Maxime Rioult
email: maxime.rioult@cea.fr

Rachid Belkhou

Hélène Magnan

Dana Stanescu

Stefan Stanescu

Francesco Maccherozzi

Cindy Rountree

Antoine Barbier

Local electronic structure and photoelectrochemical activity of partial chemically etched Ti-doped hematite

Fe 2 O 3 Ti doping PEEM Photoanode Surface etching Photoelectrolysis

The direct conversion of solar light into chemical energy or fuel through photoelectrochemical water splitting is promising as a clean hydrogen production solution. Ti-doped hematite (Ti:α-Fe 2 O 3 ) is a potential key photoanode material, which despite its optimal band gap, excellent chemical stability, abundance, non-toxicity and low cost, still has to be improved. Here we give evidence of a drastic improvement of the water splitting performances of Ti-doped hematite photoanodes upon a HCl wet-etching. In addition to the topography investigation by atomic force microscopy, a detailed determination of the local electronic structure has been carried out in order to understand the phenomenon and to provide new insights in the understanding of solar water splitting. Using synchrotron radiation based spectromicroscopy (X-PEEM), we investigated the X-ray absorption spectral features at the L 3 Fe edge of the as grown surface and of the wet-etched surface on the very same sample thanks to patterning. We show that HCl wet etching leads to substantial surface modifications of the oxide layer including increased roughness and chemical reduction (presence of Fe 2+ ) without changing the band gap. We demonstrate that these changes are profitable and correlated to the drastic changes of the photocatalytic activity.

Introduction

Within the framework of renewable energy development and solar fuels, an increasing attention is paid to the sun light assisted water splitting as a clean method for hydrogen production. As a matter of fact, hydrogen is an energy carrier of choice which does not lead to any greenhouse gas production if produced with solar light. If the idea of producing hydrogen using water splitting assisted by solar light is very seductive, it remains unfortunately also very challenging and many material science issues have to be solved. During the process, electron-hole pairs are generated in a semiconductor electrode, upon solar light absorption, and are subsequently used to promote the oxido-reduction reactions of water leading to oxygen production at the photoanode and hydrogen production at the photocathode [1,2].

Since the pioneering discovery of water photoassisted electrolysis using semiconducting TiO 2 in 1972 by Fujushima and Honda [1], several materials were investigated as photoanodes [3] where the water oxidation occurs (2OH

-þ 2h þ → 1 2 O 2 þ H 2 O
). Hematite, i.e. the α-Fe 2 O 3 iron oxide, is one of the most promising materials regarding its characteristics. It is an n-type semiconductor with a quasi-ideal band gap (~2.2 eV) for solar water splitting applications. Indeed this material is able to absorb ca. 40% of the solar light spectrum and its theoretical solar-to-hydrogen conversion yield reaches 13% [3]. It is abundant on earth and very stable in aqueous environments, which makes it a serious candidate in the framework of environment friendly energy production [4]. Moreover, the valence band edge of hematite is located below the H 2 O/O 2 redox potential which favors the water oxidation reaction [5]. Unfortunately, hematite has not only advantages. It has been demonstrated that it has poor transport properties [6,7] (low conductivity and low carrier lifetime) and surface kinetics [8]. Besides this, the conduction band edge of α-Fe 2 O 3 is not well positioned with respect to the potential of water reduction reaction (2H 2 O + 2e -→ H 2 + 2OH -), thus an external bias is necessary to promote water splitting [9].

Ti-doping of hematite is a seductive idea to improve bare hematite properties and has been the subject of numerous studies in recent years [7,[10][11][12][13][14][15]. We have recently demonstrated that Ti-doping greatly enhances single crystalline hematite films water splitting performances by increasing both electrons and holes mobilities and also the free carrier concentration by one order of magnitude [10,11]. Surface engineering of hematite has also been a widely explored route over the last few years. Major paths concern the deposition of co-catalysts designed to improve the chemical kinetics (e.g. cobalt oxides) [16,17] or the use of overlayers to passivate surface defects where photogenerated charges can recombine (e.g. alumina) [18]. Surface chemical treatments (e.g. with HF or HCl) are inexpensive experimental processes usually used for sample preparation. These processes are also commonly used to remove native and/or amorphous oxide on semiconductors' surfaces (like Si wafer cleaning). In parallel, it has been reported that such wet chemical etching can also improve the surface conductivity and semiconductors' photocurrent. For instance, Ziegler et al. [19] reported that a concentrated HCl etching of p-type gallium phosphide removes the native gallium oxide, inducing a 0.2 V reduction of the onset potential. Chernomordik et al. [20] reported that a HF treatment on hematite nanowires removes the amorphous iron oxide phase at the surface of the nanowires, resulting in a single crystalline surface and in a reduction of the onset potential by 0.3 V. Moreover, Basilio et al. [21] reported that a H 3 PO 4 crystallographic wet etching on GaN photocathodes (water reduction) (i) generated a greater surface area (+14%), (ii) decreased the surface carrier concentration by cleaning up the surface defects and (iii) caused the appearance of stepped edges and etched facets with better photocatalytic activity, resulting in doubling the energy conversion efficiency.

Nonetheless, chemical wet-etching of oxides may induce additional deep morphology and electronic structure changes. In the present work, we examine the effect of HCl etching on the surface of Ti-doped hematite epitaxial films and the improved photoelectrochemical properties of these films. The films were deposited by atomic oxygen assisted molecular beam epitaxy (AO-MBE) on Pt (111) single crystals. AO-MBE allows a high control of the stoichiometry and of the crystalline quality of the thin films. Working on oxide epitaxial thin films deposited on single crystals prevents the formation of a native or amorphous oxide phase at the surface that could lower the water splitting performances or mingle characterizations [20]. To study the chemical wet-etching effect on our particular system, we studied Ti-doped hematite thin films patterned by optical lithography and partially HCl-etched, making possible the simultaneous investigation of as-grown and etched surfaces on the very same sample. Topography and spatial-resolved electronic structure measurements were performed by atomic force microscopy (AFM) and X-ray photoemission electron microscopy (X-PEEM) respectively. Microscopy techniques are tools of choice to investigate the local variation of a particular physical property.

Materials and methods

Sample preparation (deposition, lithography and etching)

Ti-doped hematite layers of 20 nm thickness were deposited on single crystalline Pt (111) substrates using AO-MBE, a technique that makes possible the deposition of epitaxial layers of controlled morphology, thickness and doping level. The principle and the specificities of this technique were described elsewhere as well as the growth of Ti-doped hematite thin films. [10,11,22,23] By UV lithography, we defined on the sample surface micrometric squared (10-50 μm wide) patterns of resist (S1813). After a 15 min annealing at 110 °C on a hot plate the samples were dipped in a bath of 38%-HCl for partial etching and then cleaned with distilled water. The etching time was 5 s for a 2 nm step height. In the end, the morphology of the sample consisted in as-grown Tidoped hematite patches separated from each other by a partially HCletched Ti-doped hematite surface. Detailed morphology data are given in Section 3. The resist was cleaned in acetone followed by an isopropanol bath.

Crystallographic structure and chemistry

In situ Reflexion High Energy Electron Diffraction (RHEED) patterns were observed and acquired during film growth, in order to monitor the crystal quality and structure of the samples. In situ X-ray Photoemission Spectroscopy (XPS) spectra were systematically recorded just after deposition in order to determine the stoichiometry and the electronic structure of the films. More precisely, we recorded Fe2p, Ti2p, O1s core levels and the valence band region using Al Kα radiation. The doping level in atomic percentage of Ti (at.% Ti) is defined as follows: at. % Ti = I(Ti)/[I(Ti) + I(Fe)], where I(Ti) and I(Fe) are respectively the Ti2p and Fe2p integrated intensities corrected by the corresponding Scofield factors (cross sections). For all studied samples, the measured doping level is 2 at.% (±0.3 at.%).

Photoelectrochemistry

The photoelectrochemical response of our films was studied using a three electrodes cell described previously [10,11]. All electrochemical measurements were performed at room temperature using a NaOH 0.1 M (pH = 13) solution as electrolyte, a platinum wire as counter electrode and an Ag/AgCl electrode for the potential reference (V Ag/AgCl = + 0.197 V vs. the standard hydrogen electrode (SHE)). The sample was mounted as anode (working electrode) using a dedicated sample holder that allows the contact only between the hematite surface and the electrolyte. The illumination source was a Newport 1000 W Xe Arc Lamp with an infrared filter with an incident light flux around 100 mW/cm 2 (measured with a Newport 1918-R Power Meter). Potential control and current acquisition between the three electrodes were done using a Princeton Applied Research (PAR) 263A potentiostat controlled by a computer. For current vs. voltage measurements, I(V), the potential was swept from 0 to + 0.8 V vs. Ag/AgCl at a speed of 50 mV/s. The photocurrent is defined as the difference between the current recorded under light and the one without (dark).

Atomic Force Microscopy (AFM)

We used AFM to compare the surface morphology of our samples before and after HCl-etching. The images were acquired using a Bruker™ FASTSCAN head and Nanoscope V controller. The Bruker's FastScan-B cantilevers utilize a 30 μm long triangular silicon nitride cantilever with a force constant of 4 N/m. The silicon tip used to collect the topography measurements had an extremely sharp tip radius of 5 nm.

X-ray PhotoEmission Electron Microscopy (X-PEEM)

X-PEEM is a full-field X-ray spectromicroscopy technique based on the photoemission process, in the same way as conventional XPS (photon in, electron out). Conventional XPS does not provide spatial information below the tens of μm scale, while X-PEEM gives simultaneously spectroscopic and microscopic information at scales that can reach tens of nm at best. The use of synchrotron light as primary incident excitation source enables one to use various incident photon energies, making possible real space-resolved imaging of core levels, valence band or absorption edges spectroscopies. Our X-PEEM experiments were carried out on the I06 Nanoscience beamline at the Diamond Light Source (Didcot-UK). Before analysis and in order to remove any surface contaminant due to air exposure, the samples were annealed under an O 2 atmosphere at a pressure of 10 -6 mbar and at a temperature of 300 °C for 1 h.

Results and discussion

Structure and chemistry

The growth behavior of Ti-doped and undoped hematite thin films has been described in detail in previous works [10,11,22,23]. Briefly, RHEED patterns with straight diffraction lines corresponding to the hematite surface reciprocal lattice were observed, demonstrating an epitaxial growth. The stoichiometry and the Fe and Ti oxidation states were determined by means of in situ XPS measurements. These measurements were performed just after the thin film growth prior to the UV lithography and HCl partial etching. By convention, we considered the O1s line at 530.1 eV as reference for the binding energy [10,11].

We measured binding energies of 711.2 eV for the Fe2p 3/2 core level and of 719.2 eV for the typical Fe 3+ shake-up satellite. This confirms the Fe 3+ ionic state of iron in all as-grown films and the absence of any Fe 2+ species. The Ti2p 3/2 line shows a narrow single feature at a binding energy of 458 eV which corresponds to the binding energy of Ti 4+ species included in the hematite host matrix [10,11].

Photoelectrochemical properties

Fig. 1 presents the photocurrent density vs. voltage curves for asgrown and 10 μm square patterned films (partially HCl-etched, cf. inset in Fig. 1, the etching step was about 2 nm). In both cases, only anodic photocurrent appears within the chosen range of potentials. We see that the photocurrent density obtained for the partially HCletched sample is at least 25% higher than for the as-grown one for all the bias voltages used. Moreover, for the partial HCl-etched sample the water splitting onset potential is significantly reduced by 0.1 V with respect to the as-grown one.

Atomic force microscopy

Fig. 2 shows the topography AFM picture and the height profile of a film patterned with 10 μm squares (etching step of about 2 nm). From Fig. 2a (AFM picture) and 2b (height profile), we can see that the overall roughness is increased on the HCl-etched surface. The net result is an increased active surface. More precisely, the root mean square (RMS) roughness increases from 1 nm (as-grown surface) to 4.6 nm (partially HCl-etched surface). This roughness increase likely means that the chemical etching is not uniform and that it occurs preferably in the direction perpendicular to the surface along defects of the crystallographic structure like antiphase boundaries [23].

X-PEEM

The X-PEEM results presented here were obtained on a thin film patterned with rectangles of various sizes (10 μm-50 μm) and a step height of ca. 2 nm. X-PEEM reveals the local electronic structure of iron in our patterned films on the as-grown and HCl-etched surfaces within the same region. The Fe L 3 absorption edge spectra were selectively recorded on each area and are depicted on Fig. 3a for both surface regions. Indeed, the qualitative analysis of these spectra has demonstrated being able to distinguish the electronic configurations of Fe in the different possible oxide environments [22]. The crystal field typical for oxides splits the L 3 edge in two characteristic contributions labeled L 3A and L 3B . The shape and relative intensities of these peaks give insight in the valence state of the ions and their location in the lattice. The contribution L 3A around 707.2 eV (resp. L 3B around 708.6 eV), is strengthened in the presence of Fe 2+ (resp. Fe 3+ ) ions. This interpretation can be derived from the simple comparison between reference spectra of Fe 3 O 4 (33% Fe 2+ , 67% Fe 3+ ) [24] and α-Fe 2 O 3 (100% Fe 3+ ) [22,25] (see inset of Fig. 3a) and has already been used by Droubay et al. in the framework of X-ray absorption microspectroscopy of ilmenite [26]. By comparing the two experimental spectra (Fig. 3a), we observe an increase of the L 3A component for the HCl-etched surface. In order to see if this behavior is characteristic of the etched surface we have calculated the ratio R of the intensities of the two peaks of the absorption spectrum (B/A, see Fig. 3a) on each point of the X-PEEM image. The map of the ratio R is shown in Fig. 3c (on a 20 μm square) and a Mirror Electron Microscopy (MEM) picture recorded on the same region is shown in Fig. 3b.

The MEM mode uses the same setup as X-PEEM, but instead of using photons as excitation source, low energy electrons are used. The MEM images are formed using backscattered low energy electrons and are of high surface morphology sensitivity. As a matter of fact, if the electron energy is not sufficiently high to overcome the surface potential barrier, they are simply reflected by the surface. The contrast in MEM between the as-grown surface and the HCl-etched surface in our case is linked to a work function variation between the as-grown and the etched surfaces, and/or to a difference of roughness, which is consistent with the AFM results. Lines just next to the edges are due to lithography optical interference issues. The R mapping (Fig. 3c) follows exactly the morphology with a uniform value in the HCl-etched zone. The black areas in the right center on Fig. 3c are due to inhomogeneous incident beam and channel plate saturation. These zones are not visible on the MEM pictures because we use an electron beam as primary source. Moreover R varies from 1.8 on the as-grown surface to 1.6 on the etched one, a value which is intermediate between what is expected for Fe 2 O 3 (≈1.8 for 0% Fe 2+ ) and Fe 3 O 4 (≈1.4 for 33% Fe 2+ ) (see inset of Fig. 3a). Assuming a linear variation of the R factor with the Fe 2+ percentage, we can evaluate the amount of Fe 2+ in the etched region around 15%. Using the same instrument, we also performed the valence band measurement of the two regions. The two spectra were identical even in resonant conditions, showing that the Fe 2+ introduced by etching does not reduce the band gap. Patches of sizes different from 20 μm exhibited very similar results.

Unlike Ziegler et al. [19] and Chernomordik et al. [20], the HCl etching effect on our hematite layers does not deal with any amorphous structure removal. Indeed the presence of such a structure would have been noticed by RHEED which is sensitive to the crystalline order of the very top surface layer. In our case, partial HCl wet-etching has a substantial effect on the topography of the films, but also on their electronic structure and on the iron oxidation state. The roughness increase, consistent with the results reported in reference [21], conjugated with the iron oxide reduction can account for a significant improvement of the photoelectrochemical properties. A roughness increase can easily improve the photoelectrochemical properties thanks to a larger reactive surface. Moreover, as in Ref. [21], we can reasonably assume that the chemical etching is crystalline surface plane dependent. Hence, the pure (0001) hematite surface may make appear upon etching crystalline planes of higher Miller indexes, which may have better photocatalytic activity than the basal hexagonal one. Indeed, there is a strong anisotropic conductivity in hematite, the conductivity being 4 times higher perpendicular to the (0001) direction than along it [5]. This phenomenon has also been observed in the case of the thoroughly-studied anatase TiO 2 photoanode material, where water is preferably adsorbed on (001) facets than on (101) facets [27]. Moreover, the presence of Fe 2+ can also explain the water splitting improvement because of an increased carrier mobility [28]. Interestingly, the introduction of Fe 2+ with HCl etching occurs without changing the band gap, avoiding any alteration of the semiconducting properties of our films.

Conclusions

The effect of a partial etching by hydrochloric acid on the surface topography and on the electronic structure was investigated by atomic force microscopy and X-ray photoelectron microscopy respectively. It was found that on the one hand HCl etching increases the overall roughness of the surface, providing a higher number of catalytic sites. On the other hand, the etched oxide is locally reduced, as it is shown by the presence of Fe 2+ ions. These changes in the material properties result in a substantial improvement of the water splitting performances, in particular a significant reduction of the onset potential. Those results provide new insights on the parameters governing the hematite potential for environment friendly hydrogen production.

Fig. 1 .

 1 Fig. 1. Photocurrent density vs. voltage curves for a 2 at.% Ti-doped hematite 20 nm film: as-grown ( ), and partially HCl-etched (patterned by UV lithography) ( ). The inset shows the used lithography pattern (10 μm squares). Bright zones (resp. dark) correspond to the etched surface (resp. as-grown).

Fig. 2 .

 2 Fig. 2. (a) 2D topography AFM picture of a patterned 2 at.% Ti:Fe 2 O 3 (20 nm thickness, 2 nm step height). (b) Height profile along the white dashed line (gray curve).

Fig. 3 .

 3 Fig. 3. (a) Fe L 3 absorption edge for the as-grown surface (red solid line) and the etched surface (blue dashed line). Orange items show the positions of the L 3A and L 3B contributions, as well as the R coefficient graphic determination. The inset shows experimental spectra of reference Fe 2 O 3 (gray solid line) and Fe 3 O 4 (black solid line) films. (b) MEM images of 20 μm squares (field of view 20 μm). (c) Map of the R coefficient in the same region.

M. Rioult et al. / Surface Science 641 (2015) 310-313

Acknowledgments

This work was funded by the CEA project DSM-Energie Hemaphoto and supported in part by Triangle de la Physique and Ile-de-France (C'Nano and ISC-PIF) under the IMAFMP grants.

Author contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.


Luigi De Magistris

Brice Paquette

David Orry

Olivier Facy

Giovanni Di Giacomo

Patrick Rat

Christine Binquet

Pablo Ortega-Deballon
email: pablo.ortega-deballon@chu-dijon.fr

M.D Pablo Ortega Deballon

Preoperative inflammation increases the risk of infection after elective colorectal surgery: results from a prospective cohort

Keywords: colorectal surgery, C-reactive protein, albumin, procalcitonin, anastomotic leak, postoperative complications, intra-abdominal infection, surgical site infection

Background

Septic complications after colorectal surgery are frequent and sometimes life-threatening. It is wellknown that inflammation impairs the healing process. It has been suggested that preoperative ongoing inflammation could increase the risk of postoperative infections. This study aimed to elucidate the role of preoperative inflammation on postoperative infectious complications and to understand if, through biological markers, it is possible to identify preoperatively patients at higher risk of infection.

Methods

A prospective, observational study was conducted in three centers from November 2011 to April 2014. Consecutive patients undergoing elective colorectal surgery with anastomosis were included.Any ongoing infection was an exclusion criterion. C-reactive protein, albumin, prealbumin and procalcitoninplasma levels were measured preoperatively. Postoperative infections were recorded according to the definitions of the Centers for Diseases Control. The areas under the receiver operating characteristic curve were analyzed and compared to assess the accuracy of each preoperative marker.

Results

Four-hundred and seventy two patients were analyzed. Infectious complications occurred in 118 patients (25%) and mortality in 6 patients (1.3%). In the univariate analysis, preoperative C-reactive protein andalbuminemiawere found significantly associated withpostoperative infectious complications (P = 0.008 and P = 0.0002, respectively). Areas under the ROC curve for preoperative C-reactive protein and albuminemia were 0.57and 0.62, respectively.

Conclusions

This study confirms the association between preoperative inflammatory activity, hypoalbuminemia and the onset of infections after surgery. Trials aiming to decrease the inflammatory activity before surgery in order to prevent postoperative complications are warranted.

Introduction

Septic complications are responsible for most morbidity, mortality and failure to rescue after colorectal surgery. Despite the steady progress of surgical techniques, preoperative assessment and perioperative management,the risk of overall infection after colorectal surgery is between 16 and 51% [START_REF] Alves | Postoperative mortality and morbidity in French patients undergoing colorectal surgery[END_REF][START_REF] Facy | Diagnostic accuracy of inflammatory markers as early predictors of infection after elective colorectal surgery: results from the IMACORS study[END_REF][START_REF] De Magistris | Laparoscopic sigmoidectomy in moderate and severe diverticulitis: analysis of short-term outcomes in a continuous series of 121 patients[END_REF][START_REF] Bakker | Risk factors for anastomotic leakage and leak-related mortality after colonic cancer surgery in a nationwide audit[END_REF] . In addition to their short-term clinical impact leading sometimes to life-threatening situations, postoperative infections increase the length of hospital stay, costsand decrease cancer survival.Hence, the identification of patients at risk of developing postoperative infectious complications (PIC) is essential. It has been shown by several authors that high blood levels of Creactive protein (CRP) and procalcitonin (PCT) between postoperative days 2 and 5 predict postoperative infections [START_REF] Cousin | Diagnostic accuracy of procalcitonin and C-reactive protein for the early diagnosis of intraabdominal infection after elective colorectal surgery: a meta-analysis[END_REF] . Inflammatory markers monitoring is thus becoming a standard of care in gastrointestinal surgery.

Recently, some authors suggested that preoperative inflammatory activity is related to a higher risk of PIC [START_REF] Moyes | Preoperative systemic inflammation predicts postoperative infectious complications in patients undergoing curative resections for colorectal cancer[END_REF][START_REF] Kubo | Elevated preoperative C-reactive protein levels are a risk factor for the development of postoperative infectious complications following elective colorectal surgery[END_REF][START_REF] González-Martínez | Proteínas mediadoras de la respuesta inflamatoria como predictores de resultados adversos postoperatorios en pacientes quirúrgicos octogenarios: estudio prospectivo observacional[END_REF][START_REF] Ortega-Deballon | C-reactive protein is an early predictor of septic complications after elective colorectal surgery[END_REF] . It is well known that local inflammation impairs the healing process and systemic inflammation impairs the immune response [START_REF] Nozoe | Preoperative elevation of serum C-reactive protein is related to impaired immunity in patients with colorectal cancer[END_REF] . Preoperative hypoalbuminemia and a low nutritional index are also well-known risk factors of PIC that warrant a specific nutritional support [START_REF] Hu | Preoperative malnutrition assessments as predictors of postoperative mortality and morbidity in colorectal cancer: an analysis of ACS-NSQIP[END_REF][START_REF] Hennessey | Preoperative hypoalbuminemia is an independent risk factor for the development of surgical site infection following gastrointestinal surgery: a multi-institutional study[END_REF][START_REF] Braga | Clinical evidence for pharmaconutrition in major elective surgery[END_REF] .

Whether the effect of hypoalbuminemia is more related to an impaired healing process or to an impaired immune response is not clear. Hypoalbuminemia may also be due to an increased catabolism induced by ongoing inflammation and not only to malnutrition.It has been stated that preoperative immunonutrition decreases the risk of postoperative infections, even in well-nourished patients. Immunonutrition has both, immuno-modulating and anti-inflammatory properties, as well as it reduces the insulin-resistance which has been related to the risk of PIC [START_REF] Matsuda | Preoperative oral immune-enhancing nutritional supplementation corrects TH1/TH2 imbalance in patients undergoing elective surgery for colorectal cancer[END_REF][START_REF] Okamoto | Attenuation of the systemic inflammatory response and infectious complications after gastrectomy with preoperative oral arginine and omega-3 fatty acids supplemente immunonutrition[END_REF][START_REF] Ortega-Deballon | Are adiponectin and leptin good predictors of surgical infection after colorectal surgery? A prospective study[END_REF] . These data suggest that modulating preoperative inflammation could decrease the risk of PIC.

The aim of this study was to analyze the link between preoperative inflammation, hypoalbuminemia and PIC.

Materials and methods

Design

From November 2011 to April 2014, patients who underwent elective colorectal surgery with anastomosis in the Departments of Digestive Surgery of the University Hospitals of Dijon and Besançon and the "Georges-François Leclerc Anticancer Center" in Dijon were included in a prospective database (IMACORS: Inflammatory MArkersafter COloRectal Surgery) [START_REF] Facy | Diagnostic accuracy of inflammatory markers as early predictors of infection after elective colorectal surgery: results from the IMACORS study[END_REF] .

Inclusion and exclusion criteria

Patients were eligible for the study if they met all the following criteria: age 18 years or older, a scheduled colorectal resection with anastomosis, preoperative values of CRP, PCT, albuminemia and prealbuminemia available. Patients with a previous ongoing infection, operated on in an emergency or with suspected peritoneal carcinomatosis were excluded. Before inclusion, written informed consent was obtained from all patients.The study was approved by the regional ethics committee CPP Est 1.

Clinical and laboratory assessment and follow-up

Potential patient-specific and preoperative risk factors were recorded prospectively for each patient.

The preoperative factors included age, gender, obesity, surgical indication (cancer, polyps, chronic inflammatory bowel disease, chronic diverticular disease, other diseases), laboratory data on the day before surgery (CRP, PCT, prealbuminemia, albuminemia and white blood cell count), American Society of Anesthesia score, comorbidities (cardiovascular, respiratory or hepatic diseases, presence of diabetes mellitus and immunosuppression status) and bowel preparation (decision made by operating surgeon). The surgical data recorded included the procedure and approach (laparoscopy or laparotomy), level of anastomosis and technique (mechanical or handsewn), operative time, perioperative accidents, presence of drainage, diverting stoma and perioperative blood transfusion.

During the postoperative period, patients were examined by the attending surgeon daily. The presence of fever (central temperature >38°C), abdominal signs, bowel movements, and the volume and aspect of drainage (if present) were recorded daily. The attending surgeon made the decision to carry out imaging examinations according to his/her own criteria. The rate of septic complications was calculated, including intra-abdominal infection, wound infection, central line infection, pneumonia and urinary tract infection, independently of their severity. Patients were seen in the outpatient clinic within 6 weeks after operationand all complications occurring before POD 30 were recorded.

CRP levels were determined by immunonephelometry (Dimension Vista system, Siemens Healthcare France) in all participating centers. PCT levels were measured by using the Brahms PCT Kryptor® or Vidas®assays (Thermo Fisher Scientific, Hennigsdorf, Germany). The first assay is a homogeneous one-step sandwich immunoassay using time-resolved amplified cryptate emission technology on a Kryptor® system (Thermo-scientific, Asnière, France). The second assay is a onestep immunoassay sandwich method associated with a final fluorescent detection step on a Vidas® system(Biomérieux, Marcy, France). Close concordance between the two assays has been previously established elsewhere. [START_REF] Lagabrielle | Dosage de la procalcitonine : comparaison des résultats obtenus sur Kryptor® (Brahms) et Vidas® (BioMérieux)[END_REF] Plasma levels of albumin (g/l) and prealbumin (g/l) were assayed using commercially available colorimetric (albumin-bromocresol purple complex) and immunoturbidimetric assay kits (Dimension Vista System Flex reagent cartridge, Saint-Denis, France), respectively, in all participating centers.

Assessment and definition of complications

Intra-abdominal infection was defined according to the Centres for Disease Control and Prevention. This definition includesany one of the following criteria: the presence of pus or enteric contents within the drains, the presence of abdominal or pelvic collections in the area of the anastomosis on postoperative imaging (abdominal CT scanner), leakage of contrast through the anastomosis during enema or evident anastomotic dehiscence at reoperation for postoperative peritonitis. All intraabdominal infection were considered independently of their clinical significance. Wound infection, urinary-tract infection, catheter-related infection, and pneumonia were also defined according to the Centres for Disease Control and Prevention [START_REF] Horan | CDC/NHSN surveillance definition of health careassociated infection and criteria for specific types of infections in the acute care setting[END_REF] .

Statistical analysis

Statistical analysis was performed using of SAS software, version 9.4 (SAS Institute Inc; Cary NC, USA) and Stata software version 12 (StataCorpInc, College Station, TX USA). Continuous outcomes were presented as means and standard deviations for normally distributed variables, and with median and interquartile range (IQR) otherwise, and compared with Student tests or nonparametric tests as appropriate. Categorical data were expressed with percentage frequencies and compared with Chi-2 tests.Logistic regressions were used in order to estimate associations between CRP, PCT, albuminemia and PIC, while adjusting for confounding factors. Variables associated with PIC with a p-value below 0.20 in bivariate analyses were included in the model. A backward selection strategy was then applied. The log-linearity of the relation between each continuous variable and PIC was assessed using the fractional polynomial approach [START_REF] Royston | A new approach to modelling interactions between treatment and continuous covariates in clinical trials by using fractional polynomials[END_REF] . Interactions with cancer were systematically tested.

A two-sided P value of less than 0.05 was considered to be significant. The areas under the ROC curves were calculated for the preoperative value of CRP, PCTand albumin.

Results

Patients and morbidity

Among the 510 patients included in the IMACORS cohort, 38 patients were excluded for this ancillary study due to missing preoperative data. Thus, 472 patients were retained for the analysis. The characteristics of the patients are presented in Table 1. In total, PIC occurred in 118 (25%) of the 472 patients: intra-abdominal infection occurred in 58 (49.2% of PIC, overall incidence of 12.3%), wound infection in 46 (9.7%), urinary-tract infections in 22 (4.7%), pneumonias in 5 (1%) and catheter-related infections in 9 (1.9%). Six patients (1.3%), all of whom having a PIC, died before POD 30. The median time between the operation and the diagnosis of PIC was 7 days (interquartile range: 5-11 days). The intra-abdominal infection diagnosis was established with imaging in 40 patients (69%), clinical features in 13 (22.4%), and at surgery in the remaining 5 patients (8.6%).

Preoperative risk factors

Albuminemia and CRP levels were significantly higher in patients going into develop PIC (P = 0.0002and P = 0.008, respectively); prealbuminemia did not reach significance (P = 0.0589)(Table 2). Preoperative PCT was not associated with an increased risk of developing infections (P> 0.9).

When adjusting for confounding covariates (age and cancer) in a multivariate analysis, preoperative CRP and albuminemia were still associated with a later PIC occurrence (P = 0.036 and P< 0.001, respectively).No interaction with cancer was found (P = 0.420 for CRP, P = 0.183 for PCT and P = 0.902 for albuminemia).The area under the ROC curve for preoperative albumin, CRP and PCT was respectively of 0.62, 0.57 and 0.50 (figure 1). When a cut-off value of 5 mg/L was used, preoperative CRP yielded a sensitivity of 52%, a specificity of 58% and a negative predictive value of 78% for the prediction of PIC. Among surgical factors, only operating time > 4h predisposed patients to PIC (P = 0.046). CRP was not significantly associated with intra-abdominal infection.

When patients with preoperative CRP levels higher than 100 mg/L (9 patients) were excluded from the cohort (in these cases, extreme values could have reflected undetected ongoing infections or inflammatory diseases), values of CRP remained significantly higher in those going into develop PIC(P = 0.016).

Patients operated on for cancer (330 patients) had higher preoperative values of CRP than those operated on for benign diseases (142 patients): 17.8 vs. 7.1 mg/L, p = 0.0002; but they did not have a significantly higher risk of PIC. percentages in peripheral blood, and it can be an indicator of impaired immunity in patients with colorectal cancer [START_REF] Nozoe | Preoperative elevation of serum C-reactive protein is related to impaired immunity in patients with colorectal cancer[END_REF] . The increased inflammatory activity in patients operated on for colorectal cancer has also been shown in previous studies [START_REF] Duvillard | A case-control study of pre-operative levels of serum neutrophil gelatinase-associated lipocalin and other potential inflammatory markers in colorectal cancer[END_REF] . It could be expected that this could lead to an increase in the risk of PIC or surgical-site infection in cancer patients, but this was not the case in the present series, as it was not the case in previous series analyzing the risk factors of infection after colorectal surgery [START_REF] Telem | Risk factors for anastomotic leak following colorectal surgery: a case control study[END_REF][START_REF] Mäkelä | Risk factors for anastomotic leakage after left-sided colorectal resection with rectal anastomosis[END_REF] .

Other data supporting the role of preoperative systemic inflammation on the onset of PIC come from studies having measured adipocytokines preoperatively in patients undergoing elective colorectal surgery. Leptin is a proinflammatoryadipocytokine related to systemic inflammation and, more specifically, to the inflammatory activity in the fatty tissue. This has been suggested as an explanation of the increased risk of surgical site infection in obese patients [START_REF] Frasson | Risk factors for anastomotic leak after colon resection for cancer: multivariate analysis and nomogram from a multicentric, prospective, national study with 3193 patients[END_REF][START_REF] Ortega-Deballon | Preoperative adipocytokines as a predictor of surgical infection after colorectal surgery: a prospective survey[END_REF] . Leptin is also related to insulin resistance, which is involved in the risk of infection [START_REF] Mullen | Impact of body mass index on peroperative outcomes in patients undergoing major intraabdominal cancer surgery[END_REF] , and its values are also consistently elevated in patients going into develop PIC [START_REF] Ortega-Deballon | Are adiponectin and leptin good predictors of surgical infection after colorectal surgery? A prospective study[END_REF] .

Further results supporting the role of inflammation in the onset of PIC come from clinical trials which showed a benefit of preoperative immunonutrition and perioperative anti-inflammatory drugs. Actually, immunonutrition modulates the immune and inflammatory responses after surgery and decreases significantly the risk of PIC in patients undergoing major gastrointestinal or colorectal surgery for cancer, even in well-nourished patients [START_REF] Gianotti | A randomized controlled trial of preoperative oral supplementation with a specialized diet in patients with gastrointestinal cancer[END_REF][START_REF] Akbarshahi | Perioperative nutrition in elective gastrointestinal surgery potential for improvement?[END_REF][START_REF] Marimuthu | A meta-analysis of the effect of combinations of immune modulating nutrients on outcome in patients undergoing major open gastrointestinal surgery[END_REF] . On the same way, a benefit in morbidity and length of hospital stay has been shown with the use of glucocorticoids and nonsteroidal anti-inflammatory drugs in the preoperative and early postoperative period, respectively 36 -37 . Some authors have even used preoperative glucocorticoids to decrease CRP values without increasing the infectious risk [START_REF] Suezawa | Clinical benefits of methylprednisolone in off-pump coronary artery bypass surgery[END_REF] . Another issue is the specific impact of gut microbiota in the preoperative inflammatory and the risk of anastomotic leak and PIC. All thesedata converge in showing that preoperative inflammatory activity is related to an increased risk of both, postoperative infection and intra-abdominal infection. This opens the door to clinical trials aiming at reducing preoperative inflammation combining nutritional and pharmacological approaches.

Could preoperative CRP select patients who deserve a more intensive surveillance policy or even additional antibiotic prophylaxis? We think that its ability to discriminate patient who will develop a PIC from patients who will not (AUC=0.57) does not allow its use to change either the antibiotic prophylaxis policy or the postoperative surveillance.

This study has some limits. We did not measure other inflammatory, stress and healingmarkers (e.g. interleukin-6, insulin, leptin, adiponectin, praline, etc), whose values could have been helpful for a more global overview of the relationship between preoperative inflammation, healing and PIC.Seemingly, we did not analyze the preoperative imaging results looking for an explanation to elevated CRP levels in patients with higher values. However, all patients with preoperative known ongoing complications were excluded from the IMACORS cohort. Furthermore, the exclusion of the extremely high values of CRP did not change the relationship between the marker and the incidence of PIC. This suggests that the inflammatory preoperative status has a role on the onset of PIC, even for lower values.Finally, an analysis of the visceral adipose tissue in the preoperative imaging (as it is a source of inflammation) would have been of interest.

In conclusion, preoperative ongoing inflammation and hypoalbuminemia are related to the onset of postoperative infection after elective colorectal surgery. The benefit of therapeutic approaches to reverse the inflammatory activity before surgery should be assessed in future trials in which patients could be selected according to these markers.

ACKNOWLEDGMENTS

The authors are indebted to Elisabeth Devilliers, MD, for herheartful support to this project. The authors also thank the data monitoringboard (Cassandra Porebski, Emilie Galizzi, Alexandra Felin, Amandine Martin, Fanny Lachaux, DonyaSouhielDa Costa, Joelle Fritsch, and ChrystelleCappe); the safety monitoringboard (AurélieGrandvuillemin, PharmD); the administrativesupport team (Evelyne Phu and Maud Carpentier), the staff of the participantsurgical departments for their help; Sandrine Vinaultand Sandrine Daniel for their help with the statistical analysis,and Mr Philip Bastablefor the language revision of the article.

The IMACORS study was funded by the Groupement de Coopération Sanitaire Grand-Est, the Regional Council of Burgundy, a French National Research Agency grant under the program "Investissementsd'Avenir" (ANR-11-LABX-0021), and a grant from Brahms France SAS.

The authors declare no conflicts of interest.

Discussion

Based on a prospective studydesigned toassess CRP and PCT as early predictors of postoperative infections following colorectal surgery, this analysis confirmed the relationship between preoperative systemic inflammation measured by CRP and the risk of PIC. Hypoalbuminemia was also associated with a higher risk of infections, as it has already been shown in several works [START_REF] Hennessey | Preoperative hypoalbuminemia is an independent risk factor for the development of surgical site infection following gastrointestinal surgery: a multi-institutional study[END_REF][START_REF] Ataseven | Pre-operative serum albumin is associated with post-operative complication rate and overall survival in patients with epithelial ovarian cancer undergoing cytoreductive surgery[END_REF][START_REF] Bohl | Hypoalbuminemia independently predicts surgical site infection, pneumonia, length of stay, and readmission after total joint arthroplasty[END_REF] .

Unsurprisingly, preoperative PCT was not related to the onset of PIC, as it is more a marker of ongoing bacterial infection than a marker of general inflammatory activity.

Intra-abdominal infections after colorectal surgery become clinically evident around postoperative days 6 and 7 or even later, except in case of technical default in which the onset of symptoms is earlier [START_REF] Facy | Diagnostic accuracy of inflammatory markers as early predictors of infection after elective colorectal surgery: results from the IMACORS study[END_REF][START_REF] Frasson | Risk factors for anastomotic leak after colon resection for cancer: multivariate analysis and nomogram from a multicentric, prospective, national study with 3193 patients[END_REF][START_REF] Krarup | Management of anastomotic leakage in a nationwide cohort of colonic cancer patients[END_REF][START_REF] Hyman | Anastomotic leaks after intestinal anastomosis: it's later than you think[END_REF] . Even when routine CT scan is performed for the screening of intra-abdominal infection, its sensitivity is low before POD 4 [START_REF] Kornmann | Beware of false negative CT scan for anastomotic leakage after colonic surgery[END_REF] . Overall, these data give an idea of anastomotic leakage as an event appearing at the end of the first postoperative week. However, levels of CRP and PCT rise significantly higher as early as postoperative day 1 in patients going into infection [START_REF] Cousin | Diagnostic accuracy of procalcitonin and C-reactive protein for the early diagnosis of intraabdominal infection after elective colorectal surgery: a meta-analysis[END_REF][START_REF] Ortega-Deballon | C-reactive protein is an early predictor of septic complications after elective colorectal surgery[END_REF] . This fact raises the hypothesis of inflammatory response starting at the tissue level much earlier than the onset of the clinical picture. Inflammation may be not only a consequence but also a risk factor of surgical infection and its role might even begin preoperatively.

Several authors have suggested that preoperative ongoing inflammation and hypoalbuminemia are associated with an increased risk of both, PIC and intra-abdominal infection [START_REF] Moyes | Preoperative systemic inflammation predicts postoperative infectious complications in patients undergoing curative resections for colorectal cancer[END_REF][START_REF] Kubo | Elevated preoperative C-reactive protein levels are a risk factor for the development of postoperative infectious complications following elective colorectal surgery[END_REF][START_REF] González-Martínez | Proteínas mediadoras de la respuesta inflamatoria como predictores de resultados adversos postoperatorios en pacientes quirúrgicos octogenarios: estudio prospectivo observacional[END_REF][START_REF] Ortega-Deballon | C-reactive protein is an early predictor of septic complications after elective colorectal surgery[END_REF][START_REF] Frasson | Risk factors for anastomotic leak after colon resection for cancer: multivariate analysis and nomogram from a multicentric, prospective, national study with 3193 patients[END_REF][START_REF] Telem | Risk factors for anastomotic leak following colorectal surgery: a case control study[END_REF][START_REF] Mäkelä | Risk factors for anastomotic leakage after left-sided colorectal resection with rectal anastomosis[END_REF][START_REF] Golub | A multivariate analysis of factors contributing to leakage of intestinal anastomosis[END_REF] . Our results confirm this relationship between preoperative CRP and the risk of PIC. Hypoalbuminemia may result of malnutrition, particularly in cancer patients, but is also due to the accelerated catabolism induced by systemic inflammation. The mechanism by which these factors increase the risk of complications seems to be a composite of impaired healing and impaired immune response.

Consistent results have been found recently in patients operated on for Crohn's disease: higher preoperative values of CRP (with a cutoff of 14.50 mg/L) and rising values in the 2 weeks before surgery were associated with an increased risk of intra-abdominal septic complications [START_REF] Zuo | A practical predictive index for intraabdominal septic complications after primary anastomosis for Crohn's disease: change in Creactive protein level before surgery[END_REF] .

Preoperative elevation of serum CRP has been significantly related to the reduction of lymphocyte


I García-Mata

O Giraud

B Georgeot

J Martin

R Dubertrand

G Lemarié
email: lemarie@irsamc.ups-tlse.fr

Scaling theory of the Anderson transition in random graphs: ergodicity and universality

We study the Anderson transition on a generic model of random graphs with a tunable branching parameter 1 < K ≤ 2, through a combination of large scale numerical simulations and finitesize scaling analysis. We find that a single Anderson transition separates a localized phase from an unusual delocalized phase which is ergodic at large scales but strongly non-ergodic at smaller scales. The critical regime is characterized by multifractal wavefunctions located on few branches of the graph. Two different scaling laws apply on both sides of the transition: a linear scaling as a function of the linear size of the system on the localized side, and an unusual volumic scaling on the delocalized side. The critical scalings and exponents are found to be independent of the branching parameter and should describe all infinite dimensional random graphs without boundary.

Ergodicity properties of quantum states are crucial to assess transport properties and thermalization processes in condensed matter systems. It is at the heart of the eigenstate thermalization hypothesis which has attracted enormous attention lately [1]. A paramount example of non-ergodicity is Anderson localization where the interplay between disorder and quantum interference leads to exponentially localized states [2]. In 3D, a critical value of disorder separates a localized from an ergodic delocalized phase. At the critical point eigenfunctions are multifractal, another non trivial example of non-ergodicity [START_REF] Abrahams | years of Anderson Localization[END_REF][START_REF] Evers | [END_REF]. Recently, those questions have been particularly highlighted in the problem of many-body localization [5][6][7][8][9]. Because Fock space has locally a tree-like structure, the problem of Anderson localization on different types of graphs [10][11][12][13][14][15] has attracted a renewed activity [16][17][18][19][20][21][22][23][24][25]. In particular, the existence of a delocalized phase with non-ergodic (multifractal) eigenfunctions in a broad range of disorder is debated [19-21, 23, 25].

The problem of non-ergodicity also arises in another context corresponding to glassy physics [START_REF] Mézard | Spin glass theory and beyond[END_REF]. For directed polymers on the Bethe lattice [START_REF] Derrida | [END_REF], a glass transition occurs at low temperature, leading to a phase where the directed polymer explores few branches of the tree instead of the exponential number of branches at its disposal, a strongly non-ergodic behavior. As there is a mapping to directed polymer models in the Anderson-localized phase [10,[28][29][30], it has been recently proposed that this type of non-ergodicity is also relevant in the delocalized phase [18].

In this letter, we study the Anderson transition in a family of random graphs [31][32][33], where a tunable parameter p allows us to interpolate continuously between the 1D Anderson model and the random regular graph model of infinite dimensionality. Our main tool is the single parameter scaling theory of localization [34]. It has been used as a crucial tool to interpret the numerical simulations of Anderson localization in finite dimensions [START_REF] Abrahams | years of Anderson Localization[END_REF]35,36] and to achieve the first experimental measurement of the critical exponent of the Anderson transition in 3D [37]. In our case, the infinite dimension of the graphs leads to highly non-trivial finite-size scaling properties: unusually, we find different scaling laws on each side of the transition. Our detailed analysis of extensive numerical simulations leads to the following scenario. A single Anderson transition is present which separates a localized phase from an ergodic delocalized phase. However a characteristic non-ergodicity volume Λ emerges in the latter phase. For scales below Λ, states are non-ergodic in the sense that they take significant values only on few branches, and display a multifractal behavior. For scales above Λ, this structure repeats itself and leads to large scale ergodicity. At the threshold, Λ diverges, and the behavior below Λ extends to the whole system. The critical behaviors do not depend on the graph parameter p, which strongly supports the universality of this scenario.

In order to describe the localization properties, we use two complementary approaches. First we derive recursive equations for the local Green function using a mapping to a tree [10], which we solve using the pool method from glassy physics [16,28], and analyse the critical behavior by finite-size scaling. Second, we perform exact diagonalization of very large system sizes up to N ≈ 2.10 6 , and we extract the scaling properties of eigenfunction moments. We use the box-counting method in this new context of graphs of infinite dimensionality to perform a local analysis and to extract the non-ergodicity volume Λ unambiguously.

Random graph model.-We consider a 1D lattice of N sites with periodic boundary conditions. Each site is connected to its nearest neighbors and ⌊pN ⌋ shortcut links are added (⌊.⌋ is the integer part). These shortcuts give an average distance between pairs of sites that increase logarithmically with N , so that the graph has an infinite dimensionality (see Supplemental Material and [38]). The system is described by an N -dimensional Hamiltonian

H = N i=1 ε i |i i| + i,j |i j| + ⌊pN ⌋ k=1 (|i k j k | + |j k i k |) in the position basis {|i , 1 ≤ i ≤ N }.
The first term describes on-site disorder, with ε i i.i.d. Gaussian random variables with zero mean and standard deviation W . The second term runs over nearest neighbors. The third term gives the long-range links that connect pairs (i k , j k ), randomly chosen such that |i k -j k | > 1.

The case p = 0 is the usual 1D Anderson model. At finite p, our system is a random graph with mean connectivity K = 1 + 2p, allowing us to access a regime where 1 < K ≤ 2.

Glassy physics approach.-We first use a recursive technique used to investigate localization on the Bethe lattice [10,16,[START_REF] Derrida | [END_REF]28,39]. It is exact for a Cayley tree (which has no loop), but only an approximation in the case of a generic graph. For a regular tree graph with K + 1 neighbors, the diagonal elements G ii of the Green operator follow

G ii = (ǫ i -E - K j=1 G jj ) -1
, where the sum runs over the K children j of node i [10]. The local tree-like structure of our random graph model is such that each parent node has either one or two children. This leads to three recursion equations which determine the probability distribution of G (see Supp. Mat.).

In order to probe the localization properties on the disordered graph, we use the belief propagation method (or pool method), which consists of sampling the distribution of G with a Monte-Carlo approach [16,28]. For a fixed value of E, we start from an initial pool of M pool complex values for the local variables G ii , 1 ≤ i ≤ M pool and calculate the next generation by applying the recursion relations. The important quantity is the typical value of the imaginary part ImG which goes to zero in the localized phase as ln ImG ∼ -M g /ξ l when the number of generations M g tends to infinity, (here X denotes ensemble averaging) whereas in the delocalized phase ln ImG converges to a finite value. We observed that the localization length ξ l diverges at the transition as

ξ l ∼ [W -W c (M pool )]
-ν l with the critical exponent ν l ≈ 1 and a critical disorder W c (M pool ) which depends on M pool (see also [16,28]). We determined W c (M pool ) for values of M pool up to 10 6 . The results, presented in the inset of Fig. 1, show that W c (M pool ) converges to W ∞ c ≈ 1.77 for p = 0.06 as M pool → ∞. Following [16,40], we assume that ln ImG follows a single parameter scaling law:

ln ImG = -(M g ) ρ F G (M g /ξ) , (1)
with the scaling parameter In order to sample correctly the distribution of ImG, values of M pool as large as possible are usually considered, which entails typically M g ≤ M pool (see above). However our numerical results show that the scaling behavior (1) is valid but visible only for M g ≫ M pool . Moreover, for a given initial pool, the fluctuations of ln ImG when M g is varied can be extremely large (especially at criticality).

ξ ∼ |W -W c (M pool )| -ν [41]. w pool @ ¡ ¢ ¢ £ A IXV IXT IXR IXP IH ¤ IH S IH ¥ IH Q IH ¦ § IXT IXÏ XR IX© w g a$ h n s m q i & IH S IH ¥ IH Q IH ¦ IH IH IH Q IH ¦ IH IH IH FIG.
In order to analyze the scaling behavior (1) we therefore considered values of M pool from 50 to 800, M g from 2M pool to 10 5 , and averaged additionally over 100 different realizations of the pool. The one-parameter scaling hypothesis ( 1) is confirmed by the data collapse shown in Fig. 1, which allows us to extract the scaling exponents ν ≈ 1.4 ± 0.2 and ρ ≈ 0.28 ± 0.07, that do not depend on M pool . Therefore, in the delocalized phase, the typical value of ImG vanishes at the transition with an essential singularity lim Mg →∞ ln ImG ∼ -(W c (M pool )-W ) κ with κ = ρν ≈ 0.39 ± 0.16 the critical exponent in the delocalized phase, compatible with the value 1/2 predicted analytically [14,15]. Moreover, from ν l = ν(1 -ρ) we recover the value ν l ≈ 1.0 ± 0.2 (see also [16]). Scaling analysis of eigenfunction moments.-We now describe the results of our second approach. We performed exact diagonalizations of a large number of realizations of graphs with up to N ∼ 2 × 10 6 sites and obtained for each realization 16 eigenfunctions closest to the center of the band using the Jacobi-Davidson iterative method [42]. We performed a multifractal analysis of the eigenfunctions |ψ by considering the scaling of average moments P q = N i=1 |ψ i | 2q for real q as a function of N . For a d-dimensional system of linear size L and volume N = L d , multifractal eigenfunctions have P q ∼ L -τq at large L, or equivalently P q ∼ N -χq with χ q = τ q /d, defining non-trivial multifractal dimensions D q = τ q /(q -1). In the localized case D q = 0 whereas for wavefunctions delocalized over the whole space D q = d.

Our graphs however correspond to a case of infinite dimensionality [START_REF] Dorogovtsev | Lectures on complex networks[END_REF], where the system volume N = V g (d N ) is exponential in the linear size d N ∼ log 2 N , the diameter of the system (see Supp. Mat.). Figure 2 shows that the critical behavior P 2 ∼ (log 2 N ) -τ2 ∼ d N -τ2 indeed holds with τ 2 ≈ 0.42 for p = 0.06 and W = 1.6 ≈ W c (upper right inset). One recovers the analytical predictions [START_REF] Evers | [END_REF]13,15] that χ q = 0 at infinite dimensionality. Moreover, in the light of the analogy with directed polymers, eigenfunctions are localized on few branches in the localized phase. At criticality the localization length diverges to the system size d N and one should observe multifractality at the level of few branches of the graph.

The following one-parameter scaling hypothesis should naturally follow:

P q = d -τq N F lin (d N /ξ) . (2)
It is consistent with the scaling theory for both the Anderson transition in finite dimension [START_REF] Rodriguez | [END_REF] and the glassy physics approach detailed above. A careful finite-size scaling analysis of our data shows that (2) yields a very good data collapse on the localized side of the transition, see Fig. 2, upper branch in the main panel. The scaling parameter ξ ∼ ξ l the localization length diverges as ξ ∝ (W -W c ) -ν l near the transition, with ν l ≈ 1. ± 0.1 (in agreement with the value found by our first glassy physics approach).

However, in the delocalized phase, small but systematic deviations are observed (see Supp. Mat.). This leads us to propose a different scaling in this phase. Indeed, the linear scaling (2) of the moments with d N rescaled by the length ξ is not the only possibility: the system volume N could instead be rescaled by a characteristic volume Λ (correlation volume):

P q = d -τq N F vol (N/Λ). ( 3
)
Both scaling hypotheses (2) and ( 3) are strictly equivalent in finite dimension, but lead to very different behaviors for a graph of infinite dimensionality. In the first linear scaling picture (2), the delocalized states consist of the repetition of linear critical structures of size ξ and the moments behave as P q ≈ ξ -τq N -(q-1)/ξ . It is reminiscent of the non-ergodic behavior discussed in [19][20][21]23].

In the second volumic scaling picture (3), a delocalized state consists of N/Λ volumic critical structures of size Λ, and moments behave as

P q ≈ Λ N q-1 1 -τ q ξ dN
(see Supp. Mat.). This is consistent with previous analytical results [14,15]. The finite size scaling shown in Fig. 2 clearly indicates that the volumic scaling (3) puts all the curves onto a single scaling function in the delocalized phase W < W c , with correlation volume diverging exponentially at the transition as ln Λ ≈ (W c -W ) -κ , κ ≈ 0.46 ± 0.1 (similar to the value obtained with our glassy physics approach and to the analytical prediction κ = 1/2 [14,15]).

Non-ergodicity volume.-In order to probe the local properties of localization in our system, we use the boxcounting method, which consists of investigating the scaling properties of moments of coarse-grained wavefunctions. Dividing the system of N sites into boxes of ℓ consecutive sites along the lattice (i.e. not following the longrange links) and defining a measure

µ k = i∈box k |ψ i | 2
of each box, moments are defined as P q (ℓ) = k µ q k . For a multifractal state, they are expected to scale as P q (ℓ) ∼ ℓ πq at large N with nontrivial π q [45]. In the localized case π q = 0 whereas for a wavefunction delocalized over the whole system π q = q -1. Figure 3 displays the moments P 2 (ℓ) as a function of ℓ for different W , and shows that three distinct regimes can be identified. At scales below the mean distance 1/(2p) between two long-range links, all moments have a power-law behavior with π 2 ≈ 0.53, independently of W . The value of π 2 is close to τ 2 ≈ 0.42 found above and measures the critical multifractality on few branches. At intermediate scales, the moments follow a plateau characteristic of a strongly non-ergodic behavior. Beyond a certain characteristic scale λ which depends on W and N , the moments are linear in ℓ, which corresponds to an ergodic behavior. In the localized case λ ∼ N , so that there is no ergodic behavior, while in the delocalized case, λ saturates to a finite value (see below), and states are ergodic at scales above λ, non-ergodic below: we therefore call λ the nonergodicity volume. One can extract λ from a rescaling of the local slopes πq (ℓ) ≡ d ln Pq(ℓ) d ln ℓ [45] in the ergodic regime ℓ ≫ λ (see inset of Fig. 4).

The data shown in Fig. 4 can be described by a linear

scaling λ N/ log 2 N = G lin log 2 N ξ
in the localized regime, with ξ ∼ ξ l the localization length, and by a volumic scaling λ N/ log 2 N = G vol N Λ in the delocalized regime. At the threshold, the critical behavior λ ∼ N/ log 2 N shows that wavefunctions are located on single branches of log 2 N sites. This confirms the description of the critical behavior as multifractal on few branches, an extension of the glassy non-ergodic character of the localized phase when ξ diverges. Moreover, the volumic scaling shows that in the limit N ≫ Λ the non-ergodicity volume λ saturates to the correlation volume Λ (see Supp. Mat.), which therefore can be interpreted as the non-ergodicity volume in the thermodynamic limit. This implies that the delocalized phase is ergodic in the limit of large N ≫ Λ.

For other values of q, we have checked that the scaling properties are the same for q ≥ 1 whereas for q < 1 the volumic behavior of the delocalized phase extends to the critical and localized regimes. This is to be expected [15,17,46,47]: for q < 1 all small values of the wavefunction, even outside the few localization branches, contribute to the moment.

Universality.-We checked the universality of our results by considering different values of the graph parameter from p = 0.01 to p = 0.49 (see Supp. Mat.), which changes the average branching parameter K = 1 + 2p considerably. Our data show that the critical scalings are insensitive to the value of p. Moreover, the critical exponents have universal values κ ≈ 0.5 and ν l ≈ 1.

Conclusion.-Our study strongly supports the following picture for the Anderson transition on infinite dimensional random graphs of any branching K without boundary: A single transition separates a localized phase from an ergodic delocalized phase. In the delocalized phase, the non-ergodicity volume Λ marks the threshold between a non-ergodic behavior reminiscent of glassy physics at small scales and an ergodic behavior at large scales. At the transition, Λ diverges, so that the behavior below Λ extends to the whole system. This highlights a new type of strong non-ergodicity with multifractality restricted to few branches. To detect such features has required the use of state of the art exact diagonalizations of very large size systems, along with a highly non-trivial finite-size scaling analysis. In particular, the ergodic character of the delocalized phase is controlled by the unusual volumic scaling in this phase, different from the linear scaling which applies in the localized phase and to the whole transition in the Cayley tree as found in our glassy physics approach, where boundary effects are important and the delocalized phase is believed to be non-ergodic [17,21,23,25]. It will be very interesting to study the consequences of the unusual features we found on the dynamics of such systems and to see whether the non-trivial scaling laws we observed apply to other models, for example in many-body localization where non-ergodicity is a crucial issue.

Supplemental material to "Scaling theory of the Anderson transition in random graphs: ergodicity and universality"

Topological properties of the random graph model

In this Section we want to assess the topological properties of the random graphs considered in our model more in details. In particular random graphs often refer to Erdös-Renyi random graphs, i.e. graphs having N vertices and (p + 1)N links which are chosen with a uniform probability over the set of all such graphs. In this Section it will be explained that, while the set of random graphs considered here is a subset of the set of Erdös-Renyi random graphs, they share several generic features of those graphs. This strenghtens the idea that our model for graphs posesses generic properties. We also want to stress that our random graph model coincides with the model of Random Regular Graphs (RRG) at the limit p = 0.5.

The first reason why the considered random graphs are generic, is that they have infinite dimension. The dimension refers here to the Hausdorff dimension of a graph. First one defines a distance between two vertices i and j as the number of vertices of a shortest path connecting i and j. Then one can compute the mean pair distance l as the average distance between any pair of vertices of the graph. Another quantity of interest for our study is the diameter, or the linear size, of our graph. It is denoted by d N and is defined as the largest distance between any pair of vertices of the graph. The Hausdorff dimension of the graph is evaluated by taking a sequence of graphs with increasing number of vertices N . The variation of l as a function of N leads to the definition of the Hausdorff dimension d H , see e.g. [S2]:

l ∼ N 1/dH , N → ∞ . (S1)
Erdös-Renyi random graphs have infinite dimension as the mean pair distance grows like l ∼ ln N for large N . We checked that this is also the case for our model of random graphs whenever p is positive. More precisely for p = 0.49 we found numerically an asymptotic form l ≈ 1.44 ln N , which agrees with the prediction l ∼ ln N for 3-regular graphs in RRG. Such a scaling for the mean pair distance also means that the diameter grows logarithmically as a function of N for large graphs. This is illustrated for three different values of p in Fig. S1. In particular for p = 0.49 the best fit is close to the prediction d N ∼ ln N for 3-regular graphs in RRG (see [S1]).

The second common feature between our graphs and Erdös-Renyi random graphs is the clustering coefficient. It is defined as the probability that two given neighbors of a fixed vertex are themselves connected by a link. An important property of Erdös-Renyi random graphs is that, when the vertex number N increases, the clustering coefficient grows like N -1 [S2]. For our model of random graphs, it was found numerically that the clustering coefficient obeys the following law, for any p:

C(N, p) ≃ 3 N p(1 -p) , (S2)
which agrees with the scaling for Erdös-Renyi random graphs. Another common point with Erdös-Renyi random graphs is that the graphs considered in our study locally look like a tree. More precisely the number of small loops is independent of the number of vertices [S2].

Furthermore, as we have checked that l quickly decreases as a function of p, our model of random graphs shares also the main features of smallworld networks [S3].

Recursion equations

For a regular tree graph with K + 1 neighbors, the recursion equation is obtained by considering the Green operator G(E) = (M -E I) -1 of the adjacency matrix M of the network with some vertex m removed. If i is a child node of m, the diagonal entry G ii (E) of the Green function can be expanded (see e.g. [S4]) as

G ii = (M ii -E - K j=1 G jj ) -1 ,
where the sum runs over the neighbors j of node i other than m. As the tree is self-similar, the G jj and G ii all have the same probability distribution P (G). Moreover, the G jj are independent and also are independent of the random variables M ii = ǫ i , so that at a fixed value of the energy E the above relation determines a functional equation for the probability P (G).

In order to obtain a similar recursion relation for our random graph model, we consider the local tree-like structure of the graph. The tree is such that each parent node has either one or two children, with probability respectively 1 -2p and 2p. Following the cavity method [S5], we consider a graph where some parent node (say m) has been removed. Each child node i is the root of a tree which can be of three different types: either i has one remaining neighbor (case A), or two neighbors connected by two nearest-neighbor links (case B), or two neighbors connected by one nearest-neighbor and one long-range link (case C). We thus have to distinguish between three types A i , B i and C i , of local random variables Gii (E), corresponding to the three possible local patterns. We introduce a fourth type G of random variable G i , equal to A i with probability 1 -2p and to B i with probability 2p. Local pattern of type A are those where node i has a single child j, which can itself be of type A or B (note that type C is excluded since the link between i and j is of nearest-neighbor type). That is, the child is of type G. In case B, node i has a neighbor of type C and a neighbor of type G. In case C, the two children are of type G. The analog of the recursion equation on the regular tree now takes the form of three recursion relations

A i = 1 ǫ i -E -G j , ( S3
)

B i = 1 ǫ i -E -G j1 -C j2 , (S4)
C i = 1 ǫ i -E -G j1 -G j2 , (S5)
together with the condition

G i = A i with probability 1 -2p B i with probability 2p . ( S6
)
The probability distributions for each type of random variable follow a set of functional equations that can be directly inferred from Eqs. ( S3)- (S6).

Scaling analysis

Deviations to the linear scaling hypothesis in the delocalized regime

Here we show the results of finite-size scaling of the moment P 2 for p = 0.06 following the linear scaling hypothesis (2) in the delocalized regime W < W c ≈ 1.6. In Fig. S2, the best rescaling of the data when the linear size of the graph d N ∼ log 2 N is rescaled by the scaling parameter ξ is shown. Small but systematic deviations are observed (data have been represented with lines to better see these deviations). Clearly, the lines corresponding to different values of the disorder strength do not have the right curvature to be put on each other via such a rescaling. On the contrary, we recall that the linear scaling hypothesis ( 2) is fully consistent with our data in the localized regime W > W c (see Fig. 2).

Asymptotic behaviors in the delocalized phase

The linear scaling hypothesis (2) predicts a non-ergodic delocalized phase in the sense of [S6, S7] which has its origin in the glassy non-ergodicity of the localized phase. Following the scaling law hypothesis (2), a delocalized state is built from critical structures of size ξ. As a critical structure consists of few branches, a delocalized state is to follow these branches for ξ steps, then to make connections to K branches, then follow the K branches for ξ steps that perform each K connections, etc. One thus sees that the number of critical structures constituting a delocalized state is V g (d N /ξ) where V g (X) is the volume of a graph of size X. Therefore, the asymptotic behavior of P q in the delocalized regime should be: P q ∼ ξ -τq /V g (d N /ξ) q-1 where the number V g (d N /ξ) appears on the denominator due to normalization. Because the volume V g (X) of graphs of infinite dimensionality scales exponentially with the linear size X, and using V g (d N ) = N , P q ≈ ξ -τq N -(q-1)/ξ (S7) in the delocalized phase, d N ≫ ξ. This behavior corresponds to the following asymptotic dependence of the scaling function F lin (see (2)):

F lin (X) ∼ X τq [V g (X)] q-1 , for X → ∞, W < W c . (S8)
In the volumic scaling hypothesis (3), the scaling parameter Λ(W ) plays the role of the correlation volume, and the scaling function is expressed as the ratio of the two characteristic volumes instead of the ratio of the characteristic linear sizes. In this case, a critical structure is an hypercube, and the number of such structures in a delocalized state is N/Λ. Therefore, the leading asymptotic behavior of P q when N ≫ Λ is

P q ≈ Λ N q-1 . (S9)
In order to compensate the term d N -τq in (3), the scaling function F vol should have the following asymptotic behavior: This leads to the following more precise asymptotic behavior:

F vol (X) ∼ V g -1 (X) τq X q-1 , for X → ∞, W < W c . (S10)
P q ≈ Λ N q-1 1 -τ q ξ d N . (S11)
This is precisely what is predicted by the analytical theory [S8]: P 2 ≈ C/N with ln C ∝ (W c -W ) -κ close to the threshold, and κ the critical exponent in the delocalized phase. The corrections in the parenthesis are negligible in the limit d N ≫ ξ.

Finally, the volumic scaling hypothesis for the non-ergodicity volume λ(N, W )

λ N/d N = G vol N Λ , (S12)
with d N ∼ log 2 N , implies the following asymptotic behavior:

G vol (X) ∼ V g -1 (X) X , for X → ∞, W < W c . (S13)
Because V g -1 (N/Λ) = d N -ξ, in the limit of large system volume N ≫ Λ:

λ ≈ N d N Λ N (d N -ξ) = Λ 1 - ξ d N . (S14)
Therefore, in the delocalized phase, the non-ergodicity volume λ saturates to the correlation volume in the thermodynamic limit.

Universality versus a change of the graph parameter p

Here we present the results of the same scaling analysis represented in Fig. 2 for two distinct values of the graph parameter, p = 0.01 and p = 0.49. The same critical scalings and critical exponents are observed in Fig. S3: the linear scaling law (2) where the linear system size d N ∼ log 2 N is rescaled by the scaling parameter ξ describes the data in the localized regime W > W c while a volumic scaling law (3) as a function of the ratio of the system volume N and the correlation volume Λ holds in the delocalized regime. As in the case p = 0.06 shown in the paper (see Fig. 2), the scaling parameter ξ ∼ ξ l (localization length) diverges at the transition point as ξ ∼ (W -W c ) -ν l with ν l ≈ 1, while the correlation volume Λ diverges exponentially as ln Λ ∼ (W c -W ) -κ with the critical exponent κ ≈ 0.5.

FIG. 2 .

 2 FIG.2. Scaling of the moment P2 with N for p = 0.06 and W ∈ [0.8, 3]. For W > Wc (localized phase), linear scaling of P2 /(log 2 N ) -τ 2 following Eq. (2) with τ2 = 0.42. For W < Wc (delocalized phase), volumic scaling following Eq. (3). Upper inset: The critical behavior of P2 at W = 1.6 ≈ Wc is very well fitted (black line) by P2 = A0(log 2 N ) -τ 2 with A0 a constant and τ2 ≈ 0.42. Lower inset: correlation volume Λ (circles) and localization length ξ (squares) on both sides of the transition. Solid lines are the fits: ln Λ = A1 + A2(Wc -W ) -κ and ξ = A3(W -Wc) -ν l , with Wc = 1.6 (assigned value), yielding κ ≈ 0.46 and ν l ≈ 1.

FIG. 3 .

 3 FIG. 3.P2 versus the box volume ℓ for N = 221 with p = 0.06 and W = 1.0 ( ), 1.3 (•), 1.6 (△), 2.4 (⋄) . The gray shaded area on the left delimits the small ℓ < 1/2p multifractal behavior (the dotted line has slope 0.53). The lightblue shaded area on the right delimits the large scale ℓ ≫ λ ergodic behavior (the dashed line has slope 1).

FIG. 4 .

 4 FIG.[START_REF] Evers | [END_REF]. Scaling behavior of the non-ergodicity volume λ with N = 2 10 to 221 , p = 0.06 and W ∈ [0.8, 2.4]. At the threshold W = Wc ≈ 1.6, λ ∼ N/ log 2 N . In the localized phase, the data collapse is excellent when plotted as a function of (log 2 N )/ξ (linear scaling, see text). In the delocalized regime, a volumic scaling as function of N/Λ makes the curves collapse. Inset: Determination of the non-ergodicity volume λ through the rescaling of π2(ℓ) data (see text) as a function of ℓ/λ in the large scale ℓ ≫ λ ergodic regime, for W = 1.6 and N = 2 10 ( ), 2 14 (•), 2 18 (△) and 2 21 (⋄).

  FIG. S1. (Color online) Diameter dN of our random graph model as a function of the total number N of vertices. For each value of N , 10000 random graphs were sampled. The values reported here are the center of a Gaussian fit of the distribution of their diameter. Red circles: numerical data for p = 0.01. Red dashed line: Logarithmic fit, dN ∼ 78 ln(0.015N ). Green squares: numerical data for p = 0.06. Green dashed line: Logarithmic fit, dN ∼ 13 ln(0.10N ). Blue triangles: numerical data for p = 0.49. Blue dashed line: Logarithmic fit, dN ∼ 1.8 ln(1.5N ).

FIG. S2 .

 S2 FIG.S2. Deviations to the linear scaling hypothesis (2) for the moment P2 in the delocalized regime W < Wc ≈ 1.6 for p = 0.06. The best rescaling of the data as a function of (log 2 N )/ξ shows small but systematic deviations. The data are represented by lines of different colors according to W .

FIG. S3 .

 S3 FIG.S3. Scaling of the moment P2 for the graph model with N for two values of p. (Left panel) p = 0.01, W ∈ [0.25, 1.6]. At W = 0.6 ≈ Wc, the behavior of P2 is well fitted by P2 = A0(log 2 N ) -τ 2 with A0 a constant and τ2 = 0.69 (upper inset). In the main panel, the value τ2 = 0.8 was considered because it provides the best global scaling. In the localized regime, a linear scaling (2) as a function of (log 2 N )/ξ gives a very good collapse of all the curves for different disorder strengths W > Wc. In the delocalized regime, the rescaling of the volume N by the correlation volume Λ (3) puts all the data onto each other. Lower inset: correlation volume Λ (circles) and localization length ξ (squares) on both sides of the transition. Solid lines are the fits: log 2 Λ = A1 + A2(Wc -W ) -κ and ξ = A3(W -Wc) -ν l , with Wc = 0.6 (assigned value), yielding κ = 0.48 and ν l = 0.96. (Right Panel) p = 0.49, W ∈[START_REF] Abrahams | years of Anderson Localization[END_REF]12]. At W = 6 ≈ Wc, P2 ∼ (log 2 N ) -τ 2 with τ2 ≈ 0.306 (upper inset). In the main panel the value τ2 = 0.3 gives the best global scaling. The same scaling procedure as for p = 0.01 and 0.06 has been followed here. Lower inset: Solid lines are the fits: log 2 Λ = A4 + A5(Wc -W ) -κ and ξ = A6(W -Wc) -ν l , with Wc = 6 and κ = 0.5 (assigned values), yielding ν l = 1.0.

We thank Claudio Castellani and Nicolas Laflorencie for interesting discussions. We thank CalMiP for access to its supercomputer and the Consortium des Équipements de Calcul Intensif (C ÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11. This work was supported by Programme Investissements d'Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT, by the ANR grant K-BEC No ANR-13-BS04-0001-01, by the ARC grant QUANDROPS 12/17-02 and by the CONICET-CNRS bilateral project PICS06303R.