P log n n f (B * n ) > n -ε/2 = 0, which leads to the desired result.

Reversibility and technical Lemmata for trajectories on the axis

In this section we prove technical estimates for the part of the trajectory restricted to the axis, note that we do not need the condition α > 3 here, in fact most of the time α > 1 is enough so we mention the condition for α for each statement. For typographical simplicity, for any x and y in K c , let P(x → y) be the probability of the shortest path on K c to join y from x. For instance, on figure 2:

P(x → y) = x i=1 1 - 3 4i α y-1 i=1 1 4i α . Figure 2
Note that the expression of P(x → y) has a useful consequence to compute probability of the form P • (X ρ = •). For example, one can see that there exists 0 < C < 1 such that:

P x (X ρ = (1, 1)) > C, ∀x ∈ K c . (23)
Indeed, as α > 1, P(x → (0, 0)) > ∞ i=1 1 -3 4i α := a > 0. Then:

P x (X ρ = (1, 1)) ≥ P (x → (0, 0)) p((0, 0), (0, 1))p((0, 1), (1, 1)) = a 4 2 =: C > 0.
The first statement below treats about the reversibility of the random walk on K c .

Lemma 4.1. For all x, y ∈ K c such that x + e i , y + e j ∈ K: P x (X ρ = y + e j ) = P(x → y) P(y → x)

x α y α P y (X ρ = x + e i ).

Proof. First we prove that for all x, y ∈ K c and all n ∈ N * :

P x (X n = y, n < ρ) = P(x → y) P(y → x) P y (X n = x, n < ρ). ( 24
)
This fact is a simple consequence of the reversibility of random walk X. To prove (24), take a path Γ from x to y on K c of length n, Γ := (x 0 = x, x 1 , . . . , x n-1 , x n = y). Its probability is

P x (X 1 = x 1 , . . . , X n = y) = n-1 i=0 p(x i , x i+1 ) = P(x → y)A Γ
and note that A Γ is a product such that if p(x i , x i+1 ) appears in A Γ there is also necessarily j = i, such that p(x j , x j+1 ) = p(x i+1 , x i ).

If we reverse the path (taking i → n -i), we obtain similarly

P y (X 1 = x n-1 , . . . , X n-1 = x 1 , X n = x) = n-1 i=0 p(x i+1 , x i ) = P(y → x)A Γ
as the reversion does not change the value of A Γ . As a result summing on all path Γ of length n from x to y:

P x (X n = y, n < ρ) = Γ n-1 i=0 p(x i , x i+1 ) = P(x → y) Γ A Γ = P(x → y) P(y → x) Γ P(y → x)A Γ = P(x → y) P(y → x) Γ n-1 i=0 p(x i+1 , x i ) = P(x → y) P(y → x) P y (X n = x, n < ρ).
Now, the result of the lemma follows taking x, y ∈ K c such that x + e i , y + e j ∈ K:

P x (X ρ = y + e j ) = n≥0 P x (X n = y, n < ρ, X n+1 = y + e j ) = n≥0 P x (X n = y, n < ρ)p(y, y + e j ) = P(x → y) P(y → x) p(y, y + e j ) n≥0 P y (X n = x, n < ρ) = P(x → y) P(y → x)
p(y, y + e j ) p(x, x + e i ) P y (X ρ = x + e i ) = P(x → y) P(y → x)

x α y α P y (X ρ = x + e i ).

Remark 4.2. There is a counterpart of the precedent result on K: For all w, z ∈ K and all n ∈ N * such that w + e i , z

+ e j ∈ K c P w (X η = z + e j ) = P z (X η = w + e i ) . (25)
The proof of (25) is very similar to (24) and is left to the reader (see Figure 2).

The lemma below gives an asymptotic of the distribution of the exit coordinate from the axis. We use the following notation: for any x ∈ Z 2 , let (T k x ) k≥0 be the sequence defined by T 0 x = 0 and for all k ∈ N * :

T k x = inf{k > T k-1 x , X k = x},
for simplicity we write T x instead of T 1 x . Lemma 4.3. Assume α > 1, there exists c + > c -> 0 such that for all i > 1

1 + c -i -α ≤ 4i α P (0,i) (X ρ = (1, i)) ≤ 1 + c + i -α . ( 26
)
Proof. Using the strong Markov property:

P (0,i) (X ρ = (1, i)) = k≥0 P (0,i) T (0,i) < ρ k P (0,i) ρ < T (0,i) , X ρ = (1, i) = p((0, i), (1, i)) k≥0 P (0,i) T (0,i) < ρ k = 1 4i α 1 1 -P (0,i) T (0,i) < ρ =: 1 4i α 1 1 -h(i) . (27)
In order to obtain a lower bound for h(i), we apply the Markov property several times:

h(i) = 1 4i α h(i + 1) + 1 - 3 4i α h(i -1) (28) = 1 4i α 1 - 3 4(i + 1) α + 1 4(i + 1) α h(i + 2) + 1 - 3 4i α 1 4(i -1) α + 1 - 3 4(i -1) α h(i -2) (29) ≥ 1 4i α 1 - 3 4(i + 1) α + 1 - 3 4i α 1 4(i -1) α ≥ c - i α .
The upper bound is also obtained from (28): first h(i + 1) is simply bounded from above by 1 -1 2(i+1) α using (29). We treat h(i -1) separately with a similar reasoning as the one to obtain (27) and taking a particular trajectory:

h(i -1) = p((0, i -1), (0, i)) 1 -P (0,i-1) T (0,i-1) < T (0,i) ∧ ρ = 1 4(i -1) α P (0,i-1) T (0,i-1) > T (0,i) ∧ ρ ≤ 1 4(i -1) α P (0,i-1) T (0,i-1) > ρ, X ρ ∈ {(-1, 1); (1, 1)} ≤ 1 4(i -1) α 1 1 2 i-1 k=2 1 -3 4k α , then h(i) ≤ 1 4i α 1 - 1 2(i + 1) α + 1 4(i -1) α 1 1 2 i-1 k=2 1 -3 4k α
, and as α > 1, i-1 k=2 1 -3 4k α is strictly positive constant so we get the upper bound. The following Corollary is a consequence of Lemma 4.3. Recall that for i > 0, L (0,i) = {z = (0, z 2 ), z 2 ≥ i}.

Corollary 4.4. For α > 1, all z ∈ K c and i > 0:

P(z → (0, i)) ≤ 4i α P z (X ρ = (1, i)) ≤ 1 + c + i -α . (30) Moreover, 1. if z ∈ L (0,i) : lim i→+∞ 4i α P z (X ρ = (1, i)) = 1; (31)
2. there exists C > 0 such that for all z / ∈ L (0,i) :

P z (X ρ = (1, i)) ≤ C i 2α . ( 32
)
3. there exists 0 < C < 1 such that for all z ∈ K c , P z (T z < ρ) ≤ C.

Proof. Using the strong Markov property:

P z (X ρ = (1, i)) = P z (T (0,i) < ρ, X ρ = (1, i)) = P z (T (0,i) < ρ)P (0,i) (X ρ = (1, i)), (33)
which implies (30) using Lemma 4.3 and the fact that P(z → (0, i)) ≤ P z (T (0,i) < ρ).

1. This formula is direct as

P(z → (0, i)) = z k=i+1 1 - 3 4k α ≥ ∞ k=i+1 1 - 3 4k α =: R i -→ i→+∞ 1,
as R i is the rest of a convergent infinite product. 2. With (33), the inequality P z (T (0,i) < ρ) ≤ P (0,i-1) (T (0,i) < ρ) and an obvious symmetry:

P (0,i-1) (X ρ = (1, i -1), T (0,i) > ρ) ≤ P (0,i-1) (X ρ = (1, i -1)),
and (30) implies the existence of C > 0 such that for all positive i, C ≥ 1 + c + i -α .

3. We can take z = (0, i) for i ≥ 1 without loss of generality. Using (28), the same symmetry as the previous point and (30):

P (0,i) (T (0,i) < ρ) ≤ 1 4i α + P (0,i-1) (T (0,i) < ρ) ≤ 1 4i α + P (0,i-1) (X ρ = (1, i -1)) ≤ 1 4i α 2 + c + i α .
As a result, we can find N > 0, such that for all i ≥ N , P (0,i) (T (0,i) < ρ) ≤ 1 2 , and taking C := max(max 0≤i<N P (0,i) (T (0,i) < ρ), 1

2 ), we have the claimed result.

Corollary 4.5. Let β > 0 such that α -β > 1, there exists M > 0 such that:

∀z ∈ K c , E z X β ρ ≤ M. ( 34
)
Proof. Without loss of generality, we can assume that z = (0, i) with i ≥ 1. Using the symmetry of the problem and ( 26), for j > 2:

P (0,i) X ρ = j = 2P (0,i) (X ρ = (1, j)) + 6P (0,i) (X ρ = (j, 1)) = 2P (0,i) T (0,j) < ρ P (0,j) (X ρ = (1, j)) + 6P (0,i) T (j,0) < ρ P (j,0) (X ρ = (j, 1)) ≤ 8P (0,j) (X ρ = (1, j)) ≤ C + j -α .
Consequently:

E (0,i) X β ρ = j≥1 P (0,i) X ρ = j j β = P (0,i) X ρ = 1 + j≥2 P (0,i) X ρ = j j β ≤ 1 + C + j≥2 j β-α = M.
The last Lemma is a simple consequence of Corollary 4.5 and Corollary 4.4

Lemma 4.6. Assume α > 2, there exists c and c such that for any i and x ∈ K

E[X ρi ] ≤ c and P(X ρi = x) ≤ c /x α . ( 35
)
Proof. According to Corollary 4.5, there exists M > 0 such that for any

E[X ρi ] = E[E Xη i [X ρ ]] ≤ M . Similarly, writing P(X ρi = x) = E[P Xη i (X ρ = x)],
one can obtain the second inequality with (30).

Lemma 4.7. Assume α > 1. For any a, r > 0 and m large enough

P (0,a) (ρ > m) ≤ m -r . ( 36
)
Proof. For all k ∈ N * , let T * k = inf{n ≥ 0, X n = k}. As m goes to infinity, we can assume without loss of generality that 2a < log m. Then, one can write (we have chosen to not indicate the integer parts for typographical simplicity) :

P (0,a) (ρ > m) = P (0,a) (ρ > m, T * log m < ρ) + P (0,a) (T * log m > ρ > m) := (I) + (II).
For part (I) by symmetry, the fact that P (0,a) (T (0,log m) < ρ) > P (0,a) (T (0,-log m) < ρ) and Lemma 4.1, we obtain:

(I) ≤ 4P (0,a) (T (0,log m) < ρ) = 4 P (0,a) (X ρ = (1, log m)) P (0,log m) (X ρ = (1, log m))
≤ 16(log m) α P (0,a) (X ρ = (1, log m))

= 16a α P((0, a) → (0, log m)) P((0, log m) → (0, a))

≤ 16a α log m-1 i=a 1 4i α ∞ i=1 1 -3 4i α = 4C log m-1 i=a+1 1 4i α ≤ 4C log m-1 i= log m 2 1 4 log m 2 α ≤ 4C 4 log m 2 α -log m 2 ,
which implies that for all r > 0, (I) is bounded from above by m -r for m large enough. Otherwise, for second term (II)

(II) = P (0,a) (T * log m > ρ > m) = P (0,a)   z∈K c ,|z|≤log m L (z, ρ) > m   ≤ P (0,a) ∃z ∈ K c , |z| ≤ log m, L (z, ρ) > m 4 log m + 1 ≤ z∈K c ,|z|≤log m P (0,a) L (z, ρ) > m 4 log m + 1 .
Note that for all k ≥ 2 and all z ∈ K c , using the third point of Corollary 4.4:

P (0,a) (L (z, ρ) > k) = P (0,a) (T z < ρ) P z (T z < ρ) k ≤ P z (T z < ρ) k ≤ Ck .
Consequently:

(II) ≤ z∈K c ,|z|≤log m C m 4 log m = (4 log m + 1) C m 4 log m ,
so, for all r > 0 , (II) is also bounded from above by m -r for m large enough.

In the following Lemma we obtain asymptotic (with respect to the coordinate of the starting point) of the exit time from the axis ρ and its second order.

Lemma 4.8. Assume α > 1, let 0 < β ≤ 2 and 0 < ε < 1. Then, for large i

|E (0,i) [ρ β ] -i β | = O(i β-ε ). ( 37
)
Proof. For the lower bound, just note that

E (0,i) ρ β ≥ E (0,i) ρ β 1 ρ≥i-i 1-ε ≥ (i -i 1-ε ) β P (0,i) ρ ≥ i -i 1-ε = (i -i 1-ε ) β 1 -P (0,i) ρ < i -i 1-ε =: L(i)
and with (30):

P (0,i) ρ < i -i 1-ε ≤ k≥i 1-ε P (0,i) (X ρ = (1, k)) ≤ k≥i 1-ε P (0,k) (X ρ = (1, k)) ≤ k≥i 1-ε 1 4k α 1 + c + k α ≤ C + i -(α-1)(1-ε) . This implies that for large i, L(i) = (i -i 1-ε ) β (1 + o(1)) = i β + O(i β-ε ). For the upper bound, E (0,i) ρ β = E (0,i) ρ β 1 ρ<i+i 1-ε + 1 ρ≥i+i 1-ε ≤ i + i 1-ε β + E (0,i) ρ β 1 ρ≥i+i 1-ε =: i + i 1-ε β + U (i).
Then, using (36), with r > β + 1 and i large enough:

U (i) = k≥i 1-ε (i + k) β P (0,i) (ρ = i + k) ≤ k≥i 1-ε (i + k) β k r = o(1).
This finishes the proof.

Technical lemmata for trajectories on the cone

Recall the definition of η = inf{k > 0, X k ∈ K}. In this section we obtain a (uniform) local limit result for X η (Lemma 5.1) as well as a uniform tail for η (Lemma 5.4).

Lemma 5.1. For any δ > 0, uniformly in y ≤ x 1-δ :

lim x→+∞ x 3 y P (1,x) (X η = (y, 0)) = lim x→+∞ x 3 y P (y,1) (X η = (0, x)) = 16 π . lim x→+∞ x 3 y P (1,x) (X η = (0, y)) = lim x→+∞ x 3 y P (1,y) (X η = (0, x)) = 16 π .
Proof. Note that the first equality on both above lines comes from the symmetry of the distribution of the simple random walk on the cones (see Remark 4.2).

For any sequence (Y n , n ≥ 1) of real random variables, introduce Y n := inf 1≤k≤n Y k . We start with P (1,x) (X η = (y, 0)). First, writing X n = (X 1 n , X 2 n ), we easily see that:

P (1,x) (X η = (y, 0)) = k≥x P (1,x) (X k = (y, 0), η = k) = k≥x P (1,x) (X 1 k > 0, X 1 k = y, X 2 k-1 > 0, X 2 k = 0) = 1 4 k≥x P (1,x) (X 1 k-1 > 0, X 1 k-1 = y, X 2 k-1 > 0, X 2 k-1 = 1),
as the k-th step is necessarily vertical, more precisely X 2 k-1 = 1 and X 2 k = 0. If H k j is the event {among the first k -1 steps, there is exactly j horizontal ones} and if Z is the symmetric random walk on Z, one can write:

j k :=P (1,x) (X 1 k-1 > 0, X 1 k-1 = y, X 2 k-1 > 0, X 2 k-1 = 1|H k j ) =P 1 (Z j > 0, Z j = y)P x (Z k-j-1 > 0, Z k-j-1 = 1). (38)
Thus, for 0 < ε < 1:

P (1,x) (X η = (y, 0)) = 1 4 k≥x k-1-x j=y-1 P(H k j )P 1 (Z j > 0, Z j = y)P x (Z k-j-1 > 0, Z k-j-1 = 1) = 1 4 k≥x j∈B k,ε P(H k j ) j k + 1 4 k≥x j∈B c k,ε P(H k j ) j k =: Σ 1 + Σ 2 , (39)
where

B k,ε := [ (k-1)(1-ε) /2, (k-1)(1+ε) /2] and B c k,ε its complementary in {y -1, • • • , k -1 -x}. First note that according to (54), for j in B k,ε P(H k j ) ≤ e -ε 2 k 6 , implying: Σ 2 ≤ k≥x e -ε 2 k 6 j∈B c k,ε j k ≤ k≥x ke -ε 2 k 6 ≤ ∞ x te -ε 2 t 6 dt = e -ε 2 x 6 6 ε 2 x + 6 ε 2 (40)
and as a result lim ε→0 lim x→+∞ x 3 y Σ 2 = 0. In view of what we want to prove, we only consider j ∈ B k,ε in the following and we write:

Σ 1 = 1 4 x≤k<εx 2 + εx 2 ≤k≤x 2 /ε + k>x 2 /ε j∈B k,ε P(H k j ) j k =: Σ 11 + Σ 12 + Σ 13 .
• Asymptotic behaviour of Σ 12 Applying Lemma 6.3:

Σ 12 = 2xy π x 2 /ε k=εx 2 j∈B k,ε P(H k j ) e -x 2 2(k-j) (j + 1) 3 2 (k -j) 3 2 1 + O y 2 k + o x 3 k 2 .
Using formula (53), a lower bound for Σ 12 is given by :

Σ 12 ≥ 16xy π(1 + ε) 3 1 + O 1 x 2δ x 2 /ε k=εx 2 e -x 2 k(1-ε) (k + 1) 3 j∈B k,ε P(H k j ) ≥ 16xy π(1 + ε) 3 1 + O 1 x 2δ x 2 /ε k=εx 2 e -x 2 k(1-ε) (k + 1) 3 1 + O 1 √ k = 16xy π(1 + ε) 3 1 + O 1 x + O 1 x 2δ x 2 /ε k=εx 2 e -x 2 k(1-ε) (k + 1) 3 . ( 41
)
With the substitution u = x 2 /z:

x 2 /ε k=εx 2 e -x 2 k(1-ε) (k + 1) 3 ≥ 1 - 2 εx 2 + 1 3 x 2 /ε k=εx 2 e -x 2 k(1-ε) (k -1) 3 ≥ 1 - 2 εx 2 + 1 3 x 2 /ε k=εx 2 k k-1 e -x 2 z(1-ε) z 3 dz ≥ 1 - 2 εx 2 + 1 3 x 2 ε εx 2 e -x 2 z(1-ε) z 3 dz = 1 - 2 εx 2 + 1 3 1 x 4 x 2 εx 2 x 2 x 2 ε ue -u 1-ε du =: g(x, ε).
By Lebesgue's dominated convergence theorem lim x→+∞ x 4 g(x, ε) =

1 ε ε ue -u 1-ε du and lim ε→0 lim x→+∞ x 4 g(x, ε) = +∞ 0 ue -u 1-ε du = 1, this implies lim ε→0 lim x→+∞ x 3 y Σ 12 ≥ 16 π lim ε→0 lim x→+∞ 1 (1 + ε) 3 1 + O 1 x + O 1 x 2δ x 4 y εx 2 ≤k≤x 2 /ε g(x, ε) ≥ 16 π .
Note that this last inequality implies the desired lower bound as:

lim x→+∞ x 3 y P (1,x) (X η = (y, 0)) ≥ lim ε→0 lim x→+∞ x 3 y Σ 12 ≥ 16 π .
To obtain an upper bound, as j∈B k,ε P(H k j ) ≤ 1,

Σ 12 ≤ 16xy π(1 -ε) 3 1 + O 1 x 2δ x 2 /ε k=εx 2 e -x 2 (k+1)(1+ε) k 3 , ( 42
)
the rest of the proof is similar as the one for the lower bound implying that lim ε→0 lim x→+∞ x 3 y Σ 12 ≤ 16 π . To complete the proof for the upper bound we have to show that Σ 11 and Σ 13 are negligible: • Σ 13 negligibility:

We just note, by (50) and (51), that

Σ 13 ≤ 1 4 k>x 2 /ε max j∈B k,ε j k ≤ 32 π k>x 2 /ε xy (k(1 -ε)) 3 e -x 2 (k+1)(1+ε) (43) ≤ 32xy π(1 -ε) 3 x 2 -ε x 2 (x 2 + ε) 2 ε x 2 +ε x 2 -ε 0 ue -u 1+ε du.
This finally implies lim

ε→0 lim x→+∞ x 3 y Σ 13 ≤ lim ε→0 32(1 + ε) 2 π(1 -ε) 3 ε 1+ε 0 ve -v dv = 0.
• Σ 11 negligibility : We use same first inequality as in (43) and then split again the sum : let 0 < δ < 1

Σ 11 ≤ x≤k≤x 2-δ/2 max j∈B k,ε j k + x 2-δ/2 <k≤εx 2 max j∈B k,ε j k =: Σ 11 + Σ * 11 .
According to Lemma 6.5, for 0 < ε < 1 /3 and x large enough, there exists c -> 0 such that:

Σ 11 ≤ x≤k≤x 2-δ/2 max j∈B k,ε j k ≤ x≤k≤x 2-δ/2 max j∈B k,ε P(Z k-j-1 ≥ x -1) ≤ x≤k≤x 2-δ/2 e -c-x 2 k ≤ x 2-δ/2 e -c-x δ/2 , so lim x→+∞ x 3
y Σ 11 = 0. And similarly as for Σ 13 above, using (50) and ( 51)

Σ * 11 ≤ x 2-δ/2 <k<εx 2 32xy π(k(1 -ε)) 3 e -x 2 (k+1)(1+ε) ≤ 32xy π(1 -ε) 3 x 2 -ε x 2 (x 2 + ε) 2 ∞ x 2 +ε ε(x 2 -ε) ue -u 1+ε du. ( 44
)
Implying that :

lim ε→0 lim x→+∞ x 3 y Σ * 11 ≤ lim ε→0 C + ε e -1 ε(1+ε) = 0.
So finally lim ε→0 lim x→+∞ x 3 y Σ 11 = 0. We now give some details for P (1,x) (X η = (0, y)).

As y ≤ x 1-δ , we can assume without loss of generality that y < x. Like the previous case:

P (1,x) (X η = (0, y)) = 1 4 x 2 /ε k=εx 2 k-1-(x-y) j=0 P(H k j )P 1 (Z j > 0, Z j = 1)P x (Z k-j-1 > 0, Z k-j-1 = y) = 1 4 k≥x j∈B k,ε P(H k j )h j k + 1 4 k≥x j∈B c k,ε P(H k j )h j k =: Ω 1 + Ω 2 , where B c k,ε is here the complementary of B k,ε in {0, • • • , k-1-(x-y)}.
According to (40), Ω 2 is negligible. Now, we only focus on the quantity Ω 12 := 1 4 εx 2 ≤k≤x 2 /ε j∈B k,ε P(H k j )h j k and (49) and (52) give

Ω 12 = 1 + O 1 x 2δ + o 1 x 2xy π x 2 /ε k=εx 2 j∈B k,ε P(H k j ) e -x 2
2(k-j-1)

(j + 1) 3 2 (k -j -1) 3 2
.

Using (53), with the same reasoning as the one for (y, 0):

Ω 12 ≥ 1 + O 1 x 2δ + o 1 x 16xy π(1 + ε) 3 x 2 /ε k=εx 2 e - x 2 (k-1)(1-ε) (k + 1) 3 Ω 12 ≤ 1 + O 1 x 2δ + o 1 x 16xy π(1 -ε) 3 x 2 /ε k=εx 2 e -x 2 (k+1)(1+ε) (k -1) 3 .
The upper and lower bound for Ω 12 we obtain here is the equivalent of what we obtain for Σ 12 in the previous case (see (41) and ( 42)). Then the rest of the proof is very similar than what we previously detail so we chose to let this to the reader. The only difference is for the quantity x-y≤k≤x 1+δ P(H k j )h j k , where we use again Lemma 6.5.

Remark 5.2. For any δ > 0, uniformly in y ≤ x 1-δ , there exists C > 0 such that:

P y (X η > x) ≤ C y x 2 .
Lemma 5.3. Assume α > 3. There exists C > 0 such that for all n ∈ N * , all 1 ≤ i ≤ n and 0 < ε < α-3 4 :

P(X ηi > n 1 /2+2ε ) ≤ C n 1+4ε
Proof. For i ≥ 2, using the previous remark and Lemma 4.6 (two times):

P(X ηi > n 1 /2+2ε ) = x∈K P(X ρi-1 = x)P x (X η > n 1 /2+2ε ) ≤ x≤n 1 /2+ε P(X ρi-1 = x)P x (X η > n 1 /2+2ε ) + x>n 1 /2+ε P(X ρi-1 = x) ≤ C   1 n 1+4ε x≤n 1 /2+ε xP(X ρi-1 = x) + x>n 1 /2+ε 1 x α   ≤ C E[X ρi-1 ] n 1+4ε + 4 n (α-1)( 1 /2+ε) ≤ C 1 n 1+4ε + 1 n (α-1)( 1 /2+ε) .
As 0 < ε < α-3 4 , we have 1 + 4ε < (α -1)( 1 /2 + ε). Note that for i = 1, the proof is easier using straightly Remark 5.2 and we have our claimed result.

Lemma 5.4. For any y ∈ ∂K such that y = o(k 1/2 ), lim k→+∞ k 2 y P y (η = k) = 8 π . Proof. For typographical simplicity, we only treat P (1,x) (η > n) with x > 0 (the other ones can be obtain by symmetry). Recall the definition of H .

. before (38), following the same ideas as for the proof of Lemma 5.1, for ε > 0:

P (1,x) (η = k) = m≤k-1 P (1,x) (η = k|H k m )P(H k m ) = m∈B k,ε + m∈B c k,ε P (1,x) (η = k|H k m )P(H k m ) := Σ 1 + Σ 2 ,
and we can prove that Σ 2 is negligible (see (40) for details). Thus we only study Σ 1 and write:

Σ 1 = 1 4 m∈B k,ε P(H k m )P 1 (Z m > 0)P x (Z k-m-1 > 0, Z k-m-1 = 1) + 1 4 m∈B k,ε P(H k m )P 1 (Z k-m-1 > 0, Z k-m-1 = 1)P x (Z m > 0) =: Σ 11 + Σ 12 .
For any large k and any x = o(k 1/2 ), with Corollary 6.2, (50) and (53):

Σ 11 ≥ x π m∈B k,ε P(H k m ) e -x 2 2(k-m) √ m(k -m) 3 2 1 + o x 3 (k -m) 2 + o 1 m ≥ 4x π(1 + ε) 2 e -x 2 k(1-ε) k 2 1 + o x 3 k 2 + o 1 k m∈B k,ε P(H k m ) ≥ (1 -2ε) 4x π e -x 2 k(1-ε) k 2 ≥ (1 -3ε) 4x πk 2 .
The upper-bound is simpler as:

Σ 11 4x π(1 -ε) 2 1 (k -1) 2 1 + o x 3 k 2 + o 1 k ≤ 4x πk 2 (1 + 2ε)
Thus, lim ε→0 lim n→∞ nΣ 11 /x = 4/π. Σ 12 can be treated similarly and we obtain our claimed result.

We conclude this section with two lemmata, the first one is the counterpart on K of Lemma 4.6 and the last one a useful identity: Lemma 5.5. For any 0 < β < 2 there exists C > 0 such that for all i ≥ 1:

E X β ηi ≤ C.
Proof. According to [START_REF] Terry | Exit times of n-dimensional random walks[END_REF], Theorem 1.3 page 223 (see also [START_REF] Denisov | Random walks in cones[END_REF] Lemma 10 page 1007), for any 0 < β < 2 there exists C > 0 such that for any x ∈ K:

E x max i≤η X β i ≤ C(1 + x β ). ( 45
)
Using the strong Markov Property (two times), previous inequality and Corollary 4.5 yields

E X β ηi = E E Xρ i-1 X β η ≤ CE X β ρi-1 = CE E Xη i-1 X β ρ ≤ CE[M ] = CM. (46)
Lemma 5.6. For all x ∈ K c :

y∈∂K P y (X η = x) = 2.
Proof. We can assume without loss of generality that x = (0, i) for i ≥ 1. Using the reversibility:

y∂K P y (X η = (0, i)) = P (1,1) (X η = (0, i)) + j≥2 P (j,1) (X η = (0, i)) + P (1,j) (X η = (0, i)) + P (-1,1) (X η = (0, i)) + j≥2 P (-1,j) (X η = (0, i)) + P (-j,1) (X η = (0, i)) = P (1,i) (X η = (0, 1)) + j≥2 P (1,i) (X η = (j, 0)) + P (1,i) (X η = (0, j)) + P (-1,i) (X η = (-1, 0)) + j≥2 P (-1,i) (X η = (0, j)) + P (-1,i) (X η = (-j, 0)) = 2.

Appendix

In this appendix, we give asymptotic results linked to (Z n ) n≥0 , the symmetric random walk on Z, results that we used throughout this Recall that B k,ε = [ (k-1)(1-ε) /2, (k-1)(1+ε) /2] and H k j is the event {among the first k -1 steps, there is exactly j horizontal ones} .

Lemma 6.1. Let 0 < δ < 1, assume that k is an integer such that lim x→+∞ ln k /ln x ∈ [(2 -δ), 2], then 2 -k k k-x 2 = 2 πk e -x 2 2k 1 + o x 3 k 2 . ( 47
)
If, moreover there exists y ≤ x 1-δ :

2 -k k k-y 2 = 2 πk 1 + O y 2 k . (48)
Proof. Using Stirling formula:

2 -k k k-x 2 = 2 -k √ 2π k k+ 1 2 e -k 1 + 1 12k + o 1 k k-x 2 k-x+1 2 e -k-x 2 k+x 2 k+x+1 2 e -k+x 2 1 + 1 6k + o 1 k = 2 πk 1 1 -x k k-x+1 2 1 + x k k+x+1 2 1 - 1 12k + o 1 k . Moreover A := 1 - x k k-x+1 2 1 + x k k+x+1 2 = e k-x+1 2 ln(1-x k ) e k+x+1 2 ln(1+ x k ) = exp k -x + 1 2 - x k - x 2 2k 2 - x 3 3k 3 + o x 3 k 3 + k + x + 1 2 x k - x 2 2k 2 + x 3 3k 3 + o x 3 k 3 = exp x x k + x 3 3k 3 - x 2 2k 2 (k + 1) + o x 3 k 2 = exp x 2 2k + x 4 3k 3 - x 2 2k 2 + o x 3 k 2 = exp x 2 2k + o x 3 k 2 = exp x 2 2k 1 + o x 3 k 2 .
In the first case:

2 -k k k-x 2 = 2 πk e -x 2 2k 1 + o x 3 k 2 1 - 1 12k + o 1 k = 2 πk e -x 2 2k 1 + o x 3 k 2 ,
If y ≤ x 1-δ , e -y 2 2k = 1 -y 2 2k + o y 2 2k = 1 + O y 2 k , giving the second formula. As u = o(k 1/2 ), we conclude using Lemma 6.1, Lemma 6.3. When k goes to infinity

P 1 (Z k > 0, Z k = 1) = 2 π 2 (k + 1) 3 2 1 + o 1 k 2 . ( 49
)
Let 0 < δ < 1, for any x large enough and for any k ≥ x 2-δ/2 , P x (Z k > 0, Z k = 1) = 2 π 2x (k + 1)

3 2 e -x 2 2(k+1)

1 + o x 3 k 2 . ( 50
)
If, moreover y ≤ x 1-δ ,

P 1 (Z k > 0, Z k = y) = 2 π 2y (k + 1) 3 2 1 + O y 2 k + o 1 k 2 , (51)
P x (Z k > 0, Z k = y) = 2 π 2xy k 3 2 e -x 2 2k 1 + o x 3 k 2 + O y 2 k + O (xy) 2 k 2 . ( 52
)
Proof. Using the stationarity of (Z k ) k≥0 and the Desire André's reflexion principle (see [START_REF] Feller | An Introduction to Probability Theory[END_REF], p. 72-73 and 95 problems for solution):

P x (Z k > 0, Z k = 1) = P(Z k > -x, Z k = 1 -x) = 1 2 k k k+1-x 2 - k k-x-1 2 = 1 2 k+1 2x k + 1 k + 1 k+1-x 2
, and we easily obtain (49) taking x = 1 and using Lemma 6.1. The reasonings to obtain (50) and (51) are very similar and use again Lemma 6.1.

To prove (52), we use again the Desire André's reflexion principle and formula (47) To finish we recall elementary facts use several times in Section 5.

P x (Z k > 0, Z k = y) = P(Z k > -x, Z k = y -x) = 1 2 k k k-(x-
Lemma 6.4. When k goes to infinity:

j∈B k,ε P(H k j ) ≥ 1 -O 1 √ k , (53)
and for all j / ∈ B k,ε P(H k j ) ≤ e -ε 2 k 6 .

(54)

Proof. Formulas (53) and (54) are respectively applications of the Berry-Esseen and Chernoff inequalities.

Lemma 6.5. For k > x -y with y = o(x) and 0 < ε < 1 /3, there exists c -> 0 such that for x large enough:

max j∈B k,ε P(Z j ≥ x -y) ≤ e -c -x 2 k .
Proof. Proof is elementary, we give some details for completeness. Assume y = 1 for the moment, for any α > 0:

P(Z j ≥ x -1) = P e αZj ≥ e α(x-1) ≤ E e αZj e -α(x-1) = E e αZ1 j e -α(x-1) = e j ln E[e αZ 1 ]-α(x-1) ≤ e j(E[e αZ 1 ]-1)-α(x-1) = e j(cosh α-1)-α(x-1) .

For 0 < α < 2, cosh α ≤ 1 + 3 4 α 2 implying that for such α: P(Z j ≥ x -1) ≤ e 3 4 jα 2 -α(x-1) =: e ψ(α) .

One can find that ψ reaches its minimum for α = 2(x-1) 3j (note that for 0 < ε < 1 /3 and k > x, this quantity is bounded above by 2) and for that choice of α: P(Z j ≥ x -1) ≤ e -(x-1) 2 3j ≤ e -(x-1) 2 3k ≤ e -x 2 6k , for x large enough. This finishes the proof for y = 1, for general y proof is identical just the optimal α changes in α = 2(x -y)/3j.

Figure 1 :

 1 Figure 1: probabilities of transition

Corollary 6. 2 .

 2 When k goes to infinity, for any m ∈ B k,ε any u = o(k 1/2 ), According to[START_REF] Feller | An Introduction to Probability Theory[END_REF] pp.72 and pp.88-89, P u (Z m = j) = 2 assuming m ∈ B k,ε :uP u (Z m = 1) ≤ P u (Z m > 0) = u j=1 P u (Z m = j) ≤ uP u (Z m = u) .

Acknowledgment: We would like to thank Julien Barré for introducing us the work of physicists and the problematics which inspire this paper.