Stephane Picot

Michel Cucherat

Anne-Lise Bienvenu

Systematic review and meta-analysis of diagnostic accuracy of Loop-mediated isothermal amplification (LAMP) methods compared to microscopy, PCR, and rapid diagnostic tests, for malaria diagnosis

Keywords: Malaria, Diagnosis, Meta-analysis, Loop-mediated isothermal amplification, microscopy, PCR, Rapid diagnostic tests

Background

Diagnosis is a challenging issue in the way to malaria elimination. LAMP could be an alternative to conventional methods. Then, it is of interest to evaluate the diagnostic accuracy of LAMP for malaria compared to microscopy, PCR, and RDTs.

Methods and design

We searched MEDLINE, Web of Science, and Scopus, from inception to 1st July 2019. Prospective and retrospective, randomized and non-randomized, mono-center and multi-center studies, including symptomatic or asymptomatic patients that reported one LAMP method and one comparator (microscopy, RDT, PCR) were included. PROSPERO registration number: CRD42017075186.

Results

Sixty-six studies published between 2006 and 2019 were included leading to the analysis of 30 641 LAMP tests. The pooled sensitivity of LAMP remained between 96 and 98% whatever the comparator. The pooled specificity of LAMP was around 95%, but is a little higher if best PCR studies are considered. AUC were found to be above 0.98 whatever the subgroup of studies considered and DOR to be around 1000 for all subgroups except for Plasmodium vivax.

Conclusion

This meta-analysis confirms that LAMP method is robust for malaria diagnosis both in symptomatic and asymptomatic people. Thus, the impact of LAMP for the control of malaria is expected to be important.

Background

The global malaria elimination appears to be more difficult to reach than expected (WHO 2018). It will require highly sensitive, reliable and easy-to-perform methods for the diagnosis of malaria [START_REF] Landier | The role of early detection and treatment in malaria elimination[END_REF]. Malaria diagnosis can be conducted using microscopy, Rapid Diagnosis Tests (RDT), Polymerase Chain Reaction (PCR), or a combination of these methods. Microscopic examination of Giemsa stained thick or thin smears is the reference standard providing its quality can be maintained and a good expertise is available [START_REF] Billo | Inter-observer agreement according to malaria parasite density[END_REF]). However, expert malaria microscopists are less common than needed leading to impaired microscopy-based diagnosis, especially for non-falciparum species [START_REF] Ruas | Non-falciparum malaria imported mainly from Africa: a review from a Portuguese hospital[END_REF]. Limit of detection (LOD) may vary substantially according to the experience and training of microscopists, from a range of 5 to 100 parasites/µL [START_REF] Zimmerman | Malaria diagnosis for malaria elimination[END_REF]. Rapid Diagnosis Tests (RDTs) for malaria [START_REF] Odaga | Rapid diagnostic tests versus clinical diagnosis for managing people with fever in malaria endemic settings[END_REF]) have increased the availability of reliable diagnosis in remote areas [START_REF] Wilson | Laboratory diagnosis of malaria: conventional and rapid diagnostic methods[END_REF]. RDTs are widely used in endemic areas with a positive effect on the management of fever cases, whereas RDTs have definitive limitations. While the sensitivity and specificity of the majority of RDTs are good for Plasmodium falciparum, this is not the case for other human malaria species. Besides, RDTs LOD is too high (between 100 to 200 parasites/µL, depending on the test, with the exception of recent Ultrasensitive RDTs) to be useful to detect asymptomatic carriers in case of gametocyte carriage which are the main reservoir of Plasmodium parasites responsible for continuing transmission. Lastly, most of RDTs have been designed for the detection of HRP2, leading to an emerging selection of parasite clones deleted in HRP2/3 proteins in South-America and more recently in sub-Saharan Africa, with a risk of false-negative results [START_REF] Menegon | Identification of Plasmodium falciparum isolates lacking histidine-rich protein 2 and 3 in Eritrea[END_REF] .

In this context, molecular detection of malaria would be extremely useful to overcome these limitations. The molecular detection of malaria parasites using PCR or Q-PCR has indeed proved its capacity to detect low parasitemia with LOD of 0.05 parasites/µL or less [START_REF] Roth | Molecular malaria diagnostics: A systematic review and meta-analysis[END_REF]. A wide range of PCR methods has been developed for the malaria diagnosis, providing valuable information on the parasite biology and pyrogenic threshold. Authors tried to simplify the method to allow large utilization in the malaria endemic areas, but one should consider that these methods are unavailable for point-of-care diagnosis without a subsequent training and equipment [START_REF] Amir | Diagnostic tools in childhood malaria[END_REF]. PCR is still limited to well-equipped centers mostly far from remote endemic areas inconsistent with the need of prompt initiation of a documented treatment of malaria cases.

Loop-mediated isothermal amplification of DNA (LAMP) has been firstly described in 2000 [START_REF] Notomi | Loop-mediated isothermal amplification of DNA[END_REF] and its first application to malaria diagnosis was in 2006 [START_REF] Poon | Sensitive and inexpensive molecular test for falciparum malaria: detecting Plasmodium falciparum DNA directly from heattreated blood by loop-mediated isothermal amplification[END_REF]. LAMP is an isothermal molecular method using a DNA polymerase from Bacillus stearothermophilus which has strand displacement activity leading to DNA auto-cycling without temperature changes. The different tests available are mostly targeting the mitochondrial genome of Plasmodium parasite. This method of DNA amplification is easier and cheaper to perform than PCR since it requires less equipment and lab facilities [START_REF] Lucchi | Expanding the malaria molecular diagnostic options: opportunities and challenges for loop-mediated isothermal amplification tests for malaria control and elimination[END_REF]. It is expected to have a higher sensitivity than microscopy and RDT and excellent likelihood negative value [START_REF] Ponce | Diagnostic accuracy of loopmediated isothermal amplification (LAMP) for screening patients with imported malaria in a nonendemic setting[END_REF]. Recent technical developments led to consider LAMP as a suitable tool for diagnosis in endemic and non-endemic areas (Polley et al. 2013a). Among the questions that remain are the place of LAMP method in a diagnosis process according to available resources, malaria transmission level, lab equipment and staff training.

The aim of this systematic review was to evaluate the diagnostic accuracy of Loop-mediated isothermal amplification (LAMP) methods compared to microscopy, PCR and RDTs for malaria diagnosis in symptomatic and asymptomatic patients from endemic and non-endemic areas.

To this end, this systematic review will answer the following questions: (1) when compared to the microscopic examination of blood smears or RDTs (reference standards) and to PCR (standard of truth) what are the performances (sensitivity, specificity, positive predictive value, negative predictive value) of LAMP methods (index method) for detection of human Plasmodium parasites? (2) Is there a difference among main Plasmodium species when LAMP tests are used for detection? (3) Is there a difference in LAMP tests performances among symptomatic and non-symptomatic patients? (4) Is there a difference in LAMP tests performance among the available methods?

Methods

This systematic review and meta-analysis followed the preferred reporting items for systematic review and meta-analyses (PRISMA) guidelines and the Cochrane handbook for systematic reviews of diagnostic test accuracy [START_REF] Mcgrath | Recommendations for reporting of systematic reviews and meta-analyses of diagnostic test accuracy: a systematic review[END_REF]). The study protocol was registered with the international Prospective Register of Systematic reviews (PROSPERO) on 24 August, 2017 and was last updated on 25 September, 2017 (registration number CRD42017075186).

Search strategy

Literature search strategies was developed using medical subject headings (MESH) in the National Library of Medicine and text word related to malaria diagnosis and LAMP. We searched MEDLINE, (("Lamp"[Journal] OR "lamp"[All Fields]) AND ("malaria"[MeSH Terms] OR "malaria"[All Fields]))

Web of Science, and Scopus from inception to 1st July 2019. To ensure literature saturation, we scanned the reference list of included and excluded studies.

Eligibility criteria

Prospective and retrospective, randomized and non-randomized, mono-center and multi-center studies from symptomatic or asymptomatic patients without age limitation that reported at least one LAMP method for malaria diagnosis (Plasmodium genus or species) and one comparator (microscopy, RDT, PCR) were included up to July 2019. Studies from malaria endemic or nonendemic areas, and from research, hospital, and tertiary care settings, were included. If a 2x2 table of true positive, false negative, true negative, and false positive counts, was not available, data were extracted from the original article. There was no language limitation for inclusion. Case reports, animal (including mosquitoes) studies, studies using parasites clones from culture or parasite DNA form collection, and methodological studies without clinical samples, were excluded.

Study selection

After searches in the proposed databases have been completed, a single library was created in Zotero (Version 5.0.18, Corporation for Digital Scholarship, Virginia, USA). The duplicates were removed using Zotero and then, checked manually. A Microsoft Excel spreadsheet (Microsoft Corp., Redmond, WA, USA) was used to manage data and literature search. Two identical copies of Excel spreadsheets were created and ascribed to two independent reviewers. The first step was to screen titles and abstracts of selected articles to exclude studies according to the exclusion criteria. Reviewers must justify exclusion of any items in the spreadsheet. A consensus meeting allowed to avoid any divergence between the two spreadsheets. Studies were included for next step if there is any uncertainty. Based on the first selection list, the next step was to read the full texts to exclude studies that did not meet inclusion criteria. This second step was validated by another consensus meeting to select which article will be considered eligible for review.

Disagreements were resolved by discussion and reasons for excluding studies were recorded.

A data extraction form was built to compile data from included studies.

Data extraction and checking

Variables such as title, references, authors, country of the study (endemic vs non-endemic), participants (symptomatic vs non-symptomatic), interventions (genus and/or specie specific detection, commercial or in-house method, prospective or retrospective study), comparators (PCR, microscopy, RDT), and number of participants, were recorded. To ensure reproducibility and completeness of data extraction, a predesigned Excel spreadsheet (Microsoft Corp., Redmond, WA, USA) compiling all variables to be extracted was used. Duplicate, overlapping, or companion studies, detected during the data extraction process were excluded. Two investigators (ALB, SP) independently collected the true positive, false positive, true negative, and false negative counts.

Disagreements over data extraction were resolved by discussion. Data were centrally checked by an independent operator for completeness, plausibility, and integrity, before synthesis.

Quality assessment

The methodological quality of included studies was evaluated to reduce systematic biases and inferential error of the extracted data. Risk of bias of the including studies was independently assessed by two reviewers (ALB, SP) using the quality assessment of diagnostic accuracy studies (QUADAS-2) [START_REF] Whiting | QUADAS-2: a revised tool for the quality assessment of diagnostic accuracy studies[END_REF]. QUADAS-2 assesses four areas of bias including patient selection, index test, reference standards, and flow/timing. Risk of bias summary and graph were generated in Review Manager 5 (RevMan version 5.3.5, http://community.cochrane.org/tools/reviewproduction-tools/revman-5) to provide a synthetic analysis of bias. Moreover, predefined sensitivity analyses after excluding studies judged to have potential risk of bias were performed. Studies without high risk of bias for any of the defined parameters were considered for "best studies" subgroup.

Data Synthesis

RevMan 5 was the software used for meta-analysis. Values of test accuracy compared to microscopic examination of blood smears and non-isothermal PCR methods including sensitivity, specificity, positive likelihood ratio (PLR), negative likelihood ratio (NLR), and 95% confidence intervals, using data extracted from sources (True positives, True Negative, False Positive, False negative) or calculated from the data available were computed. Given the general use of microscopic examination of thin or thick blood smears as reference standard, microscopy was considered as the clinical practice comparator. PCR was considered as the standard of truth as it was the most relevant to compare with LAMP; thus, PCR was considered as the direct comparator.

RDTs are widely used in endemic malaria areas, but not frequently used as a comparator for sensitive diagnostic tests; thus, RDTs was considered as a second clinical practice comparator.

Results of individual study were presented graphically on forest plot, as well as on receiver operating characteristic (ROC) curve. To evaluate overall performances of the LAMP tests, a bivariate (bivariate diagnostic random effect meta-analysis) and univariate analyses (DerSimonian and Laird random effect model) were conducted using R 3.5.2 software (R Foundation for Statistical Computing, Vienna, Austria); summary point estimates of the pairs of sensitivity and specificity with their 95% confidence interval, diagnostic odd ratios with their 95% confidence interval, and receiver operating characteristic (SROC) curves, were obtained.

Results

Search results

The literature search identified 526 records through three different sources, including 158 from MEDLINE, 168 from SCOPUS, and 200 from Web of sciences (Figure 1). After removal of the duplicates, the remaining studies (n=158) were screened. Fifty-four records (n=54) were excluded as they did not meet the inclusion criteria. The 104 eligible studies were assessed in details. Sixtysix studies were included (Table 1-3). The reasons for exclusion of the 38 studies were the lack of data allowing direct method comparison or the impossibility to extract those data, LAMP targeting gametocytes only or non-human plasmodium species (P. chabaudi), detection of single nucleotide polymorphism, non-human samples (monkeys, mosquitos) and reviews.

Study characteristic

Sixty-six studies published between 2006 and 2019 were included leading to 30641 individual LAMP tests compared either to microscopy, RDT or PCR or a combination of these comparators. 51% (34/66) of the studies were retrospective and 10.6% (7/66) were using samples from asymptomatic people. Most of them (41%, 27/66) were conducted in South-East Asian countries, 27% (18/66) in Africa, 24% (16/66) in Europa or North America from malaria imported cases, and 7% in South-America (5/66). The reference methods used for comparison with LAMP was microscopy for 53 studies (19418 tests), PCR for 55 studies (17479 tests) and RDTs for 17 studies (2065 tests). Most of the LAMP tests (33/66) were performed using LoopAmp (Eiken, Tochigi, Japan), 39% (26/66) using an in-house method or other commercial test, and 10% (7/66) using Illumigene Malaria (Meridian Biosciences, Cincinatti, USA). Three methods were used to read LAMP tests averaged among naked-eyes, turbidimetry, or fluorescence. Among these studies, 11 were designed to detect P. falciparum, 13 to detect P. vivax, and 10 to detect P. falciparum and P. vivax. Others studies were designed to detect different combinations of Plasmodium species. A vast majority of studies were using blood samples, but saliva and urines were tested in three studies.

Quality and heterogeneity of diagnostic studies

Risk of bias for patient selection was considered low in 56% of the diagnostic studies and high in 36% (Figure 2). Reporting and execution of LAMP was adequate in 97% of the studies (Figure 2).

The quality of verification with a reference standard was good in 100% of the studies (Figure 2). Statistical analysis of heterogeneity found no substantial or moderate heterogeneity between studies included in the analysis of LAMP compared to microscopy and/or PCR, as well as for the sub-group analysis of P. vivax. The value of I² was 0% according to the bivariate analysis (REML) and <60% according to the univariate analysis (DOR).

Results of synthesis and sensitivity analysis

Table 4 shows the pooled sensitivity and specificity, AUC, DOR and I² according to the bivariate analysis (REML) and OR and I² according to the univariate analysis (DOR).

LAMP compared to microscopy

Fifty-five studies comparing LAMP with microscopy were included, for a total of 19687 tests. Most of the studies showed a sensitivity > 0.95 (43/55, 78%) and a specificity > 0.95 (33/55, 60%) (Figure 3). There was no difference between pooled sensitivity of all included studies (0.971 [0.965-0.985]) compared to a selection of the best studies (0.974 [0.960-0.984]). Few studies showed lower sensitivity or specificity values mainly due to the use of non-blood samples or Plasmodium speciesspecific tests. Two studies used samples from patients during pregnancy, and six studies tested asymptomatic people. While a decrease in sensitivity and specificity may be expected for pregnant women and asymptomatic people considering the lower parasitemia, no specific impact of pregnancy or asymptomatic malaria was detected (Figure 4). There is no evidence for a positive or negative impact of LAMP providers (either Eiken, Meridian, or in-house) (Figure 5) and the reader methods (either naked-eyes, turbidimeter, fluorescence) (Figure 6).

LAMP compared to PCR

Fifty-seven studies comparing LAMP with PCR were included, for a total of 17996 tests. Most of the studies showed a sensitivity > 0.95 (48/57, 84%) and a specificity > 0.95 (32/57, 56%) (Figure 7).

Half of the studies (28/57) have a prospective design and 29 a retrospective design. Ten of these studies were detecting P. vivax only (17%) and 9 were detecting P. falciparum only (15.8%). Most of the other studies were designed to detect two or more Plasmodium species. Eight studies (14%) were designed to test asymptomatic people and 2 (3.5%), to test pregnant women; there was no evidence of an impact of these conditions on LAMP results (Figure 8).

LAMP compared to RDTs

Seventeen studies comparing LAMP with RDTs were included, for a total of 15848 tests. Most of the samples were collected from asymptomatic volunteers (13585 tests (68%), 7/17 studies). The sensitivity was 1.0 in 14/17 (82%) and specificity was > 0.95 in 10/17 studies (59%) (Figure 9). The study designs were prospective for 13/17 studies (76%). The majority of samples were collected from different studies conducted in Zanzibar (9329/15848, 58.8%) between 2014 and 2017.

Overall performance of LAMP tests

The pooled sensitivity of LAMP remained between 96 and 98% whatever the comparator (microscopy PCR, or RDT). The pooled specificity of LAMP was around 95%, but is little higher (98%) if best PCR studies are considered. If the analysis is performed for LAMP tests extracted from a selection of best included studies for microscopy defined as studies presenting the lower risk of bias, the accuracy measures of LAMP were closed to the analysis of the overall population for pooled microscopy and PCR (0.974 [0.960-0.984] compared to 0.977 [0.965-0.985]). AUC were found to be above 0.98 whatever the subgroup of studies considered. For all subgroups of studies, DOR were demonstrated to be around 1 000 using a bivariate analysis. Using univariate analysis, OR were above 1 000 for all subgroups. Interestingly, higher DOR or OR were observed if LAMP was compared to PCR. If the subgroup of studies "vivax" was considered, DOR and OR were much lower, around 500 confirming the difficulty of vivax malaria diagnosis.

Discussion

Strengthening malaria diagnosis should be a priority in endemic and non-endemic areas [START_REF] Nema | Strengthening diagnosis is key to eliminating malaria in India[END_REF]. Indeed, malaria elimination is expected by 2030 in many endemic countries, leading to substantial decrease in patient's parasitemia and increase in the proportion of asymptomatic malaria parasites carriers. Microscopy is an appropriate tool when the number of patients suffering malaria is high enough to allow permanent training and experience of microscopists, whereas RDTs are appropriate when the mean parasitemia is over the RDTs' limit of detection. New accurate diagnosis methods are needed to address the new issues related to changing malaria epidemiology. Among these, high sensitivity is a prerequisite. The LAMP method is probably one of the most interesting diagnosis test recently introduced despite its limitation to well-equipped labs with high levels of resources. This method has been tested both in endemic and non-endemic areas in different conditions. The purpose of this study was to compared and analyze the performance of LAMP method versus microscopy, RDTs, and PCR.

We performed an extensive search of all the available studies, without limitation in reference methods, study area, and clinical presentation of tested people, including both symptomatic and asymptomatic. The vast majority of these studies were conducted between 2015 and 2019, while only four studies were conducted before 2010, demonstrating the very recent interest in this diagnosis method. Statistical heterogeneity was tested using the I 2 statistic, which measures the variation across studies due to inter-study heterogeneity. Indeed, heterogeneity were expected to be related to the LAMP method (in-house or commercial, specie-specific detection or not, DNA target used), the method of test reading (automate, naked-eye), the patients (immune or non-immune, area of contamination, levels of parasitemia), and the comparators (microscopy, PCR, RDT). Of note, there were no significant heterogeneity among the studies included in the meta-analysis thus rendering the results conclusive. Moreover, there were no evidence of significant publication biases according to the QUADAS-2. Then, applicability of the test accuracy estimates generated in this study is high due to the high number of studies included leading to approximately 20 000 tests performed for each comparator, to the low risk of bias, and to the absence of heterogeneity of the included studies.

AUC above 0.98 demonstrated that LAMP is a test with an excellent specificity and sensitivity.

Whatever the group of studies considered, DOR around 1000 reflected a test with an excellent discriminant power. Concordant OR were found using univariate analysis, thus supporting that the results were robust whatever the method of analysis used. Higher DOR or OR observed if LAMP was compared to PCR may be explained by the fact that LAMP and PCR are both molecular tests whereas microscopy is not; PCR is moreover considered as the direct comparator in this metaanalysis and microscopy as the clinical practice comparator. If the subgroup of studies "vivax" was considered, DOR and OR were much lower indicating that the test is a little less efficient to detect vivax malaria. Among the included studies, LAMP method varied depending on the targeted Plasmodium species, the primers sequences, the amplification time, the method for signal detection, the commercial or in-house method, and the samples used (blood, saliva, urine). Despite this, no significant changes in the test performances were observed, except for non-blood samples (saliva, urine) showing lower diagnosis accuracy, as previously demonstrated by biological evidence.

The primary audience of this meta-analysis is clinicians, biologists, public health policy makers, and stakeholders. The results of this study may help them to choose the most appropriate first line diagnosis method depending on the local requirements and availabilities. Microscopic analysis of stained blood smears is the reference for malaria diagnosis since decades. However, its limitations linked to the required skills and the recent availability of new methods may provide an opportunity for changing the malaria diagnosis paradigm. This study gathers evidence for the future of malaria diagnosis and the implementation of LAMP as a reference test for malaria diagnosis.

This meta-analysis has some limitations. First, the search strategy was performed without using search filters, but unpublished data were not taken into account as well as abstract from meetings since all the requested information related to the study quality were not available. Considering the high number of included studies and the absence of heterogeneity among the included studies, it is expected that potential missing studies should not have a significant impact on the meta-analysis results. Second, studies from endemic and non-endemic areas were included and pooled leading to the possibility of bias linked to the level of circulating parasitemia in the studied population.

However, the aim of this study was to evaluate the overall diagnosis accuracy of LAMP for malaria and this question is not related to the level of malaria endemicity. The potential impact of transmission area could be detected by the difference between symptomatic and asymptomatic people. The data analysis showed that there was no difference in diagnosis accuracy of LAMP using blood samples from these two populations. Third, LAMP method was compared to microscopy, RDTs and PCR. None of these methods should be considered as a standard of truth since methodological difference may lead to significant difference in limit of detection or performances.

Moreover, the training and experience of microscopy operators have a direct impact on the quality of results. Considering this limitation, PCR is certainly the best comparator for LAMP.

Conclusion

The challenge of malaria diagnosis is moving forward detection of asymptomatic carriers presenting low parasitemia in endemic areas where the transmission is declining. Microscopy and RDTs are highly efficient for the diagnosis of febrile malaria patients, and PCR shows high sensitivity and specificity while its practicability is low. The positioning of malaria LAMP in this context needs to be clarified thus the first step is to compare its performances to the historical methods for both symptomatic and asymptomatic patients. This meta-analysis clearly confirms a very high DOR (>1000) of LAMP compared to microscopy, RDT, and PCR. Malaria LAMP is now widely recommended as a first line method for the diagnosis of imported malaria cases in non-endemic countries due to its high negative predictive value. Same performances are observed testing symptomatic and asymptomatic patients, then the impact of LAMP for the control of malaria in the near future is expected to be important.

Figure 1 :

 1 Figure 1: Flow chart

Figure 2 :

 2 Figure 2: Risk of bias graph of studies included in the meta-analysis

Table 1 :

 1 List of included studies comparing LAMP to microscopy

		Years	Authors	TP	FP	FN	TN
	1	2014	(Aydin-Schmidt et al. 2014)	115	1	0	749
	2	2016	(Britton et al. 2016b)	59	3	39	44
	3	2018	(Cheaveau et al. 2018)	51	7	1	290
	4	2010	(Chen et al. 2010)	115	0	2	30
	5	2015	(Cuadros et al. 2015)	0	11	0	151
	6	2017	(De Koninck et al. 2017)	11	1	0	18
	7	2014	(Dinzouna-Boutamba et al. 2014)	96	41	0	40
	8	2018	(Frickmann et al. 2018)	235	3	3	759
	9	2014	(Ghayour Najafabadi et al. 2014)	68	0	0	40
	10	2018	(Girma et al. 2018)	118	7	0	437
	11	2007	(Han et al. 2007)	67	3	1	50
	12	2019	(Hartmeyer et al. 2019)	18	9	0	11
	13	2018	(Kaur et al. 2018)	145	3	0	17
	14	2018	(Kollenda et al. 2018)	177	57	0	289
	15	2019	(Kudyba et al. 2019)	33	6	0	52
	16	2011	(Lau et al. 2011)	13	0	2	59
	17	2016	(Lau et al. 2016)	132	0	0	69
	18	2012	(Lee et al. 2012)	46	19	1	62
	19	2012	(Lu et al. 2012)	160	0	4	50
	20	2010	(Lucchi et al. 2010)	89	1	5	11
	21	2016	(Lucchi et al. 2016)	135	9	1	64
	22	2015	(Marti et al. 2015)	43	4	0	158
	23	2018	(McCreesh et al. 2018)	16	27	15	1861
	24	2014	(Mohon et al. 2014)	105	1	1	104
	25	2016	(Mohon et al. 2016)	69	4	0	67
	26	2015	(Morris et al. 2015)	19	46	0	3918
	27	2016	(Ocker et al. 2016)	116	0	2	15
	28	2015	(Oriero et al. 2015a)	104	23	0	214
	29	2014	(Patel et al. 2014)	91	0	5	45
	30	2017	(Piera et al. 2017)	73	0	0	19

Table 2 :

 2 List of included studies comparing LAMP to PCR

		Years	Authors	TP	FP	FN	TN
	1	2014	(Aydin-Schmidt et al. 2014)	105	2	10	748
	2	2017	(Aydin-Schmidt et al. 2017)	20	2	29	2957
	3	2016	(Britton et al. 2016a)	62	3	36	44
	4	2018	(Cheaveau et al. 2018)	57	0	0	291
	5	2010	(Chen et al. 2010)	116	0	1	30
	6	2015	(Cook et al. 2015)	10	3	0	984
	7	2017	(Cuadros et al. 2017)	29	11	0	387
	8	2014	(Dinzouna-Boutamba et al. 2014)	128	9	0	40
	9	2014	(Ghayour Najafabadi et al. 2014)	68	2	0	40
	10	2018	(Girma et al. 2018)	121	4	0	562
	11	2007	(Han et al. 2007)	67	3	1	50
	12	2019	(Hartmeyer et al. 2019)	27	0	1	10
	13	2017	(Hayashida et al. 2017)	27	0	3	31
	14	2013	(Hopkins et al. 2013)	178	3	21	70
	15	2018	(Kaur et al. 2018)	148	0	0	17
	16	2018	(Kollenda et al. 2018)	234	0	9	280
	17	2019	(Kudyba et al. 2019)	39	0	1	51
	18	2011	(Lau et al. 2011)	12	1	0	61
	19	2016	(Lau et al. 2016)	132	2	0	67
	20	2012	(Lee et al. 2012)	64	1	0	63
	21	2012	(Lu et al. 2012)	160	0	2	50
	22	2010	(Lucchi et al. 2010)	89	1	1	14
	23	2016	(Lucchi et al. 2016)	140	8	4	57
	24	2015	(Marti et al. 2015)	43	0	0	158
	25	2014	(Mohon et al. 2014)	106	0	2	103
	26	2016	(Mohon et al. 2016)	71	2	0	67
	27	2016	(Ocker et al. 2016)	116	0	1	15
	28	2015	(Oriero et al. 2015a)	100	5	7	72
	29	2015	(Oriero et al. 2015b)	120	7	11	203
	30	2013	(Patel et al. 2013)	70	0	4	46
	31	2014	(Patel et al. 2014)	91	0	5	45
	32	2017	(Perera et al. 2017)	67	0	2	630
	33	2017	(Piera et al. 2017)	73	0	0	19
	34	2010	(Polley et al. 2010)	28	0	2	113
	35	2013	(Polley et al. 2013a)	62	14	1	624
	36	2017	(Ponce et al. 2017)	85	4	0	210
	37	2006	(Poon et al. 2006)	96	1	5	100
	38	2010	(Pöschl et al. 2010)	72	0	1	32
	39	2017	(Rypien et al. 2017)	78	2	0	60
	40	2014	(Sattabongkot et al. 2014)	67	0	5	147

Table 3 :

 3 List of included studies comparing LAMP to RDTs

		Years	Authors	TP	FP	FN	TN
	1	2014	(Aydin-Schmidt et al. 2014)
				13	37	0	415
			(Asymptomatic)
	2	2014	(Aydin-Schmidt et al. 2014) (Suspected)	115	6	0	744
	3	2017	(Aydin-Schmidt et al. 2017)	13	9	0	2997
	4	2015	(Cook et al. 2015)	10	8	1	978
	5	2015	(Cuadros et al. 2015)	3	8	0	159
	6	2017	(De Koninck et al. 2017)	8	4	0	18
	7	2014	(Dinzouna-Boutamba et al. 2014)	137	0	0	40
	8	2018	(Girma et al. 2018)	20	105	0	437
	9	2019	(Hartmeyer et al. 2019)	18	9	0	10
	10	2018	(Kaur et al. 2018)	143	5	0	17
	11	2018	(McCreesh et al. 2018)	16	27	15	1861
	12	2016	(Mohon et al. 2016)	61	12	0	79
	13	2015	(Morris et al. 2015)	19	46	0	3918
	14	2015	(Oriero et al. 2015b)	126	1	0	236
	15	2014	(Sattabongkot et al. 2014)	65	0	5	147
	16	2018	(Tambo et al. 2018)	23	24	0	2593
	17	2017	(Tegegne et al. 2017)	9	6	0	72

Table 4 :

 4 Data synthesis of LAMP results compared to microscopy, and/or PCR, or RDT

		LAMP vs microscopy	LAMP vs microscopy LAMP vs microscopy	LAMP vs PCR	LAMP vs PCR	LAMP vs PCR	LAMP vs RDT
		and PCR		(best studies)		(best studies)	(vivax)
	Bivariate
	analysis
	(REML)
	Sensitivity	0.977 (0.965-0.985)	0.971 (0.957-0.980)	0.974 (0.960-0.984)	0.971 (0.957-	0.961 (0.935-	0.948 (0.800-	0.966 (0.921-
	(95%CI.lb-				0.980)	0.976)	0.988)	0.986)
	95%CI.ub)
	Specificity	0.947 (0.924-0.964)	0.956 (0.938-0.968)	0.951 (0.927-0.967)	0.956 (0.938-	0.984 (0.968-	0.956 (0.865-	0.964 (0.921-
	(95%CI.lb-				0.968)	0.988)	0.987)	0.984)
	95%CI.ub)
	AUC	0.985	0.985	0.985	0.985	0.987	0.982	0.984
	DOR	1061.447 (595.199-	913.979 (542.306-	959.506 (511.141-	913.979	1658.503	462.891 (76.554-	911.208
	(2.5%-97.5%)	1892.929)	1540.381)	1801.169)	(542.306-	(824.983-	2798.903)	(295.097-
					1540.381)	3334.165)		2813.651)
	I 2 (%)	0	0	0	0	0	0	0
	Univariate
	analysis (DOR)
	OR	1221.186 (695.726-	1102.801 (651.748-	1158.573 (619.187-	1102.801	1772.017	504.757 (84.814-	943.033
	(95% CI)	2143.512)	1866.015)	2167.826)	(651.748-	(890.516-	3003.993)	(301.015-
					1866.015)	3526.096)		2954.371)
	I 2 (%)	50.0	63.1	58.6	63.1	64.8	83.9	70.7

Funding

None

ALB developed the search strategy. SP provided expertise on malaria diagnosis and MC expertise on meta-analysis. All authors read, provided feed-back and approved the final manuscript.

Conflict of interest

None

Ethical Approval

Not applicable


Athanasios Batakis

Pierre Debs

Dorian Le

D Le Peutrec

Power of the Crowd

Keywords: Galton Watson, Dynamical System, Elections, Random Walk

published or not. The documents may come

Introduction

First let us recall the definition of a Galton Watson tree (GW) and give a few notations. Assume that N is a N-valued random variable following a distribution q: P(N = i) = q i for i ∈ N. In order to have a meaningful probabilistic setting, we assume that q 0 + q 1 = 0 (Bötcher case). Let φ be the root of the tree and N φ an independent copy of N . Then, we draw N φ children of φ: these individuals are the first generation. In the following we write N for N φ for typographical simplicity. At the m-th generation, for each individual x we pick N x an independent copy of N where N x is the number of children of x and so on. The set T, consisting of the root and its descendants, forms a GW of offspring distribution q. We denote by |x| the generation of x and for m ∈ N, T m = {x ∈ T, |x| ≤ m} the GW cut at height m and the leaves of T m are the elements of T m \T m-1 .

Here we want to represent the propagation of an opinion in a population represented by a GW of height m. More precisely consider the set of probability vectors P k defined by P k := p = (p 0 , . . . , p k ) ∈ (R + ) k+1 :

k i=0 p i = 1 and p 0 < 1 ⊂ R k+1 , (1.1)
and fix p ∈ P k . Each node of T m has the opinion {1, . . . , k} according to the following rules:

• Independently of the others, each leaf has an opinion according to p: P(leaf thinks i) = p i , P(the leaf is undecided) = p 0 .

• The opinions spread to nodes at generation m -1 this way (see Figure 1):

(R1) the undecided children have no influence, except when the children are all undecided, in that case their ancestor has no opinion; (R3) if several opinions are equally represented and the others are less, then the ancestor is undecided.

• We repeat this step for level m -2 and so on (see Figure 2).

As claimed, we want to determine the asymptotic behavior of the distribution p(m) of the state at the root of T m when m goes to infinity:

It is hence sufficient to study the behavior of H when acting on P i . If there exists i ∈ {1, . . . , k -1} such that p 1 = • • • = p i > p i+1 ≥ • • • ≥ p k > 0, then i + 1, . . . , k are called minor opinions and otherwise, i.e. if p 1 = p 2 = • • • = p k > 0, we say that we are in the uniform case.

In Section 2, we prove that the major opinions do not vanish when m → ∞, contrary to the minor opinions, and we state in Proposition 2.8 a sufficient criterion to reduce the analysis to the uniform case. The biggest advantage of the uniform case is to study the fixed points of a function defined on (a subset of) R instead of those of a function on P k . It naturally follows that if there is only one major opinion, regardless of the law of reproduction of N , this opinion spreads a.s. to the root asymptotically. Although we have stated a very general problem, our main results below are available in more restrictive cases: we only consider n-ary trees for n ≥ 2 or GW trees supported in 2N + 1 and two major opinions. This includes in particular a binary ("for-against") referendum, an election with two candidates. In the case of a n-ary tree for n ≥ 2, we obtain the following Theorem 1.1 For every k ≥ 2 and p ∈ P k such that p 1 = p 2 > p 3 ≥ • • • ≥ p k and p 1 < 1 2 , p(m) converges to α n , 1-αn 2 , 1-αn 2 , 0 k-2 when m → ∞, where α n is the unique fixed point in (0, 1) of the function

f n : t ∈ [0, 1] -→ k,0≤2k≤n n 2k 2k k 1 -t 2 2k t n-2k ∈ [0, 1].
Moreover, the above convergence remains true when p 1 = p 2 = 1 2 and n is even, whereas p(m) = (0, 1 2 , 1 2 , 0 k-2 ) for every m ≥ 0 when p 1 = p 2 = 1 2 and n is odd. The following result on the GW trees is "just" a corollary as it needs to add a tricky argument to the proof of Theorem 1.1 in the odd case.

Corollary 1.2 Taking a GW tree whose support is included in 2N + 1 and such that E[N 2 ] < ∞, the result of Theorem 1.1 for odd n remains true, replacing α n by α, the unique fixed point in (0, 1)

of f : t ∈ [0, 1] -→ n≥1 q 2n+1 f 2n+1 (t).
As claimed, we make a brief list of the differences in [START_REF] Benjamini | Annihilation and coalescence on binary trees[END_REF] and [START_REF] Debs | Diseases transmission in a z-ary tree[END_REF]. In [START_REF] Benjamini | Annihilation and coalescence on binary trees[END_REF], one can see that for n = 2, the result of Theorem 1.1 is still true for any number of major opinions and the limit is explicit, in other words for all i ≥ 2 and p 1 = p i > p 3 ≥ • • • ≥ p k , p(m) converges to i-1 2i-1 , 1 2i-1 , . . . , 1 2i-1 , 0 k-i . With the rules explained in [START_REF] Debs | Diseases transmission in a z-ary tree[END_REF], the function studied in the uniform case with a n-ary tree and i major opinions is the following:

h n : t ∈ [0, 1 /i] -→ (1 -(i -1)t) n -(1 -it) n
and one can see this as the probability that the first opinion spreads. This function admits a unique fixed point x in (0, 1 /i] and the authors show that for n ∈ {3, 4, 5}, p(m) converges to (1 -ix, x, . . . , x, 0 k-i ). For n ≥ 6, stranger things happen: for instance for n = 6, x is a repelling fixed point of h n and if i = 2, the authors show that there is a unique attracting orbit of prime period 2. Moreover numerical simulations suggest the existence of a unique attracting orbit for every n and i.

Let us come back to the organization of the paper: in Section 3, we give the proof of Theorem 1.1 and Corollary 1.2. If the support of the GW tree is a subset of 2N, we have just succeeded to prove that if there is convergence, it does to α,

1-α 2 , 1-α 2 , 0 k-2 , where α is the unique fixed point in [0, 1) of f : t ∈ [0, 1] -→ n≥1 q 2n f 2n (t).
We have the attractivity but do not succeed to prove that we have convergence for all p ∈ P k . In this section, we also provide bounds for the values of the fixed points α n of the functions f n . The general case seems to be unreachable for the moment, we just have proved the existence of a non-repulsive fixed point, not even its uniqueness. Nevertheless, we give an example where everything works, the geometric law, in Section 4. Finally, in Section 5 we make some remarks and give open questions and Section 6 is an Appendix.

Reduction to the Uniform case

As stated in the introduction, the aim of the present paper is the study of a particular type of dynamical systems: more precisely, given the function H : P k → P k defined below (see (2.2)), we are interested in the behavior of the orbits H ℓ (x) of the elements x ∈ P k (see (1.1)) when ℓ goes to infinity. In this section we give sufficient conditions to reduce the problem to a subfamily of functions H corresponding to the uniform case, namely the functions h k defined below (see (2.4)).

First, we need to specify the function H. Summing on the number of children of a "typical" node and on the number of children with a neutral opinion, the probability that, for i = 1, . . . , k, the i-th opinion spreads to their parent is equal to

n≥2 q n n-1 m0=0 n m 0 p m0 0 S i n-m 0 n -m 0 m 1 , m 2 , . . . , m k k j=1 p mj j , (2.1)
where

S i n = {(m 1 , . . . , m k ) ∈ N k , ∀j = i, m j < m i ∈ N,
k j=1 m j = n} and n m1,m2,...,m k is the multinomial coefficient. Our problem then requires to study the fixed points in P k of the function H : P k → R k+1 defined by:

H i (p 0 , . . . , p k ) =          n≥2 q n n-1 m0=0 n m0 p m0 0 S i n-m 0 n-m0 m1,m2,...,m k k j=1 p mj j when i = 0, 1 - k j=1 H j (p 0 , . . . , p k ).
when i = 0.

(2.

p i+1 = • • • = p k = 0, then H i+1 (p) = • • • = H k (p) = 0
and it is thus sufficient to study H :

P i ⊂ R i+1 → P i .
In what follows we denote, for k ∈ N,

Q k := {p ∈ P k , p 1 ≥ • • • ≥ p k } (2.3)
and according to the previous remark we only need to consider the action of the function H on Q k . In the uniform case, i.e. when p

1 = • • • = p k ∈ (0, 1 k ], one has simply H(1 -kp 1 , p 1 , . . . , p 1 ) = (1 -kh k (p 1 ), h k (p 1 ), . . . , h k (p 1 )), where h k is the real function defined on [0, 1 k ] by h k (x) = n≥2 q n n-1 m0=0 n m 0 (1 -kx) m0 S 1 n-m 0 n -m 0 m 1 , m 2 , . . . , m k x n-m0 ∈ 0, 1 k . (2.4)
The study of the fixed points in Q k of H in the uniform case is thus reduced to the study of the fixed points in (0, 1 k ] of h k . Note also here that 0 is a fixed point of h k which is repulsive, since

h ′ k (0) = n≥2 q n n n -1 S 1 1 1 m 1 , m 2 , . . . , m k = n≥2 n q n = E[N ] ≥ 2 .
Let us also recall that the generating function G of N is defined by

∀s ∈ [-1, 1], G(s) = E s N = n≥0 s n P(N = n) = n≥0 s n q n . (2.5) On (-1, 1), G is C ∞ and: ∀k ∈ N, G (k) (s) = E[N (N -1) . . . (N -k + 1)s N -k ] , (2.6)
implying that

∀k ∈ N, G (k) (1 -) = E[N (N -1) . . . (N -k + 1)] and G (k) (0) = k! q k . (2.7)
In particular, we have here G(0) = P(N = 0) = 0, G ′ (0) = P(N = 1) = 0, and

G ′ (1 -) = E[N ] ≥ 2.
Lemma 2.2 Assume that p ∈ Q k and that G (2) (1) is finite. Then, there exists η > 0 such that

∀m ∈ N, p 1 (m) ≥ β := min{η a q a , p 1 } ,
where a := inf{n ≥ 2, q n = 0}.

Proof. For p ∈ Q k , we have p 1 > 0, p 1 ≥ • • • ≥ p k , and p 0 = 1 - k i=1 p i . We get H 1 (p) > n≥2 q n n n -1 p n-1 0 S 1 1 1 m 1 , m 2 , . . . , m k k j=1 p mj j = n≥2 q n np 1 p n-1 0 = p 1 G ′ 1 - k i=1 p i . (2.8) Since G (2) (1) = E[N (N -1)] ∈ R * + , one can write G ′ (1 -t) = G ′ (1) -tG (2) (1) + ε(t),
where ε(t) t -→ t→0 0.

As a result, there exists 0 < η ′ < G ′ (1) /3G (2) (1) such that |ε(t)| ≤ tG (2) (1)

2 when 0 ≤ t ≤ η ′ . Then, for 0 ≤ x ≤ η := η ′ k and 0 ≤ y ≤ (k -1)x, xG ′ (1 -x -y) ≥ x G ′ (1) -(x + y)G (2) (1) -(x + y) G (2) (1) 2 = x G ′ (1) -3(x + y) G (2) (1) 2
≥ x G ′ (1) - G ′ (1) 2 = x G ′ (1) 2 ≥ x,
where the last inequality follows from

G ′ (1) = E[N ] ≥ 2.
Thus, according to (2.8), H 1 (p) ≥ p 1 when p 1 ≤ η. In addition, when p 1 > η :

H 1 (p) ≥ n≥a q n n n -a p n-a 0 p a 1 = p a 1 a! G (a) (p 0 ) ≥ p a 1 a! G (a) (0) ≥ η a q a > 0.
An obvious recurrence gives the claimed result.

Remark 2.3 Applying the relation (2.8) to a fixed point (p 0 , p 1 , . . . , p k ) ∈ Q k of H we get

p 1 = H 1 (p) > n≥2 q n n n -1 p n-1 0 p 1 = p 1 G ′ (p 0 ) , then, since p 1 > 0, G ′ (p 0 ) < 1.
The following elementary lemma ensures the validity of further results using the differentiability of H on P k . Lemma 2.4 Suppose that G ′ (1) is finite. Let J k := p = (p 0 , . . . , p k ) ∈ (R + ) k+1 : k i=0 p i ≤ 1 and still denote by H i the functions defined by (2.2) on J k for 1 ≤ i ≤ k. Then, these functions are of class C 1 on J k .

Proof. Note first that by definition, for every 1 ≤ i ≤ k and p ∈ J k , H i (p) writes:

H i (p) := n≥2 q n H i,n (p) := n≥2 q n m∈N k+1 ,|m|=n a n,m p m ,
where a n,m ∈ R + and, for every m = (m 0 , . . . , m k ) ∈ N k+1 , |m| := k i=0 m i and p m := k i=0 p mi i . More precisely, for every n ≥ 2 and m ∈ N k+1 satisfying |m| = n,

0 ≤ a n,m ≤ n m and hence H i,n (p) ≤ m∈N k+1 ,|m|=n n m p m = (p 0 + • • • + p k ) n ≤ 1.
It follows that, for every 1

≤ i ≤ k, H i is continuous on J k and satisfies ∀p ∈ J k , H i (p) = n≥2 q n H i,n (p) ≤ n≥2 q n (p 0 + • • • + p k ) n = G(p 0 + • • • + p k ).
Moreover, for every ℓ ∈ {0, . . . , k}, p ∈ J k , and n ≥ 2:

0 ≤ H i,n ∂p ℓ (p) ≤ ∂(p 0 + • • • + p k ) n ∂p ℓ = n(p 0 + • • • + p k ) n-1 ≤ n.
This implies the claimed result, since G ′ (1) = n≥2 nq n is finite.

The following lemma ensures that the minor opinions can not spread to the root asymptotically:

Lemma 2.5 Assume that G (2) (1) is finite. In the (non uniform) case with i < k ∈ N * and p ∈ Q k such that

p 1 = • • • = p i > p i+1 ≥ • • • ≥ p k ≥ 0, it holds p j (m) →
m→∞ 0 for every j ∈ {i + 1, . . . , k}.

Proof. Note that we just have to prove that lim n→∞ p i+1 (n) = 0 when p i+1 > 0. In this case, writing w n = pi+1(n) p1(n) > 0, we can easily see that for every n ≥ 0, w n+1 = w n u n , where:

u n := z≥2 q z z-1 m0=0 z m 0 p m0 0 (n) S 1 z-m 0 z -m 0 m 1 , m 2 , . . . , m k p m1-1 i+1 (n)p mi+1 1 (n) k j=2,j =i+1 p mj j (n) z≥2 q z z-1 m0=0 z m 0 p m0 0 (n) S 1 z-m 0 z -m 0 m 1 , m 2 , . . . , m k p m1-1 1 (n)p mi+1 i+1 (n) k j=2,j =i+1 p mj j (n)
.

For every (m 1 , . . . , m k ) ∈ S 1 z-m0 , since p 1 > p i+1 > 0, we have that

p m1-1 i+1 (n)p mi+1 1 (n) < p m1-1 1 (n)p mi+1 i+1 (n) when m i+1 < m 1 -1,
implying that 0 < w n+1 < w n . Thus, (w n ) is a positive decreasing sequence, and consequently converges to some ℓ ≥ 0. Since w 0 < 1, note that ℓ < 1. By compacity, there exists moreover a subsequence n m such that lim m→∞ p j (n m ) = a j for every j = 0, . . . , k. From Lemma 2.2, we have a 1 > 0. Now, assume that a i+1 > 0. Since ℓ < 1, we have a 1 > a i+1 > 0 and, using the definition of u n :

lim m→∞ u nm = z≥2 q z z-1 m0=0 z m 0 a m0 0 S 1 z-m 0 z -m 0 m 1 , m 2 , . . . , m k a m1-1 i+1 a mi+1 1 k j=2,j =i+1 a mj j z≥2 q z z-1 m0=0 z m 0 a m0 0 S 1 z-m 0 z -m 0 m 1 , m 2 , . . . , m k a m1-1 1 a mi+1 i+1 k j=2,j =i+1 a mj j = ℓ ′ < 1.
This permits us to easily conclude:

ℓ = lim m→∞ w nm+1 = lim m→∞ w nm u nm = ℓℓ ′ < ℓ ,
which is a contradiction. Then a i+1 = 0 and consequently p i+1 (n) → n→∞ 0.

Remark 2.6 Actually, for every j ∈ {i + 1, . . . , k}, the convergence p j (m)

→ m→∞ 0 is exponential, i.e. ∃ a ∈ (0, 1) , ∃ C > 0 , ∀ m ∈ N : 0 ≤ p k (m) ≤ • • • ≤ p i+1 (m) ≤ Ca m .
(2.9)

Note that to prove (2.9), it suffices to show that lim sup n→∞ u n < 1, where (u n ) n≥0 is the positive sequence introduced in the preceding proof. But since p i+1 (m) → m→∞ 0, there exists a sequence (ε n ) n≥0 converging to 0 such that:

u n = z≥2 q z z z -1 p z-1 0 (n) + ε n z≥2 q z z-1 m0=0 z m 0 p m0 0 (n) S 1 z-m 0 z -m 0 m 1 , m 2 , . . . , m k p m1-1 1 (n)p mi+1 i+1 (n) k j=2,j =i+1 p mj j (n) ≤ z≥2 q z z z -1 p z-1 0 (n) + ε n z≥2 q z z z -1 p z-1 0 (n) + z≥2 q z z 0 p z-1 1 (n) = G ′ (p 0 (n)) + ε n G ′ (p 0 (n)) + p -1 1 (n)G(p 1 (n)) ≤ G ′ (1 -β) G ′ (1 -β) + G(β) + ε n G(β) -→ n→∞ G ′ (1 -β) G ′ (1 -β) + G(β) < 1 ,
which implies lim sup n→∞ u n < 1 and then (2.9).

In what follows, given a real function f , we say that a fixed point x of f is linearly attracting for f when f is differentiable at

x and |f ′ (x)| < 1. Proposition 2.7 Assume that G ′ (1) is finite. Let i ≤ k ∈ N * and assume that xi ∈ (0, 1 i ] is a lin- early attracting fixed point for the function h i defined in (2.4). Then, x = (1 -ix i , xi , . . . , xi , 0 k-i ) is an attracting fixed point for H : Q k,i := {p ∈ Q k , p 1 = • • • = p i > p i+1 ≥ • • • ≥ p k } → Q k,i .
Proof. To prove that x is attracting, note that it is sufficient to show that all the eigenvalues of the matrix A := ∂ H ∂x (y) are in (-1, 1), where y = (x i , 0 k-i ) and H = ( H1 , . . . ,

Hk-i+1 ) = (H 1 , H i+1 , . . . , H k ) is a truncated version of H. For ℓ ∈ {1, . . . , k -i + 1}, Hℓ is then defined by n≥2 q n n-1 m0=0 n m 0   1 -ix 1 - k j=i+1 x j   m0 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj 1 k j=i+1 x mj j ,
where l = 1 when ℓ = 1 and l = ℓ + i -1 when ℓ ∈ {2, . . . , k -i + 1}. Let us prove that the matrix A is upper triangular, which will immediately lead to the knowledge of its spectrum. For this purpose, let us compute ∂ Hℓ ∂xr (y) when l ≥ r ∈ {1, i + 1, . . . , k}. First, ∂ Hℓ ∂xr (x 1 , x i+1 , . . . , x k ) equals, when r = 1,

- n≥2 q n n-1 m0=0 n m 0 im 0   1 -ix 1 - k j=i+1 x j   m0-1 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj 1 k j=i+1 x mj j + n≥2 q n n-1 m0=0 n m 0   1 -ix 1 - k j=i+1 x j   m0 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k   i j=1 m j   x i j=1 mj-1 1 k j=i+1 x mj j ,
and, when r ∈ {i + 1, . . . , k},

- n≥2 q n n-1 m0=0 n m 0 m 0   1 -ix 1 - k j=i+1 x j   m0-1 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj 1 k j=i+1 x mj j + n≥2 q n n-1 m0=0 n m 0   1 -ix 1 - k j=i+1 x j   m0 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj 1 m r x mr -1 r k j=i+1, =r x mj j .
Thus, by evaluating at y = (x i , 0 k-i ):

• When ℓ = r = 1, ∂ H1 ∂x 1 (y) = - n≥2 q n n-1 m0=0 n m 0 im 0 (1 -ix i ) m0-1 S 1 n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj i k j=i+1 0 mj + n≥2 q n n-1 m0=0 n m 0 (1 -ix i ) m0 S 1 n-m 0 n -m 0 m 1 , m 2 , . . . , m k   i j=1 m j   x i j=1 mj -1 i k j=i+1 0 mj = - n≥2 q n n-1 m0=0 n m 0 im 0 (1 -ix i ) m0-1 S 1 n-m 0 n -m 0 m 1 , . . . , m i , 0 k-i xn-m0 i + n≥2 q n n-1 m0=0 n m 0 (1 -ix i ) m0 S 1 n-m 0 n -m 0 m 1 , . . . , m i , 0 k-i (n -m 0 )x n-m0-1 i = h ′ i (x i ) ,
where the function h i has been defined in (2.4).

• When l > r = 1,

∂ Hℓ ∂x 1 (y) = - n≥2 q n n-1 m0=0 n m 0 im 0 (1 -ix i ) m0-1 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj i k j=i+1 0 mj + n≥2 q n n-1 m0=0 n m 0 (1 -ix i ) m0 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k   i j=1 m j   x i j=1 mj -1 i k j=i+1 0 mj = 0 . • Lastly, when l ≥ r > 1, ∂ Hℓ ∂x r (y) = - n≥2 q n n-1 m0=0 n m 0 m 0 (1 -ix i ) m0-1 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj i k j=i+1 0 mj + n≥2 q n n-1 m0=0 n m 0 (1 -ix i ) m0 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj i m r 0 mr-1 k j=i+1, =r 0 mj = n≥2 q n n-1 m0=0 n m 0 (1 -ix i ) m0 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj i m r 0 mr-1 k j=i+1, =r 0 mj
which, when l > r, equals 0 and, when l = r, equals

n≥2 q n n-1 m0=0 n m 0 (1 -ix i ) m0 n -m 0 0 l-1 , 1, 0 k-l = n≥2 nq n (1 -ix i ) n-1 = G ′ (1 -ix i ).
As claimed, A is thus upper triangular and its spectrum is

{h ′ i (x i ), G ′ (1 -ix i )}. Moreover, h ′ i (x i ) belongs to (-1, 1)
by assumption and, according to Remark 2.3, G ′ (1 -ix i ) also belongs to (-1, 1) (since H(x) = x ∈ Q k ). The statement of Proposition 2.7 follows.

The following proposition is an adaptation of Proposition 3.11 in [START_REF] Debs | Diseases transmission in a z-ary tree[END_REF].

Proposition 2.8 Assume that G (2) (1) is finite. Let i ≤ k ∈ N * and assume that xi ∈ [0, 1 i ] is a linearly attracting fixed point for the function h i defined in (2.4) whose basin of attraction contains (0,

1 i ]. Then, xi ∈ (0, 1 i ] and x = (1 -ix i , xi , . . . , xi , 0 k-i ) is a globally attracting fixed point for H : Q k,i = {p ∈ Q k , p 1 = • • • = p i > p i+1 ≥ • • • ≥ p k } → Q k,i .
Proof. Note first that when k ≥ i, p 1 = • • • = p i > 0, and p j = 0 for j = i + 1, . . . , k, then p(m) converges to (1 -ix i , xi , . . . , xi , 0 k-i ) by hypothesis. According to Lemma 2.2, it thus holds xi > 0.

We have now to extend this result when k > i,

p 1 = • • • = p i > p i+1 ≥ • • • ≥ p k ≥ 0, and p i+1 > 0.
Let us again consider the truncated version of H, H = ( H1 , . . . ,

Hk-i+1 ) = (H 1 , H i+1 , . . . , H k ),
where, for ℓ ∈ {1, . . . , k + i -1}, Hℓ is defined by

n≥2 q n n-1 m0=0 n m 0   1 -ix 1 - k j=i+1 x j   m0 S l n-m 0 n -m 0 m 1 , m 2 , . . . , m k x i j=1 mj 1 k j=i+1 x mj j ,
where l = 1 when ℓ = 1 and l = ℓ + i -1 when ℓ ∈ {2, . . . , k -i + 1}. Let us also define the set

Qk,i :=    (x 1 , . . . , x k-i+1 ) ∈ R k-i+1 , 1 i ≥ x 1 > x 2 ≥ x 3 ≥ • • • ≥ x k-i+1 ≥ 0, ix 1 + k-i+1 j=2 x j ≤ 1    .
Let us show that, for every p ∈ Qk,i , Hm (p) converges to x = (x i , 0 k-i ), which is equivalent to the convergence result stated in Proposition 2.8. We fix p = (p 1 , p i+1 , p i+2 , . . . , p k ) ∈ Qk,i and recall from Lemma 2.2 that, for every m ∈ N, Hm (p

) 1 = p 1 (m) ≥ β := min{η a q a , p 1 }.
As xi is a linearly attracting fixed point of h i , for every ε > 0 small enough, B(x i , ε /2) is h i -invariant. Now, while noting that Hm (x, 0 k-i ) = (h m i (x), 0 k-i ) for every m ∈ N, we define

E m := {x ∈ [0, 1] : h m i (x) ∈ B(x i , ε /
2)} for some arbitrarily small ε > 0. The sequence (E m ) m≥0 is an ascending chain of sets and from the convergence in the uniform case,

[β, 1 /i] ⊂ m≥0 E m .
As the inverse image of an open set of R by a continuous function, E m is an open set for all m ∈ N.

Since (E m ) m≥0 is an increasing sequence of open sets covering the compact [β,

1 /i], there exists N ∈ N such that [β, 1 /i] ⊂ N m=0 E m = E N , implying that: ∀x ∈ [β, 1 /i], HN (x, 0 k-i ) ∈ B(x i , ε /2) × {0} k-i ⊂ B(x, ε /2).
On the closed bounded set

G := [β, 1 /i] × R k-i + ∩ Qk,i
, HN is uniformly continuous and thus there exists δ > 0 such that

∀(x, y), (x ′ , y ′ ) ∈ G , (x, y) -(x ′ , y ′ ) ≤ δ ⇒ HN (x, y) -HN (x ′ , y ′ ) ≤ ε /2.
According to Lemma 2.5, p j (m) = H m (p) j → 0 for every j ∈ {i + 1, . . . , k}. Consequently, there exists

N 1 ∈ N such that, for every m ∈ N: m ≥ N 1 implies (p i+1 (m), . . . , p k (m)) ≤ δ and then HN (p 1 (m), p i+1 (m), . . . , p k (m)) -HN (p 1 (m), 0 k-i ) ≤ ε /2.
Thus, for every m ≥ N 1 , the fact that p 1 (m) ∈ [β, 1 /i] implies:

A : = (p 1 (m + N ), p i+1 (m + N ), . . . , p k (m + N )) -x = HN (p 1 (m), p i+1 (m), . . . , p k (m)) -x ≤ HN (p 1 (m), p i+1 (m), . . . , p k (m)) -HN (p 1 (m), 0 k-i ) + HN (p 1 (m), 0 k-i ) -x ≤ ε /2 + ε /2 = ε,
which concludes the proof of Proposition 2.8 since ε > 0 is arbitrarily small. Remark 2.9

1. In the statement of Proposition 2.8, we can actually lighten the hypothesis |h ′ i (x i )| < 1 using the fact that h i is C 1 on [0, 1 i ] (which follows from Lemma 2.4) and a monotonicity argument. However, this would require unnecessary extra work as this hypothesis is satisfied in all our examples. 2. It is not difficult to prove that if we have just one major opinion, it spreads almost surely to the root. Indeed, in the "uniform" case with only one opinion, according to the rules, the probability that in a GW of height m the unique opinion does not spread to the root is equal to:

p 0 (m) = n≥2 q n p n 0 (m -1) = G(p 0 (m -1)) = G m (p 0 ).
Since N ≥ 2 a.s., G is strictly convex on [0, 1] with 0 and 1 as sole fixed points. It follows that :

∀ p 0 ∈ [0, 1) , lim m→∞ G m (p 0 ) = 0 and thus ∀ p 1 ∈ (0, 1] , lim m→∞ p 1 (m) = 1 .
Proposition 2.8 then ensures the convergence in the non uniform case with one major opinion.

Conclusion :

From the above results, one deduces that:

• For any p ∈ Q k , defining i := max ℓ ∈ {1, . . . , k} , p ℓ = p 1 ∈ {1, . . . , k}, we have p j (m) → m→∞ 0 for every j ∈ {i + 1, . . . , k}.

The accumulation points of the sequence (H ℓ (p)) ℓ≥0 have thus the form (1-ix i , xi , . . . , xi , 0 k-i ) where, according to Lemma 2.2, xi ∈ (0, 1 i ]. In particular, the fixed points (resp. the m-cycles) of H in Q k are the (1-ix i , xi , . . . , xi , 0 k-i ), where i ∈ {1, . . . , k} and xi is a fixed point (resp. a m-cycle) of h i in (0,

1 i ]. • Recall that, for i ≤ k ∈ N * , Q k,i = {p ∈ Q k , p 1 = • • • = p i > p i+1 } and H(Q k,i ) ⊂ Q k,i .
Proposition 2.7 implies that the fixed point (1

-ix i , xi , . . . , xi , 0 k-i ) of H : Q k,i → Q k,i is attracting when xi ∈ (0, 1 i ] is linearly attracting for h i . Conversely, if (1-ix i , xi , . . . , xi , 0 k-i ) is attracting for H : Q k,i → Q k,i , then xi ∈ (0, 1
i ] is obviously attracting for h i . Finally, according to Proposition 2.8, if the basin of attraction of a fixed point xi of h i in (0, 1 i ] is (0, 1 i ], then the basin of attraction of (1

-ix i , xi , . . . , xi , 0 k-i ) for H : Q k → Q k is Q k,i
, and the converse is clearly true.

The 2 major opinions case or the second run of an election

In this section, we consider only two major opinions. Moreover, contrary to the previous section, we study the probability that the "neutral" opinion spreads, i.e. that in a group of n individuals, no opinion has a majority. With in mind the results of the preceding section, we focus on the uniform case: if t ∈ [0, 1] is the probability that a given individual gives a white vote, the probability of each opinion is 1-t 2 and the probability of the group to come up undecided is then given by

H 0 t, 1 -t 2 , 1 -t 2 = 1 -2h 2 1 -t 2 := f n (t) = k,0≤2k≤n n 2k 2k k 1 -t 2 2k t n-2k .
We will thus study the fixed points of f n in [0, 1), or equivalently the fixed point of h 2 in (0, 1 2 ].

We start by a crucial remark providing an integral formula for the functions f n .

Lemma 3.1 For all 0 ≤ t ≤ 1:

f n (t) = 1 π π 0 ((1 -t) cos x + t) n dx. (3.1)
Proof. Recall the Wallis integral for all k ≥ 0:

π 2 0 cos 2k xdx = π 2 (2k)! (k!2 k ) 2 = π 2 2k+1 2k k ⇔ 1 2π π -π cos 2k xdx = 1 2 2k 2k k (3.2)
and note that, using the substitution u = π 2 -x :

π 0 cos 2k+1 xdx = π 2 -π 2 sin 2k+1 udu = 0. (3.3)
Then, using (3.2) and (3.3), we can write:

f n (t) = k,0≤2k≤n n 2k (1 -t) 2k t n-2k 1 2 2k 2k k = k,0≤2k≤n n 2k (1 -t) 2k t n-2k 1 2π π -π cos 2k xdx = k,0≤k≤n n k (1 -t) k t n-k 1 2π π -π cos k xdx = 1 2π π -π k,0≤k≤n n k (1 -t) k t n-k cos k xdx = 1 2π π -π ((1 -t) cos x + t) n dx,
and we easily conclude using the parity.

There is a more elegant way to prove Lemma 3.1 by using Fourier series: let us consider the random walk on L 2 (S 1 ), the space of square integrable functions on the circle, defined on its usual (e k = e ikx ) k∈Z basis by

Z 0 = 1 and P(Z n+1 = e k |Z n = e ℓ ) = 1 -p 2 1 k=ℓ-1 + 1 -p 2 1 k=ℓ+1 + p1 k=ℓ ,
where 0 ≤ p ≤ 1.

Using that P Zn+1 /Zn = e ix = P Zn+1 /Zn = e -ix = 1-p 2 , that P ( Zn+1 /Zn = 1) = p, and the independence of the random variables Zn+1 /Zn, we get

f n (p) = P(Z n = 1).
The (infinite) matrix associated to the walk is

A =           . . . . . . . . . . . . . . . . . . • • • . . . 0 • • • p 1-p 2 0 0 • • • 0 0 • • • 1-p 2 p 1-p 2 0 • • • 0 0 • • • 0 1-p 2 p 1-p 2 • • • 0 0 • • • 0 0 1-p 2 p • • • 0 . . . • • • . . . . . . . . . . . . . . . . . .          
, so that A applied to e ℓ equals 1-p 2 e ℓ-1 + 1-p 2 e ℓ+1 + pe ℓ . Let L be the associated linear operator on L 2 (S 1 ). A straightforward easy computation shows that L(e iℓx ) = ((1 -p) cos x + p) e iℓx , which implies that L is a scalar operator :

L : L 2 (S 1 ) ∋ h → ((1 -p) cos x + p) h ∈ L 2 (S 1 )
and therefore the iterated operator L n is given by

L n (h) = ((1 -p) cos x + p) n h.
On the other hand,

P(Z n = 1) =< A n e 0 , e 0 >=< L n 1, 1 >= 1 2π 2π 0 ((1 -p) cos x + p) n dx. Remark 3.2 As a polynomial of degree n, f n ∈ C ∞ ([0, 1]
) and, for every k ∈ {0, . . . , n}: Proof. To prove the unicity, we have to distinguish two cases according to the parity of n.

f (k) n (t) = 1 π n! (n -k)! π 0 (1 -cos x) k (t(1 -cos x) + cos x) n-k dx. ( 3
• Odd case (see the orange graph of Figure 3): Using Lemma 3.1, Remark 3.2 and (3.2) , we have:

f n (0) = 1 π π 0 cos n xdx = 0 and f n (1) = 1 π π 0 dx = 1, (3.5)
f ′ n (0) = n π π 0 cos n-1 xdx - π 0 cos n xdx = n 2 n-1 n -1 n-1 2 > 1, (3.6)
f ′ n (1) = n π π 0 (1 -cos x)dx = n > 1, (3.7)
implying that f n has at least one fixed point in (0, 1). The inequality in (3.6) follows from (see (6.6))

∀n ≥ 1, 2 2π(2n + 1) < ξ 2n := 1 2 2n 2n n (3.8)
Note also that the formulas (3.5) are direct with the spreading rules. Moreover, Remark 3.2 with k = 3 gives (note that n ≥ 3 since n is odd):

∀t ∈ [0, 1], f (3) n (t) = 1 π n! (n -3)! π 0 (1 -cos x) 3 (t(1 -cos x) + cos x) n-3 dx > 0,
implying that f n has at most three fixed points in [0, 1]. As a result, f n has a unique fixed point α n in (0, 1).

• Even case (see the blue graph of Figure 3):

Since

f n (0) = n 2 n n n 2 > 0, f n (1) = 1 and f ′ n (1) = n > 1,
we deduce that f n has at least one fixed point in (0, 1). Using again Remark 3.2, f n is strictly convex in [0, 1] and has thus at most two fixed points in this interval. As a result, f n has a unique fixed point α n in (0, 1).

Thanks to the unicity of α n , we just have to show f ( 1 /2) < 1 /2 to obtain α n < 1 /2. According to the formula (6.3) of Lemma 6.2:

f n 1 2 = 1 2 n k,0≤2k≤n 2 -2k n 2k 2k k = 1 2 2n 2n n = ξ 2n . (3.9)
As ξ 2 = 1 /2 and (ξ 2n ) n≥0 is a strictly decreasing sequence according to Lemma 6.4, ξ 2n < 1 /2 for all n ≥ 2.

Remark 3.4 If we look at the GW case, we have to study the fixed points in [0, 1) of:

f : t ∈ [0, 1] → n≥2 q n f n (t).
With similar arguments as those of the previous proof, it is not difficult to prove the existence of a fixed point α ∈ (0, 1 /2), since

f (1) = 1, f ′ (1) = E[N ] > 1, and f ( 1 /2) < 1 /2.
Indeed, if f (0) = n≥1 q 2n f 2n (0) > 0, we have our result and, otherwise, q 2n = 0 for every n ≥ 1, so f ′ (0) = n≥1 q 2n+1 f ′ 2n+1 (0) > 1 and we can easily conclude. Moreover, if the support of N is a subset of 2N or of 2N + 1, we have the unicity of α by the arguments used in the previous proof.

The odd case

u m+1 = f n (u m ) for m ≥ 0. Since ∀t ∈ [0, 1] , f ′ n (t) = n π π 0 (1 -cos x)(t(1 -cos x) + cos x) n-1 dx > 0,
the function f n is strictly increasing on [0, 1] and a simple reasoning shows that if x 0 ∈ (0, α n ), (u m ) is strictly increasing and bounded above by α, and if x 0 ∈ (α, 1), (u m ) is strictly decreasing and bounded below by α n . As a consequence, for all x 0 ∈ (0, 1):

lim m→∞ u m = α n .
Remark 3.6

1. Note that the fixed point α n of f n is a linearly attracting, i.e. f ′ n (α n ) ∈ (-1, 1). Indeed, it holds obviously f ′ n (α n ) ∈ (0, 1] by the preceding proof. Moreover, the equality f ′ n (α n ) = 1 would imply that α n is an inflection point of f n and then that f

(2) n (α n ) = 0, which would lead to f n (t) > t on (α n , 1] since f (2)
n is strictly increasing on [0, 1], a contradiction.

2. The reasoning here can be applied for a GW with a reproduction law whose support is a subset of 2N + 1. Indeed, the studied function f = n≥1 q 2n+1 f 2n+1 is then strictly increasing on [0, 1] and admits a unique fixed point on this interval. The case of a n-ary tree when n ≥ 3 is odd Note first that in this case, the statement of Theorem 1.1 is obvious when p 1 = p 2 = 1 2 , since 0 is a fixed point of f n and thus (0, 1 2 , 1 2 , 0 k-2 ) is a fixed point of H. It thus remains to prove Theorem 1.1 in this case when p 1 = p 2 ∈ (0, 1 2 ). To this end, let us fix p ∈ Q k (with k ≥ 2) such that p 1 = p 2 ∈ (0, 1 2 ) and p 2 > p 3 ≥ • • • ≥ p k ≥ 0, and let us assume for a moment that there exists β ′ > 0 such that p 0 (m) ≥ β ′ for every m > 0. It then holds 0

< β ≤ p 1 (m) ≤ 1-β ′ 2 < 1
2 for every m > 0. Thus, with the same arguments as those used in the proof of Proposition 2.8, but working now with the compact set [β,

1-β ′ 2 ] ⊂ (0, 1 2 ) instead of [β, 1 2 ], one shows that p(m) -→ α n , 1 -α n 2 , 1 -α n 2 , 0 k-2 when m → ∞.
To conclude, let us then prove that when p 1 = p 2 ∈ (0, 1 2 ), there exists β ′ > 0 such that p 0 (m) ≥ β ′ for every m > 0.

First, let us observe from the spreading rules that if p 0 (ℓ) > 0 for some ℓ ∈ N, then p 0 (m) > 0 for every m ≥ ℓ. In particular, p 0 (m) > 0 for every m ∈ N when p 0 > 0 and, when p 0 = 0, then k ≥ 3 and p 3 > 0, which implies p 0 (1) > 0 (also from the spreading rules, since n ≥ 3 is odd). Consequently: p 0 (m) > 0 for every m > 0.

Moreover, note from the spreading rules that for every m > 0, Using now Remark 2.6 and 1p 0 (m) ≥ 2β, note also that there exist C = 2βD > 0 and a ∈ (0, 1) such that for every m > 0, k ℓ=3 p ℓ (m) ≤ Ca m and thus

2p 1 (m) = 1 -p 0 (m) - k ℓ=3 p ℓ (m) ≥ 1 -p 0 (m) 1 - 1 2β k ℓ=3 p ℓ (m) ≥ 1 -p 0 (m) 1 -Da m .
Take m 0 ∈ N * and b ∈ (a, 1) such that Da m ≤ b m for every m ≥ m 0 . It then follows from (3.10) and (3.11) that:

∀m ≥ m 0 , p 0 (m + 1) ≥ (1 -b m ) n k,0≤2k≤n n 2k 2k k 1 -p 0 (m) 2 2k p n-2k 0 (m) = (1 -b m ) n f n (p 0 (m)) ≥ (1 -b m ) n min{p 0 (m), α n }.
Reasoning by induction thus leads to:

∀m ≥ m 0 , p 0 (m) ≥ min{p 0 (m 0 ), α n } m-1 ℓ=m0 (1 -b ℓ ) n ≥ min{p 0 (m 0 ), α n } A n ,
where A := +∞ ℓ=m0 (1-b ℓ ) is positive since the convergence of the Neumann series ℓ≥m0 b ℓ implies the one of ℓ≥m0 ln(1 -b ℓ ) to the real negative number B = ln(A). It follows that for every m > 0, p 0 (m) ≥ β ′ := min{A n α n , A n p 0 (m 0 ), p 0 (m 0 -1), . . . , p 0 (1)} > 0 , which concludes the proof of Theorem 1.1 in the case of a n-ary tree when n ≥ 3 is odd.

The general case of a GW tree supported in 2N + 1

We now look at the function f = n≥1 q 2n+1 f 2n+1 and at the corresponding function H. As above, the statement of Corollary 1.2 is obvious when p 1 = p 2 = 1 2 , since 0 is a fixed point of f and thus (0, 1 2 , 1 2 , 0 k-2 ) is a fixed point of H. It thus just remains to prove it when p 1 = p 2 ∈ (0, 1 2 ), so we fix

p ∈ Q k (with k ≥ 2) such that p 1 = p 2 ∈ (0, 1 2 ) and p 2 > p 3 ≥ • • • ≥ p k ≥ 0.
Reasoning as we did above with a n-ary tree when n ≥ 3 is odd, it is sufficient to show that there exists β ′ > 0 such that p 0 (m) ≥ β ′ for every m > 0.

To this end, note first that the relation

n≥1 q 2n+1 f ′ 2n+1 (0) > 1 (see (3.6)) implies the existence of n * ∈ N * such that 1≤n≤n * q 2n+1 f ′ 2n+1 (0) > 1. The function f := 1≤n≤n * q 2n+1 f 2n+1 hence satisfies f (0) = 0, f ′ (0) > 1, and f ≤ f on [0, 1], implying f (1) ≤ f (1) = 1.
It thus admits at least one fixed point in (0, 1) and we define α * as the smallest one. It follows that f (x) > x on (0, α * ) and, since f is increasing on [0, 1], the function f satisfies f (x) ≥ min{x, α * } for every x ∈ [0, 1]. We can then conclude by following the same lines as above for a n-ary tree: again, the spreading rules imply that p 0 (m) > 0 for every m > 0 and that

p 0 (m + 1) = H 0 (p 0 (m), . . . , p k (m)) ≥ n≥1 q 2n+1 k,0≤2k≤2n+1 2n + 1 2k 2k k p 2k 1 (m)p 2n+1-2k 0 (m) ≥ n * n=1 q 2n+1 k,0≤2k≤2n+1 2n + 1 2k 2k k p 2k 1 (m)p 2n+1-2k 0 (m) .
Reasoning as in the lines following (3.11) then implies the existence of m 0 ∈ N * and of b ∈ (0, 1) such that, for every m ≥ m 0 ,

p 0 (m + 1) ≥ n * n=1 q 2n+1 (1 -b m ) 2n+1 k,0≤2k≤2n+1 2n + 1 2k 2k k 1 -p 0 (m) 2 2k p 2n+1-2k 0 (m) ≥ (1 -b m ) 2n * +1 f (p 0 (m)) ≥ (1 -b m ) 2n * +1 min{p 0 (m), α * } and then p 0 (m) ≥ min{p 0 (m 0 ), α * } m-1 ℓ=m0 (1 -b ℓ ) 2n * +1 ≥ min{p 0 (m 0 ), α n } +∞ ℓ=m0 (1 -b ℓ ) 2n * +1 > 0 .
This implies the existence of β ′ > 0 such that p 0 (m) ≥ β ′ for every m > 0 and then concludes the proof of Corollary 1.2. The only difficulty to obtain this statement is to prove that

The even case

f ′ n (α n ) > -1. Indeed, since f n (0) > 0 and f ′ n (1) > 1 = f n (1)
, the unicity of α n leads to

f n (x) > x, ∀x ∈ (0, α n ) and f n (x) < x, ∀x ∈ (α n , 1)
,

and hence f ′ (α n ) ≤ 1. Since moreover f is strictly convex on [0, 1], we have f ′ (α n ) < 1 since the equality f ′ (α n ) = 1 would imply f n (x) > x on [0, 1], a contradiction.
A direct proof of Proposition 3.7 using integral estimates relying on the relation (3.1) is proposed in the following subsection. Nevertheless, we would like to point out that we have come up with a totally independent proof using Budan's theorem: Theorem 3.8 (of Budan-Fourier) [START_REF] Conkwright | An Elementary Proof of the Budan-Fourier Theorem[END_REF] Let P (x) = 0 be a polynomial equation with real coefficients of degree n and let a < b be any two real numbers. Then, there exists k ∈ N such that the number of roots (counted with multiplicity) of this equation in the interval (a, b] is equal to

V a (P ) -V b (P ) -2k ,
where, for c ∈ R, V c (P ) is the number of sign variations in the sequence P (c), P ′ (c), . . . , P (n) (c). This proof has its own interest since it can be applied to prove the attractivity of a fixed point in the more general setting of GW, see Remark 3.10. It relies on the Lemma 3.9 For all even n ≥ 2, the function

γ : t ∈ [0, 1] -→ tf n (t) = k,0≤2k≤n n 2k 2k k 1 -t 2 2k t n+1-2k ∈ R,
is strictly increasing on (0, 1 /2).

Proof. Writing:

(1 -t) 2k t n+1-2k ′ = t n 1 -t t 2k-1 1 t (n + 1 -2k) -(n + 1) , the inequality γ ′ (t) > 0 for t ∈ (0, 1 /2) is equivalent to: ∀ t ∈ (0, 1 /2) , k,0≤2k≤n 2 -2k n 2k 2k k 1 -t t 2k-1 1 t (n + 1 -2k) -(n + 1) > 0.
Using the substitution s = 1-t t ⇔ t = 1 1+s , it is equivalent to prove on (1, +∞):

g(s) : = k,0≤2k≤n 2 -2k n 2k 2k k s 2k-1 ((1 + s)(n + 1 -2k) -(n + 1)) = k,0≤2k≤n 2 -2k n 2k 2k k s 2k-1 (s(n + 1 -2k) -2k) > 0.
In order to use Theorem 3.8, we need the ℓ-th derivatives of the function g for 0 ≤ ℓ ≤ n and their values at the limits of the interval (1, +∞). As:

d ℓ ds ℓ (s 2k ) = (2k)! (2k -ℓ)! s 2k-ℓ , d ℓ ds ℓ (2ks 2k-1 ) = (2k)! (2k -ℓ)! (2k -ℓ)s 2k-ℓ-1 , n 2k = n! (n -ℓ)! n -ℓ 2k -ℓ (2k -ℓ)! (2k)! ,
we obtain:

g (ℓ) (s) = n! (n -ℓ)! k,ℓ≤2k≤n 2 -2k 2k k n -ℓ 2k -ℓ s 2k-ℓ-1 (s(n + 1 -2k) -(2k -ℓ)) . (3.12)
Since n is even and 2k ≤ n for every k considered in the sum in (3.12),

g (ℓ) (s) ∼ s→∞ s n-ℓ 2 -n n n 2 n! (n -ℓ)! > 0, ∀ℓ ∈ 0, n .
Moreover, according again to (3.12), for every ℓ ∈ 0, n :

g (ℓ) (1) = n! (n -ℓ)! k,ℓ≤2k≤n 2 -2k 2k k n -ℓ 2k -ℓ (n + 1 + ℓ -4k) .
The sign of g (ℓ) (1), and thus of k,ℓ≤2k≤n

2 -2k 2k k n -ℓ 2k -ℓ (n + 1 + ℓ -4k) := k,ℓ≤2k≤n µ k α k ,
where α k := n + 1 + ℓ -4k, is difficult to obtain directly. As (α k ) k≥0 is a decreasing sequence, the main idea is to use Lemma 6.1 to bound below this sum with a quantity that we are able to compute. Defining

ν k := n -ℓ 2k -ℓ , it holds µ k ν k = 2 -2k 2k k = ξ 2k ,
and, as (ξ 2k ) k≥0 is a decreasing sequence (see Lemma 6.4), we can apply Lemma 6.1 which gives:

k,ℓ≤2k≤n µ k α k k,ℓ≤2k≤n µ k ≥ k,ℓ≤2k≤n ν k α k k,ℓ≤2k≤n ν k = k,ℓ≤2k≤n n-ℓ 2k-ℓ (n + 1 + ℓ -4k) k,ℓ≤2k≤n n-ℓ 2k-ℓ .
Moreover, according to (6.2) and to (6.4):

k,ℓ≤2k≤n

ν k α k = (n -ℓ + 1) k,ℓ≤2k≤n n -ℓ 2k -ℓ -2 k,ℓ≤2k≤n n -ℓ 2k -ℓ (2k -ℓ) =      (n -ℓ + 1)2 n-ℓ-1 -2(n -ℓ)2 n-ℓ-2 if ℓ ∈ {0, . . . , n -2}, (n -ℓ + 1)2 n-ℓ-1 -2 if ℓ = n -1, 1 if ℓ = n, =      2 n-ℓ-1 if ℓ ∈ {0, . . . , n -2}, 0 if ℓ = n -1, 1 if ℓ = n.
It follows that g (ℓ) (1) ≥ 0 for every ℓ ∈ 0, n , so we have proved that g (ℓ) (1) and g (ℓ) (+∞) have always the same sign. According to Theorem 3.8, the number of roots of g in (1, +∞) is thus zero and hence g > 0 on (1, +∞) since lim t→+∞ g(t) = +∞.

Proof of Proposition 3.7 Using f n (α n ) < 1 and Lemma 3.9, the proof of Proposition 3.7 is straightforward:

∀ t ∈ (0, 1 /2) , γ ′ (t) = f n (t) + tf ′ n (t) > 0 ⇔ f ′ n (t) > - f n (t) t ,
and taking t = α n leads to f ′ n (α n ) > -1.
Remark 3.10 The statement of Lemma 3.9 remains actually true when n ≥ 2 is odd. It follows that in the GW case, the function

t → tf (t) = n≥2 q n tf n (t)
is strictly increasing on (0, 1 /2). In particular, we have f ′ (α) > -1 for every fixed point α ∈ (0, 1 2 ). Recall moreover (Remark 3.4) that f admits at least one fixed point in (0, 1 2 ) and that, either f (0) = 0 and f ′ (0) > 1, or f (0) > 0. Hence, denoting by α the smallest fixed point in (0, 1 2 ), we have necessarily -1 < f ′ (α) ≤ 1, which almost implies the linear attractivity of α. Furthermore, when the support of q is included in 2N, the convexity of f implies the attractivity of its unique fixed point.

Basin of attraction of the fixed point α n

The attractivity of α n is not enough to obtain the even case in Theorem 1.1. In order to apply Proposition 2.8, we have to prove that the basin of attraction of α n is [0, 1).

The proof is carried out in two steps. First, we prove the existence of n 0 ∈ N such that f ′ n (α n ) > 0 for every even n > n 0 , which implies that the basin of attraction of α n is [0, 1) when n > n 0 . Secondly, we prove numerically that the basin of attraction of α n is also [0, 1) for every even 2 ≤ n ≤ n 0 . Moreover, we estimate the constants appearing in the computations with precision in order to minimize n 0 and then the number of values of n for which we have to check the result numerically.

Lemma 3.11 For all even n ∈ N * and all t ∈ [0, 1],

f n (t) ≥ 1 2π(n + 1) . (3.13)
Proof. Let us simply note that for every even n ∈ N * and every t ∈ [0, 1],

f n (t) = 1 π π 2 0 ((1 -t) cos x + t) n dx + 1 π π π 2 ((1 -t) cos x + t) n dx ≥ 1 π π 2 0 cos n xdx ≥ 1 2π(n + 1)
,

where the last inequality follows from standard estimates on Wallis integrals, see (6.5).

Remark 3.12 Lemma 3.11 implies in particular

α n ≥ 1 √ 2π(n+1)
for every even n ∈ N * .

Let us now recall two classical results, obtained by integration by parts, which will be useful in the sequel: if X follows the standard normal distribution, then

∀x > 0 , P(X ≥ x) = 1 √ 2π ∞ x e -u 2 2 du ≤ e -x 2 2 x √ 2π , (3.14)
and ∀n ∈ N , E X 2n = (2n)! 2 n n! . (3.15)
On one hand, using (3.14):

1 π 1 0 e -n x 2 2 dx = 2 nπ 1 √ 2π √ n 0 e -u 2 2 du = 2 nπ 1 2 -P(X ≥ √ n) ≥ 2 nπ 1 2 - e -n 2 √ 2πn = 1 √ 2πn - e -n 2 πn .
Lemma 3.13 There exists n 0 ∈ N such that for all even n > n 0 , we have

f ′ n 1 √ 2π (n+1)
> 0.

Proof. Recall that:

f ′ n (x) = n π π 0 (1 -cos t) ((1 -x) cos t + x) n-1 dt. Let ζ n = 1 √ 2π (n+1)
. We have

π n f ′ n (ζ n ) = 1 0 (1 -cos t) ((1 -ζ n ) cos t + ζ n ) n-1 dt + π 1 (1 -cos t) ((1 -ζ n ) cos t + ζ n ) n-1 dt = A + B.
Using successively that cos u ≤ 1 -

u 2 /2 + u 4 /4!, cos u ≥ 1 -u 2 /2, and ln(1 -u 2 /2) + u 2 /2 ≥ -u 4 /4 on [0, 1]
, and e u -1 ≥ u on R, we obtain

A = 1 0 (1 -cos t) ((1 -ζ n ) cos t + ζ n ) n-1 dt ≥ 1 0 t 2 2 - t 4 4! e (n-1) ln(cos t) dt ≥ 1 0 t 2 2 - t 4 4! e -(n-1) t 2 2 dt -(n -1) 1 0 t 2 2 - t 4 4! t 4 4 e -(n-1) t 2 2 dt ≥ 1 0 t 2 2 - t 4 4! e -(n-1) t 2 2 dt -(n -1) 1 0 t 6 8 e -(n-1) t 2 2 dt ≥ 1 0 t 2 2 e -(n-1) t 2 2 dt - 1 0 t 4 4! e -(n-1) t 2 2 dt -(n -1) 1 0 t 6 8 e -(n-1) t 2 2 dt =: D -(E + F ).
It is an easy task to see that with the substitution u = t √ n -1 and (3.15):

E + F √ 2π ≤ (n -1) -5 /2 E[X 4 ] 2 × 4! + E[X 6 ] 16 = (n -1) -5 /2 . (3.16)
In order to bound below D, we use integration by parts, (3.15) and (3.14):

D √ 2π = (n -1) -3 /2 1 2 √ 2π √ n-1 0 u 2 e -u 2 2 du = (n -1) -3 /2 E[X 2 ] 4 - 1 2 √ 2π ∞ √ n-1 u 2 e -u 2 2 du = (n -1) -3 /2 1 4 - 1 2 √ 2π √ n -1e -n-1 2 + ∞ √ n-1 e -u 2 2 du ≥ (n -1) -3 /2 1 4 - 1 2 √ 2π √ n -1e -n-1 2 + e -n-1 2 √ n -1 = (n -1) -3 /2 1 4 - n 2 2π(n -1) e -n-1 2 .
(3.17)

Let us now turn to B and denote by

M := {t ∈ [1, π], (1 -ζ n ) cos t + ζ n < 0}.
Since n -1 is odd, we have:

B = π 1 (1 -cos t) ((1 -ζ n ) cos t + ζ n ) n-1 dt ≥ M (1 -cos t) ((1 -ζ n ) cos t + ζ n ) n-1 dt ≥ π π 2 (1 -cos t)(1 -ζ n ) n-1 (cos t) n-1 dt = -(1 -ζ n ) n-1 (W n + W n-1 )
,

where W n is the Wallis integral. As (1 -ζ n ) n-1 ≤ e - n-1 √ 2π(n+1) and (W n ) is a decreasing sequence such that for all n ≥ 1, W n ≤ π 2n : B ≥ -2W n-1 e - n-1 √ 2π(n+1) ≥ - 2π n -1 e - n-1 √ 2π(n+1
π n f ′ n (ζ n ) ≥ √ 2π(n -1) -3 /2 1 4 - n 2 2π(n -1) e -n-1 2 - 1 (n -1) -(n -1)e - n-1 √ 2π(n+1) = √ 2π(n -1) -3 /2 1 4 -w n .
The statement of Proposition 3.13 follows, since lim n→∞ w n = 0. We introduce the following notation for every even n ≥ 2: we denote by xn := arg min

x∈[0,1] f n (x) (3.19)
the global minimum of the function f n on [0, 1]. Note that the strict convexity of f n (as n is even) together with f ′ n (0) < 0 < f ′ n (1) ensures that xn is unique and belong to (0, 1), with f ′ n (x n ) = 0.

Lemma 3.14 For every even n ∈ N * such that f ′ n (α n ) > 0, the basin of attraction of α n is [0, 1). This is in particular the case for every even n sufficiently large.

21

In particular, taking any n 0 ≥ 26 such that 1 4 -w n0 > 0, one has f ′ n (ζ n ) > 0 and then f ′ n (α n ) > 0 for every even n ≥ n 0 . It is also easy to check numerically that 1 4 -w 350 > 0 (see the right graph in Figure 6), and thus f ′ n (α n ) > 0 for every even n ≥ 350. Moreover, computer assisted estimates show that f ′ n (α n ) > 0 for every even 26 < n < 350, see Figure 7 below. Thus, f ′ n (α n ) > 0 for every even n > 26 and it follows from Lemma 3.14 that the basin of attraction of α n is [0, 1) for every even n > 26. The statement of Theorem 1.1 in this case is then a consequence of Proposition 2.8. Let us consider the case n even and f ′ n (α n ) < 0. The function f n being strictly convex on [0, 1], the inverse image f -1 n (x n ) of its minimum xn is composed by at most two elements, a n < b n ∈ [0, 1] (see Figure 8). Now, if x ∈ (b, 1), there exists N ∈ N * such that f N (x) ≤ b and for all 0 ≤ q < N, f q (x) ∈ (b, 1). Indeed, since f (y) < y on (b, 1), if N does not exist, the sequence (f q (x)) q≥0 is decreasing and bounded below by b, so tends toward a fixed point of f in [b, 1), which raises a contradiction. As a result, the definition of N and the monotonicity of f on (b, 1) imply

a ≤ x = f (b) ≤ f N (x) ≤ b
and we conclude that (b, 1) is included in the basin of attraction of α since [a, b] is. In particular, the basin of attraction of α contains [a, 1) and, as f is decreasing on [0, a), f ([0, a)) = (x, f (0)] ⊂ [a, 1), implying that [0, a) and thus [0, 1) is included in the basin of attraction of α.

The rest of the proof of Theorem 1.1 is obtained using computer assistance to find good approximations for the quantities xn ,

f n (x n ), α n , f ′ n (α n ), a n , b n , f ′ n (a n ) and f ′ n (b n ):
for every even 4 ≤ n ≤ 26, a n and b n exist and satisfy the assumptions of Lemma 3.16, see the following Table .  Proposition 2.8 thus implies the statement of Theorem 1.1 in this case.

n xn f n (x n ) α n f ′ n (α n ) a n b n f ′ n (a n ) f ′ n (b n ) 4 0,2531
0,228 0,2288 -0,0659 0,1308 0,4264 -0,4724 0,2431 6 0,207 0,1818 0,1825 -0,0674 0,0936 0,414 -0,5641 0,1894 8 0,1766 0,1548 0,1554 -0,0595 0,0773 0,3889 -0,5877 0,1519 10 0,1547 0,1369 0,1373 -0,05 0,0685 0,359 -0,576 0,1254 12 0,1382 0,1238 0,1241 -0,0408 0,0631 0,3273 -0,5429 0,1059 14 0,1252 0,1138 0,114 -0,0324 0,0597 0,2948 -0,4959 0,091 16 0,1146 0,1059 0,106 -0,0251 0,0576 0,2621 -0,4392 0,0793 18 0,1059 0,0994 0,0994 -0,0187 0,0566 0,2296 -0,3753 0,0698 20 0,0985 0,0939 0,0939 -0,0131 0,0565 0,1973 -0,3055 0,0621 22 0,0922 0,0892 0,0892 -0,0083 0,0577 0,1652 -0,2301 0,0552 24 0,0867 0,0852 0,0852 -0,0042 0,0607 0,133 -0,1474 0,0476 26 0,0818 0,0816 0,0816 -0,0007 0,0702 0,0967 -0,0452 0,0278 28 0,0776 0,0785 0,0785 0,0024 30 0,0738 0,0757 0,0757 0,0051 Remark 3.17

1. Note that for n ∈ {28, 30}, the cells corresponding to a n and b n are empty since no pre-image of xn exists in these cases.

2. The strategy in this section can not be used for the case of a GW, whereas Lemma 3.9 implies the non-repulsivity of the fixed point which is a big step to achieve our goal if we succeed to prove the unicity of the fixed point.

Estimates on the fixed points α n

In this section we obtain bounds for the fixed points of f n depending on n. As previously, we denote for n ∈ N, ξ 2n = 2 -2n 2n n .

To obtain the upper bound of (3.20), let us write for n ≥ 2:

f n (ξ n # ) = k,0≤2k≤n n 2k ξ 2k (1 -ξ n # ) 2k ξ n-2k n # =: k,0≤2k≤n ν k α k ,
where α k := ξ 2k , and let µ k := n 2k . The positive sequence (α k ) k≥0 is decreasing according to Lemma 6.4, and writing

ν k µ k = 1 ξ n # -1 2k ξ n n # ,
the positive sequence ( ν k/µ k ) k≥0 is increasing. Then, Lemma 6.1 and the formulas (3.9), (6.4) give:

f n (ξ n # ) ≤ k,0≤2k≤n µ k α k k,0≤2k≤n µ k k,0≤2k≤n ν k = 2 -n 2n n 2 n-1 1 + (-1) n (1 -2ξ n # ) n 2 ≤ ξ 2n (1 + (1 -2ξ n # ) n ) .
Consequently, using in addition and (1 -x) n ≤ e -nx for all x ∈ [0, 1] and the lower bound in (3.21):

(

1 -2ξ n # ) n ≤ e -2nξ n # ≤ e - 4n √ 2π(n # +1) .
Then, with (3.22):

f n (ξ n # ) ≤ ξ 2n # 1 + e - 4n √ 2π(n # +1) ≤ ξ n # 1 √ 2 e 1 2n # 1 + e - 4n √ 2π(n # +1) =: ξ n # w n .
Since n # = n when n is even and n # = n -1 when n is odd, the sequences (w 2k ) k≥1 and (w 2k+1 ) k≥1 are clearly decreasing and, as w 3 , w 4 ≤ 1, we have for every n ≥ 3 :

f n (ξ n # ) ≤ ξ n # and thus α n ≤ ξ n # .

An Example of GW

All the simulations with a GW seem to show that there is a unique fixed point in (0, 1) and its basin of attraction is (0, 1). As we have already said, we have not been able to adapt the techniques of Section 3 to prove the uniqueness of the fixed point in a general framework. Nevertheless, we propose an example in which we are able to prove everything. In this section, we assume that the reproduction law N follows a shifted geometric distribution with parameter p ∈ (0, 1), in other words:

q n = P(N = n) = p(1 -p) n-2 , ∀n ≥ 2.
This example is very satisfying as we can obtain explicit formulas. More precisely, we have the following: Lemma 4.1 If N = X + 1 where X follows a geometric distribution with parameter p ∈ (0, 1), we have: Proof. We have:

f (t) = p (1 -p) 2 -((1 -p)t + 1) + 1 (p(2 -p + 2t(p -1))) 1 2 . ( 4
f (t) = n≥2 q n f n (t) = n≥2 p(1 -p) n-2 1 π π 0 ((1 -t) cos x + t) n dx = p π π 0 ((1 -t) cos x + t) 2 n≥2 (1 -p) n-2 ((1 -t) cos x + t) n-2 dx = p π π 0 ((1 -t) cos x + t) 2 1 -(1 -p)((1 -t) cos x + t) dx = p π(1 -p) 2 π 0 (1 -p) 2 ((1 -t) cos x + t) 2 -1 + 1 1 -(1 -p)((1 -t) cos x + t) dx = p π(1 -p) 2 π 0 -(1 -p)((1 -t) cos x + t) -1 + 1 1 -(1 -p)((1 -t) cos x + t) dx = -p((1 -p)t + 1) (1 -p) 2 + p π(1 -p) 2 π 0 dx 1 -(1 -p)((1 -t) cos x + t)
With the substitution u = tan x 2 , we obtain:

π 0 dx 1 -(1 -p)((1 -t) cos x + t) = 2 +∞ 0 du p + (2 -p + 2t(p -1))u 2 = 2 p +∞ 0 du 1 + (2-p+2t(p-1)) p u 2 = π (p(2 -p + 2t(p -1))) 1 2
.

Then

f (t) = p (1 -p) 2 -((1 -p)t + 1) + 1 (p(2 -p + 2t(p -1))) 1 2
.

In Figure 9, we can see that f seems to have one fixed point on (0, 1). It is not difficult to find this point, resolving:

f (t) = t ⇔ -2(1 -p)t 3 + (2 -5p)t 2 + 4pt -p = 0 ⇔ (t -1)(-2(1 -p)t 2 -3pt + p) = 0 ⇔ t = 1 or t = -3p ± (p(p + 8)) 1 2

4(1 -p) .

And we can easily see that the only root that interests us is α = -3p+(p(p+8))

1 2

4(1-p)

.

Lemma 4.2 The basin of attraction of α is [0, 1).

Proof. The first and second derivative of f are given by:

f ′ (t) = p 1 -p -1 + p (p(2 -p + 2t(p -1)) 3 2
and f (2) (t) = 3p 3 (p(2 -p + 2t(p -1))

3 2
.

As stated in Remark 3.15 since f is strictly convex, it is sufficient to show that f ′ (α) ≥ 0. One can see that: As g(1) = 2, if we prove that g is an increasing function on [0, 1], we obtain formula (4.2). As:

g ′ (p) = (4 + p -(p(p + 8)) 1
2 )((p(p + 8))

1 2 -3p) 3p is obviously positive, our proof is complete.

To conclude, according to Proposition 2.8, for every k ≥ 2 and p ∈ P k such that p 1 = p 2 > p 3 ≥ • • • ≥ p k , p(m) converges to α, 1-α 2 , 1-α 2 , 0 k-2 when m → ∞.

Remark 4.3 Considering the n-ary tree as a GW tree with reproduction law N = n a.s., we have E[N ] = n. In order to obtain the same mean in the geometric case, it suffices to take p = 1 n-1 . With this choice of p, α ∼ 1 / √ 2n when n goes to infinity, which is consistent with the bounds found in Proposition 3.18.

Open questions and variant case

As a conclusion we make some remarks on the properties of the main objects studied in this work and discuss possible generalisations of our results.

1. One can notice in the figure 3, that the red curve of f 3 , seems to cut the blue one f 4 at its minimum. In fact, that is true for all n ≥ 2, that is:

f 2n (x 2n ) = f 2n-1 (x 2n ) . (5.1)
Indeed, according to (3.4): With an obvious induction reasoning, the formula (5.2) gives:

f ′ n (t) =
= nf ′ n-1 (t) - n π π 0 cos x(1 -t) + t -t 1 -t (cos x(1 -t) + t) n-1 dx = n 1 + t 1 -t f n-1 (t) - n 1 -t f n (t) ⇔ t -1 n f ′ n (t) = f n (t) -f n-1 (t) (5.
f n (t) -t = f n (t) -f 1 (t) = (t -1) n k=2 1 k f ′ k (t).
(5.3)

We think that these equalities have a probabilistic meaning, but did not manage to come up with an explanation.

2. The general GW case for two opinions seems for the moment out of reach, even though our simulations suggest that our results stay valid. Contrary to the article [START_REF] Debs | Diseases transmission in a z-ary tree[END_REF], the mean of the reproduction law E[N ] does not seem to play a particular role: there seems to be always convergence.

We have to study cases with more than two opinions: nevertheless, even in the case of a n-ary tree, using links with random walks in order to obtain formulas like (3.1) for a number i > 2 of major opinions is not clear to us. Moreover, we have seen that even in the case with two opinions, parity plays an important role; already for i = 3, the calculations become devilish and it seems to us that we need to find much finer methods than direct computations. For instance in Figure 10, we can see that the shape of the graph is linked to the remainder of the Euclidean division of n by 3 and the equivalent formula for f n is:

f n (t) =

Appendix

In this appendix, we recall some classical definitions and results used throughout this paper.

The following result is crucial to prove the stability statement of Section 3.2.1.

Lemma 6.1 Consider two positive sequences (µ k ) k≥0 and (ν k ) k≥0 such that ( ν k/µ k ) k≥0 is increasing. Then, for every decreasing (resp. increasing) sequence (α k ) k≥0 and for every 0 ≤ ℓ ≤ n, Proof. Let us prove the well-known formula (6.5) for the sake of completeness. For n ≥ 0, as 0 ≤ sin t ≤ 1 in [0, π /2] (and 0 < sin t < 1 in (0, π /2)):

W n > 0 and W n+1 -W n = π /2 0 sin n t(sin t -1)dt < 0, implying the (strict) monotonicity of (W n ) n≥0 . Moreover, for every n ∈ N: Consequently, the sequence (n + 1)W n+1 W n n≥0 is constant and then:

W n+2 =
∀n ∈ N * , nW n W n-1 = W 1 W 0 = π 2 .
Using the monotonicity of (W n ), we obtain the formula (6.5) since:

∀n ≥ 0 , nW 2 n < π 2 < (n + 1)W 2 n .
Recall now that for all n ≥ 0 (see (3.2)),

W 2n = π 2 (2n)! 2 2n (n!) 2 = π 2 ξ 2n .
Thus:

1. The (strict) monotonicity of the sequence (ξ 2n ) n≥0 follows from the one of (W n ) n≥0 .

2. Using (6.5), we obtain:

Figure 1 :Figure 2 :

 12 Figure 1: The rules

Figure 3 :Lemma 3 . 3

 333 Figure 3: The graphs of f n for n = 4 (blue), n = 3 (orange)

3. 1 . 1 Figure 4 : 5 ] 3 . 5

 114535 Figure 4: The graphs of f n for n = 3 (blue), n = 5 (orange), n = 7 (green) and n = 9 (red), on [0, 1] and [0, 1 /5]

3. 1 . 2

 12 Proof of Theorem 1.1 and Corollary 1.2

p 0

 0 (m + 1) = H 0 (p 0 (m), . . . , p k (m)) ≥

3. 2 . 1 3 . 7 Figure 5 :

 21375 Figure 5: The graphs of f n for n = 2 (blue), n = 4 (orange), n = 6 (green) and n = 10 (red)

  f ′ 2n (α 2n ) for n in [2, 250]Proof of Theorem 1.1 in the even case when 2 ≤ n ≤ 26.

Figure 8 :

 8 Figure 8: The graph of f 2We haveα n < xn (since f ′ n (α n ) < 0 and f ′ n (x n ) = 0) and f n (x n ) < xn (since xn > α n, the unique fixed point in (0, 1)). Note also that in the case of existence of a n and b n , we have 0 ≤ a n < α n < xn < b n < 1. We have moreover in this case the following Lemma 3.16 Assume the existence of a n and b n and that k := max |f ′ n (a n )|, |f ′ n (b n )| < 1.Then, the basin of attraction of α n is [0, 1).

Figure 9 :

 9 Figure9: The graphs of f for p = 1 /2 (blue), p = 1 /4 (orange), p = 9 /10 (green)

2 3

 2 (p(p + 8)) 1 2

n π π 0 (t( 1 -

 01 cos x) + cos x) n-1 dx -n π π 0 cos x(t(1 -cos x) + cos x) n-1 dx = nf ′ n-1 (t) -cos x(1 -t) + t) n-1 dx

  2n ) = 0 = f 2n (x 2n ) -f 2n-1 (x 2n ).

Figure 10 :

 10 Figure 10: The graphs of f n for n = 7 (blue), n = 9 (orange), n = 11 (green)

π / 2 0 2 0cos 2 t

 222 sin t sin n+1 tdt = (n + 1)π /sin n tdt = (n + 1)(W n -W n+2 ) ⇔ (n + 2)W n+2 = (n + 1)W n ⇔ (n + 2)W n+2 W n+1 = (n + 1)W n+1 W n .

Acknowledgments

We would like to thank our colleagues Luc Hillairet for his fruitful ideas and conversations, and Thomas Haberkorn for his help with simulations. We also thank the anonymous referee who pointed out an argument permitting to state Corollary 1.2 for a GW tree with support included in 2N + 1 without assuming its finiteness.

Proof. The second part of Lemma 3.14 is an immediate consequence of its first part and of the convexity of f n together with Lemmas 3.11 and 3.13, which imply f ′ n (α n ) > 0 for every even n large enough.

Let us then prove the first part of this lemma. Thanks to the convexity of f n , we have 0 < xn < α n and then f n (x n ) > xn . We drop the subscript n in the rest of the proof below to lighten the notation and we define the recursive sequence (u m ) m≥0 by u 0 = x 0 ∈ [0, 1) and [START_REF] Athreya | Branching processes[END_REF], and f (x) ≤ x. Thus, if x 0 ∈ [α, 1), the sequence (u m ) is decreasing and bounded below by α, implying that (u m ) tends to α, the fixed point of f in [α, 1).

We can thus conclude with the two previous cases when x 0 ∈ [x, f (x)). On [0, x): f is decreasing and f ([0, x)) = (f (x), f (0)] ⊂ [f (x), 1). We can thus again conclude with the two first cases when x 0 ∈ [0, x).

Remark 3.15 In the general GW setting, if the function f = n q n f n is strictly convex, using previous arguments we get existence and unicity of the fixed point α in [0, 1). If in addition f ′ (α) ≥ 0, the basin of attraction of α is [0, 1) with a similar reasoning as in the proof of Lemma 3.14.

Proof of Theorem 1.1 in the even case

Proof in the even case when n > 26.

Let us observe that the sequence (w n ) converging to 0 defined at the end of the proof of Lemma 3.13 is decreasing for n large enough. More precisely,

where w ′ n is decreasing for all n > 1. On the other hand, the derivative of the function x → (x -1)e

2 π and negative at x = 26 and hence (w n ) is decreasing for n ≥ 26. Proposition 3.18 We have:

where n # := 2 n 2 .

Proof. According to (see (6.6))

we have just to prove lower bound ξ 4n ≤ α n for n large enough and the upper bound

Using moreover the monotonicity of the sequence (ξ 2n ) and (6.4), we have for every n ≥ 2:

, where the last inequality arises from using twice (see (6.7))

Using now the relations e x < 1 + 4 3 x for x ≤ 1 2 , (1 -x) n ≤ e -nx for x ∈ [0, 1], and (3.21), which implies

n+1 n

Finally we can check that

for all n sufficiently large, and computer assisted calculations show that n ≥ 536 is sufficient. Hence, we have f n (ξ 4n ) ≥ ξ 4n and thus α n ≥ ξ 4n for every n ≥ 536.

Proof. Assume that the sequence ( ν k/µ k ) k≥0 is increasing, which is equivalent to:

When the sequence (α k ) k≥0 is decreasing, the formula (6.1) is equivalent to:

which is true by hypothesis.

In the following lemma, we state classical results on binomial coefficients. Lemma 6.2 For all n ∈ N:

and

Proof.

1. For all x, y ∈ R and n ≥ 0, the relation

Taking x = y = 1 and n = 1, we obtain the right equality of (6.2). Moreover, with a very similar reasoning:

and taking x = y = 1, we obtain

We conclude by using

2. In [START_REF] Gould | Combinatorial Identities: Table I: Intermediate Techniques for Summing Finite Series[END_REF], the author uses an expansion of (x 2 +2x) n to prove (6.3) (see the formula (1.65) there). We will use here the following series expansion, for ℓ ∈ N and x ∈ [0, 1) (which also permit to prove the generalization of (6.3) stated in Remark 6.3 below):

The first one can be obtained by induction and the second one is classical. Thus:

Identifying the coefficients, we obtain (6.3).

Remark 6.3 Adapting the above proof of (6.3), we can show that for all n ∈ N and all ℓ ∈ N * :

We conclude this appendix with this last lemma, giving some properties of the Wallis integrals and of the strongly related quantities ξ 2n = 1 The sequence (W n ) n≥0 is positive and strictly decreasing and, for all n ≥ 1: π 2(n + 1) < W n < π 2n (6.5)

We have moreover the following properties:

1. The sequence (ξ 2n ) n≥0 is strictly decreasing.

For all