MD Nadjib Hammoudi
email: nadjib.hammoudi@psl.aphp.fr.

MD Dimitri Arangalage

MD Morad Djebbar

MD Katia Stankovic Stojanovic

MD Magali Charbonnier

MD Richard Isnard

MD Robert Girot

MD Pierre-Louis Michel

MD François Lionnet

Subclinical Left Ventricular Systolic Impairment in Steady State Young Adult Patients with Sickle-cell Anemia

Keywords: echocardiography, left ventricular function, speckle-tracking, strain, Sickle-cell anemia

Purpose: Chronic volume overload in sickle-cell anemia (SCA) is associated with left ventricular (LV) enlargement and hypertrophy. The effect of the disease on LV systolic function remains debated. The aim of our study was to investigate LV systolic function in SCA patients using 2D speckle-tracking imaging.

Methods:

We compared 30 steady state asymptomatic adult SCA patients (17 women, mean age 24.7±5.1 years) with 30 age and sex-matched healthy subjects (17 women, mean age 25.0±4.9 years). In addition to conventional echocardiographic parameters including LV ejection fraction (EF) and LV mass index (LVMi), global longitudinal strain (GLS) and strain rate (GLSR) were measured.

Results: GLS (-17.9±2.0% vs. -19.7±2.5%, p=0.004) and GLSR (-0.92±0.09s -1 vs. -1.07±0.17s -1 , p<0.0001) values were lower in SCA patients while LVEF values (60.1±3.8% vs. 61.7±4.7%, p=0.30) were not different. LVMi was increased in SCA patients (100.7±23.5g/m² vs. 72.4±15.2g/m², p0.0001) and GLSR was significantly lower in the subgroup of patients with LV hypertrophy (-0.88±0.09s -1 vs. -0.96±0.08s -1 , p=0.02). In SCA patients LVMi was correlated to GLS (r=0.58, p=0.001) and GLSR (r=0.45, p=0.015) pleading in favor of a pathological LV remodeling.

Conclusions:

Asymptomatic SCA patients exhibited a subclinical alteration of LV systolic function. Myocardial dysfunction appears to be linked to the degree of LV hypertrophy. 2D speckle-tracking imaging might be useful for long-term follow-up and to study the natural course of LV dysfunction in SCA patients.

BACKGROUND

Sickle-cell disease is one of the most common inherited blood disorders worldwide [START_REF] Weatherall | A case for developing North-South partnerships for research in sickle cell disease[END_REF].

Besides chronic anemia, many pathophysiological processes contribute to the complexity of the disease including hemolysis and repeated vaso-occlusive events with ischemia-reperfusion injury leading to endothelial cell dysfunction [START_REF] Weatherall | A case for developing North-South partnerships for research in sickle cell disease[END_REF][START_REF] Rees | Sickle-cell disease[END_REF]. Concomitantly to the life expectancy improvement observed over the past years, the prevalence of heart disease in adult patients has also increased representing now up to one fourth of all deaths [START_REF] Fitzhugh | Cardiopulmonary complications leading to premature deaths in adult patients with sickle cell disease[END_REF].

In patients with homozygous sickle cell disease, also called sickle-cell anemia (SCA), a diastolic dysfunction, as well as left ventricular (LV) remodelling, including dilation and hypertrophy, have been previously described [START_REF] Gladwin | Cardiovascular abnormalities in sickle cell disease[END_REF]. These modifications are best explained by an adaptative response to the volume overload effect of chronic anemia [START_REF] Batra | Cardiac abnormalities in children with sickle cell anemia[END_REF][START_REF] Lester | Cardiac abnormalities in children with sickle cell anemia[END_REF].

While chronic volume overload in valvular heart disease such as mitral or aortic regurgitation [START_REF] Vahanian | Guidelines on the management of valvular heart disease (version 2012): the Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS)[END_REF] and prolonged anemia induced by iron deficiency in rat models [START_REF] Naito | Adaptive response of the heart to longterm anemia induced by iron deficiency[END_REF] induce LV systolic dysfunction, its' occurrence in SCA disease is still debated with conflicting results in past studies [START_REF] Batra | Cardiac abnormalities in children with sickle cell anemia[END_REF][START_REF] Lester | Cardiac abnormalities in children with sickle cell anemia[END_REF][START_REF] Voskaridou | Sickle-cell disease and the heart: review of the current literature[END_REF][START_REF] Eddine | Ventricular structure and function in children with sickle cell disease using conventional and tissue Doppler echocardiography[END_REF][START_REF] Lamers | Evaluation of left ventricular systolic function in pediatric sickle cell anemia patients using the end-systolic wall stressvelocity of circumferential fiber shortening relationship[END_REF][START_REF] Covitz | The heart in sickle cell anemia. The Cooperative Study of Sickle Cell Disease (CSSCD)[END_REF]. Recently, ultrasound speckle-tracking imaging has emerged as a strong and sensible tool that allows early diagnosis of LV systolic dysfunction [START_REF] Mor-Avi | Current and evolving echocardiographic techniques for the quantitative evaluation of cardiac mechanics: ASE/EAE consensus statement on methodology and indications endorsed by the Japanese Society of Echocardiography[END_REF].

We hypothesized that a systolic LV dysfunction that could be diagnosed by speckle-tracking imaging may exist during steady-state SCA disease in adult patients.

METHODS

Study population

We enrolled 30 patients with SCA, aged 18 years and older, in stable condition and in sinus rhythm. All these patients were referred to our echocardiography laboratory for routine outpatient evaluation of cardiac function and/or systematic screening for pulmonary arterial hypertension. The diagnosis of homozygous sickle-cell disease was based on molecular genetic techniques. Patients who had developed acute chest syndrome, vaso-occlusive crisis or an acute complication within the previous 4 months, including fever, surgery, blood transfusion or hospital admission whatever the reason, were excluded in order to specifically focus on a group of steady state patients free of confounding factors that could be linked to an impaired LV function. The other exclusion criteria were the use of cardiovascular medication at the time of enrollment, hypertension, history of heart failure, moderate or severe valvular heart disease, atrial fibrillation, pregnancy and the presence of an associated comorbidity including autoimmune diseases, HCV or HIV infections and kidney failure defined by a glomerular filtration rate <60ml/mn/1.73m². Clinical and biological data were collected from records of the reference centre for sickle-cell disease (Tenon Hospital, Paris, France). From the general population, 30 age-and sex-matched healthy subjects without history of cardiac or pulmonary disease and with normal electrocardiograms were recruited as control subjects.

Blood pressure was measured in patients and controls in supine position at the end of the echocardiographic examination using a Dash 3000 monitor (GE Healthcare; Horten, Norway).

Informed consent was obtained from each patient. The study was approved by the institutional committee on human research.

Standard echocardiography

Transthoracic echocardiography was performed in all patients with the use of the Vivid 7 system (GE Healthcare; Horten, Norway) and transferred to a workstation equipped with the Echopac PC software (GE Vingmed Ultrasound; Horten, Norway) for offline analysis. All exams were acquired and analyzed off-line blinded to the clinical data by a senior cardiologist (NH). All measurements were averaged over 3 consecutive cardiac cycles. All projections were obtained according to the recommendations of the American Society of Echocardiography [START_REF] Quiñones | Recommendations for quantification of Doppler echocardiography: a report from the Doppler Quantification Task Force of the Nomenclature and Standards Committee of the American Society of Echocardiography[END_REF][START_REF] Lang | Recommendations for chamber quantification: a report from the American Society of Echocardiography's Guidelines and Standards Committee and the Chamber Quantification Writing Group, developed in conjunction with the European Association of Echocardiography, a branch of the European Society of Cardiology[END_REF]. From M-mode, the following measurements were made at end diastole: LV internal diameter, inter-ventricular septal and posterior wall thicknesses. LV mass was derived and indexed to body surface area (LVMi), relative wall thickness was also calculated (posterior wall thicknesses × 2 / LV internal diameter) and LV remodeling was categorized as recommended [START_REF] Lang | Recommendations for chamber quantification: a report from the American Society of Echocardiography's Guidelines and Standards Committee and the Chamber Quantification Writing Group, developed in conjunction with the European Association of Echocardiography, a branch of the European Society of Cardiology[END_REF]. LV hypertrophy was defined by an LVMi > 95g/m² in women and >115g/m² in men. Further classification as either concentric hypertrophy (relative wall thickness > 0.42) or eccentric hypertrophy (relative wall thickness ≤ 0.42) was made.

From 2-dimensional mode, end systolic left and right atrial areas were measured, LV volumes and ejection fraction (LVEF) were derived from Simpson's modified biplane method. From pulsed wave Doppler mode, LV outflow tract time-velocity integral, early and late peak diastolic velocities of the mitral (E and A) inflow and the E-wave deceleration time were measured. LV ejection volume and output were calculated and indexed to body surface area as recommended [START_REF] Quiñones | Recommendations for quantification of Doppler echocardiography: a report from the Doppler Quantification Task Force of the Nomenclature and Standards Committee of the American Society of Echocardiography[END_REF]. The peak e' velocity was used to calculate the E/e' ratio using pulsed tissue Doppler imaging of the lateral mitral annulus [START_REF] Nagueh | Recommendations for the evaluation of left ventricular diastolic function by echocardiography[END_REF]. Diastolic dysfunction was defined by E/A ratio <1.0 and/or a deceleration time >240 ms; E/A ratio ≥1.0 and E/e'ratio >10; E/A ratio higher than the 95 th percentile for age or deceleration time <140 ms and E/e' >10. This classification of LV diastolic function has a prognostic value on mortality in patients with SCA [START_REF] Sachdev | Diastolic dysfunction is an independent risk factor for death in patients with sickle cell disease[END_REF].

From continuous wave Doppler, peak tricuspid regurgitation was recorded in multiple views and the highest level of velocity was selected. Elevated pulmonary systolic pressure was defined by a peak tricuspid regurgitation jet velocity ≥ 2.5m/s, severe elevation by a peak tricuspid regurgitation jet velocity ≥ 2.9m/s [START_REF] Parent | A hemodynamic study of pulmonary hypertension in sickle cell disease[END_REF].

Speckle-tracking imaging

For 2D speckle-tracking imaging, sector size and depth were adjusted to achieve optimal visualization of all LV myocardium at the highest possible frame rate (mean value 87 ±14 frames/s). Multiple consecutive cardiac cycles of the 3 standard apical views (4, 2 and 3 chambers views) were acquired during breath holding. In each view, the myocardium was automatically divided by the software into 6 segments. The analyzed values within the middle points for all resulting 18 segments were averaged to obtain the global longitudinal strain (GLS) and the global longitudinal peak systolic strain rate (GLSR); in accordance with current conventions GLS and GLSR values were presented as negative (segment shortening) [START_REF] Mor-Avi | Current and evolving echocardiographic techniques for the quantitative evaluation of cardiac mechanics: ASE/EAE consensus statement on methodology and indications endorsed by the Japanese Society of Echocardiography[END_REF]. The adequacy of tracking was verified manually, and the region of interest was readjusted to achieve optimal tracking. The segment was excluded if no acceptable border was traced. GLS and GLSR values were not calculated if > 1 segment per view were excluded.

Statistical analysis

All quantitative data are expressed as mean ± standard deviation (SD); qualitative data are expressed as number and as percentage. For case-control analysis, the Wilcoxon signed rank test was used for univariate analysis of continuous variables and the McNemar's test was used to compare categorical data.

Using the Mann-Whitney U test and the chi-square test a secondary analysis compared the subgroup of patients with and without LV hypertrophy. The Pearson's correlation test was used to assess the univariate relations between variables.

Intraobserver and interobserver variabilities for LV deformation parameters, LV volume, LVEF and LVMi measurements were assessed in 20 studies by 2 independent observers (NH and DA). Using the Bland-Altman approach, the 95% limit of agreement was calculated. In addition, the coefficient of variation (CV) defined as the ratio of the SD of the difference of paired samples to the average of the paired samples and the intra-class correlation coefficient (ICC) were calculated. For intraobserver variability the CV and the ICC were respectively 4.1% and 0.93 for GLS, 4.3% and 0.92 for GLSR, 8.1% and 0.96 for LV end diastolic volume index, 4% and 0.83 for LVEF and 5.5% and 0.99 for LVMi. The 95% limits of agreement were -1.63 to 2.35% for GLS, -0.10 to 0.13 for GLSR, -17.1 to 14.7 ml/m² for LV end diastolic volume index, -7.1 to 6.8% for LVEF and -15.4 to 12.2g/m² for LVMi. For interobserver variability the CV and the ICC were respectively 6.6% and 0.88 for GLS, 6.2% and 0.91 for GLSR, 7.6% and 0.96 for LV end diastolic volume index, 3.6% and 0.85 for LVEF and 7.8% and 0.98 for LVMi. The 95% limits of agreement were -1.2 to 3.7% for GLS, -0.05 to 0.18 for GLSR, -11.0 to 16.5 ml/m² for LV end diastolic volume index, -5.5 to 6.8% for LVEF and -15.1 to 21.7 g/m² for LVMi. SPSS software version 17.0 (SPSS, Inc, Chicago, IL) was used for calculation. A p<0.05 indicated statistical significance.

RESULTS

Patients' mean age was 24.7±5.1 years and 17 patients (57%) out of a total of 30 were women.

Percentages of patients with histories of thoracic vaso-occlusive events or cerebral vasculopathy were 30% and 7% respectively. The mean hemoglobin level was 8.5±0.9 g/dl and in 11 cases (36%) it was ≤ 8g/dl. Patients and controls were matched for age and gender and there was no significant difference between the 2 groups for body surface area and body mass index. Diastolic and mean blood pressures were significantly lower in patients than in controls; the heart rate was not higher in patients (Table 1), only 3 patients had heart rates > 90/min and none >100/min. Morphological and hemodynamic characteristics are summarized in Table 2. LV dimensions and mass were greater in patients than in controls. In patients, the LV hypertrophy was mainly eccentric; only 1 patient had concentric hypertrophy. The mean tricuspid regurgitant jet velocity was higher in the patient group including 13 patients (43%) with elevated pulmonary systolic pressures and none with severe elevation. LVEF measurements were > 50% in all patients. Despite similar LVEF between the 2 groups, the GLS and the GLSR values were significantly decreased in the SCA group (Table 2, Figure 1). LV speckle analysis was not feasible in 1 patient and 1 control subject, because of poor image quality. Patients presented a diastolic dysfunction and had a higher E/e' ratio (Table 2).

In order to assess the potential impact of blood pressure or heart rate on the relationship between GLS and SCA, 3 linear regression analyses including respectively, mean arterial blood pressure, diastolic blood pressure and heart rate were done. The results of the 3 analyses were similar and confirmed the independent link between GLS and SCA (r²=0.15, p=0.002).

The GLSR was significantly lower in patients with LV hypertrophy (Table 3). A correlation was found between LVMi and GLS (r=0.58, p=0.001) as well as with GLSR (r=0.45, p=0.015). There was a significant inverse correlation between the hemoglobin level and LVMi (r=-0.43, p=0.017) as well as LV end-diastolic diameter index (r=-0.36, p=0.047). The hemoglobin level was not statistically correlated to LVEF (r=-0.04, p=0.83), E/e' ratio (r=-0.28, p=0.13), GLS (r=0.15, p=0.44) and GLSR (r=0. 28, p=0.14).

DISCUSSION

In this study we found that in addition to morphological modifications including hypertrophy and dilation, LV systolic function assessed by 2D speckle tracking imaging was impaired in a population of steady state SCA young asymptomatic adult patients with preserved LVEF. The severity of LV hypertrophy was linked to the impairment of global longitudinal LV deformation parameters. The GLS was linked to SCA independently to blood pressure and to heart rate. LV remodeling was inversely associated to the hemoglobin level.

Despite not being a pure form of anemia [START_REF] Sun | New insights into sickle cell disease: a disease of hypoxia[END_REF], the LV remodeling observed in homozygous sickle-cell disease is commonly attributed to chronic anemia as reduced blood oxygencarrying capacity induces an increased cardiac output [START_REF] Gladwin | Cardiovascular abnormalities in sickle cell disease[END_REF]. As described in this study and noted previously by others, an important stroke volume increase associated with a relatively mild heart rate elevation contribute to the cardiac output increase [START_REF] Covitz | The heart in sickle cell anemia. The Cooperative Study of Sickle Cell Disease (CSSCD)[END_REF]. Modifications in LV preload and afterload could explain the stroke volume increase; the significant reduction of the mean arterial blood pressure while the cardiac index is increased underlines the decrease in peripheral vascular resistances [START_REF] Denenberg | Cardiac function in sickle cell anemia[END_REF]. Volume overload contributes to the increased cardiac output by inducing an increase in LV preload and a substantial LV enlargement [START_REF] Varat | Cardiovascular effects of anemia[END_REF][START_REF] Takahashi | Studies of hyperkinetic circulatory state in chronic anemia][END_REF].

According to Laplace's Law, this volume overload increases the peak systolic wall stress resulting in wall thickening to normalize the systolic stress. Wall thickening and fiber elongation contribute to the pattern of eccentric hypertrophy in which the ratio of wall thickness to chamber radius remains normal [START_REF] Grossman | Wall stress and patterns of hypertrophy in the human left ventricle[END_REF].

Regarding the LV systolic function assessment, conflicting results in previous studies are partly explained by the multitude of different echocardiographic parameters used to assess myocardial contractility including load-dependant measures such as the LVEF calculation [START_REF] Lester | Cardiac abnormalities in children with sickle cell anemia[END_REF][START_REF] Covitz | The heart in sickle cell anemia. The Cooperative Study of Sickle Cell Disease (CSSCD)[END_REF][START_REF] Balfour | Cardiac size and function in children with sickle cell anemia[END_REF][START_REF] Covarrubias | Left ventricular function in sickle cell anemia: a noninvasive evaluation[END_REF][START_REF] Gerry | Evaluation of left ventricular function in patients with sickle cell anemia[END_REF]. In our study, deformation parameters were significantly reduced while the LVEF was similar in both groups. Myocardial deformation assessed by strain and strain-rate is the result of a complex interaction between the intrinsic contractile force and the extrinsic loading conditions applied to a tissue with variable elastic properties [START_REF] Sutherland | Strain and strain rate imaging: a new clinical approach to quantifying regional myocardial function[END_REF]. Therefore, it could be argued that the observed modifications of LV longitudinal deformation may not only be the result of an alteration in contractility but may reflect modifications of loading conditions or in myocardial stiffness. In our study the afterload was lower and the preload higher in the SCA group and it is admitted that under these conditions deformation values should increase [START_REF] Burns | Left ventricular strain and strain rate: characterization of the effect of load in human subjects[END_REF][START_REF] Rösner | Left ventricular size determines tissue Doppler-derived longitudinal strain and strain rate[END_REF]. This argument pleads in favor of the occurrence of a LV systolic impairment in young adult SCA patients. Similarly, a recent meta-analysis reported an impairment of a relatively load independent LV systolic function assessment parameter (end-systolic stress to endsystolic volume index) despite preservation of load dependant parameters in SCA patients [START_REF] Poludasu | Left ventricular systolic function in sickle cell anemia: a meta-analysis[END_REF].

An interesting parallel can be made concerning LV remodeling between patients suffering from chronic anemia and endurance athletes in whom a comparable cardiac remodeling has been reported [START_REF] Pelliccia | Assessment of left ventricular hypertrophy in a trained athlete: differential diagnosis of physiologic athlete's heart from pathologic hypertrophy[END_REF]. In contrast with the results of our study, LV strain and strain rate values in endurance athletes have been reported to be normal or increased [START_REF] Galderisi | Differences of myocardial systolic deformation and correlates of diastolic function in competitive rowers and young hypertensives: a speckle-tracking echocardiography study[END_REF][START_REF] Baggish | The impact of endurance exercise training on left ventricular systolic mechanics[END_REF]. Besides, while increased LV mass is associated with improved deformation indices in endurance athletes [START_REF] Galderisi | Differences of myocardial systolic deformation and correlates of diastolic function in competitive rowers and young hypertensives: a speckle-tracking echocardiography study[END_REF], LV hypertrophy is associated with an impairment of LV function in our population further supporting the hypothesis that a LV myocardial impairment exists in sickle-cell disease patients. Similar findings have been reported in the course of hypertrophic cardiomyopathy [START_REF] Galderisi | Differences of myocardial systolic deformation and correlates of diastolic function in competitive rowers and young hypertensives: a speckle-tracking echocardiography study[END_REF] and myocardial fibrosis is associated with depressed strain values in these patients [START_REF] Popović | Association between regional ventricular function and myocardial fibrosis in hypertrophic cardiomyopathy assessed by speckle tracking echocardiography and delayed hyperenhancement magnetic resonance imaging[END_REF]. Moreover, as morphological and functional LV parameters are linked, LV mass may be helpful in routine practice for the identification of a subgroup of SCA patients athigh risk of LV dysfunction.

In case of severe organic mitral regurgitation, prolonged burden due to volume overload might result in LV dysfunction and irreversible myocardial damage. If a significant LV myocardial morphological and/or functional involvement is/are identified, current guidelines recommend the correction of the regurgitation in asymptomatic patients [START_REF] Vahanian | Guidelines on the management of valvular heart disease (version 2012): the Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS)[END_REF]. The LV abnormalities observed in SCA patients are multifactorial including the combination of chronic anemia and microvascular dysfunction [START_REF] Gladwin | Cardiovascular abnormalities in sickle cell disease[END_REF]. In our study, LV deformation parameters revealed a LV impairment in a population of young adult SCA patients with normal LVEF, suggesting a promising role of these parameters for the detection of LV dysfunction at an early stage before major and irreversible damage of the myocardium occurs. As cardiovascular mortality in adult SCA patients is high [START_REF] Fitzhugh | Cardiopulmonary complications leading to premature deaths in adult patients with sickle cell disease[END_REF] our findings may provide clinicians a useful method of identification of a category of patients at high risk of developing longterm congestive heart failure and/or premature death. Moreover these patients may benefit from more regular follow up and more aggressive clinical interventions in order to improve their prognosis.

A similar study using deformation imaging was recently conducted on a pediatric population of sickle cell disease patients with contrasting results as deformation indices were not modified [START_REF] Blanc | Right ventricular systolic strain is altered in children with sickle cell disease[END_REF]. It could be hypothesized that the LV systolic impairment, as found in our population of adults, occurs later in the course of the disease. Age related deterioration of LV systolic function was previously described [START_REF] Poludasu | Left ventricular systolic function in sickle cell anemia: a meta-analysis[END_REF]. Moreover, in a population of adult patients presenting with sickle-cell crisis, the usefulness of deformation indices assessed by speckle tracking for detecting LV systolic function impairment has already been reported [START_REF] Sengupta | Left ventricular myocardial performance assessed by 2-dimensional speckle tracking echocardiography in patients with sickle cell crisis[END_REF]; our study identified LV dysfunction in a population of steady-state patients. Contrastingly, no impairment in LV systolic function was reported in a study based on 3-dimensional LV speckle tracking echocardiography [START_REF] Ahmad | Evaluation of myocardial deformation in patients with sickle cell disease and preserved ejection fraction using threedimensional speckle tracking echocardiography[END_REF]. This promising new method performed on a precommercial prototype software still suffers from technical limitations such as low temporal resolution. Besides, in addition to SCA patients, hemoglobin sickle-cell (SC) patients were also included in this study. It has already been described that SC patients have very different clinical and biological characteristics [START_REF] Lionnet | Hemoglobin sickle cell disease complications: a clinical study of 179 cases[END_REF]. Therefore, the results must be compared to ours with caution.

Consistently with previous studies [START_REF] Eddine | Ventricular structure and function in children with sickle cell disease using conventional and tissue Doppler echocardiography[END_REF][START_REF] Sachdev | Diastolic dysfunction is an independent risk factor for death in patients with sickle cell disease[END_REF][START_REF] Caldas | Evaluation of 107 patients with sickle cell anemia through tissue Doppler and myocardial performance index[END_REF][START_REF] Johnson | Left ventricular hypertrophy and diastolic dysfunction in children with sickle cell disease are related to asleep and waking oxygen desaturation[END_REF], we also found a LV diastolic function impairment. Diastolic dysfunction is known to be an independent risk factor for mortality in SCA [START_REF] Sachdev | Diastolic dysfunction is an independent risk factor for death in patients with sickle cell disease[END_REF]. It is worth noting that the E/e' ratio is normal in healthy endurance athletes [START_REF] Galderisi | Differences of myocardial systolic deformation and correlates of diastolic function in competitive rowers and young hypertensives: a speckle-tracking echocardiography study[END_REF][START_REF] Baggish | The impact of endurance exercise training on left ventricular systolic mechanics[END_REF].

Several limitations of our study need to be acknowledged. First, our population was relatively small as we decided to focus on strict inclusion criteria in order to reduce any bias by selecting only steady state homozygous sickle-cell disease patients free of confounding factors that could be linked to LV systolic impairment. Despite this limitation, we found a significant impairment of deformation indices. Second, we only evaluated the LV longitudinal deformation parameters as they appear to be the most efficient and reproducible parameters in speckle tracking technology [START_REF] Feigenbaum | A practical approach to using strain echocardiography to evaluate the left ventricle[END_REF]. Moreover, it has been earlier described that the natural course of myocardial diseases is characterized by the impairment of the longitudinal function that occurs before the onset of circumferential and radial alterations [START_REF] Kalogeropoulos | Echocardiographic evaluation of left ventricular structure and function: new modalities and potential applications in clinical trials[END_REF]. Finally, the prognostic implications of LV deformation parameters were not assessed in this study and need further investigations.

Steady state young adult patients suffering from homozygous sickle-cell anemia exhibited a subclinical impairment of myocardial contractility. 2D speckle-tracking imaging might constitute a useful tool for long-term follow-up and to study the natural course of LV dysfunction in these patients. Additional studies are required to further understand the prognostic implications of deformation parameters.

DISCLOSURES

The authors have no disclosures.

FUNDING, GRANT OR OTHER SUPPORT

None

ACKNOWLEDGEMENT None

Figure 1 .

 1 Figure 1. Comparison of LV global longitudinal strain measures between a sickle-cell anemia

Table 1 :

 1 General characteristics of the study subjects

	SCA patients	Controls	p value
	(n=30)	(n=30)

Table 2 :

 2 Echocardiographic measurements in patients and controls 1

	SCA patients	Controls	p value
	(n=30)	(n=30)

Table 3 :

 3 Comparison of clinical and echocardiographic data according to LV Mass index in

	Sickle cell Anemia patients
		LV	Normal	p value
		hypertrophy	LV morphology
		(n=14)	(n=16)
	Age (years)	24.3 ± 5.3	25.0 ± 5.1	0.80
	Women	10 (71)	7 (44)	0.13
	Systolic Blood Pressure (mmHg)	114.4 ± 10.0	113.9 ± 9.0	0.88
	Diastolic Blood Pressure (mmHg)	64.1 ± 7.8	66.7 ± 9.3	0.51
	Heart rate (beats/min)	70.9 ± 10.6	73.9 ± 10.1	0.37
	Hemoglobin (g/dl)	8.3 ± 0.6	8.8 ± 1.1	0.21
	LV mass index (g/m²)	119.2±17.0	84.6±14.8	< 0.0001
	Cardiac index (L/min/m²)	4.2 ± 0.6	3.9 ± 0.8	0.13
	TR velocity (m/s)	2.4 ± 0.2	2.4 ±0.2	0.52
	E (cm/s)	100.2 ± 13.3	89.6 ± 17.0	0.05
	A (cm/s)	48.8 ± 13.0	50.1 ± 10.7	0.49
	E/A	2.2 ± 0.5	1.9 ± 0.4	0.15
	E-wave deceleration time (ms)	163.6 ± 21.9	174.6 ± 42.9	0.65
	e' (cm/s)	16.6 ± 4.0*	18.0 ± 3.4	0.38
	E/e'	6.7 ± 3.4*	5.1 ± 1.3	0.10
	Diastolic dysfunction	5 (36)	1 (6)	0.04
	LV Ejection Fraction (%)	59.4 ± 3.8	60.8 ± 3.8	0.42
	Global longitudinal strain (%)	-17.1 ± 2.4*	-18.5 ± 1.5	0.07
	Global longitudinal strain rate (s -1)	-0.88 ± 0.09*	-0.96 ± 0.08	0.02
	*n=13.

Data are expressed as mean ± SD or as number (%) TR velocity = tricuspid regurgitant maximal velocity; E = Early peak diastolic velocity of the mitral inflow; A= Late peak diastolic velocity of the mitral inflow; e' = early diastolic mitral annular tissue Doppler velocity, E/e' = ratio between peak velocities of mitral E wave and early-diastolic mitral annulus; SCA = Sickle Cell Anemia


Sebastian Müller
email: mueller@cmi.univ-mrs.fr

INTERACTING GROWTH PROCESSES AND INVARIANT PERCOLATION

Keywords: AMS 2000 subject classifications: Primary 60J10, 60J80; secondary 05C80 survival, interacting branching random walk, invariant percolation, unimodular random networks

The aim of this paper is to underline the relation between reversible growth processes and invariant percolation. We present two models of interacting branching random walks (BRWs), truncated BRWs and competing BRWs, where survival of the growth process can be formulated as the existence of an infinite cluster in an invariant percolation on a tree. Our approach is fairly conceptual and allows generalizations to a wider set of "reversible" growth processes.

1. Introduction. We discuss two different interacting branching random walks (BRWs) in discrete time. In the first model, called BRW N , only a finite number N of particles are allowed per site. A natural question is whether the process BRW N may survive with positive probability. Partially answers to this question were given by Zucca [START_REF] Zucca | Survival, extinction and approximation of discrete-time branching random walks[END_REF]. We complete these results for symmetric BRWs on Cayley graphs in Theorem 1.1: BRW N survives with positive probability for sufficiently large N .

The second model concerns competing BRWs. Suppose there are two different types or species of particles: invasive and non-invasive particles. The invasive particles behave like particles in a usual BRW and are not influenced by the non-invasive particles. These latter however die once they share a site with an invasive particle. We prove, see Theorem 1.2, that for weakly surviving (or transient) BRWs on Cayley graphs both processes may coexist with positive probability.

Our proofs are based on a connection between the survival of reversible growth processes and the existence of infinite clusters in percolation on trees. This connection was used previously by Schramm [START_REF] Kozma | Percolation on a product of two trees[END_REF] and Benjamini and Mueller [START_REF] Benjamini | On the trace of branching random walks[END_REF]. In the first reference BRWs are used to study connectivity properties of Bernoulli percolation on non-amenable Cayley graphs. Benjamini and Mueller [START_REF] Benjamini | On the trace of branching random walks[END_REF] use results on invariant percolation (on trees) to study BRWs on unimodular random graphs.

In general the study of interacting growth processes or particles systems is challenging and a general treatment seems to be out of reach (at least at the moment). The case-by-case study often involves a high amount of technical effort. The approach given here is more conceptual using soft proofs. While we concentrate on two concrete examples in this paper we want to underline that our approach is fairly general and only relies on two steps: the formulation of the process as a unimodular random network and the control of the marginal of the corresponding invariant percolation.

1.1. Motivation. Besides highlighting the connection between growth processes and percolation there are several other motivations for the underlying work. One of these motivations is to propose models for spatial interaction and competition of growth processes. One of the earliest and simplest models of growth processes is the Galton-Watson branching process where particles branch independently of the history of the process. However, this may not be very realistic when there is competition for limited resources such as space and food in the habitat. A considerable effort was made to introduce dependence in the sense that the individual reproduction may be influenced by the history of the population. While many of these models consider dependence only on the total population size, we refer to Kersting [START_REF] Kersting | On recurrence and transience of growth models[END_REF] and the paper referring to it for a mathematical introduction, the study of models with local interactions is perhaps even more challenging. Models in this direction are for example contact processes or restrained BRWs. Here, particles breed depending on the local configurations of the particles and one is interested in extinction, equilibrium and explosion of the process. We refer to Bertacchi et al. [START_REF] Bertacchi | Ecological equilibrium for restrained branching random walks[END_REF] and references therein. A natural way to model local dependencies between particles is also the truncated BRW N that was introduced in Bertacchi and Zucca [START_REF] Bertacchi | Approximating critical parameters of branching random walks[END_REF] in continuous and in Zucca [START_REF] Zucca | Survival, extinction and approximation of discrete-time branching random walks[END_REF] in discrete time.

As a byproduct of our approach we can also control the following processes. In a first model dependencies are not only between particles of the same generation but also between particles of different generations. Suppose that each vertex has a finite amount of resources that allow at most N particles to branch; once the resources are used any other particle visiting this site will die without producing any offspring. For this model an analogue result of Theorem 1.1 holds for the weakly surviving regime, see Section 3. Another model of annihilating BRW that can be treated is the process where the probability that two particles, meeting at a same vertex, annihilate each other is a function of their distance in the family tree. For instance, particles annihilate each other if and only if their distance in the family tree is larger than some constant M . Despite the non-monotonicity of the model one can prove that for M large enough the process survives in the weakly surviving regime.

Our second model describes two species competing for resources and studies whether the weaker (or non-invasive) species has a chance of survival. Models for competing spatial growth attracted a lot of attention in the last decades. Perhaps the most common models in the probability community are the voter model, the Richardson model, and mixtures of these two. We refer to Häggström and Pemantle [START_REF] Häggström | First passage percolation and a model for competing spatial growth[END_REF], Kordzakhia and Lalley [START_REF] Kordzakhia | A two-species competition model on Z d[END_REF], and Blair-Stahn [START_REF] Blair-Stahn | First passage percolation and competition models[END_REF] for an introduction and more references. Let us note that most of the results are restricted to Z d and make strong use of a connection with first passage percolation. Our model constitutes, to our knowledge, one of the first models beyond Z d and is more realistic for models where the space of possible habitat grows exponentially (is expanding) in time. In particular, this is relevant for models at the early stage of competing populations. Moreover, it provides a stochastic model for the so called dominance displacement competition, we refer to Amarasekare [START_REF] Amarasekare | Competitive coexistence in spatially structured environments: a synthesis[END_REF] for more details and references. In these kind of models superior competitors can displace inferior competitors but not vice-versa. However, the inferior competitors can establish "patches or niches" where the superior competitor do not colonize. This latter phenomomen is highlighted by our theoretical result: as long as the superior competitor does not colonize the whole space the free patches are large enough to allow the inferior competitor to survive.

A somehow completely different motivation originates from a structure theoretic question. Classification of groups in terms of the behavior of random processes attracted a lot of attention. In particular, a consequence of Gromov's famous theorem on groups of polynomial growth is that a finitelygenerated group admits a recurrent random walk if and only if it contains a finite-index subgroup isomorphic to Z or Z 2 , e.g., see Chapter 3 in Woess [START_REF] Woess | Random walks on infinite graphs and groups[END_REF]. Kesten's criterion for amenability says that a finitely-generated group is amenable if and only if the spectral radius for all (or some) symmetric random walks is equal to 1. This phenomenon is also underlined by phase transitions on non-amenable graphs whose study underwent a rapid development, e.g., see Lyons [START_REF] Lyons | Phase transitions on nonamenable graphs[END_REF]. Moreover, Benjamini [START_REF] Benjamini | Survival of the weak in hyperbolic spaces, a remark on competition and geometry[END_REF] proposed a deterministic competition model that admits coexistence on hyperbolic groups but not on Z d . A motivation for this is to find a stochastic process (or a system of interacting processes) that shows an additional phase-transition on (oneended) hyperbolic groups compared to non-hyperbolic groups. Theorem 1.2 shows that coexistence of competing BRWs, at least in the weakly surviving regime, occurs regardless of the hyperbolicity of the underlying graph.

1.2. Structure of the paper. We formulate our models and corresponding main results, Theorem 1.1 and Theorem 1.2, in the rest of this section. In Section 2 we introduce the notations and basic results of random unimodular networks (URNs) and present two preliminary results on percolation of URNs, Lemma 2.1 and Theorem 2.3. Section 3 contains the proof of Theorem 1.1 and Section 4 the one of Theorem 1.2. 1.3. Branching random walks. The definition of a branching random walk (BRW) requires a probability distribution, µ = (µ k ) k≥0 , describing the branching and a transition kernel, P = (p(x, y)) x,y∈V describing the movement of the particles on some underlying discrete space V . The BRW starts at some initial position o with one particle and then at each (discrete) time step each particle splits into k particles with probability µ k and each of the resulting particles moves one step according to the transition kernel P . Both splitting and movement of a particle at time n are independent of the previous history of the process and the behavior of the other particles at time n.

The expected number of offspring is denoted by m = k kµ k and we will always assume that the corresponding Galton-Watson process is supercritical, i.e., m > 1. Furthermore, we assume that P is the transition kernel of an irreducible random walk. Therefore, the spectral radius ρ = ρ(P ) = lim sup n→∞ (p (n) (x, x)) 1/n , x ∈ V , of the underlying random walk is well defined.

There is an alternative description of BRWs that uses the concept of treeindexed random walks introduced in [START_REF] Benjamini | Markov chains indexed by trees[END_REF]. Let (T, r) be a rooted infinite tree. Denote by v the vertices of T and let |v| be the (graph) distance from v to the root r. For any vertex v denote v -the unique predecessor of v, i.e., v -∼ v and |v -| = |v| -1. We denote by G = (V, E) a graph with vertex set V and edge set E and write (G, o) for a rooted graph. The Cayley graph of a finitely generated group Γ with respect to some generating set S will also be denoted by G; in this case V = Γ.

Let (G, o) be a rooted graph. The tree-indexed process (S v ) v∈T is defined inductively such that S r = o and for vertices x, y ∈ V we have

P S v = x|S v -= y, {S w : w / ∈ {v, v -}, |w| ≤ n} = P(S v = x|S v -= y) = p(x,

y).

A tree-indexed random walk becomes a BRW if the underlying tree T is a realization of a Galton-Watson process. We call T the family tree and G the base graph of the BRW. The symbol T will sometimes stand for random variables taking values in the space of trees and sometimes for their realizations.

If G is a Cayley graph the BRW can be described as a marked (or labeled) Galton-Watson tree. Let Γ be a finitely generated group with group identity o and write the group operations multiplicatively. Let q be a symmetric probability measure on a finite symmetric generating set of Γ. The corresponding random walk on Γ is the Markov chain with state space Γ = V and transition probabilities p(x, y) = q(x -1 y) for x, y ∈ Γ. Equivalently, the random walk (starting in x) can be described as

S n = xX 1 • • • X n , n ≥ 1,
where the X i are i.i.d. random variables with distribution q. In order to define the BRW we label the edges of T with i.i.d. random variables X v with distribution q; the random variable X v is the label of the edge (v -, v). These labels correspond to the steps of the BRW and positions of particles are given by

S v = o • i X v i where v 0 = r, v 1 , . . . , v n = v is the unique geodesic from r to v at level n.
A BRW is said to survive strongly (or locally) if every vertex is visited infinitely many times with positive probability and to survive weakly if the process survives with positive probability but every finite subset is eventually free of particles. In formulae:

strong survival ⇔ ∀x ∈ G : P(|{v : S v = x}| = ∞) > 0 weak survival ⇔ P(|T| = ∞) > 0 and ∀x ∈ G : P({v : S v = x}| = ∞) = 0.
Important to note that several authors speak sometimes of transience and recurrence of BRWs instead of weak and strong survival. A consequence of the classification of recurrent groups and Kesten's amenability criterion is that a BRW on a Cayley graph survives strongly if and only if mρ(P ) > 1, see also [START_REF] Gantert | The critical Branching Markov Chain is transient[END_REF] for an alternative proof.

We make the following standing assumptions on the underlying probability measures.

Assumption 1.1.

• The underlying Galton-Watson process is supercritical, m = k kµ k > 1, and the offspring distribution µ is of finite support, i.e., there exists some d such that d-1 k=0 µ k = 1. Furthermore, we assume that µ 1 > 0. • Let G be a finitely generated group with symmetric finite generating set S. The distribution q of the random walk on G is symmetric and such that supp(q) = S and q(e) > 0.

Remark 1.1. While the assumptions on supercriticality of the Galton-Watson process and symmetry and irreducibility of the random walk are crucial, the other assumptions are made for sake of a better presentation and to avoid periodic subtleties. In particular, the assumption that the genealogy of the invasive BRW is a Galton-Watson process with finite support is not needed anywhere. Moreover, the assumption on finite support of the offspring distribution can be removed from the BRW N and the non-invasive process by adding an additional coupling of the Galton-Watson process with a "truncated" version. For instance, denote by µ (M ) a truncated version of µ, i.e., µ

(M ) k = µ k for all k < M and µ (M ) M = ∞ k=M µ k , where M is chosen sufficiently large such that m (M ) > 1. Theorem 1.1 guarantees the existence of some N (M ) c such that the BRW (M ) N with underlying offspring distribution µ (M ) may survive if N ≥ N (M ) c . Since BRW (M ) N is stochastically dominated by BRW N this implies survival of the latter if N ≥ N (M ) c
. The argument for the offspring distribution of the non-invasive process is similar.

1.4. Truncated branching random walk. Branching random walks may be used to describe the evolution of a population or particle system at early stage with no restrictions on resources. In order to refine the model one might introduce a limit of particle at each site: for some N ∈ N at most N particles are allowed at a same site at the same time. While most of the existing models describing variants of this models are in continuous time, we prefer a description in discrete time since our proof technique is more suitable to this setting. Let N ∈ N and W be a finite set and denote by C(W, N ) a random variable that chooses uniformly N elements from the set W with the convention that if N > |W | then all |W | elements are chosen.

We define an auxiliary process: let S aux v be a BRW with offspring distribution µ and transition kernel P and denote by T aux the corresponding family tree. For every x ∈ X and n ∈ N we denote by W n,x = {w : |w| = n, S aux w = x} the particles at generation n that are in position x. We add a special state † to the state space V and define the process BRW N on V ∪{ †} as

(1.1) S N v = S aux v if v ∈ C(W |v|,S aux v , N ) † otherwise, for all v ∈ T aux ,
where {C(W n,x , N ), n ∈ N, x ∈ V } is a family of independent random variables independent of S aux v . The state † induces a site percolation on the family tree T aux in the following way: declare a vertex v closed if S aux v = † and open otherwise. Configurations of this percolation are denoted by η † , where η † (v) = 1 corresponds to the fact that the site v is open and η † (v) = 0 to the fact that the site v is closed. We denote by T r the connected component containing the root.

The truncated process BRW N can therefore be denoted by (S N v ) v∈Tr . We say that BRW N survives if |T r | = ∞. It is easy to see that survival is a monotone property: the process survives for N 2 > N 1 if it survives for N 1 .

The following results asserts that there exists a non-trivial phase transition.

Theorem 1.1. Let P be the transition kernel of a symmetric irreducible random walk on an infinite finitely generated group Γ and let µ be an offspring distribution with m > 1. Then, there exists a critical value N c = N c (µ, P ) < ∞ such that if N ≤ N c the process dies out a.s. and if N > N c the process survives with positive probability.

Remark 1.2. Some of the results have been proved by Zucca [START_REF] Zucca | Survival, extinction and approximation of discrete-time branching random walks[END_REF] in Theorem 6.5: the case m > 1/ρ(P ) was settled completely but the case m ≤ 1/ρ(P ) was only proved for some BRWs on Z d and on the homogeneous tree. Zucca's results are presented in the more general context of quasitransitivity and treat some BRWs with drift on Z d that are not covered by our result. While his proof technique is different to ours, it is interesting to note that he uses as well a percolation argument for the case m > 1/ρ(P ); this time directed percolation on products of N.

Remark 1.3. The notion of weak and strong survival can be adapted to the truncated BRW in a natural way. If the underlying BRW survives weakly then BRW N survives weakly if N > N c . If the underlying BRW survives strongly then Theorem 6.5 in [START_REF] Zucca | Survival, extinction and approximation of discrete-time branching random walks[END_REF] implies that there exists some

N (s) c such that BRW N survives strongly if N > N (s)
c . However, it is not known if there is an additional regime of weak survival in this case, i.e., there exists one

N (w) c < N (s) c such that BRW N survives weakly if N (w) c < N ≤ N (s) c .
1.5. Competing branching random walks. We consider two competing BRWs that interact in the following way. One BRW is invasive, i.e., the particles are not influenced by the other particles, and the second is noninvasive in the sense that once a particle shares a site (at the same time) with an invasive particle it dies without having any offspring. The particles live on an infinite finitely generated group Γ and we note (µ i , P i ), (µ n , P n ) for their offspring distribution and transition kernels. Moreover, denote by m i and m n their expected number of offspring.

We give a formal definition of a slightly different process in the following. The branching distributions µ i and µ n give rise to two family trees T i and T n . The non-invasive BRW will start in o and the invasive in some point x = o.

The invasive BRW (S i v ) v∈T i is defined as an ordinary BRW. In order to define the non-invasive BRW we first construct an intermediate version. Let S aux v be an ordinary BRW with (µ n , P n ) and introduce an additional state †. Denote by T i k = {v ∈ T i : |v| = k} the (random) collection of vertices of T i at distance k from the root.

The non-invasive BRW on V ∪ { †} is defined together with (S i v ) v∈T i on a joint probability space such that

(1.2) S n v = S aux v if S i w = S aux v ∀w ∈ T i |v| , † otherwise, for v ∈ T aux .
We denote by P = P o,x the canonical probability measure describing both processes on a same probability space.

We introduce a percolation of the family tree T aux by declaring a vertex v ∈ T aux closed if and only if S aux v = †. Configurations of this percolation are denoted by η † . We denote by T n r the connected component of T aux containing the root r.

We say that there is coexistence if with positive probability both processes survive, i.e.,

P o,x (|T n r | = ∞, |T i | = ∞) > 0.
Using the assumptions µ i,1 > 1 and q n (e) > 0 together with the strong Markov property one sees that if

P o,x (|T n r | = ∞, |T i | = ∞)
> 0 holds for some x then it holds for all x = o. Theorem 1.2. Let P i and P n transition kernels of random walks on a infinite finitely generated group G and let µ i and µ n satisfying Assumption 1.1. Then, there is coexistence of the invasive and the non-invasive process if m i ρ i < 1.

Remark 1.4 (Strongly surviving regime). Theorem 1.2 states that there is always coexistence if the invasive BRW is weakly surviving. Since we assume the underlying random walk to be symmetric the result does not apply to BRW on Z d . This is because Kesten's amenability criterion implies that there is no (symmetric) weakly surviving BRW on amenable groups (including Z d ). However, on Z d one can show that there is no coexistence if m i > m n . This can be seen by proving a shape theorem using large deviation estimates of the underlying random walks. We refer to [START_REF] Comets | Shape and local growth for multidimensional branching random walks in random environment[END_REF] where a shape theorem was established even in random environment. However, this approach fails for groups beyond Z d since large deviations for random walks on groups are up to now not sufficiently studied. Moreover, there is no reason why the shape of the particles should be a "convex set"; see also [START_REF] Hueter | Anisotropic branching random walks on homogeneous trees[END_REF] and [START_REF] Candellero | Branching random walks on free products of groups[END_REF] for results on groups with infinitely many ends. Hence, one may ask in the flavor of Benjamini [START_REF] Benjamini | Survival of the weak in hyperbolic spaces, a remark on competition and geometry[END_REF]: does coexistence in the strongly surviving regime, m i ρ i > 1 and m n ρ n > 1, depend on the hyperbolicity of the base graph?

Remark 1.5 (Critical case). We can not treat the critical case, m i ρ i = 1, in general since we do not know for which groups and walks the Green function G(x, y|ρ -1 i ) decays exponentially in d(x, y). However, this is true for finite range symmetric random walks on hyperbolic groups, see [START_REF] Gouëzel | Local limit theorem for symmetric random walks in Gromovhyperbolic groups[END_REF], and hence our methods also cover the critical case in this setting.

Remark 1.6. On groups with infinitely many ends we have that on the event of co-existence not every non-invasive particle has an offspring that will be killed. This is due to the fact that the invasive BRW leaves some neighborhoods of the boundary unvisited where the non-invasive process may live in peace, see [START_REF] Hueter | Anisotropic branching random walks on homogeneous trees[END_REF] and [START_REF] Candellero | Branching random walks on free products of groups[END_REF]. The results in [START_REF] Lalley | Hyperbolic branching Brownian motion[END_REF] strongly suggests that this is still true for Fuchsian groups and one is tempted to ask if this phenomenon holds true for general groups. Since the shape of a single BRW is connected to the question of co-existence for competing BRWs, see also Remark 1.4, an answer to this question seems to be related to the conjecture that the trace of a weakly surviving BRW has infinitely many ends, see [START_REF] Benjamini | On the trace of branching random walks[END_REF].

2. Preliminaries. Unimodular random graphs (URGs) or stochastic homogeneous graphs have several motivations and origins. We concentrate in this note on the probabilistic point of view since it gives rise to the tools we are going to use. For more details on the probabilistic viewpoints we refer to [START_REF] Aldous | Processes on unimodular random networks[END_REF], [START_REF] Benjamini | Ergodic theory on stationary random graphs[END_REF], [START_REF] Benjamini | Unimodular random trees[END_REF] and to [START_REF] Kaimanovich | Stochastic homogenization of horospheric tree products[END_REF] for an introduction to the ergodic and measure theoretical origins.

One of our motivation to consider unimodular random graphs is the use of a general Mass-Transport Principle (MTP) which was established in [START_REF] Benjamini | Recurrence of Distributional Limits of Finite Planar Graphs[END_REF] under the name of "Intrinsic Mass-Transport Principle" and is basically (2.1). It was motivated by the fact that the Mass-Transport Principle is heavily used in percolation theory and therefore lifts many results on unimodular graphs to a more general class of graphs. In [START_REF] Aldous | Processes on unimodular random networks[END_REF] a probability measure on rooted graphs is called unimodular if this general form of the MTP holds. Another motivation to consider URGs is the fact that unimoduar random trees (URTs) can be seen as connected components in an invariant percolation on trees, see [6, Theorem 4.2] or Theorem 2.2 in this paper.

Let us define URGs properly. Recall that we write (G, o) for a graph G = (V, E) with root o. A rooted isomorphism between two rooted graphs (G, o) and (G ′ , o ′ ) is an isomorphism of G onto G ′ which takes o to o ′ . We denote by G * the space of isomorphism classes of rooted graphs and write [G, o] for the equivalence class that contains (G, o). The space G * is equipped with a metric that is induced by the following distance between two rooted graphs (G, o) and (G ′ , o ′ ). Let α be the supremum of those r > 0 such that there exists some rooted isomorphism of the balls of radius ⌊r⌋ (in graph distance) around the roots of G and G ′ and define d((G, o), (G ′ , o ′ )) = 1/(1 + α). This metric turns G * into a separable and complete space. In the same way one defines the space G * * of isomorphism classes of graphs with an ordered pair of distinguished vertices. A Borel probability measure ν on G * is called unimodular if it obeys the Mass-Transport Principle: for all Borel function f :

G * * → [0, ∞], we have (2.1) x∈V f (G, o, x)dν([G, o]) = x∈V f (G, x, o)dν([G, o]).
Observe that this definition can be extended to networks. A network is a graph G = (V, E) together with maps from V and E to some complete separable metric space Ξ. These maps will serve as marks (sometimes called labels) of the vertices and edges of the graph and may a priori be unrelated. Edges are considered as directed so that each edge is given two marks. While the definition of the above equivalence classes for networks is straightforward, one has to adapt the metric between two networks as follows: α is chosen as the supremum of those r > 0 such that there is some rooted isomorphism of the balls of radius ⌊r⌋ around the roots of G and G ′ and such that each pair of corresponding marks has distance at most 1/r. A probability measure on rooted networks is called unimodular if Equation (2.1) holds. Realizations of these measures or denoted as unimodular random networks (URN). Following the existing literature we use the notation (G, o) as well for networks and specify the marks of a network only when it is necessary.

Let us illustrate this definition with the very important examples of Galton-Watson measures. Let µ = {µ k } k∈N be a probability distribution on the integers. The Galton-Watson tree is defined inductively: start with one vertex, the root of the tree. Then, the number of offspring of each particle (vertex) is distributed according to µ. Edges are between vertices and their offspring. We denote by GW the corresponding measure on the space of rooted trees. In this construction the root clearly plays a special role. For this reason, in the unimodular Galton-Watson measure (UGW) the root has a biased distribution: the probability that the root has degree k + 1 is proportional to µ k k+1 . In cases where we use the UGW measure instead of the standard GW measure to define the family tree of the BRW we denote the BRW by UBRW.

It will be important to change the marks of a URN in a way that the network remains unimodular. For instance, let ξ : V → Ξ be a mark of a URN with measure ν. Let φ be a measurable map on rooted networks that takes each network to an element of the mark space Ξ. Define Φ as the map on rooted networks that takes a network (G, o) to another network on the same underlying graph, but replaces the mark ξ(x) by φ(G, x) for all vertices x ∈ V . Then, by Lemma 4.1 in [START_REF] Benjamini | Unimodular random trees[END_REF] the push forward measure Φ * ν is also unimodular. One can also add i.i.d. marks to existing networks. Let ξ 1 : V → Ξ be a mark and define a new mark ξ(x) = (ξ 1 (x), ξ 2 (x)) where the (ξ 2 (x)) x∈V are realizations of i.i.d. random variables with distribution p. Denote by ν p the resulting measure. Again, Lemma 4.1 in [START_REF] Benjamini | Unimodular random trees[END_REF] states that ν p is unimodular.

Edge marks are maps from V × V → Ξ and therefore the above transformation can be stated analogously for edge marks. If an edge mark ξ is symmetric, i.e., ξ(x, y) = ξ(y, x) ∀x, y ∈ V , it can be seen as a map from E → Ξ. Now, consider a UGW-tree, add i.i.d. edge marks with distribution q and denote the resulting measure by UGW q . The latter marks correspond to the steps of the BRW and we can interpret, using the definition of the UBRW as a tree-indexed random walk, the UBRW as a URN of measure UGW q .

Let ν be a unimodular measure on rooted networks (G, o) and suppose that the mark space Ξ contains a particular mark †. This special mark induces a natural percolation on the rooted network: a vertex is closed if its mark equals to † and open otherwise. We refer to Section 6 in [START_REF] Aldous | Processes on unimodular random networks[END_REF] for more formal definitions and some background on percolation on URNs.

Lemma 2.1. Let ν be a unimodular measure on rooted networks. Let † be a particular element of the mark space that induces a percolation. Denote by (C, o) the connected (marked) component containing the origin and denote by ν † its corresponding measure. Then, the measure ν † is again a unimodular measure on rooted networks.

Proof. The proof is a check of the Mass-Transport Principle (2.1). Define Φ as the map that takes (G, o) to the connected component (C, o). The measure ν † is given as the push forward measure Φ * ν. We denote by V C the vertex set of (C, o) and by V G the vertex set of (G, o). For any positive borel function f : G * * → [0, ∞] define its "restrictions"

f C (G, x, y) = f (C, x, y), if x, y ∈ V C , 0, otherwise,
where (C, o) = Φ((G, o)). By the change of variables formula for the push forward measure we obtain

x∈V C f (C, o, x)dν † ([C, o]) = x∈Φ([G,o]) f (Φ([G, o]), o, x)dν([G, o]) = x∈V G f Φ([G,o]) (Φ([G, o]), o, x)dν([G, o]).
Unimodularity of ν implies that the latter term is

x∈V G f Φ([G,x]) (Φ([G, x]), x, o)dν([G, o]),
which equals

x∈V C f (Φ([G, x]), x, o)dν([G, o]) = x∈V C f (C, x, o)dν † ([C, o])
We make strong use of the following connection between unimodular measures and invariant percolation. Theorem 2.2 (Theorem 4.2, [START_REF] Benjamini | Unimodular random trees[END_REF]). Let ν be a probability measure on rooted networks whose underlying graphs are trees of degree at most d. Then ν is unimodular iff ν is the law of the open component of the root in a labeled percolation on a d-regular tree whose law is invariant under all automorphisms of the tree.

In an invariant percolation the probability that an edge is open is well defined and is called the marginal of the percolation. There are results by Adams and Lyons [START_REF] Adams | Amenability, Kazhdan's property and percolation for trees, groups and equivalence relations[END_REF] and Häggström [START_REF] Häggström | Infinite clusters in dependent automorphism invariant percolation on trees[END_REF] that state that for invariant percolation on homogeneous trees a sufficiently high marginal guarantees (with positive probability) the existence of infinite clusters. We generalize this result to "invariant percolation" on supercritical Galton-Watson trees. This is done by adapting the proof of Theorem 1.6 in [START_REF] Häggström | Infinite clusters in dependent automorphism invariant percolation on trees[END_REF]. Theorem 2.3. Let UGW be a supercritical unimodular Galton-Watson measure of maximal degree d. Then, there exists some c UGW < 1 such that for any unimodular labeling and any particular element † of the mark space the induced percolation UGW † assigns positive probability to the existence of infinite clusters if the marginal is greater than c UGW .

Proof. Lemma 2.1 and Theorem 2.2 imply that UGW † defines an invariant (site) percolation of the homogeneous tree of degree d. As we have to treat two interlaced percolations, we denote by η UGW the configurations of the percolation induced by UGW and by η † the configurations induced by UGW † . From the definition of UGW † we have that components that are connected in η † are also connected in η UGW . For any vertex v in T d and a given configuration η we write C η (v) for the connected component containing v in η. We denote by ∂C the outer (vertex) boundary of a vertex set C. We can now adapt the proof of Theorem 1.6 in [START_REF] Häggström | Infinite clusters in dependent automorphism invariant percolation on trees[END_REF]. Given a configuration η † we define a function ψ η † on the vertex set of T d . For a vertex v denote by v 1 , . . . , v d its adjacent vertices in T d and let (2.2)

ψ(v) =              1 if η † (v) = 1 and |C η † (v)| = ∞, 1 if η † (v) = 1, |C η † (v)| < ∞, and |Cη † (v)| |∂Cη † (v)| ≥ K, 0 if η † (v) = 1, |C η † (v)| < ∞, and |Cη † (v)| |∂Cη † (v)| < K, 1 + d i=1 f (v i ) if η † (v) = 0,
where

f (w) = |Cη † (w)| |∂Cη † (w)| if |C η † (w)| < ∞ and |Cη † (w)| |∂Cη † (w)| < K, 0 otherwise,
for some positive constant K to be chosen later. We can now, as in the proof Theorem 1.6 in [START_REF] Häggström | Infinite clusters in dependent automorphism invariant percolation on trees[END_REF], interpret ψ as a distribution of mass over the vertices. Originally every vertex has mass 1. For vertices v in an infinite cluster or vertices v in finite clusters with

|Cη † (v)| |∂Cη † (v)| ≥ K the mass in v remains unchanged. If v is in a finite cluster such that |Cη † (v)|
|∂Cη † (v)| < K then v distributes its mass equally to the closed vertices incident to C η † (v). If η † (v) = 0, then v receives additional mass from the distributing vertices. For two vertices v and w we write △ψ(v, w) for the flow of mass from v to w (using the above interpretation) and obtain

ψ(v) = 1 + w △ψ(w, v).
Consider random configurations X UGW and X † that are distributed according to UGW and UGW † . Since UGW † is unimodular we have for any pair of vertices v and w that

E UGW † [△ψ(v, w)] = 0.
Since ψ is bounded we obtain that

(2.3) E UGW † [ψ(v)] = 1 + w E UGW † [△ψ(w, v)] = 1.
For the sake of typesetting we write {

|C X † (v)| |∂C X † (v)| ≥ K} for the event {|C X † (v)| < ∞, |C X † (v)| |∂C X † (v)| ≥ K}. Using Equation (2.
3) with the definition of ψ in Equation (2.2) we obtain that

UGW † (X † (v) = 1, |C X † (v)| = ∞) is greater or equal than 1 -UGW † X † (v) = 1, |C X † (v)| |∂C X † (v)| ≥ K -c K UGW † (X † (v) = 0),
where c K = 1 + dK. In order to adjust the value of K recall the following. The anchored (vertex) isoperimetric constant for a graph G is defined as

i(G, v) = inf S∋v |∂S| |S| ,
where S ranges over all connected vertex sets containing a fixed vertex v.

Note that i(G, v) does not depend on the choice of the edge v. Corollory 1.3 in [START_REF] Chen | Anchored expansion, percolation and speed[END_REF] states that i(T, v) > 0 a.s. on the event that T is infinite. Now, since

UGW † X † (v) = 1, |C X † (v)| |∂C X † (v)| ≥ K is bounded above by UGW i(T) -1 > K, |C X UGW (v)| = ∞ ,
we can choose K sufficiently large such that

UGW † X † (v) = 1, |C X † (v)| |∂C X † (v)| ≥ K < UGW (|C X UGW (v)| = ∞) .
Eventually, there exists some constant c > 0 such that

UGW † (|C † (v)| = ∞ | |C X UGW (v)| = ∞) > c - c K UGW † (X † (v) = 0)) UGW(|C X UGW (v)| = ∞) .
Hence, choosing the marginal UGW † (X † (v) = 1) sufficiently high assures that UGW † (|C X † (v)| = ∞) > 0.

Remark 2.1. An inspection of the proof above reveals that Theorem 2.3 holds true for unimodular measures on rooted networks whose underlying graphs are trees of bounded degree and that give positive weight to infinite networks such that almost all infinite realizations have positive anchored isoperimetric constant.

3. Truncated BRW -proof of Theorem 1.1. Since the case m > 1/ρ(P ) was proven in [START_REF] Zucca | Survival, extinction and approximation of discrete-time branching random walks[END_REF] let us assume in the following that m ≤ 1/ρ(P ). For the case µ 0 = 0 the proof is essentially given in the proof of Theorem 3.1 in [START_REF] Benjamini | On the trace of branching random walks[END_REF]. We give a concise proof using the results of [START_REF] Benjamini | Unimodular random trees[END_REF] and Theorem 2.3. Moreover, we hope that the example of truncated BRW will serve as an introduction of our approach "interacting growth process and invariant percolation" and hence is useful for a better understanding of the proof of Theorem 1.2. Our approach consists of two steps: an adaptation of the model such that the family tree is a URT and the control of the marginal of the corresponding invariant percolation.

3.1. Adapting the model. The aim is to identify an invariant percolation (or unimodular measure) of the family tree. Since the percolation induced by † is in not an invariant percolation, there is need for a reformulation of our problem. We will define a new process in a way such that vertices that were visited more than N times become "deadly" for all instances of times. In other words, if x is a vertex of the base graph such that |{v : S v = x}| > N , then we set S new v = † for all v such that S v = x. More formally, let (T, r) be the labeled UGW-tree (the BRW) and define

(3.1) φ(T, x) = †, |{v : S v = x}| > N, •, |{v : S v = x}| ≤ N.
The corresponding push forward measure Φ * UGW q is again unimodular, see Lemma 4.1 in [START_REF] Benjamini | Unimodular random trees[END_REF].

3.2.

Control of the marginal. The underlying BRW is supposed to be weakly surviving, i.e., P(|{v : S v = S r }| < ∞) = 1. Hence, we can apply Theorem 2.3 and choose N u sufficiently large such that the marginal P(|{v : S v = S r }| ≤ N u ) is sufficiently high. This guarantees that with positive probability the cluster containing r is infinite and that the process S new v survives with positive probability. Since S new v is stochastically dominated by the truncated BRW, we obtain that BRW N survives with positive probability for sufficiently large N . This yields, together with the monotonicity of the model, the existence of a critical value N c given in Theorem 1.1.

4.

Competing BRWs -proof of Theorem 1.2. We proceed in two steps as in the previous section. In this section we suppose that ρ i m i < 1 and assume without loss of generality that ρ n m n < 1.

4.1. Adapting the model. The family tree of the non-invasive process is in general not a URT. We invite the reader to the following informal description of the situation.

Let us start both processes in neighboring sites, then the offspring of the starting particles are very likely to be killed by those of the invasive process. However, if we consider some non-invasive particle very late in the genealogical process, then, given the fact that the particle exists (or is alive), one might expect that its ancestors never have been very close to invasive particles. Hence, the chances of its children to survive are high as well. As a conclusion we have to adapt the invasive process in a way that every particle of the auxiliary process (of the non-invasive process) has the same probability to encounter an invasive particle. For this purpose we will not start just one invasive process but infinitely many.

In the following we describe a first approach that gives the right idea but does not lead to a good control of the marginal. First of all, there is a natural mapping v → S v from the family tree of the auxiliary process to the base graph that we denote by Ψ. Now, on the base graph we start infinitely many independent BRWs according to (µ i , P i ) as follows. Let N ∈ N (to be chosen later) and start independent copies of invasive BRWs on each x with |Ψ -1 (x)| = N . Here it is important that the underlying BRW survives weakly; otherwise the latter set would be empty. Using these BRWs we define a random labeling of the base graph G: a vertex is labeled † if it is visited by some invasive particle at some time and • otherwise. In [START_REF] Benjamini | On the trace of branching random walks[END_REF] it was shown that the trace of a (weakly surviving) BRW is a URG and moreover that the above labeling defines a URN. We use now the map Ψ to retrieve this labeling; label a vertex v ∈ T aux with † if Ψ(v) is labeled by † and label it with • otherwise. Each of the above steps is invariant under re-rooting and so is the new labeled version of T aux . Finally, due to Lemma 2.1 the connected component of T aux with respect to the percolation induced by † is a URT. It remains to prove that the non-invasive BRW survives with positive probability when being confronted with an infinity of invasive BRWs. This would imply coexistence of the two original processes, since coexistence does not depend on the starting position of the processes. As the underlying random walks are supposed to be symmetric random walks we have, see [START_REF] Woess | Random walks on infinite graphs and groups[END_REF]Lemma 8.1], that p (n) (x, y) ≤ ρ n for all x, y ∈ Γ and all n ∈ N. Two consequences of this fact on BRWs are given in the following lemma.

Lemma 4.1. Let (µ, P ) be a BRW on a non-amenable Cayley graph with ρm < 1. Then,

1) G(x, y|m) := ∞ n=0 p (n) (x, y)m n ≤ (mρ) d(x,y) /(1 -ρm); 2) there exists some constant ℓ such that lim sup n→∞ E[|{v : S v ∈ B(o, n)}|]/m ℓn = 0.
Proof. 1) Since the random walk is nearest neighbor, i.e., supp(q) = S, we have that

G(x, y|m) = ∞ n=d(x,y) p (n) (x, y)m n ≤ ∞ n=0 (ρm) n+d(x,y) ≤ (mρ) d(x,y) 1 1 -ρm .
2) Denote R n = inf{k ≥ 0 : S v ∈ B(o, n) ∀|v| ≥ k}. In the following denote by C a constant that is always chosen sufficiently large and may change from formula to formula. For some b > 0 (to be chosen in a moment) we obtain using the Markov inequality

P(R n > bn) = P(∃v : |v| ≥ bn : S v ∈ B(o, n)) ≤ k≥bn y∈B(o,n) m k p (k) (o, y) ≤ C k≥bn m k ρ k g n ≤ C(g(mρ) b ) n .
Hence, we can choose b sufficiently large such that the latter probability is summable and lim sup R n /n ≤ b by the Lemma of Borel-Cantelli. Finally, for n sufficiently large

E[|{v : S v ∈ B(o, n)}|] ≤ E[|{v : |v| ≤ R n }|] ≤ E[|{v : |v| ≤ (b + 1)n}|] ≤ m (b+1)n+1 -1 m -1 ≤ m (b+1)n+1 ,
which yields the result for some sufficiently large ℓ.

The first part of Lemma 4.1 is used to control each of the invasive processes and the second part to adjust the "number" of these invasive processes. In order to start with the adjustment let us consider a non-invasive process with less branching. For any constant γ ∈ (0, 1], to be chosen later, we define the truncated Galton-Watson process by

µ (γ) k =    γµ n,k , for k ≥ 2, µ n,1 + (1 -γ) ∞ k=2 µ n,k , for k = 1, µ n,0 , for k = 0.
and denote its mean by m γ . This construction is made to ensure two main properties: m γ ց 1 -µ n,0 ≤ 1 as γ ց 0 and µ

(γ 1 ) k < µ (γ 2 ) k
for all γ 1 < γ 2 and k ≥ 2. This latter property allows to construct a natural coupling of the original and the "γ-processes". Hence, denote by S γ the BRW corresponding to the family tree T γ . Due to the coupling, it remains to show that the "γprocess" has positive probability of survival for some γ > 0. Let γ c be such that m γc = 1.

Recall the definition of Ψ and † in Subsection 4.1 and start independent copies of invasive BRWs on each x with |Ψ -1 (x)| = N . (The constant N is still to be chosen.) We will also denote by P the probability measure describing the non-invasive process together with the infinite number of invasive processes. Denote by A ⊂ G the (random) set where invasive processes are started.

E |{v : S i,x v = S γ r }| | x ∈ A P(x ∈ A) ≤ ∞ n=0 1 1 -ρ i m i (m i ρ i ) n x∈S(o,n) P(x ∈ A) ≤ ∞ n=0 1 1 -ρ i m i (m i ρ i ) n C N C γ m ℓn γ .
We can choose γ ∈ (γ c , 1] sufficiently small such that m ℓ γ m i ρ i < 1. Let c UGW γ be the constant from Theorem 2.3 for the Galton-Watson with offspring distribution µ (γ) . Now, choose N sufficiently large (which makes C N sufficiently small) such that the marginal P(Ξ(r) = †) > c UGWγ . This in turn implies that the non-invasive BRW with offspring distribution µ (γ) survives with positive probability if confronted with an infinite number of invasive BRWs. Hence, for some γ c ′ ∈ (γ c , 1] there is coexistence of one invasive and one non-invasive BRW since coexistence does not depend on the choice of the starting positions of the processes. Eventually, using the monotonicity in γ a standard coupling argument implies coexistence for all γ ∈ [γ c ′ , 1].

4. 2 . 1 n

 21 Control of the marginal. In general it is not possible to control the marginal of the invariant percolation above. In fact, we need a better control of the "number" of invasive processes. Denote by B(n, o) = {x : d(o, x) ≤ n} the ball of radius n around the origin o and denote by S(o, n) = {x : d(o, x) = n} the corresponding sphere. The growth rate g of the group Γ is defined as g = lim n→∞ log(|B(n, o)|).

∞

  Since A ∩ S(o, n) ⊂ {x ∈ S(o, n) : ∃v ∈ S γ v = x} and vertices in A are labeled by † we have x∈S(o,n)P(x ∈ A) ≤ E[|{v : S γ v ∈ S(o, n), ξ(v) = †}|],where ξ(v) denotes the mark of the vertex v induced by Ψ. For x ∈ A we denote by S i,x v the invasive BRW started in x with family tree T i,x . Due to Lemma 4.1 for any γ ∈ (γ c , 1) there exists some constants C γ and ℓ γ such that E[|{v :S γ v ∈ S(o, n)}|] ≤ C γ m ℓγn γ .Since the trace of the BRW is unimodular we have that there exists a constantC N → 0 (as N → ∞) such that E[|{v : S γ v ∈ S(o, n), ξ(v) = †}|] ≤ C N C γ m ℓγn γ .Moreover, the proof of Lemma 4.1 gives that the constant ℓ γ can be chosen uniform with respect to γ since there is a natural coupling for the last exit times R n of different "γ-processes". Hence, there exists some constant ℓ such that for all γ ∈ (γ c , 1]x∈S(o,n) P(x ∈ A) ≤ C N C γ m ℓn γ .Using this together with a union bound and part 1) of Lemma 4.1 we obtainP(ξ(r) = †) ≤ P ∃x ∈ A, ∃v ∈ T i,x : S i,x v = S γr ≤ x∈G P x ∈ A, ∃v ∈ T i,x : S i,x v = S γ r = n=0 x∈S(o,n)

Acknowledgment. The idea behind this paper came up during the focussed meeting "Branching Random Walks and Related Topics" in Graz. The author thanks TU-Graz and RGLIS for organization and funding. Moreover, the author appreciated the hospitality of the Department of Mathematical Structure Theory of TU-Graz during the write up. The author thanks Itai Benjamini for helpful comments on a first version of this note and is grateful to the referee for various valuable comments.