M Scohy

C Montella

F Claudel

S Abbou

L Dubau

Frédéric Maillard

E Sibert

S Sunde

Investigating the oxygen evolution reaction on Ir(111) electrode in acidic medium using conventional and dynamic electrochemical impedance spectroscopy

Introduction

The need to increase the energy production from renewable sources and their intermittent nature currently boost the development of energy storage systems such as proton exchange membrane water electrolysers (PEMWEs) [START_REF] Jacobson | Review of solutions to global warming, air pollution, and energy security[END_REF]. These systems are perceived as one of the most promising technology currently available owing to the possibility of fast start-up, to operate with higher efficiency at higher current density and to sustain larger current variations compared to alkaline water electrolysers [START_REF] Carmo | A comprehensive review on PEM water electrolysis[END_REF]. However, the sluggish oxygen evolution reaction (OER, anodic reaction) kinetics and the poor stability of the anodic material still limit their widespread development [START_REF] Aricò | Polymer electrolyte membrane water electrolysis: Status of technologies and potential applications in combination with renewable power sources[END_REF].

According to volcano plots established more than 30 years ago [START_REF] Man | Universality in oxygen evolution electrocatalysis on oxide surfaces[END_REF][START_REF] Trasatti | Electrocatalysis in the anodic evolution of oxygen and chlorine[END_REF][START_REF] Trasatti | Electrocatalysis by oxides-Attempt at a unifying approach[END_REF][START_REF] Miles | Periodic variations of overvoltages for water electrolysis in acid solutions from cyclic voltammetric studies[END_REF], iridium and ruthenium oxides (IrO2 and RuO2, respectively) are the most active electrocatalysts for the OER, but only IrO2 can maintain high electrocatalytic activity on the long term due to the instability of RuO2 under the harsh operating conditions of a PEMWE anode (acidic pH, highly oxidizing electrochemical potential and atmosphere) reaction conditions [START_REF] Hutchings | A structural investigation of stabilized oxygen evolution catalysts[END_REF] [START_REF] Antolini | Iridium as catalyst and cocatalyst for oxygen evolution/reduction in acidic polymer electrolyte membrane electrolyzers and fuel cells[END_REF]. Considering the low abundance of iridium (Ir) on the Earth's crust and its high cost [START_REF] Katsounaros | Oxygen electrochemistry as a cornerstone for sustainable energy conversion[END_REF], designing and synthesizing tailored OER nanocatalysts is required to minimize the economic and geological constraints associated with its use. Nevertheless, before switching to nanometer-sized catalysts, it is of pivotal importance to understand how Ir surfaces electrochemically oxidize and to establish structure-activity-stability relationships on atomically smooth surfaces such as Ir single crystals [START_REF] Danilovic | Activity-stability trends for the oxygen evolution reaction on monometallic oxides in acidic environments[END_REF].

To get answers to the questions, combining cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) techniques appears a promising strategy. However, there is an obvious conflict between the stationary environment required for EIS measurements and the well-known fact that metal surfaces oxidize, buckle, roughen -or even dissolve -under OER conditions, thus questioning the suitability of this technique to the studying of the electrodeelectrolyte interface in this potential domain [START_REF] Bandarenka | Exploring the interfaces between metal electrodes and aqueous electrolytes with electrochemical impedance spectroscopy[END_REF]. In the early 1970s Smith et al. developed and improved Fourier Transform (FT) methods [START_REF] Creason | Fourier transform faradaic admittance measurements III. Comparison of measurement efficiency for various test signal waveforms[END_REF][START_REF] Smith | The Acquisition of electrochemical response spectra by on-line Fast Fourier Transform: Data processing in electrochemistry[END_REF][START_REF] Schwall | On-line Fast Fourier Transform Faradaic admittance measurements: Real-time deconvolution of heterogeneous charge transfer kinetic effects for thermodynamic and analytical measurements[END_REF][START_REF] Bond | Cyclic voltammetry: A digital simulation study of the slow scan limit condition for a reversible electrode process[END_REF][START_REF] O'halloran | Rapid drop time on-line Fast Fourier Transform Faradaic admittance measurements[END_REF]. Later, the introduction and optimization of new algorithms [START_REF] Popkirov | A new impedance spectrometer for the investigation of electrochemical systems[END_REF][START_REF] Popkirov | Optimization of the perturbation signal for electrochemical impedance spectroscopy in the time domain[END_REF][START_REF] Darowicki | Theoretical description of the measuring method of instantaneous impedance spectra[END_REF] combined with Fast FT (FFT) methods led to the development of FFT-EIS [START_REF] Creason | Fourier transform faradaic admittance measurements III. Comparison of measurement efficiency for various test signal waveforms[END_REF][START_REF] Hazi | Microcomputer-based instrumentation for multi-frequency Fourier transform alternating current ( admittance and impedance) voltammetry[END_REF][START_REF] Walters | Weak adsorption of anions on gold: Measurement of partial charge transfer using Fast Fourier Transform electrochemical impedance spectroscopy[END_REF], and alternating current (AC) voltammetry [START_REF] Schwall | On-line Fast Fourier Transform Faradaic admittance measurements: Real-time deconvolution of heterogeneous charge transfer kinetic effects for thermodynamic and analytical measurements[END_REF][START_REF] Bond | Cyclic voltammetry: A digital simulation study of the slow scan limit condition for a reversible electrode process[END_REF][START_REF] Van Der Geest | An ac voltammetry study of Pt oxide growth[END_REF][START_REF] Harrington | Ac voltammetry for measurement of surface kinetics[END_REF]. An alternative to frequency response acquisition with FT technique is the application of a stream of wavelets in Hz-kHz frequency range, so-called potentiodynamic EIS or PDEIS [START_REF] Ragoisha | Potentiodynamic electrochemical impedance spectroscopy[END_REF][START_REF] Ragoisha | Potentiodynamic electrochemical impedance spectroscopy of silver on platinum in underpotential and overpotential deposition[END_REF][START_REF] Ragoisha | Potentiodynamic electrochemical impedance spectroscopy. Copper underpotential deposition on gold[END_REF]. More recently, non-stationary electrochemical processes have been studied using this technique [START_REF] Hazi | Microcomputer-based instrumentation for multi-frequency Fourier transform alternating current ( admittance and impedance) voltammetry[END_REF][START_REF] Garland | Analysis of experimental constraints and variables for time resolved detection of Fourier transform electrochemical impedance spectra[END_REF][START_REF] Ragoisha | Characterisation of the electrochemical redox behaviour of Pt electrodes by potentiodynamic electrochemical impedance spectroscopy[END_REF][START_REF] Pettit | Measurement of diffrential capacitance for 28 faradaic systems under potentiodaynamic conditions: considerations of Fourier transform and phase-selective techniques[END_REF]. In the past few years, multisine signals have been added to this technique, allowing the introduction of dynamic electrochemical impedance spectroscopy (DEIS) [START_REF] Sacci | Dynamic electrochemical impedance spectroscopy[END_REF][START_REF] Sacci | Dynamic electrochemical impedance spectroscopy, for electrocatalytic reactions[END_REF]. In a DEIS measurement, a multisine signal, i.e. the sum of sinusoidal signals with different frequencies, is superimposed onto a slow potential ramp and applied to the working electrode. Similar to AC voltammetry, the signal is modulated but, in the latter, only a single sine is applied to the electrode. In DEIS, both amplitudes and phases of the multisine signals are optimized in such a way that the overall amplitude of the potential perturbation is small enough for linearity requirements to be fulfilled, as described by Roy et al. [START_REF] Garland | Analysis of experimental constraints and variables for time resolved detection of Fourier transform electrochemical impedance spectra[END_REF] and Sacci and Harrington [START_REF] Sacci | Dynamic electrochemical impedance spectroscopy[END_REF]. FFT is subsequently applied to the current recorded during the experiment, in order to separate the responses corresponding to each frequency component in the applied potential and thus monitor the electrochemical impedance for each of these frequencies. The typical time required to acquire a DEIS spectra is less than 1 second, which allows recording electrochemical impedance under dynamic conditions. This technique has been used in corrosion science to study the evolution of pitting processes [START_REF] Darowicki | Evaluation of pitting corrosion by means of dynamic electrochemical impedance spectroscopy[END_REF][START_REF] Krakowiak | Impedance investigation of passive 304 stainless steel in the pit pre-initiation state[END_REF] or to investigate the efficiency of corrosion inhibitors [START_REF] Gerengi | Evaluation of corrosion inhibition of brass-118 in artificial seawater by benzotriazole using Dynamic EIS[END_REF]. More recently, DEIS has also been employed to unravel the degradation mechanisms of proton-exchange membrane fuel cell (PEMFC) electrodes [START_REF] Darab | Dynamic electrochemical impedance spectroscopy of Pt/C-based membrane-electrode assemblies subjected to cycling protocols[END_REF]. In particular, DEIS proved efficient to monitor agglomeration of platinum nanoparticles and crystallite growth.

Herein, we used a similar approach to get insights into the early stages of surface oxidation and the OER mechanism on Ir(111) single crystal electrode and electrochemically activated Ir(111).

Impedance measurements were also performed on OER catalysts that are closer to real-life application, such as Ir nanoparticles supported on carbon. Similarities and differences between the EIS and DEIS results are rationalized using cyclic voltammetry and inductively-coupled plasma mass spectrometry experiments.

Experimental

Single crystal electrode preparation

An Ir(111) single crystal provided by MaTecK (purity 99.999%, oriented to <0.1°, cylinder with a diameter of 5 mm and a height of 4 mm) was used in this study. Before each experiment, the Ir(111) surface was prepared by annealing in an oxygen-hydrogen flame at 2000°C, cooled in an argon + 5% hydrogen (Ar + 5 % H2) mixture and protected by a water droplet saturated with this gas mixture. The electrode was then transferred to the electrochemical cell and put in contact with the solution under potential control, at E = +0.8 V vs. the reversible hydrogen electrode (RHE). All potentials are referred to the RHE.

Synthesis of Ir/C nanoparticles

A colloidal suspension of Ir nanoparticles was synthesized using dihydrogen hexachloroiridate (IV) hydrate (H2Cl6Ir, xH2O, 99 at. %, Alfa Aesar) and ethylene glycol (EG, Rotipuran ≥ 99.9%, Roth). A 6.95 gIr L -1 solution was first prepared by dissolving H2Cl6Ir, xH2O in Milli-Q grade water (Millipore, 18.2 MΩ cm, total organic compounds <3 ppb). Then, 2.876 ml of this solution were added to 40 ml of EG and 20 ml deionized water under magnetic stirring. The pH was adjusted to 12 by dropwise addition of a 0.5 M NaOH (Suprapur 99.99%, Merck) solution diluted in 1:1 H2O:EG. The mixture was then heated under reflux and Ar atmosphere at 160°C for 3 hours under constant magnetic stirring, and then allowed to cool down at room temperature under air atmosphere and constant stirring. The second step of the synthesis consisted of adding a high-surface area carbon support to the colloidal suspension. 80 mg of Vulcan XC72 (Cabot) was dispersed in 40 ml of 1:1 H2O:EG solution and added to the colloidal suspension to obtain a nominal mass fraction of 20 wt.% Ir. The pH was then set to 3 by addition of a 0.5 M H2SO4 (Suprapur 96%, Merck) solution diluted in 1:1 H2O:EG. The solution was stirred for 20 hours before being filtered, washed several times with deionized water and dried in air at 110°C. Catalytic suspensions were made by mixing 1.8 ml of MQgrade water, 5.0 mg of catalyst powder, 723 µl of iPrOH and 27 µl of a 5.0 wt % Nafion solution (Electrochem. Inc.). The inks were first sonicated in an ultrasonic bath for 10 min. A 10 µl droplet of ink was deposited on the surface of the glassy carbon electrode (diameter 5 mm) under an N2 flux until complete evaporation of the solvents, leading to a targeted Ir loading of 20 µgIr cmgeo -2 .

Electrochemical measurements

Electrochemical experiments were carried out at room temperature in a cell made of Pyrex glass. All glassware used in this study was initially cleaned with Caro's acid (98% H2SO4 mixed with 30% H2O2) and carefully rinsed with Milli-Q grade water (Millipore, 18.2 MΩ cm, total organic compounds <3 ppb). The working electrode was either Ir(111) single crystal or a glassycarbon tip coated with Ir nanoparticles (NPs) supported onto Vulcan XC 72 in fixed mode. The single crystal was used in the hanging meniscus configuration, following the protocols described in Refs. [START_REF] Clavilier | Preparation of monocrystalline Pt microelectrodes and electrochemical study of the plane surfaces cut in the direction of the {111} and {110} planes[END_REF][START_REF] Motoo | Electrochemistry of platinum single crystal surfaces. Part I. Structural change of the Pt (111) surface followed by an electrochemical method[END_REF][START_REF] Dickertmann | Eine methode zum ausschluss von randeffekten bei elektrochemischen messungen an einkristallen[END_REF]. A platinum grid was used as the counter electrode. The reference electrode was a freshly-prepared reversible hydrogen electrode (RHE) connected to the main cell compartment by a Luggin capillary. All the potentials mentioned in this study are referred to that of the RHE.

The electrolyte was a freshly prepared 0.05 M H2SO4 solution obtained by diluting Suprapur 96 wt. % H2SO4 (Merck) with Milli-Q water. The solution was degassed with Ar (Ar >99.999 %, Messer) before each experiment. Cyclic voltammetry and EIS measurements were realized using an Autolab PGSTAT302N potentiostat. The DEIS measurements were performed using the experimental setup presented elsewhere [START_REF] Darab | Dynamic electrochemical impedance spectroscopy of Pt/C-based membrane-electrode assemblies subjected to cycling protocols[END_REF]. The ratio of the potential sweep rate (vb) on the minimum frequency (fmin) was vb/fmin = 2 mV (typically Fvb/RT << 2πfmin [START_REF] Sacci | Dynamic electrochemical impedance spectroscopy[END_REF] where F, R and T have their usual meaning) to minimize the parasitic effect of potential ramp on the ac response at low frequency. We used the baseline correction implemented by Sacci et al. in Ref. [START_REF] Sacci | Dynamic electrochemical impedance spectroscopy, for electrocatalytic reactions[END_REF].

Before any EIS or DEIS measurement, five cyclic voltammograms (CVs) between 0.05 V and 0.95 V at a sweep rate of 50 mV s -1 were performed to control the cleanliness of the Ir(111) single crystal surface. Between the control CVs and the EIS/DEIS measurements, the electrode potential was maintained at 0.8 V. The EIS and DEIS measurements directly followed the control CVs. EIS and DEIS measurements were also performed on an 'electrochemicallyactivated' Ir(111) surface formed by cycling 15 times the potential between 0.05 to 1.60 V at 50 mV s -1 . The same EIS and DEIS procedures were applied to Ir NPs supported on Vulcan XC72 after an activation step composed of 100 CVs between 0.05 and 1.40 V at 500 mV s -1 (leading to a stable electrochemical signature of the surface [START_REF] Reksten | The oxygen evolution reaction mechanism at IrxRu1-xO2powders produced by hydrolysis synthesis[END_REF]).

The EIS measurements were performed from 0.05 V to 0.95 V at intervals of 20 mV in the frequency range from 10 kHz to 10 mHz applying AC amplitude of 10 mVRMS. (VRMS is the square root of the mean square of the values for one period of the sinusoid). The frequency range of the DEIS measurements was imposed by the sweep rate. It was ranging from 10 kHz to 5 Hz and from 10 kHz to 2.5 Hz for CVs performed at 10 mV s -1 or 5 mVs -1 , respectively.

A homemade Matlab program based on the Levenberg-Marquardt algorithm [START_REF] Press | Numerical recipes[END_REF] was used to fit the EIS and DEIS data.

Inductively-coupled plasma mass spectrometry (ICP-MS) and X-ray photoelectron spectroscopy (XPS) measurements

Aliquots of the electrolyte after EIS and DEIS measurements were analyzed using a PerkinElmer NexION 2000 ICP-MS (PerkinElmer, Inc., Waltham, Massachussetts, USA). A 5 µg L -1 solution of 115 In was used as internal standard for detection of 193 Ir. 5 standard solutions were prepared with the following concentrations of Ir: 0.01, 0.05, 0.1, 0.2, 0.5 µg L -1 .

XPS analyses were also performed on as-prepared and electrochemically-activated Ir(111) electrode using a Thermo Scientific K-alpha spectrometer with a monochromatized Al X-ray source (hν = 1486.6 eV; spot size = 400 µm). Pass energies of 30 and 100 eV were used to record the core level and the survey spectra, respectively. All spectra were acquired using an electron flood gun to compensate possible positive charge accumulation during measurements.

The obtained spectra were fitted using Thermo Scientific™ Advantage Software.

Results and discussion

EIS and DEIS measurements were performed in two different potential regions: from 0.05 V to 0.95 V or from 0.05 V to 1.45 V on as-prepared (fresh) Ir(111) and on 'electrochemicallyactivated' Ir(111) single crystal electrodes, respectively. In the first potential region, only adsorption reactions occur, [START_REF] Pajkossy | Voltammetry and impedance measurements of Ir(111) electrodes in aqueous solutions[END_REF] thus establishing the relevance of the DEIS measurements. In the potential region ranging from 1.45 V to 1.60 V, oxygen evolution rendered EIS measurements challenging due to the formation of gas bubbles, and chemical, morphological and structural changes of the electrode surface [START_REF] Cherevko | Oxygen evolution activity and stability of iridium in acidic media. Part 2. -Electrochemically grown hydrous iridium oxide[END_REF][START_REF] Cherevko | Oxygen evolution activity and stability of iridium in acidic media. Part 1. -Metallic iridium[END_REF]. We will show that DEIS measurements do not suffer from these limitations.

3.1.Potentials below oxygen evolution

As-prepared Ir(111) single crystal electrode (0.05 < E < 0.95 V)

We first compare the results of the EIS and DEIS measurements in the potential region 0.05 < E < 0.95 V. After the preparation of the Ir(111) single crystal, CVs between 0.05 V and 0.95 V at a sweep rate of 50 mV s -1 were performed to control the cleanliness of the surface. The features displayed in Figure 1a are similar to those reported in previous works [START_REF] Motoo | Electrochemistry of platinum single crystal surfaces. Part I. Structural change of the Pt (111) surface followed by an electrochemical method[END_REF][START_REF] Motoo | Hydrogen and oxygen adsorption on Ir (111), (100) and (110) planes[END_REF][START_REF] Motoo | Effect of anions on hydrogen and oxygen adsorption on iridium single cyrstal surfaces[END_REF][START_REF] Wan | In situ scanning tunneling microscopy of welldefined Ir(111) surface: high-resolution imaging of adsorbed sulfate[END_REF]. The peaks related to adsorption/desorption of under-potentially-deposited hydrogen are observed at 0.075/0.095 V, respectively. The double layer region ranges from 0.2 to 0.95 V. Given the strong surface sensitivity of CVs to the surface state of the electrode, no change of the CV features during potential cycling indicates that the electrode surface is stable in this potential domain. Following these control CVs, EIS and DEIS measurements were performed at 0.05, 0.25, 0.45, 0.65, 0.85 and 0.95 V and from 0.05 to 0.95 V at 10 mV s -1 , respectively. The data obtained are represented in Figure 1b and In the potential region 0.05 < E < 0.95 V, only adsorption (e.g. sulfate anions) [Eq. ( 1)] or electrosorption (e.g. H-UPD) [Eq. ( 2)] reactions take place:

A + s ↔ A,s (1)
H + + s + e -↔ H,s (2)
where s is an adsorption site and A,s or H,s are adsorbed species.

In the case of an ideally polarizable interface (i.e. the electrode is not subject to Faradaic reaction), the electrode impedance results from the series association of the double-layer capacitance and the electrolyte resistance. On the other hand, the impedance for electrosorption reactions can be modeled by a capacitance (Cads) in series with a charge transfer resistance (Rct)

representing the kinetic of the adsorption. The double-layer charging is associated with a single capacitance (Cdl) in parallel to the adsorption. Finally, a resistance (Rel) is added in series for the electrolyte Ohmic drop. The expression of the impedance (Z) is then [START_REF] Diard | Calculation, simulation and interpretation of electrochemical impedance diagrams Part IV. Second-order electrochemical impedances[END_REF]:

Z(p) = Rel + 1 pC dl + 1 Z f (p) (3)
with Zf the Faradaic impedance and p Laplace variable

Zf(p) = Rct + 1/pCads (4)
Indeed, if the kinetics of electrosorption is too fast versus the probing tool, it cannot be estimated and the associated transfer resistance become negligible [START_REF] Sibert | High frequency impedance measurements on Pt(111) in sulphuric and perchlorique acids[END_REF]. In that case, the Equation 3 simplifies as:

Z(p) = Rel + 1/pCt (5)
where Ct = Cads + Cdl Since only one time constant is observable in the Bode plots (Figure 1d and Figure 1e), this indicates that the charge transfer resistance of electrosorption cannot be estimated in the present experimental conditions. Therefore, we will later only use Equation 5 to fit data at each EIS measurement's potential. For DEIS experiments, although the potential is continuously scanned, the scan rate is low enough, compared to minimum frequency (see experimental [START_REF] Sacci | Dynamic electrochemical impedance spectroscopy[END_REF])

to consider that the system is at steady state and the fit procedure is used.

As shown by Figure 1b and Figure 1c, due to the small deviation from a perfect vertical line, it was not possible to fit properly the measured data with Equation 5. A non-ideal capacitance, also called Constant Phase Element (CPE) (ZCPE =1/Q(jω) α where j = (-1) 1/2 here, Q is the CPE capacitance parameter and α the CPE exponent) was thus used instead of the ideal capacitance predicted by the theory. Several explanations have been suggested to justify this non-ideal behavior, such as the presence of traces of chemical impurities in the solution [START_REF] Martin | Influence of experimental factors on the constant phase element behavior of Pt electrodes[END_REF] or local variations of the impedance resulting from the hanging meniscus configuration used in the electrochemical experiments [START_REF] Martin | Influence of experimental factors on the constant phase element behavior of Pt electrodes[END_REF][START_REF] Huang | The global and local impedance response of a blocking disk electrode with local constant-phase-element behavior[END_REF].

Figure 2

As shown in Figure 2c, an excellent agreement was obtained between the theoretical and the experimental results. Note that, for the sake of clarity, the data are presented in the complex capacitance representation, where C = . By applying this model to the entire series of DEIS measurements, the values of Q and α could be extracted for all the electrode potentials. The obtained CPE exponent values, plotted in Figure 2b, are close to 1 except for the underpotentially adsorbed hydrogen desorption region, validating the nearly capacitive behavior. An equivalent capacitance was calculated from the Q formula proposed by Brug et al. [START_REF] Brug | The analysis of electrode impedances complicated by the presence of a constant phase element[END_REF].

C t = Q 1 α (R el -1 ) 1-1 α (6)
Figure 2d also outlines the good consistency between the calculated capacitances from the EIS and DEIS spectra (using Equation 6) and the capacitances extracted from the typical CV of the as-prepared Ir(111) electrode using Equation 7:

Ct = ( 7
)
where vb is the potential sweep rate.

Hence, in summary, the results presented in this section show that DEIS can be used to determine the impedance of the electrochemical system under consideration.

Electrochemically-activated Ir

(111) electrode (0.05 < E < 1.45 V)
The same series of experiments were performed on an 'electrochemically-activated' Ir(111)

electrode prepared by sweeping the potential of the as-prepared Ir(111) electrode 15 times at 50 mV s -1 between 0.05 to 1.60 V. The electrochemical activation protocol is aimed at forming an oxy-hydroxide layer that is more active than metallic iridium towards the OER [START_REF] Cherevko | Oxygen evolution activity and stability of iridium in acidic media. Part 1. -Metallic iridium[END_REF][START_REF] Gottesfeld | Electrochemical and optical studies of thick oxide layers on iridium and their electrocatalytic activities for the oxygen evolution reaction[END_REF]. It is documented in the literature [START_REF] Augustynski | ESCA study of the state of iridium and oxygen in electrochemically and thermally formed iridium oxide films[END_REF][START_REF] Hall | X-ray photoelectron spectroscopic studies of the iridium electrode system[END_REF][START_REF] Mozota | Surface and bulk processes at oxidized iridium electrodes-I. Monolayer stage and transition to reversible multilayer oxide film behaviour[END_REF][START_REF] Frazer | The oxygen evolution reaction on cycled iridium electrodes[END_REF] that potential cycling above 1.4 V leads to the growth of an oxide film [START_REF] Augustynski | ESCA study of the state of iridium and oxygen in electrochemically and thermally formed iridium oxide films[END_REF][START_REF] Conway | Surface and bulk processes at oxidized iridium electrodes-II. Conductivity-switched behaviour of thick oxide films[END_REF]. In this study, sweeping the electrode potential up to 1.6 V resulted in a more active but also more stable surface state relative to Ir(111) in agreement with literature [START_REF] Özer | Iridium(111), Iridium(110), and Ruthenium(0001) single crystals as model catalysts for the oxygen evolution reaction: insights into the electrochemical oxide formation and electrocatalytic activity[END_REF]. The Ir4f spectra measured on the fresh and 'electrochemically-activated' Ir(111) surfaces are displayed in Figure 3. The XPS spectrum on the fresh Ir(111) surface features on contribution 60.9 eV and another at 63.9 eV associated to metallic iridium [START_REF] Hall | X-ray photoelectron spectroscopic studies of the iridium electrode system[END_REF][START_REF] Freakley | The X-ray photoelectron spectra of Ir, IrO2 and IrCl3 revisited[END_REF]. These peaks are shifted by ca. 0.3 eV (61.2 eV) and a small shoulder emerges at 62 eV thus reflecting the presence of Ir(IV) [START_REF] Freakley | The X-ray photoelectron spectra of Ir, IrO2 and IrCl3 revisited[END_REF][START_REF] Pfeifer | The electronic structure of iridium oxide electrodes active in water splitting[END_REF][START_REF] Saveleva | Operando evidence for a universal oxygen evolution mechanism on thermal and electrochemical iridium oxides[END_REF][START_REF] Peuckert | XPS study on thermally and electrochemically prepared oxidic adlayers on iridium[END_REF] on electrochemically-activated Ir(111) surface (a poorer fit was obtained upon including an Ir(III) component).

3.2.Potentials above the oxygen evolution (1.45 V < E < 1.60 V)

The second part of this study focuses on the potential region comprised between 1.45 V to 1.60 V, a potential region in which the Ir(111) surface oxidizes very quickly (similar to what was found for 0.05 V < E < 1.45 V). In addition, the Ir(111) surface can dissolve electrochemically and the OER takes place in this potential domian, [START_REF] Cherevko | Oxygen evolution activity and stability of iridium in acidic media. Part 2. -Electrochemically grown hydrous iridium oxide[END_REF][START_REF] Cherevko | Oxygen evolution activity and stability of iridium in acidic media. Part 1. -Metallic iridium[END_REF] thus rendering EIS measurements highly challenging (no steady-state is achieved). We will show that DEIS is an appropriate technique to obtain information about the electrode | electrolyte interface while avoiding problems linked to these transient phenomena.

As-prepared Ir(111) single crystal electrode

Figure 5c displays the LSV recorded on the as-prepared Ir(111) electrode during potential cycling between 1.45 and 1.60 V. We first remark an increase of the geometrical current density at potentials located above 1.47 V, thus signing the onset of a new Faradaic process. Although this potential is very close to the thermoneutral voltage of the OER, we notice that such increase may also correspond to the O(II-)/O(I-) redox couple [START_REF] Pfeifer | Reactive oxygen species in iridium-based OER catalysts[END_REF] or to the dissolution of Ir atoms [START_REF] Kim | Balancing activity, stability and conductivity of nanoporous core-shell iridium/iridium oxide oxygen evolution catalysts[END_REF].

Impedance measurements were performed from 1.45 to 1.60 V with a step increase of 5 mV for each EIS measurement and with a potential sweep rate of 5 mV s -1 for DEIS measurements.

The results are plotted in the Nyquist representation in Figure 5a, 5b, 5d and 5e. The appearance of a semi-circle in DEIS and EIS measurements in the potential region 1.45 < E < 1.50 V vs. RHE confirms that a charge transfer process (electrosorption-desorption reaction in this case) takes place at the electrode surface. This semi-circle becomes more pronounced at E > 1.55 V in agreement with the increase of the current density shown in Figure 5c.

Figure 5

In contrast to the former case (0.05 V < E < 1.45 V), significant differences emerge from the comparison of the EIS and DEIS data. Based on the expression of the impedance for an electrosorption-desorption process [START_REF] Diard | Calculation, simulation and interpretation of electrochemical impedance diagrams Part IV. Second-order electrochemical impedances[END_REF][START_REF] Diard | Cinétique Electrochimique[END_REF], this can be rationalized assuming different surface coverages or concentrations of species at the interface, which result from the large time required for EIS measurements (ca. 5 min). Figure 6a shows the current-time transients recorded during stepping the potential from 0.8 V to 1.4, 1.5 or 1.6 V, thus simulating the potential variations during EIS measurements. For E ≤ 1.50 V, a fast rise in current is observed, then the current decays reaching a value close to 0 mA cm -2 . In contrast, at E = 1.60 V, after a fast decrease (~3 s), a current maximum is observed after which the current stabilizes around 1 mA cm -2 .

Based on the work of Özer et al. [START_REF] Özer | Iridium(111), Iridium(110), and Ruthenium(0001) single crystals as model catalysts for the oxygen evolution reaction: insights into the electrochemical oxide formation and electrocatalytic activity[END_REF], the electrode surface is believed to roughen in these potential conditions. This atomic roughening enhances the intrinsic OER activity of Ir(111), and may account partially for the gain in OER current. The progressive oxidation of nonstoichiometric Ir-O species, hydroxy-groups and water molecules of the topmost and nearsurface layers formed at the beginning of the OER in the experimental conditions of EIS measurements may also account for the observed catalytic trend [START_REF] Li | Atomic-scale insights into surface species of electrocatalysts in three dimensions[END_REF], as well as the oxidation of oxygen from O(II-) to O(I-) species [START_REF] Pfeifer | In situ observation of reactive oxygen species forming on oxygenevolving iridium surfaces[END_REF].

Figure 6b

compares the CVs recorded on the as-prepared Ir(111) electrode before and after the EIS (dashed black line) or the DEIS (plain red line) measurements (note that the potential was swept negatively after the (D)EIS measurements). We first remark that the underpotentiallyadsorbed/desorbed hydrogen features are strongly distorted, and their intensity is depreciated after the EIS measurement, thus confirming buckling and atomic roughening of the electrode surface. Due to the shorter acquisition time (ca. 40 s for this potential range), these phenomena are less pronounced after the DEIS measurement. Furthermore, the higher double-layer current observed after the EIS measurement suggests that the growth of the hydrous oxide layer on Ir(111) was much less important during the DEIS measurement. An oxygen reduction reaction (ORR) current also manifests in the negative-going potential scan recorded after the DEIS or the EIS measurements. Here again, the ORR current is less pronounced after the DEIS measurement, and this is in line with the shorter time required associated with the measurement.

Finally, Figure 6c shows that the as prepared Ir(111) surface is much more active toward the OER after the EIS relative to after the DEIS measurement. This agrees with the non-monotonic dependence of the OER activity as a function of time recently reported by Li et al. [START_REF] Li | Atomic-scale insights into surface species of electrocatalysts in three dimensions[END_REF] and others [START_REF] Özer | Iridium(111), Iridium(110), and Ruthenium(0001) single crystals as model catalysts for the oxygen evolution reaction: insights into the electrochemical oxide formation and electrocatalytic activity[END_REF], and again should be associated with continuous growth of an oxy-hydroxide film on the metal surface. The enhanced OER activity of the Ir(111) electrode after the EIS measurement thus confirms that the electrode surface was in average more oxidized after the EIS measurement than after the DEIS measurement [START_REF] Pfeifer | In situ observation of reactive oxygen species forming on oxygenevolving iridium surfaces[END_REF][START_REF] Kasian | Electrochemical on-line ICP-MS in electrocatalysis research[END_REF]. The large differences of the voltammograms recorded in the OER region before and after EIS and DEIS measurements on as-prepared Ir(111) and electrochemically-activated Ir(111), represented in Figure 6c and In summary, thanks to its faster acquisition process, the DEIS technique proved capable to dynamically measure the electrochemical impedance of metallic Ir single crystal with only minimal changes of the electrode structure and chemistry (mild oxidation of the surface). In contrast, the long acquisition time required for EIS measurements leads to important changes in the structure and the chemistry of the single crystal surface due to the formation of an oxyhydroxide film and the early stages of the OER.

Electrochemically-activated Ir(111) electrode

To confirm our claims, a fresh Ir(111) electrode was electrochemically-activated using the protocol discussed previously. Impedance measurements were then performed from 1.45 to Ir electrodes upon polarization in the OER potential region [START_REF] Cherevko | Oxygen evolution activity and stability of iridium in acidic media. Part 2. -Electrochemically grown hydrous iridium oxide[END_REF][START_REF] Özer | Iridium(111), Iridium(110), and Ruthenium(0001) single crystals as model catalysts for the oxygen evolution reaction: insights into the electrochemical oxide formation and electrocatalytic activity[END_REF].

Figure 7

To get insights into the chemical changes occurring on the Ir electrodes during EIS or DEIS measurements, ICP-MS measurements were performed on electrolytes used for the impedance experiments on as-prepared Ir(111) and 'electrochemically-activated' Ir(111) electrodes. In agreement with all the above, whatever the nature of the electrode, lower Ir concentration was found after DEIS compared to EIS measurements (Table 1). Nevertheless, it is noteworthy that higher Ir concentrations were monitored on the 'electrochemically-activated' Ir(111) electrode in comparison to the as-prepared Ir(111) electrode. This trend confirms former literature reports [START_REF] Cherevko | Oxygen evolution activity and stability of iridium in acidic media. Part 1. -Metallic iridium[END_REF] and can be rationalized by considering the presence of unstable Ir-species and lattice defects in the oxy-hydroxide layer as well the higher surface area of the electrochemicallyactivated Ir(111) electrode [START_REF] Kasian | The common intermediates of oxygen ovolution and dissolution reactions during water electrolysis on iridium[END_REF][START_REF] Geiger | The stability number as a metric for electrocatalyst stability benchmarking[END_REF].

Table 1

Combined with the CV results, the ICP-MS measurements thus confirm that lower chemical and structural changes of the single crystal surface were observed during the DEIS measurement, corroborating the interest of DEIS to perform impedance measurements in the OER region.

Ir nanoparticles supported on high surface area carbon

Finally, for the sake of generalizing our findings to real-life OER catalysts, we extended our approach to Ir nanoparticles supported on high surface area carbon (Ir/C) synthetized by a modified polyol method (note that similar results were obtained on Ir nanoparticles supported on antimony-doped tin oxide, not shown). Figure 8 shows that the EIS (on the left side) and DEIS (on the right side) spectra recorded on Ir/C resemble those obtained on as-prepared Ir(111) and electrochemically-activated Ir(111) electrode at low frequencies, starting from a bent line to semi-circles. In the high frequency region of both spectra, we however noticed a change of diagram shape attributed to the porosity of both carbon support and Ir NPs. The data at potentials above 1.50 V were fitted using the model of porous electrode proposed by Meyers et al. [START_REF] Meyers | The impedance response of a porous electrode composed of intercalation particles[END_REF] in the case where the solid conductivity is very large compared to the electrolyte conductivity (Equation 8).

Z(p) = A Zloc(p) 1/2 coth(B/Zloc(p) 1/2 ) (8)
with

A = / and B = L(Sp/Km) 1/2
where S is the geometric surface of the electrode, Sp is the electroactive surface of the pores per unit volume, Km the conductivity of the electrolyte filled with bubbles in the pores, L the thickness of the porous layer and Zloc the local impedance at the pores surface.

The expression of the local impedance Zloc is in this case (Equation 9):

Zloc(p) = Rct /(1 + (τ p) α ) (9)
with τ = (Rct Qdl) 1/α The result of the fit obtained at 1.6 V for EIS data is presented in Figure 8c, indicating a good adequacy between the model used and the experimental data. An increase in the high-frequency resistance with potential is apparent in the impedance-plane plots, which we associate with the presence of oxygen bubbles at the interface [START_REF] Da Silva | Oxygen evolution in acid solution on IrO2 + TiO2 ceramic films. A study by impedance, voltammetry and SEM[END_REF]. This issue was clearly avoided when the DEIS technique was used. 2. In general, a good agreement was obtained between the values derived from OER polarization curves and the DEIS measurements, but higher Tafel slopes were obtained from EIS measurements. Based on the calculations described above, the higher Tafel slopes values obtained by EIS may be rationalized by considering that the surface was more strongly oxidized in these experimental conditions (higher oxygen coverage at the surface during the measurement [START_REF] Diard | Calculation, simulation and interpretation of electrochemical impedance diagrams Part IV. Second-order electrochemical impedances[END_REF], which can be seen in Figure 6b as well).

Whereas stationary impedance spectra featured an increased series resistance due to bubble formation at the Ir nanoparticles, the series resistance in the impedance data collected through DEIS were not affected. This indicates that the DEIS data were in general less affected by gas evolution. The shorter collection time will mitigate any adverse effects on the analysis from the presence of gas at and in the vicinity of the electrode. Tafel slopes calculated from the apparent charge-transfer from the DEIS data increase at high currents in a similar fashion as the Tafel slope from the polarization curve. We thus concluded that the higher Tafel slope at high current density is related to the OER mechanism and not to the formation of oxygen bubbles. Indeed, it is well-established that the OER involves multiple reaction steps. Several OER mechanisms have been proposed in the 1950-1960s by Bockris [START_REF] Bockris | Kinetics of activation controlled consecutive electrochemical reactions: Anodic evolution of oxygen[END_REF] and Krasil'shchikov [START_REF] Krasilshchikov | Intermediate stages of oxygen anodic evolution[END_REF] They are all based on an electrosorption step followed by other steps which might be a combination, or other electrosorption/desorption steps [START_REF] Fabbri | Developments and perspectives of oxide-based catalysts for the oxygen evolution reaction[END_REF]. The theoretical Tafel slopes for the 'oxide-path' and 'electrochemical-oxide' paths are presented in Table 2. The Tafel slopes obtained from OER polarization curves seem to advocate for the electrochemical path, the rate-determining step being the second step at low current densities. In contrast, the Tafel slopes obtained by DEIS were in favor of the oxide path, the rate-determining step being one of the two chemical steps.

The ca. 20 mV difference between the measured values renders any quantitative interpretation risky, if experimental incertitude and data dispersion are taken into account. These differences in Tafel slopes are also a consequence of different dissolution mechanisms [START_REF] Cherevko | Oxygen evolution activity and stability of iridium in acidic media. Part 1. -Metallic iridium[END_REF][START_REF] Kasian | The common intermediates of oxygen ovolution and dissolution reactions during water electrolysis on iridium[END_REF][START_REF] Jovanovič | Electrochemical dissolution of iridium and iridium oxide particles in acidic media: transmission electron microscopy, electrochemical flow cell coupled to inductively coupled plasma mass spectrometry, and X-ray absorption spectroscopy study[END_REF][START_REF] Geiger | The stability number as a metric for electrocatalyst stability benchmarking[END_REF], since this experiment cannot deconvolute dissolution of Ir from OER. Furthermore, for both mechanisms, when the first step becomes rate-determining, the Tafel coefficient increases to 119 mV dec -1 . This trend is clearly visible at higher current densities on Figure 9c and was already reported by others [START_REF] Rasten | Electrocatalysis in water electrolysis with solid polymer electrolyte[END_REF]. Note that an 'electrochemicaly-activated' Ir(111) electrode was used for each EIS measurement.

  Figure 1c and in Figure 1d and Figure 1e in the Nyquist plot and in the Bode modulus representations, respectively.

Figure 1

 1 Figure 1

Figure 3 Figure

 3 Figure 3

Figure 4 14

 4 Figure 4

Figure 6d ,

 6d Figure 6d, support these hypotheses. In contrast, almost no difference in OER activity can be noticed after EIS and DEIS measurements on the electrochemically-activated Ir(111) electrode because a more stable surface state was previously achieved during the potential cycling protocol (Figure6d).

Figure 6

 6 Figure 6

1. 60 V

 60 Figure 7eshow that the Nyquist plots are composed of a semi-circle arc with a frequency shift as the electrode potential is increased. Due to the minimum frequency used in this work, fmin = 2.5 Hz, only the high-frequency part (bent line) of the semi-circle graph is observed at the lower electrode potentials (Figure7aand Figure7d). We, however, notice a decrease in the impedance values measured on 'electrochemically-activated' Ir(111) electrode compared to asprepared Ir(111) that is consistent with the higher OER activity of the 'electrochemically-aged' Ir(111) electrode in comparison with as-prepared Ir(111). Unlike what was observed on asprepared Ir(111) electrode in this potential range, the results of the DEIS and EIS data are nearly similar. Figure6dconfirms this similarity by exhibiting identical CVs before and after the EIS measurements. These findings agree with previous reports pointing toward less pronounced changes of the OER activity for electrochemically oxidized Ir electrodes compared to metallic

Figure 8 20 3 . 3 .In Figure 9 ,

 8339 Figure 8

Figure 9 The

 9 Figure 9

Figure 9d displays the

  Figure 9d displays the reciprocal of the charge transfer resistance obtained by fitting the EIS and DEIS spectra as a function of the electrode potential. A linear behavior was observed. The Tafel slopes determined from the OER polarization curves in the low current density region and calculated from the EIS and DEIS spectra are shown inTable 2. In general, a good agreement

  crystalline surface) and chemical (surface oxidation and Ir dissolution) changes. The presence of oxygen bubbles that form and detach adds another difficulty to EIS measurements. Thanks to DEIS, a more accurate picture of the phenomena occurring at the Ir(111) electrode | electrolyte interface was achieved. While the EIS and DEIS signature were similar for both asprepared and electrochemically-activated Ir(111) electrodes at low electrode potentials, only DEIS proved capable to capture the evolution of electrochemical impedance of the Ir(111) surface in OER conditions while minimizing the structural and chemical changes occurring at high potentials. Thanks to the shorter collection time of DEIS, any adverse effect of the presence of gas at and in the vicinity of the electrode was mitigated. Nearly similar values for the Tafel slopes were obtained by DEIS and polarization curves, demonstrating that this technique can be accurately used in transient conditions.

Figure 1 .

 1 Figure 1. (a) Cyclic voltammogram of as-prepared Ir(111) electrode in 0.05 M H2SO4 (potential sweep rate = 50 mV s -1 ). Red crosses display the potential at which EIS measurements were

Figure 2 .Figure 3 .

 23 Figure 2. (a) Schematic representation of the equivalent circuit used to fit the impedance data

Figure 4 .

 4 Figure 4. (a) Cyclic voltammogram of 'electrochemically-activated' Ir(111) in 0.05 M H2SO4 (potential sweep rate = 50 mV s -1 ). Red crosses indicate the potential at which EIS

Figure 5 .Figure 6 .

 56 Figure 5. (c) Linear sweep voltammogram of as-prepared Ir(111) in 0.05 M H2SO4 (potential sweep rate = 5 mV s -1 ). Red crosses display the potential at which EIS measurements were

Figure 7 .

 7 Figure 7. (c) Linear sweep voltammogram of electrochemically-activated Ir(111) in 0.05M H2SO4 at 5 mV s -1 . Red crosses display the potential at which EIS measurements were

Figure 8 .

 8 Figure 8. Nyquist plots recorded on Ir nanoparticles supported on Vulcan XC72 at different

Figure 9 .

 9 Figure 9. (a) Schematic representation of the equivalent circuit used to fit the data. (b) Nyquist

Table 2

 2

	4. Conclusions
	In this study, impedance spectra were measured by conventional and dynamic impedance
	spectroscopy (EIS and DEIS, respectively) on as-prepared Ir(111), 'electrochemically-
	activated' Ir(111) and Ir nanoparticles supported onto high surface area carbon. The EIS and
	DEIS results were comparable but DEIS allowed an important gain of time with respect to
	conventional EIS. DEIS technique also revealed as powerful tool to study dynamic (transient)

systems and circumvent some limitations of EIS. We demonstrated this by investigating oxygen evolution reaction (OER) on a freshly prepared Ir(111) electrode. In OER conditions, the Ir surface undergoes structural (formation of surface defects, buckling and roughening of the

Table 1 .

 1 Ir concentrations measured by ICP-MS in electrolytes after EIS and DEIS measurements on as-prepared Ir(111) and 'electrochemically-activated' Ir(111) electrodes.

		As-prepared Ir(111)	Electrochemically-activated Ir(111)
	EIS	0.198 ± 7.10 -3 µg L -1	0.361 ± 0.02 µg L -1
	DEIS	0.076 ± 3.10 -3 µg L -1	0.278 ± 0.01 µg L -1

Table 2 .

 2 Tafel slopes obtained from the polarization curve at low current densities and calculated from the charge transfer resistance obtained from the EIS/DEIS fits.

		Polarization curves (low current
	Tafel coefficients		EIS	DEIS
		densities)
	As-prepared Ir(111)	46 mV dec -1	75 mV dec -1	60 dec -1	mV
	Electrochemically-activated Ir(111)	46 mV dec -1	85 mV dec -1	56 dec -1	mV
	Theoretical value
	Electrochemical oxide path	40 mV dec -1 /24 mV dec -1
	(rds: 2 nd electrochemical
	step/chemical step)

Theoretical value Oxide path (rds: chemical steps)

Acknowledgments

This work was performed within the framework of the Centre of Excellence of Multifunctional Architectured Materials "CEMAM" n• AN-10-LABX-44-01. The French National Research Agency (MOISE project, grant number ANR-17-CE05-0033) and the PHC AURORA 2017 program (n° 38483VD) financially supported this study.


Kay Joerg Wiese

Andrei A Fedorenko

Kay Jörg Wiese

Field Theories for Loop-Erased Random Walks

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

Introduction

Random walks (RWs) which are not allowed to self-intersect play an important role in combinatorics, statistical physics and quantum field theory. The two most prominent models are selfavoiding walks (SAWs) and loop-erased random walks (LERWs). The SAW was first introduced in polymer physics to model long polymer chains with self-repulsion due to excluded-volume effects. It is defined as the uniform measure on RW paths of a given length conditioned on having no self-intersection. Though this model is difficult to analyze rigorously, it was discovered by de Gennes [START_REF] Gennes | Exponents for the excluded volume problem as derived by the Wilson method[END_REF] that its scaling behavior in d dimensions is given by the O(n) symmetric φ 4 theory in the unusual limit of n → 0. A loop-erased random walk (LERW) is defined as the trajectory of a random walk (RW) in which any loop is erased as soon as it is formed [START_REF] Lawler | A self-avoiding random walk[END_REF]. An example is shown on figure 1, where the underlying RW is drawn in red, and the LERW remaining after erasure in blue. Similar to a self-avoiding walk it has a scaling limit in all dimensions, e.g. the end-to-end distance R scales with the intrinsic length as R ∼ 1/z , where z is the fractal dimension [START_REF] Kozma | The scaling limit of loop-erased random walk in three dimensions[END_REF]. It is crucial to note that while both LERWs and SAWs are non-self-intersecting, their fractal dimensions do not agree since they have a different statistics on the same set of allowed trajectories. LERWs appear in many combinatorial problems, e.g. the shortest path on a uniform spanning tree is a LERW. While LERWs are non-Markovian RWs, their traces are equivalent to those of the Laplacian Random Walk [START_REF] Lyklema | The Laplacian random walk[END_REF][START_REF] Gregory | The Laplacian-b random walk and the Schramm-Loewner evolution[END_REF], which is Markovian, if one considers the whole trace as state variable. It is constructed on the lattice by solving the Laplace equation ∇ 2 Φ(x) = 0 with boundary conditions Φ(x) = 0 on the already constructed curve, while Φ(x) = 1 at the destination of the walk, either a chosen point, or infinity. The walk then advances from its tip x to a neighbouring point y, with probability proportional to Φ(y). In a variant of this model growth is allowed not only from the tip, but from any point on the already constructed object. This is known as the dielectric breakdown model [START_REF] Niemeyer | Fractal dimension of dielectric breakdown[END_REF], the simplest model for lightning. The same construction pertains to diffusion-limited aggregation [START_REF] Witten | Diffusion-limited aggregation, a kinetic critical phenomenon[END_REF]. In contrast to SAWs, LERWs have no obvious field-theoretic description. In three dimensions LERWs have been studied numerically [START_REF] Guttmann | Critical exponent for the loop erased self-avoiding walk by monte carlo methods[END_REF][START_REF] Agrawal | Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions[END_REF][START_REF] Grassberger | Scaling of loop-erased walks in 2 to 4 dimensions[END_REF][START_REF] Wilson | Dimension of the loop-erased random walk in three dimensions[END_REF], while in two dimensions they are described by SLE with κ = 2 [START_REF] Schramm | Scaling limits of loop-erased random walks and uniform spanning trees[END_REF][START_REF] Lawler | Conformal invariance of planar loop-erased random walks and uniform spanning trees[END_REF], predicting a fractal dimension z LERW (d = 2) = 5 4 . Coulomb-gas techniques link this to the 2d O(n)-model at n = -2 [START_REF] Nienhuis | Exact critical point and critical exponents of O(n) models in two dimensions[END_REF][START_REF] Duplantier | Loop-erased self-avoiding walks in two dimensions: exact critical exponents and winding numbers[END_REF]. Below, we give perturbative arguments that this construction can be done in any dimension d via the O(n)-symmetric φ 4 theory at n = -2.

Coming from a different direction, it was conjectured in [START_REF] Fedorenko | Field theory conjecture for loop-erased random walks[END_REF] that the field theory of the depinning transition of charge-density waves (CDWs) pinned by disorder is a field theory for LERWs. This equivalence is based on the conjecture of Narayan and Middleton [START_REF] Narayan | Avalanches and the renormalization-group for pinned charge-density waves[END_REF] that pinned CDWs can be mapped onto the Abelian sandpile model. The connection of Abelian sandpiles with uniform spanning trees, and thus with LERWs, was earlier established by Majumdar [START_REF] Majumdar | Exact fractal dimension of the loop-erased self-avoiding walk in two dimensions[END_REF][START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF]. Despite the lack of a proof of this equivalence, the corresponding 2-loop predictions agree with rigorous φ 4 -theory at n = -2 1 boson and 2 fermions (complex) one family of complex fermions CDWs at depinning bounds [START_REF] Fedorenko | Field theory conjecture for loop-erased random walks[END_REF] and have been tested against numerical simulations at the upper critical dimension d = 4 in [START_REF] Grassberger | Scaling of loop-erased walks in 2 to 4 dimensions[END_REF] where they correctly reproduce the leading and subleading logarithmic corrections of [START_REF] Fedorenko | Field theory conjecture for loop-erased random walks[END_REF]. The depinning transition of CDWs is described by the functional RG (FRG) fixed point for random periodic systems, proposed by Narayan and Fisher [START_REF] Fisher | Sliding charge-density waves as a dynamical critical phenomena[END_REF][START_REF] Narayan | Critical behavior of sliding charge-density waves in 4-epsilon dimensions[END_REF][START_REF] Narayan | Dynamics of sliding charge-density waves in 4-epsilon dimensions[END_REF], and developed in [START_REF] Leschhorn | Driven interface depinning in a disordered medium[END_REF][START_REF] Nattermann | Dynamics of interface depinning in a disordered medium[END_REF][START_REF] Le Doussal | Functional renormalization group and the field theory of disordered elastic systems[END_REF][START_REF] Le Doussal | 2-loop functional renormalization group analysis of the depinning transition[END_REF][START_REF] Chauve | Renormalization of pinned elastic systems: How does it work beyond one loop?[END_REF][START_REF] Wiese | Field theory of disordered elastic interfaces at 3-loop order: The β-function[END_REF][START_REF] Husemann | Field theory of disordered elastic interfaces to 3-loop order: Results[END_REF].

If true, both φ 4 -theory at n = -2 and the FRG for CDWs must agree, at least for observables related to LERWs. We show below that both the β-function and the fractal dimension of LERWs coincide. This is done using (i) graph-theoretical arguments valid at all orders in perturbation theory, (ii) non-perturbative supersymmetry techniques, and (iii) an explicit 4-loop calculation. This does not mean that the theories are identical: for example, at depinning CDWs exhibit avalanches which are seemingly absent in the φ 4 model. Our statement is that in the sector in which we can compare the two theories, they agree. This is illustrated on figure 2. It does not exclude that one of the theories can handle observables the other cannot. A classical example for this behavior is the solution of the 2d Ising model via the conformal bootstrap, as proposed by Belavin, Polyakov and Zamolodchikov (BPZ) [START_REF] Belavin | Infinite conformal symmetry in two-dimensional quantum field theory[END_REF]. Here a theory with three operators, the energy ε, the spin σ, and the identity 1 1, closes under OPE. However, other observables can be constructed from the Ising model on a lattice. Examples in case are domain walls, at criticality described by SLE. On the other hand, SLE does not (at least obviously) describe the operators of the original BPZ construction. Thus, SLE and BPZ describe different sectors of the same theory.

Mapping of LERWs onto

O(n) φ 4 -theory at n = -2
We now map LERWs onto the n-component φ 4 theory at n = -2. The latter is defined by the action

S[ φ] := x 1 2 [∇ φ(x)] 2 + m 2 2 φ(x) 2 + g 8 [ φ(x) 2 ] 2 . ( 1
)
One checks that, for n = -2, the full 2-point correlation function is given in Fourier space by the free-theory result

φ i (k)φ j (k) = φ i (k)φ j (k) 0 = δ ij k 2 + m 2 , (2)
independent of g. This fact is well known perturbatively [START_REF] Zinn-Justin | Phase Transitions and Renormalization Group[END_REF][START_REF] Kleinert | Critical Properties of φ 4 -Theories[END_REF][START_REF] Kleinert | Five-loop renormalization group functions of O(N )symmetric φ 4 -theory and ε-expansion of critical exponents up to ε 5[END_REF][START_REF] Kompaniets | Minimally subtracted six-loop renormalization of O(n)-symmetric φ 4 theory and critical exponents[END_REF]. A non-perturbative derivation is given below, by mapping onto complex fermions. Eq. ( 2) is the Laplace transform of the k-dependent Green function for a random walk (RW),

φ i (k)φ j (k) = ∞ 0 dt e -m 2 t × e -k 2 t . (3)
Here t ≥ is the time of the RW used to construct the LERW of length , which scales as ∼ t z/2 ∼ m -z , and z is the fractal dimension of the LERW. Let us convene that we draw the trajectory of a random walk in blue, and when it hits itself we do not erase the emerging loop, but color it in red; see figure 1 for an example. We claim that we can reconstruct these colored trajectories from φ 4 theory. To this aim, we first reformulate the theory (1) in terms of N = n/2 complex bosons Φ and Φ * , with

Φ i (x) := 1 √ 2 [φ 2i-1 (x) + iφ 2i (x)], i = 1, ..., N = n/2. Its action reads S[ Φ] := x ∇ Φ * (x)∇ Φ(x) + m 2 Φ * (x) Φ(x) + g 2 [ Φ * (x) Φ(x)] 2 . ( 4
)
This reformulation gives propagators a direction, making it easier to interpret them as RWs. Consider a specific path with s intersections in the path-integral representation. This conditioning introduces a small-length cutoff a [START_REF] Cloizeaux | Polymers in Solution, Their Modelling and Structure[END_REF][START_REF] Cloizeaux | Polymers in solutions: Principles and applications of a direct renormalization method[END_REF]: One may think of putting a grid of box-size a, and introduce for each box a Θ-function, which is one if the path passes through twice, and zero otherwise. It is crucial that for the self-intersections encoded in the quartic term we use the same prescription, that it squares to one, and is symmetric under exchange of the momenta. The contributions to paths with s = 1 self-intersections are (

) 5
The drawing on the l.h.s. of equation ( 5) is a LERW starting at x, ending in z, and passing through the segments numbered 1 to 3. Due to the crossing at y, the loop labeled 2 is erased; we draw it in red. The r.h.s of equation ( 5) gives all diagrams of φ 4 theory up to order g s . The first term is the free-theory result, proportional to g 0 . The second term ∼ g cancels the first term, if one puts g → 1.

Here it is crucial to have the same regularization for the interaction as for the conditioning. The third term is proportional to N , due to the loop, indicated in red. Setting N → -1 compensates for the subtracted second term. Thus setting g → 1 and N → -1, the probability to go from x to z remains unchanged as compared to the free theory. This is a necessary condition to be satisfied. Since the first two terms cancel, what remains is the last diagram, corresponding to the drawing for the trajectory of the LERW we started with.

Let us consider how this continues for s = 2 intersections. Once a first loop has been formed, there are two possibilities: The walk can either hit a blue or a red part of its own trace. Let us first assume it hits a blue part. Then a second loop is formed, and the mapping at g = 1, and N = -1 reads

Here nothing should happen, since the walk does not see the erased part of its trace. The appropriate cancelation is between "no interaction" and "reconnecting", since the latter would result in the erased loop appearing again in blue in perturbation theory. Thus also in this case we map onto the appropriate diagram of φ 4 -theory. Continuing these arguments inductively allows us to prove this for any number of intersections s.

We have thus established a one-to-one correspondence between traces of LERWs and diagrams in perturbation theory. We still need an observable which is 1 when inserted into a blue part of the trace, and 0 within a red part. This can be achieved by modifying the probability to diffuse from x to z, given by the expectation of

Φ * 1 (x)Φ 1 (z), to O(x, y, z) := Φ * 1 (x)Φ 1 (z) [Φ * 1 (y)Φ 1 (y) -Φ * 2 (y)Φ 2 (y)] . (8)
As shown in appendix Appendix A this observable can be simplified to

O(x, y, z) = Φ * 1 (x)Φ 1 (y)Φ * 2 (y)Φ 2 (z) . (9)
The fractal dimension z of a LERW is extracted from the length of the walk after erasure (blue part)

y,z O(x, y, z) z Φ * 1 (x)Φ 1 (z) ≡ m 2 y,z O(x, y, z) ∼ m -z . ( 10
)
We recall that the fractal dimension of SAWs is given by z SAW = 1/ν where the critical exponent ν is taken in the limit of n → 0. Remarkably, we can deform the measure on the RWs in a continuous way by tuning the parameter n. This interpolates between SAWs and LERWs, and more generally gives critical curves in the n-component vector model. Indeed, the operator in square brackets of ( 8) is related to the so-called crossover exponent [START_REF] Amit | Field Theory, the Renormalization Group, and Critical Phenomena[END_REF][START_REF] Kirkham | Calculation of crossover exponent from Heisenberg to Ising behaviour using the fourth-order ε expansion[END_REF][START_REF] Shimada | Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3d conformal bootstrap[END_REF]. It is defined as (see e.g. [START_REF] Kirkham | Calculation of crossover exponent from Heisenberg to Ising behaviour using the fourth-order ε expansion[END_REF], Eq. ( 8))

φ c = 2 + γ φφ -η 2 + γ φ 2 -η ≡ (2 + γ φφ -η) ν . ( 11
)
Here γ φφ is the RG function for the operator Φ 1 Φ * 2 , and η is the anomalous exponent for the dimension of the fields. For the LERWs considered here, η = 0 and ν = 1/2. More generally [START_REF] Amit | Field Theory, the Renormalization Group, and Critical Phenomena[END_REF][START_REF] Kirkham | Calculation of crossover exponent from Heisenberg to Ising behaviour using the fourth-order ε expansion[END_REF][START_REF] Kiskis | The Hausdorff dimension of random walks and the correlation length critical exponent in Euclidean field theory[END_REF],

z = d f = 2 + γ φφ -η (12)
is the fractal dimension of a curve in the critical O(n) symmetric φ 4 theory. Let us discuss what this means for LERWs and SAWs. First, for n = -2, the 2-point function between two points x and z in the φ 4 theory is the sum over all LERWs from x to z, weighted by the chemical potential m 2 conjugate to its construction time t. It equals the free propagator from x to z, since coloring loops in red or erasing them does not change the propagator. On the other hand, if we apply our picture to SAWs, red loops carry a factor of N = 0, and only configurations without selfintersections survive. In this case the ratio (10) equals 1, and the fractal dimension can be inferred from the 2-point function alone, i.e. φ c (n = 0) = 1 and d f = 1/ν(n = 0) = z SAW . Thus, the 2-point function of φ 4 theory with n = 0 equals the sum over all SAWs connecting the two points.

Let us now turn our attention to d = 2: There it is known that SLE κ for 0 ≤ κ ≤ 4 is a simple curve. RWs generated by the 2-dimensional O(n)-symmetric φ 4 theory with n = -2 cos(4π/κ) are non-self-intersecting for -2 ≤ n ≤ 2, corresponding to 2 ≤ κ ≤ 4. For κ > 4 the curves are self-intersecting. For |n| > 2 the field theory becomes massive, and the connection breaks down. As we now know that n = -2 and n = 0 correspond to LERWs and SAWs in arbitrary dimension d, we conjecture that there is a family of non-selfintersecting random walks in arbitrary dimension d, parameterized by n, such that their measure can be continuously deformed from that of LERWs at n = -2 to that of SAWs at n = 0, and beyond to n > 0. Interestingly, the barrier of |n| = 2 in two dimensions is seemingly absent in higher dimensions. This may be understood from the fact that the return probability of a random walk to the origin decreases with increasing dimension.

φ 4 -theory at N = -1, and fermions

Up to now, we worked with N = -1 families of complex bosons. We show below that instead one can consider the limit of N f → 1 in a theory with N f complex fermions, or more generally with N b bosons and N f fermions, where N = N b -N f . Among others, this provides a non-perturbative proof that the propagator at N = -1 is independent of g.

The correspondence is based on the observation that N b -component bosons Φ carry a factor of N b per loop, while N f -component fermions Ψ yield a factor of -N f . On a more formal level, this can be inferred from the path integrals for both theories. Setting H 0 := -∇ 2 + τ (x) then

Z b 0 = D[ Φ * ]D[ Φ] e -x Φ * (x)H 0 Φ(x) = e -N b tr ln(H 0 ) , (13)
Z f 0 = D[ Ψ * ]D[ Ψ] e -x Ψ * (x)H 0 Ψ(x) = e N f tr ln(H 0 ) . ( 14
)
The bosonic correlation function is given by

Φ i (x)Φ * j (y) = δ ij H -1 0 x,y = 1 Z b 0 D[ Φ * ]D[ Φ] Φ i (x)Φ * j (y) e -x Φ * (x)H 0 Φ(x) . ( 15
)
For fermions an equivalent expression holds, and Ψ i (x)Ψ * j (y) = Φ i (x)Φ * j (y) . Setting τ (x) = m 2 + iρ(x), a Hubbard Stratonovich transformation allows us to decouple the quartic interaction in a theory of N b bosons Φ i and

N f fermions Ψ i , e -g 2 x [ Φ * (x) Φ(x)+ Ψ * (x) Ψ(x)] 2 = D[ρ] e -1 2g x ρ(x) 2 -iρ(x)[ Φ * (x) Φ(x)+ Ψ * (x) Ψ(x)] . (16)
As a consequence, a system of N b bosons and N f fermions with the interaction (16) has partition function

Z b+f = D[ Φ * ]D[ Φ]D[ Ψ * ]D[ Ψ] e -x Φ * (x)(-∇ 2 +m 2 ) Φ(x)+ Ψ * (x)(-∇ 2 +m 2 ) Ψ(x) × e -g 2 x [ Φ * (x) Φ(x)+ Ψ * (x) Ψ(x)] 2 = D[ρ] e (N f -N b ) tr ln(H 0 )-1 2g x ρ(x) 2 . ( 17
)
More generally, the correlations Φ 1 (y)Φ * 1 (x) in complex N -component Φ 4 theory can be calculated from a theory with N b bosons and N f fermions, where N = N b -N f . For N b = 0, N f = 1 one gets N = -1, and the interaction is [ Ψ * (x) Ψ(x)] 2 . It vanishes due to the squares of Grassmann variables: This theory of complex fermions,

S N f =1 N b =0 [Ψ * , Ψ] = x ∇Ψ * (x)∇Ψ(x) + m 2 Ψ * (x)Ψ(x) , (18)
is a free theory. It provides a non-perturbative proof that correlation functions of complex Φ 4theory at N = -1 (n = -2 real fields) are equivalent to those of the free theory. In d = 2 this is also known from lattice models [START_REF] Nienhuis | Exact critical point and critical exponents of O(n) models in two dimensions[END_REF]. However, it does not yield the renormalization of the coupling g at N = -1. To obtain the latter, one has to study N f = 1, and take the limit of N f → 1 at the end. Or one uses one family of complex bosons N b = 1 and two families of complex fermions N f = 2, a formulation onto which we will map CDWs at depinning later.

Finally, care has to be taken in identifying observables in both theories: While the 2-point functions of bosonic fields are symmetric under their exchange, those of the fermionic theory are antisymmetric. As a consequence φ 1 φ 1 = 0, whereas ψ 1 ψ 1 = 0.

Equivalence between φ 4 -theory at N = -1 and CDWs at depinning

Charge-density waves are ground states of solids, where the charge density is varying spatially, with a period set to 1. Coupling these elastic objects to quenched disorder leads after averaging over disorder to the dynamic field theory [START_REF] Fukuyama | Dynamics of the charge-density wave. I. Impurity pinning in a single chain[END_REF][START_REF] Lee | Electric-field depinning of charge-density waves[END_REF][START_REF] Fisher | Sliding charge-density waves as a dynamical critical phenomena[END_REF][START_REF] Narayan | Critical behavior of sliding charge-density waves in 4-epsilon dimensions[END_REF][START_REF] Narayan | Dynamics of sliding charge-density waves in 4-epsilon dimensions[END_REF][START_REF] Nattermann | Vortex-glass phases in type-II superconductors[END_REF]]

S CDW = x,t ũ(x, t)(∂ t -∇ 2 + m 2 )u(x, t) - 1 2 x,t,t ũ(x, t)ũ(x, t )∆ u(x, t) -u(x, t ) . ( 19
)
The function ∆(u) is an even function with period 1. Its renormalization can be studied using functional RG (FRG) [START_REF] Fisher | Sliding charge-density waves as a dynamical critical phenomena[END_REF][START_REF] Narayan | Dynamics of sliding charge-density waves in 4-epsilon dimensions[END_REF][START_REF] Narayan | Critical behavior of sliding charge-density waves in 4-epsilon dimensions[END_REF][START_REF] Leschhorn | Driven interface depinning in a disordered medium[END_REF][START_REF] Nattermann | Dynamics of interface depinning in a disordered medium[END_REF][START_REF] Le Doussal | Functional renormalization group and the field theory of disordered elastic systems[END_REF][START_REF] Le Doussal | 2-loop functional renormalization group analysis of the depinning transition[END_REF][START_REF] Chauve | Renormalization of pinned elastic systems: How does it work beyond one loop?[END_REF]. The analysis of the FRG flow for the function ∆(u) shows that the fixed point has the form

∆(u) = ∆(0) - g 2 u(1 -u) . (20)
In the absence of higher-order terms in u, the RG flow closes in the space of polynomials of degree 2, and for the quadratic term one is left with the renormalization of a single coupling constant g.

Thus, as long as one is interested in 2-point correlation functions, or avalanches, the fixed-point function can be replaced by ∆(u) → g 2 u 2 , which generates the simpler field theory,

S CDW simp := x,t ũ(x, t)(∂ t -∇ 2 + m 2 )u(x, t) - g 4 x,t,t ũ(x, t)ũ(x, t ) u(x, t) -u(x, t ) 2 . ( 21
)
Let us define a further variant which retains from ∆ u(x, t )-u(x, t) only the term u(x, t)u(x, t ),

S SAW := x,t ũ(x, t)(∂ t -∇ 2 + m 2 )u(x, t) + g 2 x,t,t ũ(x, t)u(x, t)ũ(x, t )u(x, t ) . ( 22
)
Perturbation expansion in this theory looks exactly like the one in Eq. ( 4), with a different propagator to be compared with Eq. ( 3),

R(k, t) := ũ(k, 0)u(-k, t) = Θ(t) e -t(k 2 +m 2 ) . (23)
In this theory closed loops have weight zero, as they are acausal in Itô discretization. If one can integrate freely over all times, diagrams in the dynamic theory reduce to those in the complex scalar theory. Thus the theory defined by Eq. ( 22) can be mapped onto the action (4) with n = N = 0, i.e. a self-avoiding walk. This is well-known due to de Gennes [START_REF] Gennes | Exponents for the excluded volume problem as derived by the Wilson method[END_REF]. We now show that the action ( 21) has the same effective coupling as the action (4) at N = -1. We first remark that the renormalized coupling is extracted from diagrams with times on which they depend taken far apart. An example is given by the first diagram in Eq. ( 24) below. To show the equivalence, we start by drawing all diagrams present in the SAW-theory [START_REF] Narayan | Dynamics of sliding charge-density waves in 4-epsilon dimensions[END_REF], complementing them by the missing diagrams originating from the additional vertices of (21) as compared to [START_REF] Narayan | Dynamics of sliding charge-density waves in 4-epsilon dimensions[END_REF]. These missing diagrams, a.k.a. children, can be generated from the diagrams for SAWs by moving one arrow from one side of the vertex into which it enters to the other side,

-→ - . (24)
We then extract contributions to the effective coupling; this is cleverly done by remarking that (i) the form of the effective interaction is proportional to the second line of Eq. ( 21), and (ii) it is extracted by retaining only terms of the form present in Eq. ( 22). This implies that the second diagram in Eq. ( 24) does not contribute. Indeed it comes with two other ones,

- + 1 2 + 1 2 . ( 25
)
After time-integration, the two fields at the left-most vertex cancel, thus the above sum vanishes. The next diagram [START_REF] Le Doussal | 2-loop functional renormalization group analysis of the depinning transition[END_REF] has two children,

- - → 0 . ( 27
)
both vanish, since the first is an acausal loop, thus zero, and the second has only one connected component, thus does not contribute to the renormalization of g. Now consider [START_REF] Wiese | Field theory of disordered elastic interfaces at 3-loop order: The β-function[END_REF] This diagram contains an acausal loop, thus does not contribute to the SAW theory [START_REF] Narayan | Dynamics of sliding charge-density waves in 4-epsilon dimensions[END_REF] where it vanishes due to a factor of N = 0. The diagram however has children; together they are

- - + . ( 29
)
The modified lines are in red. We first remark that none of them restricts the time-difference between the left and right-most vertices, and they all contribute to the effective coupling. Their coefficients are 0×1-1-1+1 = -1. Integrating over times, the result is the same as in φ 4 -theory at N = -1, graphically represented as

. ( 30
)
Noting the pairwise cancelations in loops of the form [START_REF] Husemann | Field theory of disordered elastic interfaces to 3-loop order: Results[END_REF], this can be interpreted as the missing contribution of the first (acausal) diagram. To summarize: We showed that at 1-loop order the action [START_REF] Narayan | Critical behavior of sliding charge-density waves in 4-epsilon dimensions[END_REF] has the same effective coupling as the action (4), diagram by diagram (after time integration). These considerations can be formalized to higher orders, and we checked them explicitly up to 4 loops. The theory (21) has a second renormalization, namely of friction, or time, which shows up in a renormalization of x,t ũ(x, t) u(x, t). The standard route to study this is to write down all diagrams constructed from ( 21), in which one field ũ and one field u remain [START_REF] Le Doussal | 2-loop functional renormalization group analysis of the depinning transition[END_REF]. Due to the structure of the action, the latter has the form u(x, t) -u(x, t ) and can be expanded as u(x, t)(t -t ). The additional time difference, when appearing together with a response function, acts by inserting an additional point into the latter, as can be seen from the definition [START_REF] Leschhorn | Driven interface depinning in a disordered medium[END_REF], and the relation

tR(k, t) = t 0 dt R(k, t )R(k, t -t ) . ( 31
)
Following this strategy, we checked that up to 4-loop order all diagrams appearing after timeintegration are equivalent to those encountered in expectations of O, defined in Eq. ( 8). Graphically, this is proven by first realizing that one can alternatively study the renormalization of friction by considering insertions of x,t ũ(x, t) u(x, t). Doing this, the time derivative on u can be passed through a closed string of response functions to either the earliest time in the diagram, and will then act on the remaining uncontracted field u, or it will end on a vertex with no further u field, and thus vanish. The same argument can be by moving the time derivative to the field ũ. These operations restrict the class of diagrams. Graphically, inserting x,t ũ(x, t) u(x, t) corresponds again to inserting a point into diagrams correcting expectations of ũ(x, t)u(x , t ). The final step of the proof is to realize that this is equivalent to insertions of the crossover operator Φ 1 (y)Φ * 2 (y) in the theory [START_REF] Lyklema | The Laplacian random walk[END_REF].

Finally note that the absence of a renormalization of -∇ 2 + m 2 in Eqs. ( 19) and ( 21) implied by the statistical tilt symmetry is equivalent to the absence of a renormalization of the theory (18).

A non-perturbative proof for the equivalance of φ 4 -theory at N = -1 and CDWs

The method introduced in [START_REF] Parisi | Random magnetic fields, supersymmetry, and negative dimensions[END_REF][START_REF] Parisi | Supersymmetric field theories and stochastic differential equations[END_REF] allows one to write the disorder average of any observable

O[u i ] as O[u i ] = r a=1 D[ũ a ]D[u a ]D[ ψa ]D[ψ a ] O[u i ]× × exp - x ũa (x) δH[u a ] δu a (x) + ψa (x) δ 2 H[u a ] δu a (x)δu a (y) ψ a (y) . (32)
The integral over ũa ensures that u a is at a minimum. ψa and ψ a are fermionic degrees of freedom (Grassmann variables), which compensate for the functional determinant appearing in the integration over u, yielding a partition function Z = 1. Averaging over disorder gives an effective action [START_REF] Wiese | Supersymmetry breaking in disordered systems and relation to functional renormalization and replica-symmetry breaking[END_REF] S[ũ a , u a , ψa , ψ a ] =

a x ũa (x)(-∇ 2 +m 2 )u a (x) + ψa (x)(-∇ 2 +m 2 )ψ a (x) - a,b x 1 2 ũa (x)∆ u a (x) -u b (x) ũb (x) -ũa (x)∆ u a (x) -u b (x) ψb (x)ψ b (x) - 1 2 ψa (x)ψ a (x)∆ u a (x) -u b (x) ψb (x)ψ b (x) . (33)
The function ∆(u) is the same as in Eq. [START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF]. Note that we allow for an arbitrary number of replicas r. In the work [START_REF] Parisi | Random magnetic fields, supersymmetry, and negative dimensions[END_REF] the focus was on r = 1, which does not allow to extract the second cumulant of the disorder, i.e. its correlations. To do the latter, one needs at least r = 2 copies, to which we specify now. We introduce center-of-mass coordinates,

u 1 (x) = u(x) + 1 2 φ(x) , u 2 (x) = u(x) - 1 2 φ(x) , (34)
ũ1 (x) = 1 2 ũ(x) + φ(x) , ũ2 (x) = 1 2 ũ(x) -φ(x) . (35)
action [START_REF] Kleinert | Five-loop renormalization group functions of O(N )symmetric φ 4 -theory and ε-expansion of critical exponents up to ε 5[END_REF] can then be written as

S = x φ(x)(-∇ 2 + m 2 )φ(x) + ũ(x)(-∇ 2 + m 2 )u(x) + 2 a=1 ψa (x)(-∇ 2 + m 2 )ψ a (x) + φ(x) 2 ∆ φ(x) -∆(0) - 1 4 ũ(x) 2 ∆ φ(x) + ∆(0) + 1 2 ũ(x)∆ φ(x) ψ2 (x)ψ 2 (x) -ψ1 (x)ψ 1 (x) + φ(x)∆ φ(x) ψ2 (x)ψ 2 (x) + ψ1 (x)ψ 1 (x) + ψ2 (x)ψ 2 (x) ψ1 (x)ψ 1 (x)∆ φ(x) . (36)
As in the derivation of the action (21) replacing ∆(u) → g 2 u 2 , the action (36) takes the form

S = x φ(x)(-∇ 2 + m 2 )φ(x) + ũ(x)(-∇ 2 + m 2 )u(x) + 2 a=1 ψa (x)(-∇ 2 + m 2 )ψ a (x) + g 2 ũ(x)φ(x) ψ2 (x)ψ 2 (x) -ψ1 (x)ψ 1 (x) - g 8 ũ(x) 2 φ(x) 2 + g 2 φ(x)φ(x) + ψ1 (x)ψ 1 (x) + ψ2 (x)ψ 2 (x) 2 . (37)
Note that only ũ(x), but not the center-of-mass position u(x) appear in the interaction. While u(x) may have non-trivial expectations, it does not contribute to the renormalization of g, and the latter can be obtained by dropping the third line of Eq. [START_REF] Amit | Field Theory, the Renormalization Group, and Critical Phenomena[END_REF]. What remains is a φ 4 -type theory as in Eq. ( 17), with one complex boson, and two complex fermions. It can equivalently be viewed as complex φ 4 -theory at N = -1, or real φ 4 -theory at n = -2.

What is yet missing is information about the exponent z. One can use the operator O defined in Eqs. [START_REF] Guttmann | Critical exponent for the loop erased self-avoiding walk by monte carlo methods[END_REF] or [START_REF] Agrawal | Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions[END_REF], replacing Φ i by ψ i , and Φ * i by ψi . Another possibility is to introduce time, adding a time argument to all fields, and replacing -∇ 2 + m 2 by ∂ t -∇ 2 + m 2 . The interaction part, i.e. the last line of Eq. ( 37), then becomes bilocal in time, i.e. the time integral appears inside the square bracket. The tricky part is to ensure that time-integrated vacuum bubbles retain their static expectations. This can be done by specifying an initial condition, once again adding the action [START_REF] Amit | Field Theory, the Renormalization Group, and Critical Phenomena[END_REF] where all fields are evaluated at some initial time t 0 . This implies that

R(k; t , t) = φ(-k, t ) φ(k, t) = ψ i (-k, t ) ψi (k, t) = Θ(t -t)e -(k 2 +m 2 )(t -t) + δ t,t 0 δ t ,t 0 k 2 + m 2 . ( 38
)
The δ-functions are to be understood s.t.

t R(k 1 , t, t)...R(k n , t, t) = 1 (k 2 1 + m 2 )...(k 2 n + m 2 ) . ( 39
)
We explicitly checked to 3-loop order that terms proportional to ∂ t receive the same renormalization as at depinning. Furthermore we can analyze the renormalization of φ(x, t)∂ t φ(x, t) as an insertion. Contributing diagrams carry two external fields, one φ, and one φ. Passing the time derivative ∂ t φ(x, t) of insertion to this external field, what remains is the insertion of a single point in the line connecting the external fields φ and φ, but no contribution from insertions into loops. After integration over times, this is equivalent to the insertion of O defined above in Eqs. [START_REF] Guttmann | Critical exponent for the loop erased self-avoiding walk by monte carlo methods[END_REF] or (9).

Fractal dimension of LERWs at 5-loop order

We generated all diagrams entering into O(x, y, z) at 5-loop order, and into the renormalization of the coupling constant at 4-loop order, supplemented by the diagrams of [START_REF] Kompaniets | Minimally subtracted six-loop renormalization of O(n)-symmetric φ 4 theory and critical exponents[END_REF] at 5-loop order. At 3-loop order we obtain the fractal dimension z of LERWs using the massive diagrams of Ref. [START_REF] Wiese | Field theory of disordered elastic interfaces at 3-loop order: The β-function[END_REF]. To 4-and 5-loop order, we use diagrams in a massless minimal subtraction scheme obtained in [START_REF] Kleinert | Five-loop renormalization group functions of O(N )symmetric φ 4 -theory and ε-expansion of critical exponents up to ε 5[END_REF][START_REF] Kompaniets | Minimally subtracted six-loop renormalization of O(n)-symmetric φ 4 theory and critical exponents[END_REF]. The result reproduces at 4-loop order the one given for the crossover exponent in Ref. [START_REF] Kirkham | Calculation of crossover exponent from Heisenberg to Ising behaviour using the fourth-order ε expansion[END_REF], setting there n = -2. This yields for the fractal dimension z of LERWs in dimension d = 4 -ε, equivalent to the dynamical exponent of CDWs at depinning,

z = 2 - ε 3 - ε 2 9 + 2ζ(3) 9 - 1 18 ε 3 - 70ζ (
ε 5 + O(ε 6 ) = 2 - ε 3 - ε 2 9 + 0.211568ε 3 -0.611186ε 4 + 2.43354ε 5 + O(ε 6 ) . (40)
Using Borel resummation of [START_REF] Kiskis | The Hausdorff dimension of random walks and the correlation length critical exponent in Euclidean field theory[END_REF], where to improve the precision we have included the 6-loop term yields [START_REF] Kompaniets | Crossover exponent for O(n)-symmetric φ 4 -theory, and fractal dimension of loop-erased random walks at 6-loop order[END_REF] z(d = 2) = 1.244 ± 0.01, (41) z(d = 3) = 1.6243 ± 0.001. [START_REF] Fukuyama | Dynamics of the charge-density wave. I. Impurity pinning in a single chain[END_REF] This can be compared to the exact value z(d = 2) = 5/4 [START_REF] Schramm | Scaling limits of loop-erased random walks and uniform spanning trees[END_REF][START_REF] Lawler | Conformal invariance of planar loop-erased random walks and uniform spanning trees[END_REF] and the most precise numerical simulations to date by David Wilson [START_REF] Wilson | Dimension of the loop-erased random walk in three dimensions[END_REF],

z(d = 3) = 1.62400 ± 0.00005 . (43)

Summary and Perspectives

We presented evidence that both φ 4 theory with O(n)-symmetry at n = -2, as well as the field theory for CDWs at depinning describe loop-erased random walks. We sketched a proof of this equivalence, based on a diagrammatic expansion, and gave an algebraic proof for the latter. All claims were explicitly checked to 4-loop order. This equivalence gives a strong support for the Narayan-Middleton conjecture [START_REF] Narayan | Avalanches and the renormalization-group for pinned charge-density waves[END_REF] that CDWs pinned by disorder can be mapped onto the Abelian sandpile model, and thus for the conjecture of [START_REF] Fedorenko | Field theory conjecture for loop-erased random walks[END_REF]. Remarkably, while CDWs at depinning map onto Abelian sandpiles, disordered elastic interfaces at depinning map onto Manna [START_REF] Doussal | An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media[END_REF][START_REF] Wiese | Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles[END_REF]. Thus each main universality class at depinning corresponds to a specific sandpile model.

The result is surprising, since a simple φ 4 -type theory contains all necessary information to obtain the FRG fixed point of CDWs, a disordered system. It does not directly yield the renormalized 2-point function, or the physics of avalanches. As sketched on Fig. 2, our understanding is that the different field theories are not equivalent, but when restricted to the same physical sector make the same predictions. This opens a path to eventually tackle other systems which necessitate FRG via a simpler scalar field theory.

Finally, the mapping of φ 4 -theory at n = -2 onto LERWs was done at a microscopic coupling g = 1. Changing the latter to p < 1 can be interpreted as a random walk where loops are erased with probability p. Since the RG fixed point is reached for any 0 < p < 1, we conjecture that these partially loop-erased random walks are in the same universality class as LERWs. We propose taking p close to 1 to measure the correction-to-scaling exponent ω precisely. While its ε expansion is known to 6-loop order [START_REF] Kompaniets | Minimally subtracted six-loop renormalization of O(n)-symmetric φ 4 theory and critical exponents[END_REF], it is only slowly converging, and we estimate ω = 0.83 ± 0.01.

Acknowledgements

It is a pleasure to thank E. Brézin, J. Cardy, F. David, K. Gawedzki, P. Grassberger, J. Jacobsen, M.V. Kompaniets, A. Nahum, S. Rychkov, D. Wilson and J. Zinn-Justin for valuable discussions.

Then expectations of the observable O(x, z) can be rewritten as

O(x, y, z) = φ * -(x)φ + (y)φ * -(y)φ + (z) . (A.6)
Eq. (A.5) is a rotation. Relabeling φ -→ φ 1 and φ + → φ 2 , this proves the equivalence of the definitions ( 8) and ( 9).

Appendix B. Renormalization group functions and critical exponents

The β-function is defined as This result agrees with the literature on the O(n)-model [START_REF] Kleinert | Five-loop renormalization group functions of O(N )symmetric φ 4 -theory and ε-expansion of critical exponents up to ε 5[END_REF][START_REF] Kompaniets | Minimally subtracted six-loop renormalization of O(n)-symmetric φ 4 theory and critical exponents[END_REF] and for CDWs [START_REF] Husemann | Field theory of disordered elastic interfaces to 3-loop order: Results[END_REF] (where ω := β (g * )). To obtain the exponent z, we use that The result is given in Eq. ( 40) in the main text. Note that when using the diagrams of Ref. [START_REF] Kompaniets | Minimally subtracted six-loop renormalization of O(n)-symmetric φ 4 theory and critical exponents[END_REF], the mass m gets replaced by a momentum scale. Due to universality, the critical exponents are independent of the scheme.

β(g) := -m∂ m g(g 0 ) g 0 =g 0 (g) . ( B
C. The effective coupling at 5-loop order g = g 0 -3g 2 0 + 3g There are (2n -1)!! diagrams at n-loop order. At 5-loop order the vertices are sitting on a circle, and crossings inside this circle are not vertices.

Figure 1 :

 1 Figure 1: Trace of a LERW in blue, with the erased loops in red, on a 2-dimensional Honeycomb lattice.

Figure 2 :

 2 Figure 2: The different field theories for LERWs.

  result of multiple cancelations, which we can analyze vertex by vertex. The tricky part is what happens at point y: We can either not use any interaction, use the interaction following the lines of the original walk, or reconnect the lines of the walk to form a loop. Here we used cancelation of the first two terms, retaining the last one, which resulted in the given drawing. (Note that the drawn red trace has the statistics of Brownian motion, as the two possible interaction terms mutually cancel.)The other possibility is to hit a red part of the trace, say at point y

2 xO

 2 O(x, y, z)z φ * 1 (x)φ 1 (z) ≡ m (x, y, z) ∼ m -z . (B.3)With the help of Eqs. (C.1) and (D.1), this yields z

  Note that at 5-loop order the are sitting on a circle, and crossings inside this circle are not vertices. There are (2n + 1)!! terms at n-loop order.Appendix D. The observable O at 5-loop orderAmputating the external points of the observable O, and integrating over all but one point, we find e γ φφ (g 0 ) = m 4

				3 0	+ 4
	-3g 4 0	+ 4		+ 2	+ 16			+ 4			+ 4	+ 4
	+3g 5 0 1	+ 4 x,z	+ 2 O(x, y, z)	+ 8			+ 4			+ 16	+ 4	+ 8
	+8 = 1 -g 0	+ 16 + g 2 0	+ 8 + 2	+ 24		+ 32		+ 32	+ 16
	+48 -g 3 0	+ 8 + 2	+ 32 + 2	+ 4 + 8		+ 8 + 2		+ 4	+ 28
	0 8 -3g 6 +g 4 0	+ 32 + 2	+ 16 + 4	+ 112 + 8	+ 2	+ 96	+ 4	+ 4	+ 8	+ 4	+ 16
	+16	+ 8	+ 2	+ 8		+ 2		+ 4	+ 4
	+16	+ 24	+ 32		+ 8			+ 32		+ 4	+ 8
	+4	+ 12
	+1	+ 2		+ 4	+ 8		+ 16			+ 32	+ 32
	+16 -g 5 0	+ 32 + 2	+ 36 + 4	+ 8	+ 56		+ 2	+ 96	+ 4	+ 2	+ 4	+ 4	+ 16
	+8 +4	+ 16 + 2	+ 32 + 8	+ 4 + 4		+ 8 + 16		+ 32 + 16	+ 32 + 16
	+32 +8	+ 16 + 8	+ 16 + 2	+ 16 + 16		+ 96 + 4		+ 32 + 16	+ 16 + 4
	+32 +8	+ 48 + 16	+ 64 + 16		+ 32 + 8			+ 32 + 4		+ 32 + 8	+ 32 + 8
	+16 +4	+ 4 + 12	+ 4 + 48	+ 4 + 12		+ 4	+ 32	+ 8	+ 32	+ 32	+ 32
	+16 +16	+ 4 + 32	+ 8 + 16	+ 64 + 32	+ 32 + 16	+ 48 + 8	+ 48 + 8
	+96 +16	+ 224 + 48	+ 32 + 24		+ 48 + 24			+ 16 + 48		+ 32 + 16	+ 272 + 4
	+48 +4	+ 8 + 16	+ 48 + 16	+ 32 + 16			+ 24 + 16	+ 112 + 16	+ 96 + 16
	+64 +8	+ 12 + 2	+ 8 + 2	+ 32 + 4		+ 64 + 16		+ 32 + 4	+ 32 + 8
	+16	+ 8	+ 64	+ 32			+ 8		+ 16	+ 48
	+4	+ 28	+ 56		+ ...							(D.1)
	+32	+ 96	+ 64		+ 8			+ 28		+ 56	+ 4
	+...												(C.1)

Appendix A. Rewriting the operator O

We had defined in Eq. ( 8)

Due to the symmetry of the action under the mapping φ 2 → -φ 2 , (ibid. for φ * 2 ) with all other components fixed, it follows that

Symmetry upon exchange of components 1 and 2 further yields

3) Using Eqs. (A.2) and (A.3) we deduce that

Define (ibid. for the complex conjugate fields) .5)