Cécile Monthus

Real-Space Renormalization for disordered systems at the level of Large Deviations

Real-space renormalization for disordered systems at the level of large deviations

Cecile Monthus

INTRODUCTION

The theory of large deviation has a long history in mathematics (see the books [START_REF] Ellis | Entropy, Large Deviations, and Statistical Mechanics[END_REF][START_REF] Deuschel | Large Deviations[END_REF][START_REF] Dembo | Large Deviations Techniques and Applications[END_REF][START_REF] Hollander | Large Deviations[END_REF][START_REF] Azencott | Large Deviations at Saint-Flour[END_REF][START_REF] Rassoul-Agha | A Course on Large Deviations with an Introduction to Gibbs Measures[END_REF] and references therein), in particular in the area of disordered systems (see the the books [START_REF] Talagrand | Spin-glasses : a challenge for mathematicians[END_REF][START_REF] Bovier | Statistical mechanics of disordered systems : a mathematical perspective[END_REF][START_REF] Comets | Directed Polymers in Random Environments[END_REF], the review [10] and references therein). In physics, the explicit use of the large deviations framework is more recent but is nowadays recognized as the unifying language for equilibrium, non-equilibrium and dynamical systems (see the reviews [11][12][13] and references therein). In particular, this point of view has turned out to be essential to formulate the statistical physics approach of nonequilibrium dynamics (see the reviews [14][15][16][START_REF] Touchette | chapter "Large deviation approach to nonequilibrium systems" of the book "Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond[END_REF][START_REF] Bertini | [END_REF][19][20] and the PhD Theses [21][START_REF] Lecomte | Thermodynamique des histoires et fluctuations hors d'équilibre[END_REF][START_REF] Chétrite | Grandes déviations et relations de fluctuation dans certains modèles de systèmes hors d'équilibre[END_REF][START_REF] Wynants | Structures of Nonequilibrium Fluctuations[END_REF] and the HDR Thesis [START_REF] Chétrite | Pérégrinations sur les phénomènes aléatoires dans la nature[END_REF]).

It is thus natural to revisit also classical and quantum disordered systems from the perspective of large deviations [START_REF] Monthus | Recent Advances in the Theory of Disordered Systems[END_REF]. In particular, in the field of real-space renormalization procedures for classical statistical physics models, the focus of previous studies has been mostly the region of typical fluctuations around typical values, but it is interesting to study now how their large deviations properties emerge from the renormalization flows. In the present paper, we have chosen to focus on the renormalization rule for the intensive energy of the ground state of the Directed Polymer on a hierarchical lattice depending on two integer parameters A and B (see section II for more details) : the new random variable x n+1 at generation (n + 1) is obtained from (AB) independent random variables x (a,b) n of generation n with a = 1, .., A and b = 1, .., B by the following maximum and sum operations

x n+1 = max 1≤b≤B 1 A A a=1
x (a,b) n [START_REF] Ellis | Entropy, Large Deviations, and Statistical Mechanics[END_REF] Our goal will be to study the renormalization flows for the tails x → ±∞ of the corresponding probability distribution P n (x) at generation n as a function of the exponents α ± characterizing the exponential decays of the initial condition at generation n = 0

P 0 (x) ∝ x→±∞ e -λ ± 0 |x| α ± (2)
Besides its physical interpretation for the Directed Polymer model, the RG rule of Eq. 1 is also interesting on its own from the general point of view of probabilities, because it mixes the basic operations 'sum over A independent variables' and 'maximum over B independent variables'. Of course, the two following degenerate cases are very well-known:

(i) in the special case B = 1, the variable

x n = 1 A A a1=1 x (a1) n-1 = 1 A 2 A a1=1 A a2=1 x (a1;a2) n-2 = ... = 1 A n A a1=1 ... A an=1
x (a1;a2;..;an) 0

(3) reduces to the empirical average of A n independent variables x (a1;a2;..;an) 0 of generation n = 0, which is the most studied problem in the whole history of probability. The typical fluctuations are classified in terms of the Gaussian distribution of the Central Limit Theorem (see [START_REF] Jonalasinio | [END_REF][28][29] for the renormalization point of view) and in terms of the Lévy stable laws (when the variance does not exist). While the standard theory for the large deviations of the empirical average focuses on the case of symmetric large deviations [11,13], the case of asymmetric large deviations (with different scalings for rare values bigger or smaller than the typical value) have also attracted a lot of attention recently [30][31][32][33][34][35]. As recalled in Appendix A, the tails properties of the empirical average of Eq. 3 strongly depend on the tail exponents α ± of the initial condition of Eq. 2 with completely different regimes associated to compressed exponentials α ± > 1, stretched exponentials 0 < α ± < 1 and simple exponentials α ± = 1. For the more general problem of Eq. 1, one thus expects that the tail exponents α ± of the initial condition of Eq. 2 will continue to play an essential role.

(ii) in the special case A = 1, the variable of generation n = 0, which is the basic problem in the field of Extreme Value Statistics [START_REF] Gumbel | Statistics of extreme[END_REF][START_REF] Galambos | The asymptotic theory of extreme order statistics[END_REF]. The typical fluctuations are classified in terms of the three universality classes Gumbel-Fréchet-Weibull [START_REF] Gumbel | Statistics of extreme[END_REF][START_REF] Galambos | The asymptotic theory of extreme order statistics[END_REF], with many applications in various physics domains (see the reviews [START_REF] Bouchaud | [END_REF][39][40] and references therein) and have been much studied from the renormalization perspective [41][42][43][44][45]. The large deviations properties of the empirical maximum have been found to be asymmetric [35,46,47], as a consequence of the following obvious asymmetry : an 'anomalously good' maximum requires only one anomalously good variable, while an 'anomalously bad' maximum requires that all variables are anomalously bad. This simple argument allows to understand why the large deviations will be also completely different for the two tails x → ±∞ in the more general problem of Eq. 1.

x n = max 1≤b1≤B x (b1) n-1 = max 1≤b1≤B;1≤b2≤B
Since the symmetric and asymmetric large deviations properties of these two special cases (i) and (ii) have been revisited in great detail recently in the companion paper [35], we will focus here on the non-degenerate cases (A > 1, B > 1) where there is really an interplay between the maximum and the sum operations. The paper is organized as follows. In section II, we recall the origin of the RG rule of Eq. 1 for the Directed Polymer model on the hierarchical lattice of parameters (A, B), and we introduce the useful notations to analyze the renormalizations of the corresponding probability distributions. The various renormalization flows that can emerge for the two tails x → ±∞ as a function of the exponents α ± of the tails of the initial condition (Eq 2) are then discussed in the following sections with their consequences for the large deviations properties. Section III describes the generic large deviation form with respect to the length L n = A n that emerges for the positive tail x → +∞ when the initial condition corresponds to some compressed exponential decay α + > 1. Similarly, section IV describes the generic large deviation form with respect to the volume L d n = A dn that emerges for the negative tail x → -∞ when the initial condition corresponds to some compressed exponential decay α -> 1 . The anomalous large deviations properties that emerge when the initial condition decays only as a stretched exponential 0 < α ± < 1 are then discussed for the tails x → +∞ and x → -∞ in sections V and VI respectively. Finally, the intermediate simple exponential decays α + = 1 and α -= 1 are considered in sections VII and VIII respectively. Our conclusions are summarized in Section IX. The Appendix A contains a reminder on the tails properties of the empirical average of independent variables.

II. REAL SPACE RENORMALIZATION AT THE LEVEL OF LARGE DEVIATIONS

A. Hierarchical diamond lattice with two parameters (A, B)

Among real-space renormalization procedures for classical statistical physics models (see the reviews [START_REF] Th | Renormalization theories for Ising spin systems[END_REF][START_REF] Burkhardt | Real-space renormalizations[END_REF][50] and references therein), Migdal-Kadanoff block renormalizations [51,52] play a special role because they can be considered in two ways, either as approximate renormalization procedures on hypercubic lattices, or as exact renormalization procedures on certain hierarchical lattices [53][54][55]. One of the most studied hierarchical lattice is the diamond lattice which is constructed recursively from the generation n = 0 that contains a single bond of unit length L n=0 = 1 by the following rule : the generation n + 1 is made of B branches, where each of these B branches contains A bonds of generation n in series. At generation n, the length L n between the two extreme sites is thus

L n = AL n-1 = A 2 L n-2 = .. = A n L 0 = A n (5)
while the volume V n (defined as the total number of bonds at generation n) grows as

V n = (AB)V n-1 = (AB) 2 V n-2 ... = (AB) n V 0 = (AB) n (6)
The effective fractal dimension d that can be defined from the volume-length scaling

V n = L d n d = ln(V n ) ln(L n ) = ln(AB) ln A = 1 + ln B ln A (7)
allows to analyze the role of the dimensionality. The special cases mentioned in the Introduction correspond to two extreme cases for the effective dimension : the case (i) where B = 1 corresponds to the dimension d = 1, and the On these diamond lattices, many disordered models have been studied, including the diluted Ising model [56], the ferromagnetic random Potts model [57][58][START_REF] Andelman | [END_REF][60], spin-glasses [61][62][63][64][65][66][67][68][69][70][71] and the directed polymer model in a random medium [72][73][74][75][76][77][78][79][80][81][82][83]. In this paper, we will focus only on the ground-state energy of this directed polymer model.

n = 0 n = 1 n = 2

B.

RG rules for the intensive ground state energy xn of the Directed Polymer Model

At generation n, the number N n of directed paths of length L n between the two extreme sites satisfies the recurrence

N n+1 = B(N n ) A (8)
Taking into account the initial condition N n=0 = 1 at generation n = 0, one obtains the solution

ln N n = ln B + A ln N n-1 = ln B + A (ln B + A ln N n-2 ) = ... = (ln B) n-1 k=0 A k = (ln B) A n -1 A -1 = (ln B) L n -1 A -1 (9)
that corresponds to the following configurational entropy per unit length

s ≡ lim n→+∞ ln N n L n = ln B A -1 (10)
which is finite for the non-degenerate cases (A > 1, B > 1).

In the Model of the Directed Polymer in a random medium, a random energy E 0 is drawn independently for each bond of generation n = 0. At generation n, each directed path of length L n between the two extreme sites will collect L n random energies of generation n = 0, and the ground-state will correspond to the Directed path of minimum energy. The hierarchical structure of the lattice yields that the extensive ground state energy follows the closed renormalization rule [72]

E n+1 = min 1≤b≤B A a=1 E (a,b) n (11)
where E (a,b) n are (AB) independent energies of generation n. In order to analyze the large deviations properties, it is more convenient to focus on the intensive variables that represent the ground-state energies per unit length (with a minus sign)

x (a,b) n ≡ - E (a,b) n L n = - E (a,b) n A n (12)
The RG rule of Eq. 11 then translates into the RG rule

x n+1 ≡ - E n+1 A n+1 = max 1≤b≤B 1 A A a=1 - E (a,b) n A n = max 1≤b≤B 1 A A a=1 x (a,b) n ( 13
)
already mentioned in Eq. 1 of the Introduction.

C. RG rules for the probability distribution Pn(x)

The RG rule of Eq. 1 concerning random variables can be translated as follows for their probability distributions. If P n (x) denotes the probability distribution of the independent intensive variables x (a,b) n at generation n, the probability distribution P n+1 (x) at the next generation (n + 1) is then obtained via the two following steps [72] :

(1) the probability distribution A n (x) of the B independent empirical averages

x (b) n ≡ 1 A A a=1 x (a,b) n ( 14
)
is given by the convolution

A n (x) = +∞ -∞ dx 1 ... +∞ -∞ dx A P n (x 1 )...P n (x A )δ x - 1 A A a=1 x a (15)
The tails properties of this convolution A n (x) depend on the tails properties of P n (x) : as recalled in Appendix A, different regimes appear for compressed exponentials α ± > 1, stretched exponentials 0 < α ± < 1 and simple exponentials α ± = 1.

(2) the probability distribution P n+1 (x) corresponds to the distribution of the maximum of B independent variables x (b) n of Eq. 14

P n+1 (x) = BA n (x) x -∞ dx A n (x ) B-1 = BA n (x) 1 - +∞ x dx A n (x ) B-1 (16)
The two tails of P n+1 (x) for x → ±∞ are thus related to the tails of A n (x) as follows

P n+1 (x) x→+∞ BA n (x) P n+1 (x) x→-∞ BA n (x) x -∞ dx A n (x ) B-1 (17)
where one sees why the two tails x → ±∞ will be governed by completely different mechanisms.

The goal of the present paper is to analyze the renormalization rules for the functions f ± n (x) that characterize the two tails x → ±∞ [START_REF] Bertini | [END_REF] in order to extract the large deviation properties for large n.

P n (x) x→±∞ e -f ± n (x)

D.

Link with the large deviations of the intensive ground-state energy

The general expectation for the Directed Polymer model in a random medium of dimension d is that the region of values bigger than the typical value (x > x typ ) should display a large deviation form with respect to the length L n

P n (x) ∝ Ln→+∞ e -LnI + (x) for x > x typ (19)
because an 'anomalously good' ground state energy requires only L n anomalously good bond energies along the polymer. The region of values smaller than the typical value (x < x typ ) should display instead a large deviation form with respect to the volume

V n = L d n P n (x) ∝ Ln→+∞ e -L d n I -(x) for x < x typ (20)
because an 'anomalously bad' ground state energy requires L d n bad bond energies in the sample. This asymmetric large deviation form has been computed exactly for the Directed Polymer in dimension d = 1 + 1 [84,85] that belongs to the Kardar-Parisi-Zhang universality class (see the various models and interpretations in the review [86]). Here our goal will be thus to derive this asymmetric large deviation form for the hierarchical lattices of arbitrary parameters (A, B), and to compute explicitly the tails x → ±∞ of the corresponding rate functions I ± (x).

E.

Special families of explicit solutions for the renormalization flows of the tails x → ±∞ While we will write the functional renormalization rules for the tail functions f ± n (x) of Eq. 18, the analysis of their general solutions in the infinite-dimensional space of all admissible tail functions clearly goes beyond the goals of the present paper. We will instead focus on special families of explicit solutions that appear for initial conditions of the following form

P 0 (x) x→±∞ K ± 0 |x| ν ± 0 -1 e -λ ± 0 |x| α ± (21)
As already stressed many times, the exponents α ± > 0 characterizing the leading exponential decays will play an essential role. These parameters α ± will turned out to be conserved by the renormalization flow, while the other parameters may be renormalized, i.e. the tails at generation n will be of the form

P n (x) x→±∞ K ± n |x| ν ± n -1 e -λ ± n |x| α ± (22)
and will thus correspond to the following special form of the tail functions f ± n (x)

f ± n (x) x→±∞ λ ± n |x| α ± + (1 -ν ± n ) ln |x| -ln K ± n (23)
For each case labelled by the possible tail exponents α ± , we will thus compute the solutions of the closed RG flows for the three other parameters (λ ± n , ν ± n , K ± n ), in order to extract the large deviations properties and the corresponding rate functions.

The following sections are devoted to the various renormalization flows that can emerge for the two tails x → ±∞ as a function of the exponents α ± of the initial condition of Eq. 21 : we will first consider the compressed exponential cases α ± > 1 that indeed lead to the expected large deviations of Eqs 19 and 20; we will then turn to the stretched exponential cases 0 < α ± < 1 that lead to anomalous large deviations with respect to Eqs 19 and 20; finally, we will discuss the intermediate cases α ± = 1 that require a special analysis.

III.

RG FLOW OF THE TAIL x → +∞ FOR THE COMPRESSED EXPONENTIAL CASES α + > 1

A.

Functional renormalization for the tail function

f + n (x)
As recalled in Appendix A, when the tail function f + n (x) of Eq. 18 satisfies the conditions of Eq. A7, the tail of the distribution of the convolution A n (x) of Eq. 15 has been studied in detail in Ref [87] and the output is the 'democratic' formula of Eq. A6

A n (x) x→+∞ e -Af + n (x) √ A 2π (f + n ) (x) A-1 2 (24)
The tail x → +∞ of Eq. 17 is simply given by

P n+1 (x) x→+∞ BA n (x) x→+∞ Be -Af + n (x) √ A 2π (f + n ) (x) A-1 2 (25)
The identification with P n+1 (x → +∞) e -f + n+1 (x) of Eq. 18 yields the functional RG rule for the tail function

f + n (x) f + n+1 (x) = Af + n (x) + (A -1) ln   (f + n ) (x) 2π   -ln B √ A (26)
B.

Explicit solution of the RG flow for the special form of Eq. 23 when α + > 1

The special form of Eq. 23

f + n (x) x→+∞ λ + n x α + + (1 -ν + n ) ln x -ln K + n (f + n ) (x) x→+∞ λ + n α + (α + -1)x α + -2 + (ν + n -1) x 2 x→+∞ λ + n α + (α + -1)x α + -2 (27)
satisfies the conditions of Eq. A7 in the region α + > 1, and remains closed under the functional RG flow of Eq. 26 with the following RG rules for the parameters

λ + n+1 = Aλ + n ν + n+1 = A ν + n - α + 2 + α + 2 ln(K + n+1 ) = A ln(K + n ) + (A -1) ln 2π λ + n α + (α + -1) + ln(B √ A) (28)
In terms of the initial condition at generation n = 0, the solution reads

λ + n = A n λ + 0 ν + n = A n ν + 0 - α + 2 + α + 2 ln(K + n ) = A n ln B A -1 + ln K + 0 2π λ + 0 α + (α + -1) + n 2 ln A - ln B A -1 -ln 2π λ + 0 α + (α + -1) (29)
Putting everything together, it is convenient to gather all the terms involving the length L n = A n to obtain the final result for the tail function of Eq. 27

f + n (x) x→+∞ A n λ + 0 x α + - ln B A -1 -ln K + 0 2π λ + 0 α + (α + -1) + α + 2 -ν + 0 ln x - n 2 ln A + ln B A -1 + ln 2π λ + 0 α + (α + -1) + 1 - α + 2 ln x (30)
C. Conclusion for the large deviations in the tail x → +∞ when α + > 1

The RG solution of Eq. 30 thus corresponds to the expected large deviation form with respect to the length L n = A n of Eq. 19. In addition, the corresponding rate function I + (x) of Eq. 19 displays the tail behavior

I + (x) x→+∞ λ + 0 x α + - ln B A -1 -ln K + 0 2π λ + 0 α + (α + -1) + α + 2 -ν + 0 ln x (31)

D. Example with the Gaussian initial condition

The special solution of Eq. 30 will not contain the logarithmic terms in (ln x) for the initial conditions satisfying

α + = 2 ν + 0 = 1 (32)
It is thus interesting to consider the normalized Gaussian initial distribution at generation n = 0

P 0 (x) = K + 0 e -λ + 0 x 2 K + 0 = λ + 0 π (33)
The special solution of Eq. 30 then simplifies into

f + n (x) x→+∞ A n λ + 0 x 2 - ln B A -1 - n 2 ln A + ln B A -1 -ln   λ + 0 π   (34)
and corresponds for the probability distribution to the tail

P n (x) x→+∞ e -f + n (x) x→+∞ λ + 0 A n π e -A n [λ + 0 x 2 -ln B A-1 ]-ln B A-1 = λ + 0 A n π e -A n [x 2 -(x + n ) 2 ] ( 35
)
with the parameter

(x + n ) 2 = ln B A -1 1 - 1 A n (36)
IV. RG FLOW OF THE TAIL x → -∞ FOR THE COMPRESSED EXPONENTIAL CASES α -> 1

A.

Functional renormalization for the tail function f - n (x)

As recalled in Appendix A, when the tail function f - n (x) of Eq. 18 satisfies the conditions of Eq. A7, the tail of the distribution of the convolution A n (x) of Eq. 15 is given by the 'democratic' formula of Eq. A6

A n (x) x→-∞ e -Af - n (x) √ A 2π (f - n ) (x) A-1 2 (37)
As a consequence, the corresponding cumulative distribution displays the tail

x -∞ dx A n (x ) x→-∞ x -∞ dx e -Af - n (x ) √ A 2π (f - n ) (x ) A-1 2 x→-∞ e -Af - n (x) √ A A[-(f - n ) (x)] 2π (f - n ) (x) A-1 2 (38)
So the tail x → -∞ of Eq. 17 is given by

P n+1 (x) x→-∞ BA n (x) x -∞ dx A n (x ) B-1 x→-∞ BA[-(f - n ) (x)] e -Af - n (x) √ A A[-(f - n ) (x)] 2π (f - n ) (x) A-1 2 B ( 39
)
The identification of the tail P n+1 (x → -∞) e -f - n+1 (x) of Eq. 18 yields the functional RG rule for the tail function f - n (x)

f - n+1 (x) = ABf - n (x) + (A -1)B ln   (f - n ) (x) 2π   + (B -1) ln[-(f - n ) (x)] + B 2 -1 ln A -ln B (40)
B.

Explicit solution of the RG flow for the special form of Eq. 23

The special form of Eq. 23

f - n (x) x→-∞ λ - n (-x) α - + (1 -ν - n ) ln(-x) -ln K - n (f - n ) (x) x→-∞ -λ - n α -(-x) α --1 + 1 -ν - n x (f - n ) (x) x→-∞ λ - n α -(α --1)(-x) α --2 + (ν - n -1) x 2 x→-∞ λ - n α -(α --1)(-x) α --2 (41)
satisfies the conditions of Eq. A7 in the region α -> 1, and remains closed under the functional RG flow of Eq. 40 with the following RG rules for the parameters

λ - n+1 = ABλ - n (42) ν - n+1 = ABν - n + α - 2 (2 -B -AB) ln(K - n+1 ) = AB ln(K - n ) -(AB -1) ln(λ - n ) + (A -1)B ln 2π α -(α --1) -(B -1) ln(α -) -(B -2) ln A 2 + ln(B)
In terms of the initial condition at generation n = 0, the solution reads

λ - n = (AB) n λ - 0 ν - n = (AB) n ν - 0 - α - 2 ω + α - 2 ω ln(K - n ) = (AB) n ln(K - 0 ) + v + n 2 ω ln(AB) -v (43)
where we have introduced the notation

ω ≡ 1 + B -1 AB -1 v ≡ -ln(λ - 0 ) + (A -1)B AB -1 ln √ B AB -1 + ln 2π α -(α --1) - B -1 AB -1 AB AB -1 ln √ A + ln(α -) (44)
Putting everything together, the tail function of Eq. 41 reads

f - n (x) x→-∞ (AB) n λ - 0 |x| α - -ln(K - 0 ) -v + α - 2 ω -ν - 0 ln |x| - n 2 ω ln(AB) + v + 1 - α - 2 ω ln |x| (45)
C.

Conclusion for the large deviations in the tail

x → -∞ when α -> 1
The RG solution of Eq. 45 thus corresponds to the expected large deviation form with respect to the volume

V n = L d n = (AB) n of Eq. 20.
The corresponding rate function I -(x) of Eq. 20 displays the tail behavior

I -(x) x→-∞ λ - 0 |x| α - + α - 2 ω -ν - 0 ln |x| -ln(K - 0 ) -v = λ - 0 |x| α - + α - 2 1 + B -1 AB -1 -ν - 0 ln |x| (46) + ln(λ - 0 ) -ln(K - 0 ) - (A -1)B AB -1 ln √ B AB -1 + ln 2π α -(α --1) + B -1 AB -1 AB AB -1 ln √ A + ln(α -) V. RG FLOW OF THE TAIL x → +∞ FOR THE STRETCHED EXPONENTIAL CASES 0 < α + < 1 A.
Functional renormalization for the tail function f

+ n (x)
As recalled in Appendix A, the tail of the distribution of the convolution A n (x) of Eq. 15 is then given by the 'monocratic formula' of Eq. A14 (47) so the tail x → +∞ of Eq. 17 becomes

A n (x) x→+∞ A 2 e -f + n (Ax)
P n+1 (x) x→+∞ BA n (x) x→+∞ BA 2 e -f + n (Ax) (48)
The identification with P n+1 (x → +∞) e -f + n+1 (x) of Eq. 18 yields the functional RG rule for the tail function

f + n (x) f + n+1 (x) = f + n (Ax) -ln BA 2 (49)
instead of the functional RG rule of Eq. 26.

B.

Explicit solution of the RG flow for the special form of Eq. 23 when 0 < α + < 1

The special form of Eq. 23

f + n (x) x→+∞ λ + n x α + + (1 -ν + n ) ln x -ln K + n ( 50
)
remains closed for the functional RG rule of Eq. 49 with the following RG rules for the parameters

λ + n+1 = A α + λ + n ν + n+1 = ν + n ln(K + n+1 ) = ln(K + n ) + (ν + n + 1) ln A + ln B ( 51
)
In terms of the initial condition at generation n = 0, the solution reads

λ + n = A nα + λ + 0 ν + n = ν + 0 ln(K + n ) = ln(K + 0 ) + n (ν + 0 + 1) ln A + ln B (52)
Putting everything together, the tail function of Eq. 50 reads

f + n (x) x→+∞ A nα + λ + 0 x α + + (1 -ν + 0 ) ln x -ln(K + 0 ) -n (ν + 0 + 1) ln A + ln B (53)
C. Conclusion for the anomalous large deviations in the tail x → +∞ when 0 < α + < 1

The solution of Eq. 53 thus corresponds to the following anomalous large deviation form with respect to the length

L n = A n P n (x) ∝ Ln→+∞ e -L α + n J + (x) for x ≥ x typ (54)
instead of the standard form of Eq. 19. The corresponding rate function J + (x) displays the tail behavior

J + (x) x→+∞ λ + 0 x α + (55)
C. Conclusion for the anomalous large deviations in the tail x → -∞ when 0 < α -< 1

The solution of Eq. 63 thus corresponds to the following anomalous large deviation form in

BA α -n = L d-1+α - n P n (x) ∝ Ln→+∞ e -L d-1+α - n J -(x) for x ≤ x typ (64)
instead of the standard form of Eq. 20. The corresponding rate function J -(x) displays the tail behavior

J -(x) x→-∞ λ - 0 |x| α - (65)
VII. RG FLOW OF THE TAIL x → +∞ FOR THE INTERMEDIATE CASES α + = 1

A.

Explicit solution of the RG flow for the special form of Eq. 23 for α + = 1 and ν + 0 > 0

In this section, we wish to analyze the closed RG flow for the special form of Eq. 22 when α + = 1

P n (x) x→+∞ K + n x ν + n -1 e -λ + n x (66)
As explained in the Appendix A, the tail of the convolution of Eq. 15 is then given by Eq. A18 if ν + n > 0

A n (x) x→+∞ A Aν + n [K + n Γ(ν + n )] A Γ(Aν + n ) x Aν + n -1 e -Aλ + n x (67)
Then Eq. 17 yields that the tail at generation (n + 1) reads

P n+1 (x) x→+∞ BA n (x) x→+∞ B A Aν + n [K + n Γ(ν + n )] A Γ(Aν + n ) x Aν + n -1 e -Aλ + n x (68)
The identification with the notations of Eq. 66 at generation (n+1) leads to the following RG rules for the parameters

λ + n+1 = Aλ + n ν + n+1 = Aν + n ln(K + n+1 ) = A ln(K + n ) + ln Γ(ν + n ) + ν + n ln A -ln Γ(Aν + n ) + ln(B) (69)
Taking into account the initial condition at generation n = 0, the solution reads

λ + n = A n λ + 0 ν + n = A n ν + 0 ln(K + n ) = A n ln B A -1 + nν + ln A + ln(K + 0 ) + ln(Γ(ν + )) -ln Γ(A n ν + ) - 1 A -1 ln B (70)
so this solution satisfies the validity condition ν + n > 0 for any n if the initial condition does ν + 0 > 0. Putting everything together, the tail function f n (x) of Eq. 23 reads

f + n (x) x→+∞ λ + n x + (1 -ν + n ) ln x -ln K + n ( 71
) x→+∞ A n λ + 0 x -ν + 0 ln x - ln B A -1 -nν + 0 ln A -ln(K + 0 ) -ln(Γ(ν + 0 )) + ln x + ln Γ(A n ν + 0 ) + 1 A -1 ln B B.
Conclusion for the large deviations in the tail x → +∞ for α + = 1 and ν + 0 > 0

To extract the large deviation form from the solution of Eq. 71, one needs to use the Stirling formula for the Gamma function of

z = A n ν + 0 1 Γ(A n ν + 0 ) n 1 √ 2π(A n ν + 0 ) A n ν + 0 -1 2 e -A n ν + 0 ( 72
)
Plugging its logarithm ln Γ(A n ν + 0 )

n 1 A n nν + 0 ln A + ν + 0 ln(ν + 0 ) -ν + 0 + ln( √ 2π) - 1 2 n ln A + ln(ν + 0 ) (73)
into Eq. 71 yields to the standard large deviation form with respect to the length L n = A n of Eq. 19 and the corresponding rate function I + (x) displays the tail behavior

I + (x) x→+∞ λ + 0 x - ln B A -1 -ln(K + 0 ) -ln(Γ(ν + 0 )) + ν + 0 ln(ν + 0 ) -ν + 0 -ν + 0 ln x (74)
instead of Eq. 31.

VIII. RG FLOW OF THE TAIL

x → -∞ FOR THE INTERMEDIATE CASES α -= 1
A.

Explicit solution of the RG flow for the special form of Eq. 23 for α -= 1 and ν - 0 ≥ B-1 AB-1

In this section, we wish to analyze the closed RG flow for the special form of Eq. 22 when α -= 1

P n (x) x→-∞ K - n |x| ν - n -1 e -λ - n |x| (75)
As explained in the Appendix A, the tail of the convolution of Eq. 15 is then given by the analog of Eq. A18 if ν - n > 0

A n (x) x→-∞ A Aν - n [K - n Γ(ν - n )] A Γ(Aν - n ) |x| Aν - n -1 e -Aλ - n |x| (76)
Then the corresponding cumulative distribution displays the tail

x -∞ dx A n (x ) x→-∞ A Aν - n [K - n Γ(ν - n )] A Aλ - n Γ(Aν - n ) |x| Aν - n -1 e -Aλ - n |x| (77)
As a consequence, the tail at generation (n + 1) of Eq. 17 reads

P n+1 (x) = BA n (x) x -∞ dx A n (x ) B-1 x→-∞ B [Aλ - n ] B-1 A Aν - n [K - n Γ(ν - n )] A Γ(Aν - n ) |x| Aν - n -1 e -Aλ - n |x| B ( 78
)
The identification with the notations of Eq. 75 at generation (n+1) leads to the following RG rules for the parameters

λ - n+1 = ABλ - n ν - n+1 = ABν - n -(B -1) ln(K - n+1 ) = AB ln(K - n ) + B A ln Γ(ν - n ) -ln Γ(Aν - n ) + ν - n AB ln A -(B -1) ln(λ - n ) + ln A + ln B ( 79
)
Taking into account the initial condition at generation n = 0, the solution reads

λ - n = (AB) n λ - 0 ν - n = (AB) n ν - 0 - B -1 AB -1 + B -1 AB -1 ln(K - n ) = (AB) n ln(K - 0 ) + n ν - 0 - B -1 AB -1 ln A + B(A -1) (AB -1) 2 ln B - B -1 AB -1 ln(λ - 0 ) +n B -1 (AB -1) ln(AB) - B(A -1) (AB -1) 2 ln B + B -1 AB -1 ln(λ - 0 ) + n-1 k=0 (AB) n-1-k B A ln Γ(ν - k ) -ln Γ(Aν - k ) (80)
so this solution satisfies the validity condition ν - n > 0 for any n if the initial condition satisfies ν - 0 ≥ B-1 AB-1 .

B.

Conclusion for the large deviations in the tail x → -∞ for α -= 1 and ν - 0 > B-1 AB-1

To extract the large deviation form from the solution of Eq. 80, one needs to use the Stirling formula for Γ(ν - k ) and Γ(Aν - k ) to obtain the asymptotic behavior of the difference

A ln Γ(ν - k ) -ln Γ(Aν - k ) k 1 -(AB) k A ν - 0 - B -1 AB -1 ln A -k A -1 2 ln(AB) + (A -1) ln( √ 2π) - A -1 2 ln ν - 0 - B -1 AB -1 + A -1 AB -1 - 1 2 ln A (81)
As a consequence, the leading terms of order (AB) n in the solution ln(K - n ) of Eq 80 is given by

ln(K - n ) n 1 (AB) n ln(K - 0 ) - B 2(AB -1) ln A - B -1 AB -1 ln(λ - 0 ) + B(A -1) AB -1 ln(AB) 2(AB -1) + ln 2π ν - 0 -B-1 AB-1 +... (82)
One thus obtains the standard large deviation form with respect to the volume L d n = (AB) n of Eq. 20 and the corresponding rate function I -(x) displays the tail behavior

I -(x) x→-∞ λ - 0 |x| -ν - 0 - B -1 AB -1 ln |x| -ln(K - 0 ) + B -1 AB -1 ln(λ - 0 ) + B 2(AB -1) ln A - B(A -1) AB -1 ln(AB) 2(AB -1) + ln 2π ν - 0 -B-1 AB-1 (83)
instead of Eq. 46.

IX. CONCLUSIONS

In this paper, we have revisited the renormalization rule for the ground-state energy of the Directed Polymer model on a hierarchical lattice of parameters (A, B) in order to analyze the renormalization flows for the tails of probability distributions as a function of the initial condition at generation n = 0. In each case, the explicit solution has allowed to extract the scalings involved in the large deviations properties and the tail behaviors of the corresponding rate functions. Our main conclusions can be summarized as follows :

(i) the generic large deviation form with respect to the length L n for the tail x → +∞ emerges only for α + ≥ 1, while the stretched exponential 0 < α + < 1 initial conditions lead to anomalous large deviations in L α + n . (ii) the generic large deviation form with respect to the volume L d n for the tail x → -∞ emerges only for α -≥ 1, while the stretched exponential 0 < α -< 1 initial conditions lead to anomalous large deviations in L d-1+α - n . This example shows that it is interesting to analyze the renormalization flows of disordered systems at the level of large deviations, in order to go beyond the region of typical fluctuations that have been much studied in the past. is distributed with the convolution

A(x) = +∞ -∞ dx 1 ... +∞ -∞ dx A P(x 1 )...P(x A )δ x - 1 A A a=1
x a (A3)

The tail behavior as x → +∞ of this convolution depends on the tail of Eq. A1. For concreteness, it will be convenient to consider the family

P(x) x→+∞ Kx ν-1 e -λx α (A4)
so that the corresponding tail function f (x) of Eq. A1 and its second derivative read

f (x) = λx α + (1 -ν) ln x -ln K f (x) = λα(α -1)x α-2 + ν -1 x 2 (A5) 1.
The 'democratic' formula for α > 1

The 'democratic' formula obtained in Ref [87] A

democratic (x) x→+∞ e -Af (x) √ A 2π f (x) A-1 2 (A6)
can be understood from two points of view.

a. 'Democratic' saddle-point analysis of Ref [87] The formula of Eq. A6 has been derived in Ref [87] from the saddle-point evaluation of the convolution of Eq. A3 around the symmetric solution x a = x for a = 1, 2.., A with the two validity conditions (see [87] for very detailed discussions and various formulations of the validity conditions)

f (x) > 0 x 2 f (x) x→+∞ +∞ (A7)
For the special family of Eq. A4, these conditions are satisfied only in the region α > 1 (A8) while they are not satisfied for 0 < α ≤ 1.

dxe kx k -f (x k )-(x-x k ) 2 2 f (x k ) = e kx k -f (x k ) 2π f (x k ) (A11)
The scaled cumulant generating function associated to the empirical average of Eq. A3 is simply given by the power A of Eq. A9 Eq A11 then yields that the tail for k → +∞ is given by

+∞ -∞ dxe kAx A n (x) k→+∞ e kx k -f (x k ) 2π f (x k ) A = e Akx k -Af (x k ) 2π f (x k ) A-1 2 2π f (x k ) (A13)
that corresponds indeed to the saddle-point evaluation of the tail of Eq. A6.

2.

The 'monocratic' formula for 0 < α < 1

The 'monocratic' formula corresponds to the cases where the tail x → +∞ of the convolution of Eq. A3 is dominated by the drawing of the anomalously large value y Ax for the maximum of the A variables (x 1 , .., x A ), while the other (A -1) values remain typical, so that one obtains the tail behavior

A monocratic (x) x→+∞ A dyP(y)δ x - y A = A 2 P(Ax) = A 2 e -f (Ax) (A14)
that indeed gives a bigger result than the 'democratic' formula of Eq. A6 for 0 < α < 1.

3.

The intermediate case α = 1 when ν > 0

For the intermediate case α = 1 of Eq. A4

P(x)

x→+∞ Kx ν-1 e -λx (A15) one can use neither the 'democratic' formula nor the 'monocratic' formula described above. For ν > 0, the cumulant generating function φ(k) of Eq. A9 exists only for k < λ and diverges as k → λ. This singularity as k → λ is then governed by the tail x → +∞ of Eq. 66 that one assumes to be valid in the region x > C (where C is some fixed large constant C > 0)

e φ(k) k→λ +∞ C dxKx ν-1 e -(λ-k)x = K (λ -k) ν +∞ C(λ-k) dtt ν-1 e -t k→λ K Γ(ν) (λ -k) ν (A16)
The scaled cumulant generating function of Eq. A12 associated to the empirical average of Eq. A3 then displays the singularity A Γ(Aν)

x Aν-1 e -Aλx (A18)

4.

Final remark on the similarities and differences with the large deviations of the empirical average

In this Appendix, we have considered as in Ref [87] the problem of the tail x → +∞ of the empirical average of a finite number A of independent variables, while the standard large deviations problem for the empirical average focuses instead on a large number A → +∞ of independent variables, while x remains finite. The two problems are

FIG. 1 :

 1 FIG.1: First generations n = 0, 1, 2 of the hierarchical lattice of parameters A = B = 2 : the lengths Ln = A n = 2 n between the two extreme points are given by L0 = 1, L1 = 2, L2 = 4, while the volumes Vn = (AB) n = 4 n (total numbers of bonds) are given by V0 = 1, V1 = 4, L2 = 16. An example of Directed Polymer of length Ln between the two extreme points is shown in red for each generation.

For α > 1 ,

 1 the tail k → +∞ of the cumulant generating function Another way to understand Eq. A6 involves the cumulant generating function φ(k) this cumulant generating function exists even for large k, and the tail for k → +∞ is determined by the tail for x → +∞ of Eq. A1 via the saddle-point evaluation of Eq. A9 around the large saddle-point value x k satisfyingf (x k ) = k (A10)that leads to the asymptotic result e φ(k) k→+∞ +∞ -∞

dxe

  Akx A(x) = e Aφ(k) k→λ [K Γ(ν)] A (λ -k) Aν (A17)that corresponds to the following tail as x → +∞A(x) x→+∞ A Aν [KΓ(ν)]

thus clearly different, but they nevertheless display some similarities as discussed in detail in Ref [87], and the two democratic/monocratic behaviors have also been much studied in the large deviation regime [30][31][32][33][34][35].

VI.

RG FLOW OF THE TAIL x → -∞ FOR THE STRETCHED EXPONENTIAL CASES 0 < α -< 1

A.

Functional renormalization for the tail function f - n (x)

As recalled in Appendix A, the tail of the distribution of the convolution A n (x) of Eq. 15 is then given by the 'monocratic formula' of Eq. A14

The corresponding cumulative distribution displays the tail

and leads to the following result for the tail x → -∞ of Eq. 17

The identification with P n+1 (x → -∞) e -f - n+1 (x) of Eq. 18 yields the functional RG rule for the tail function

instead of the functional RG rule of Eq. 40.

B.

Explicit solution of the RG flow for the special form of Eq. 23 for 0 < α -< 1

The special form of Eq. 23

remains closed for the functional RG rule of Eq. 59 with the following RG rules for the parameters

In terms of the initial condition at generation n = 0, the solution reads

Putting everything together, the tail function of Eq. 60 reads

Appendix A: Tail analysis for the empirical average of a finite number A of random variables

In this Appendix, we consider a finite number A of independent random variables x a distributed with some probability distribution P(x) whose tail for x → +∞ is characterized by the function f (x)

The empirical average

x a (A2)


Oleksandra Ganzenko

Philippe Sistat

Clément Trellu

Valérie Bonniol

Matthieu Rivallin
email: matthieu.rivallin@umontpellier.fr

Marc Cretin
email: marc.cretin@umontpellier.fr

Reactive electrochemical membrane for the elimination of carbamazepine in secondary effluent from wastewater treatment plant

Keywords: Reactive electrochemical membrane, sub-stoichiometric titanium oxide, anodic oxidation, secondary effluent, carbamazepine

This study evaluated the feasibility of the electrooxidation process using a sub-stoichiometric titanium oxides (TiOX) reactive electrochemical membrane (REM) as anode for the treatment of a secondary effluent from a wastewater treatment plant. The main objectives were to analyze (i) the degradation of a pharmaceutical pollutant, carbamazepine, added at a concentration of 100 µg L -1 , (ii) the mineralization of the residual total organic carbon (TOC = 7 mg L -1 ), (iii) the formation of toxic byproducts and (iv) the energy consumption of the REM process. Two main operating parameters were tested: the current density applied to the REM (74, 149 and 290 A m -2 ) and the flux of TOC passing through the membrane. The results revealed that the efficiency of the process was maintained above 70% of mineralization and >98% of degradation of carbamazepine until a limiting value of TOC flux was reached. The higher the current density, the higher the limiting value of the TOC flux above which the efficiency decreased. This trend was ascribed to the limitation from the amount of reactive species produced on the REM surface (current limitation instead of mass transport limitation at lower TOC flux). As the energy efficiency strongly increased when applying a higher TOC flux, it is therefore crucial to apply the process around this threshold value. This study also showed that it is possible to control the formation of toxic chlorine-containing by-products (ClO3 -, ClO4 -, AOCl) by decreasing the current density. This issue is particularly important for such effluents containing a significant amount of Cl -ions. By taking into consideration both advantages and drawbacks, optimal results were obtained with the lowest current density of 73 A m -2 and a TOC flux through the membrane of 4.41 g h -1 m -2 corresponding to 1.34 m 3 h -1 m -2 . These conditions allowed for >98% degradation of carbamazepine and 70% of mineralization with an energy consumption of 0.15 kWh g -1 of TOC removed (i.e. 0.74 kWh m -3 of treated wastewater).

Introduction

Membrane filtration processes are widely applied in modern wastewater treatment plants (WWTP) all over the world. Electrochemical methods for water treatment are emerging technologies showing a promising perspective and inciting great scientific interest. Membrane processes have two major drawbacks. First, it is only a separation process, where pollutants are concentrated in the retentate.

Then, nanofiltration membranes with low pore size used for the retention of micropollutants require the use of high transmembrane pressure. Besides, large-scale application of electrochemical processes for the removal of micropollutants is strongly affected by mass transport limitations when treating low concentration of pollutants, which leads to a low energy efficiency. However, the combination of these two processes aims at being a synergetic technology enhancing the advantages of both processes. The idea is based on coupling both processes for the removal of pollutants passing through a reactive electrochemical membrane used as anode for the anodic oxidation process.

Anodic oxidation is an electrochemical advanced oxidation process allowing the removal of organic pollutants through different mechanisms including (i) direct electron transfer and (ii) mediated oxidation by generation of reactive radical species [START_REF] Panizza | Direct and Mediated Anodic Oxidation of Organic Pollutants[END_REF][START_REF] Martínez-Huitle | Single and coupled electrochemical processes and reactors for the abatement of organic water pollutants: a critical review[END_REF][START_REF] Nidheesh | An overview on the removal of synthetic dyes from water by electrochemical advanced oxidation processes[END_REF]. Particularly, the generation of strong oxidant species is possible when using anode materials with high overvoltage for oxygen evolution reaction [START_REF] Marselli | Electrogeneration of hydroxyl radicals on boron-doped diamond electrodes[END_REF][START_REF] Anglada | Boron-doped diamond anodic treatment of landfill leachate: evaluation of operating variables and formation of oxidation by-products[END_REF][START_REF] Moraleda | A comparison between flow-through cathode and mixed tank cells for the electro-Fenton process with conductive diamond anode[END_REF][START_REF] Garcia-Segura | Removal of organic contaminants from secondary effluent by anodic oxidation with a boron-doped diamond anode as tertiary treatment[END_REF][START_REF] Mousset | Advanced Electro-Oxidation with Boron-Doped Diamond for Acetaminophen Removal from Real Wastewater in a Microfluidic Reactor: Kinetics and Mass-Transfer Studies[END_REF]. In this case, hydroxyl radicals can be produced from the one-electron oxidation of water at the electrode surface M (Eq. 1). Hydroxyl radicals present a major advantage of being a non-selective and highly oxidizing agent.

M + H O ⟶ M OH

• + H + e (1)
In order to improve process efficiency, it is necessary to improve the amount of organic pollutants available for reaction with oxidant species at the anode surface where most of oxidation reactions take place. By using dense plate electrodes, the mass transport of pollutants from the bulk solution to the anode surface is strongly limited by the diffusion boundary layer close to the material surface. This limitation can be addressed by using REM with low pore size and high electrochemically active surface as detailed in mechanistic studies dedicated to flow-through electrochemical technologies [START_REF] Chaplin | The Prospect of Electrochemical Technologies Advancing Worldwide Water Treatment[END_REF][START_REF] Xu | Removal of trace naproxen from aqueous solution using a laboratory-scale reactive flow-through membrane electrode[END_REF][START_REF] Li | Engineering Interface with a One-Dimensional RuO2/TiO2 Heteronanostructure in an Electrocatalytic Membrane Electrode: Toward Highly Efficient Micropollutant Decomposition[END_REF][START_REF] Chen | Development of a moving-bed electrochemical membrane bioreactor to enhance removal of low-concentration antibiotic from wastewater[END_REF]. REM strongly decreases the diffusion time of organic compounds from the center of a pore to the electrode surface and consequently mass transport can be strongly improved by increasing the convection through the REM. Therefore, the flux of organic compounds passing through the REM becomes a crucial operating parameter [START_REF] Trellu | Electro-oxidation of organic pollutants by reactive electrochemical membranes[END_REF][START_REF] Trellu | Mineralization of organic pollutants by anodic oxidation using reactive electrochemical membrane synthesized from carbothermal reduction of TiO2[END_REF][START_REF] Skolotneva | A 2D Convection-Diffusion Model of Anodic Oxidation of Organic Compounds Mediated by Hydroxyl Radicals Using Porous Reactive Electrochemical Membrane[END_REF].

However, a great challenge remains the synthesis of suitable materials that can be used as REM. REM must provide (i) a sufficient conductivity, (ii) a high overvoltage for oxygen evolution reaction and

(iii) a suitable porosity for membrane filtration and enhanced mass transport conditions (in the microor nanometer range). In this context sub-stoichiometric titanium oxide (TiOx) is a promising ceramic material, which presents critical advantages from both cost and operational aspects with a wellestablished production technology and an acceptable price tag. Among TiOx phases, TiO1.75 (i.e.

Ti4O7

) has the highest conductivity (around 1000 S cm -1 ) and is robust from mechanical, chemical and thermal standpoints [START_REF] Walsh | The continuing development of Magnéli phase titanium sub-oxides and Ebonex® electrodes[END_REF]. It can be produced from TiO2 reduction by Ti metal as a powder [START_REF] Acha | Electrical resistivity of the Ti4O7 Magneli phase under high pressure[END_REF] or pressed pellets [START_REF] Andersson | Phase analysis studies on the titanium-oxygen system, Magnéli[END_REF] or by reduction under nitrogen [START_REF] Kolbrecka | Sub-stoichiometric titanium oxides as ceramic electrodes for oxygen evolution-structural aspects of the voltammetric behaviour of TinO2n-1[END_REF] or hydrogen [START_REF] Simpson | Method and apparatus for the manufacture of substoichiometric oxides of titanium by reduction with hydrogen[END_REF] gas flux. The combination of a sol-gel synthesis with spark plasma sintering or vaccum-carbothermic process was also reported for synthesis of nanostructurated Ti4O7 materials for applications in thermoelectricity [START_REF] Portehault | Facile General Route toward Tunable Magnéli Nanostructures and Their Use As Thermoelectric Metal Oxide/Carbon Nanocomposites[END_REF] or supercapacitor and electrocatalysis [START_REF] Huang | Synthesis of High-Performance Titanium Sub-Oxides for Electrochemical Applications Using Combination of Sol-Gel and Vacuum-Carbothermic Processes[END_REF]. In 2016, it was also reported the possibility to synthesize porous Ti4O7 from tubular asymmetric TiO2 ultrafiltration membranes for their further used as REM in water treatment, focusing on both kinetic and thermodynamic aspects [START_REF] Guo | Development and Characterization of Ultrafiltration TiO2 Magnéli Phase Reactive Electrochemical Membranes[END_REF]. Probe molecules have been used in order to highlight the hydroxyl radical mediated oxidation as well as the convection-enhanced mass transport of organic compounds [START_REF] Nayak | Fabrication and characterization of porous, conductive, monolithic Ti4O7 electrodes[END_REF]. Model compounds have also been used for investigating the production of chlorinated by-products [START_REF] Lin | Chlorinated Byproduct Formation during the Electrochemical Advanced Oxidation Process at Magne ĺi Phase Ti4O7 Electrodes[END_REF]. Density functional theory calculations have been coupled with experimental approaches in synthetic solutions in order to better understand reactions with organic compounds at the REM interface [START_REF] Zaky | Mechanism of p-Substituted Phenol Oxidation at a Ti4O7 Reactive Electrochemical Membrane[END_REF].

Ti4O7 REM anode has then proved its high energy-efficiency for electrochemical oxidation of highly stable perfluoroalkyl substances in synthetic solutions [START_REF] Le | Energy-Efficient Electrochemical Oxidation of Perfluoroalkyl Substances Using a Ti4O7 Reactive Electrochemical Membrane Anode[END_REF]. Ti4O7 REM anode was also coupled with the cathodic electro-Fenton process in a flow-through pilot to treat paracetamol as a model pollutant.

It was emphasized a synergistic effect related to the formation of carboxylic acids from paracetamol degradation in the bulk retentate that are more easily mineralized at the REM where both OHmediated oxidation and direct electron transfer occur [START_REF] Trellu | Integration of sub-stoichiometric titanium oxide reactive electrochemical membrane as anode in the electro-Fenton process[END_REF]. However, as regards to real effluents, only dense plate Ti4O7 was successfully used for mineralization and biodegradability enhancement of nanofiltration concentrate of landfill leachates [START_REF] Kateb | Electrochemical advanced oxidation processes using novel electrode materials for mineralization and biodegradability enhancement of nanofiltration concentrate of landfill leachates[END_REF]. Therefore, further application and development of this material now depends on feasibility studies related to (i) the application of these REM for the treatment of real effluents containing low concentration of biorefractory pollutants, (ii) the assessment of energy consumption and (iii) the evaluation of the formation of toxic by-products.

In this study, the feasibility of implementing Ti4O7 REM in a flow-through reactor for the treatment of a WWTP secondary effluent was assessed in terms of (i) mineralization of the organic matter (ii) degradation of a micropollutant, carbamazepine (CBZ), added at 100 µg L -1 (iii) production of toxic chlorine-containing by-products and (iv) energy consumption. The process efficiency was assessed in dependence of the two main operating parameters, which are the current density and the permeate flux.

The objective was to understand and to identify the key parameters for minimizing the energy consumption, optimizing the removal effectiveness and reducing the formation of toxic by-products.

Thus, this study investigates for the first time the suitability of using Ti4O7 REM for the treatment of a real WWTP secondary effluent polluted with a low concentration of pharmaceuticals.

Materials and methods

Secondary effluent

The pilot reactor was fed with a secondary effluent sampled from a WWTP located in La Grande Motte in the south of France. The WWTP is equipped with a Kubota ® Submerged membrane unit (SMU RW 400). The nominal pore size of the membrane sheet is 0.2 µm. All samples have been collected at stationary phase (stable operating parameters and stable removal efficiencies of the systems) and the main characteristics of the effluent are presented in Table 1. The effluent was doped with 100 µg L -1 of CBZ.

Chemicals

All chemicals used in this study were of reagent grade, purchased from Sigma Aldrich, including CBZ (CAS 298-46-4), sodium sulfate (CAS 7757-82-6), acetonitrile and formic acid. Ultrapure water was obtained from a Milli-Q system (Millipore Co. Ltd, resistivity>18.2 MΩ cm).

Experimental pilot reactor

The experimental lab pilot was made of a 3 L feed tank, a pump for circulation of the feed solution, a recirculation loop for the retentate, a DC power supply (from ELC, model AL924A) and a homemade electrochemical membrane cell (Fig. 1). The retentate was continuously recirculated and the permeate was collected for analysis. The whole setup was developed at European Membrane Institute of Montpellier. The Ti4O7 REM anode had the following dimensions: active length of 9 cm, inner diameter of 6 mm, outer diameter of 10 mm, and was used in outside-inside cross-flow filtration mode. The REM was developed and obtained from Saint Gobain CREE. Detailed characterization of this material was given in Trellu et al. [START_REF] Trellu | Mineralization of organic pollutants by anodic oxidation using reactive electrochemical membrane synthesized from carbothermal reduction of TiO2[END_REF][START_REF] Trellu | Integration of sub-stoichiometric titanium oxide reactive electrochemical membrane as anode in the electro-Fenton process[END_REF]. The anode is a mixture of Ti4O7 and Ti5O9 Magnéli phases. It presents a monodispersed pore size distribution, with a median pore size of 1.4 μm, a porosity (pore volume) of 41% and a specific surface area of 0.40 m 2 g -1 . The water permeability of the membrane is 3300 L m -2 h -1 bar -1 .

Scanning electron microscopy images, Hg intrusion porosimetry and X-ray diffraction data are provided as Supplementary Material. The cathode is a 3 mm thick carbon felt encircling the inner wall of the electrochemical cell with an interelectrode distance of 3 cm with the REM. Experiments were performed at the natural pH of the effluent (7.5 ± 0.3). Experiments were conducted in galvanostatic mode with a current density of 73, 145 and 290 A m -2 according to the external geometric surface area of the REM. The feed compartment was maintained under a nitrogen flux in order to avoid H2O2 production at the carbon felt under cathodic polarization.

Total Organic Carbon (TOC) analysis

Total organic carbon was measured using a Shimadzu TOC-L analyzer (680 °C catalytic combustion method). The relative standard deviation on TOC measurement was 2%. Calibration was performed using several dilutions of a standard from Accu SPEC (1000 µg C mL -1 ).

Ions analysis

Anions and cations were analyzed by ion chromatography. Anions (Cl -, NO2 -, ClO3 -, NO3 -, SO4 2-, PO4 3-, Br -) were followed using a chromatographic system Dionex Thermofisher ICS1000 with an eluent producer, equipped with a Dionex Thermofisher AS19 column, an ARDS 600 suppressor and a conductivity detector. The eluent was an aqueous solution of potassium hydroxide. Gradient elution was used in the following manner: 10 mM during the first 10 min, then increase up to 45 mM during 20 min and finally 10 mM during the last 10 min of analysis. The flow was fixed at 1 mL min -1 .

The analysis of cations (Na + , NH4 + , K + , Mg 2+ , Ca 2+ ) was performed by a chromatographic system Dionex Thermofisher ICS900, equipped with a Dionex Thermofisher CS12A column, a CERS 500 suppressor and a conductivity detector. A solution of 20 mM methanesulfonic acid was used as eluent in isocratic mode at 1 mL min -1 .

Monitoring of carbamazepine degradation

The degradation of CBZ was followed by high performance liquid chromatography (HPLC) coupled with mass spectrometry (MS). The HPLC system (Waters Alliance 2695 HPLC Pump and Waters Alliance 2695 Autosampler) was equipped with a C18 column (L = 50 mm, D = 2 mm, particle size = 5µm) thermostated at 20°C. The detector was a triple quadrupole mass spectrometer (Quattro Micro API by Waters). Injection volume was 5 µL. The composition of the mobile phase was 75% milliQ water and 25% acetonitrile with 0.1% (v/v) formic acid eluted in isocratic mode at 0.250 mL min -1 .

The duration of analysis was 3 min and the elution time of CBZ was 1.71 min. The calibration curve for quantification of CBZ concentration was performed with the real effluent doped with 100 µg L -1 of CBZ in order to minimize the effect of the matrix noise on the analysis. The detection limit was found to be 2 µg L -1 .

The triple quadrupole MS was operated in multiple reaction monitoring with compounds being ionized in the positive electrospray ionization mode (MRM: 237.2 → 194.2 with MH+ = 237.2, fragment Q1 = 194.2). The detection conditions were: capillary potential of 3.5 kV, cone potential of 25V, collision energy of 20V, source temperature of 120°C, desolvation temperature of 450°C, cone gas flow of 50 L h -1 , desolvation gas flow of 450 L h -1 . Nitrogen was used as nebulizer gas.

Analysis of halogen-specific adsorbable organic halogen (AOX)

Prior to the analysis, the residual free chlorine was measured by the diethyl-p-phenylenediamine (DPD) method and each sample was quenched with Na2SO3 aqueous solution in proportion 1.2 mol for 1 mol of residual chlorine. Then, 1 mL of nitrate stock solution ([NaNO3] = 0.2 mol L -1 , containing 25 mL L -1 HNO3 65%) was then added according to the ISO 9562 method. Then, the samples were passed through the carbon columns (containing activated carbon (AC) with a grain size distribution around 50-150 μm, provided by Envirosciences). Then, 10 mL of a nitrate solution was passed in order to remove any inorganic species. A blank sample was also prepared in the same way, using Milli-Q water. The activated carbon columns containing the adsorbed organic halogens were transferred to ceramic boats for combustion using a Mitsubishi AQF-2100H unit. The hydrogen halide gases produced were collected in ultra pure water contained in an absorption tube of the gas absorption unit (Mitsubishi GA-210).

The solution in the absorption tube was then analyzed on-line for chloride, bromide, and iodide using ion chromatography (Thermo Scientific Dionex Integrion HPIC system). The concentrations of chloride, bromide, and iodide obtained from the ion chromatograph analysis corresponded to the concentrations of adsorbable organic chloride (AOCl), bromide (AOBr), and iodide (AOI), respectively.

Energy consumption

Energy consumption (EC) was expressed as either kWh per g of TOC removed (Eq. 2) or kWh per m 3 of treated water (Eq. 3).

ℎ = * - * 1 ℎ " # = * 2
Where Ecell is the average cell voltage (V), I the applied current (A), Q is the permeate flow (m 3 h -1 ),

and TOCf and TOCp are the experimental TOC concentrations (g C m -3 ) at a given time in the feed and the permeate, respectively.

Result and discussion

Mineralization of the secondary effluent doped with carbamazepine

Control experiments were previously performed on those membranes [START_REF] Trellu | Mineralization of organic pollutants by anodic oxidation using reactive electrochemical membrane synthesized from carbothermal reduction of TiO2[END_REF]. Results obtained without current supply showed the absence of adsorption and retention of dissolved organic compounds. The main reasons are that (i) the specific surface area of 0.40 m 2 g _1 is low compared to materials tailored for adsorption and (ii) the mono-modal pore size distribution ranging between 0.8 and 1.9 µm with median pore diameter of 1.4 µm does not allow any retention of dissolved compounds Mineralization of the effluent was followed for three different current densities (73, 145 and 290 A m - 2 ). Different transmembrane pressures were tested for each applied current density, which resulted in the variation of the permeate flux. Electrochemical processes for the treatment of low concentrations of pollutants are usually operated under mass transport limitation, meaning that they are limited by the transport of organic compounds from the bulk to the electrode surface. Therefore, this study selected as crucial parameter the amount of organic matter that passed through the membrane (TOC flux) instead of the volumetric permeate flux. The TOC flux is the product of the initial value of TOC in the effluent and the volumetric permeate flux. The TOC flux elimination was defined similarly, by taking into consideration the amount of TOC removed by the process. Therefore, results are presented in fig.

2 as TOC flux elimination versus the TOC flux through the REM. As can be seen from fig. 2, for all current densities, mineralization can be maintained between 70 and >98% up to a maximum value of TOC flux. Then, at higher TOC flux, a plateau is observed for the TOC flux elimination (it corresponds to a drop of the percent removal of TOC). This maximum value of TOC flux strongly depends on the applied current density. The value of TOC flux at which the drop in efficacy occurred was between 5 and 6 g h -1 m -2 for the lowest current density, while for the middle value it was around 8 -9 g h -1 m -2 . For the highest current density of 290 A m -2 , this plateau is projected to be reached at values of TOC flux higher than 12 g h -1 m -2 . In this way, it was clear that the higher the current density applied, the higher the limiting flux was. This could be explained by taking into consideration both mass transport and current limitations. The use of REM instead of conventional plate electrodes has the major goal of reducing the mass transport limitation. In fact, owing to the low pore size of the REM, the diffusion characteristic time of organic compounds from the center of a pore to the electrode surface is strongly decreased compared to the diffusion characteristic time of organic compounds through the diffusion layer at the surface of dense electrodes [START_REF] Skolotneva | A 2D Convection-Diffusion Model of Anodic Oxidation of Organic Compounds Mediated by Hydroxyl Radicals Using Porous Reactive Electrochemical Membrane[END_REF]. Therefore, it is possible to increase the mass transport of organic compounds at the surface of the REM simply by increasing convection, i.e. by increasing the permeate flux of the wastewater passing through the membrane. However, this study highlights that there is a limiting permeate flux above which the REM becomes mainly operated under current limitation instead of mass transport limitation.

It means that above this threshold value, the amount of reactive species generated on the anode is not sufficient for the removal of the high TOC flux through the membrane. This current limitation is also consistent with the fact that the limiting value of the TOC flux is higher when the current intensity increases. In fact, a larger amount of organic compounds can be oxidized when increasing the amount of reactive species generated at higher current density.

Degradation of carbamazepine in the secondary effluent

The degradation of an individual micropollutant is also an important parameter in order to study the capacity of the process to maintain a high degradation rate of target pollutants. The results are presented on fig. 3 giving the normalized concentration of CBZ in the permeate as a function of TOC flux through the REM. At the lowest values of TOC flux, CBZ was not detected in the permeate (CBZ degradation above 98% because the detection limit of CBZ was found to be 2 µg L -1 ). A similar behavior was observed for the evolution of the percent of degradation of CBZ and mineralization of TOC. CBZ was fully degraded after passing through the REM only until a limiting value of TOC flux was reached. This value depends on the current density applied. For the lowest current density of 73 A m -2 , CBZ started to be detected in the permeate at the value of 4 -5 g h -1 m -2 . For the middle value of current density of 145 A m -2 , this value was twice as high as for the lowest, around 8 -9 g h -1 m -2 , while for the highest current density, this limiting TOC flux was around 15 -18 g h -1 m -2 . As explained in the previous section, this phenomenon of the threshold value can be explained by the operation of the process under current limitation when the TOC flux through the membrane becomes too high. It would be necessary to increase the applied current density in order keep a high value of degradation and mineralization rate of the process when the TOC flux exceeds this threshold value. Interestingly, the values of limiting flux for degradation and mineralization were similar. For example, for the middle current density, the limiting value of TOC flux for keeping TOC removal above 70% or for achieving >95% degradation of CBZ was in both case around 8 -9 g h -1 m -2 . Such conformity in results concerning the mineralization of the total organic pollution and degradation of CBZ supports the explanation by the change in process limitations as detailed above. Moreover, it is consistent with data reported in the literature on the anodic oxidation process. When AO is operated in batch reactor, mineralization kinetics are often only slightly slower than degradation kinetics. In fact, anodic oxidation is a heterogeneous process during which oxidant species are accumulated at the surface of the electrode material. Thus, organic pollutants reaching the electrode surface can be rapidly mineralized by this high local concentration of oxidant species instead of being only degraded. This phenomenon represents a great advantage for the application of this process since the formation of potentially toxic degradation by-products can be therefore strongly reduced.

However, the degradation and mineralization of residual organic pollutants is not the only parameter to take into consideration for the evaluation of the suitability of this process. The feasibility of the application of a new water treatment process must also take into consideration the potential formation of various toxic by-products. Particularly, it was reported in the literature [START_REF] Martínez-Huitle | Single and coupled electrochemical processes and reactors for the abatement of organic water pollutants: a critical review[END_REF][START_REF] Radjenovic | Challenges and Opportunities for Electrochemical Processes as Next-Generation Technologies for the Treatment of Contaminated Water[END_REF] that two main drawbacks of anodic oxidation are (i) the generation of organo-halogenated compounds and (ii) the formation of chlorates and most importantly perchlorates. As the effluent released by the wastewater treatment plant contained more than 550 mg L -1 of chlorides, it was crucial to follow the evolution of these parameters. One of the main goals of the application of the electrochemical treatment methods to such effluents would also be the minimization of the formation of these problematic substances [START_REF] Radjenovic | Challenges and Opportunities for Electrochemical Processes as Next-Generation Technologies for the Treatment of Contaminated Water[END_REF][START_REF] Bruguera-Casamada | Effect of electrogenerated hydroxyl radicals, active chlorine and organic matter on the electrochemical inactivation of Pseudomonas aeruginosa using BDD and dimensionally stable anodes[END_REF].

Evolution of chlorates and perchlorates during the treatment

The evolution of chlorides, chlorates and perchlorates is presented on fig. 4 in dependence of the applied current density. From the general A view of fig. 4, it is clear that chloride was oxidized in the process since its concentration in the permeate became lower than in the feed water. The B view from the same experiments (magnified view of A) gives a better visibility of the evolution of chlorates and perchlorates. All chloride removed from the solution was not transformed into chlorate and perchlorate since a large amount of chloride can also be transformed into volatile forms (Cl2, ClO2, Cl2O) and HOCl/ClO -(which might be used for disinfection). For example, chlorate and perchlorate represented only 10, 14 and 43% of the total loss of chloride for experiments at 73, 145 and 290 A m -2 , respectively.

Concentration of chlorate in the permeate was in the range 15 -50 mg L -1 depending on the applied current density. Interestingly, the increase in chlorate concentration was proportional to the increase of the current density. It means that the formation of chlorate was depending on the electric charge (A h) without threshold effect related to the current density.

Perchlorate was also detected in the permeate. For example, a concentration of 25 mg L -1 was observed at the highest current density. Perchlorate formation is a major drawback of anodic oxidation using anode materials with high overvoltage for oxygen evolution reaction. For example, Bergmann et al. [START_REF] Bergmann | The occurrence of perchlorate during drinking water electrolysis using BDD anodes[END_REF] reported the formation of 123 mg L -1 of perchlorate using boron-doped diamond (BDD) anode at 200 A m -2 and 1.0 A h L -1 . Therefore, it is not surprising to observe perchlorate in our case since the electric charge was 0.54 A h L -1 for j = 290 A m -2 and J = 540 L h -1 m -2 . However, on the contrary to chlorate, the concentration of perchlorate increased drastically by increasing the applied current density. The concentration was multiplied by 11 between 73 and 145 A m -2 and by 17 between 145 and 290 A m -2 . In this case, a threshold effect related to the current density was clearly observed.

These results are consistent with the results reported in synthetic solutions treated with BDD by Sànchez-Carretero et al. [START_REF] Sánchez-Carretero | Electrochemical production of perchlorates using conductive diamond electrolyses[END_REF]. In fact, a linear relation was reported between the electrical charge (with similar current density) and the formation of chlorate and perchlorate. However, the increase of the current density (with similar electrical charge) only increased the concentration of perchlorate. This might be ascribed to the complex mechanism of perchlorate formation involving both direct electron transfer and hydroxyl radical mediated oxidation, which is favored at high current density. Perchlorate formation was also reported to be strongly dependent on competitive ions and organic compounds. It has been reported that the formation of perchlorate might be strongly hindered when operating the process under current limitation in presence of organic compounds with high reaction rates with hydroxyl radicals [START_REF] Donaghue | Effect of select organic compounds on perchlorate formation at boron-doped diamond film anodes[END_REF]. This hydroxyl radical scavenging effect avoids the reaction of chlorates with hydroxyl radicals for the formation of perchlorates. This explanation might also be suitable for explaining the results observed in this study. In fact, results reported in fig. 4 have been obtained for experiments performed at a constant TOC flux of 3.8 g h -1 m -2 . Using low current density, this TOC flux is close to the limiting value corresponding to the operation of the process under current limitation, as observed in previous sections. For the higher current densities, this TOC flux is clearly below this threshold value and the process is clearly operated under mass transport limitation, which favors the reaction of hydroxyl radicals with chlorates for the formation of perchlorates. The results reported in this study on a real effluent are crucial for assessing the potential of application of REM. In fact, they highlight that REM should be operated at low current density and under current limitation in order to avoid perchlorate formation. At flux of 3.8 g h -1 m -2 , the perchlorate concentration was 130 µg L -1 and 1.46 mg L -1 at 73 and 149 A m -2 , which was low considering the high chloride concentration of the initial effluent (556 ± 65 mg L -1 ). It was also noticed that the increase of nitrate concentration with the applied current density was definitely linked to mineralization of organic nitrogen.

Fig. 4. Evolution of anions (perchlorates, chlorides, nitrates, bromides, chlorates, nitrates and sulfates) in dependence on the applied current density for a TOC flux of 3.8 g h -1 m -2 . A. General view B. Magnified view with a smaller scale on the y axis.

Evolution of adsorbable organo-halogenated compounds during the treatment

Apart from the chlorine-containing ions, the active chlorine available in the solution can react with the organics for forming the so-called organo-halogenated compounds, which are most of the time monitored through the amount of adsorbable organo-halogenated compounds (AOX). AOX are Feed water generally considered as unwanted compounds with potential genotoxic effects [START_REF] Emmanuel | Toxicological effects of disinfections using sodium hypochlorite on aquatic organisms and its contribution to AOX formation in hospital wastewater[END_REF][START_REF] Grøn | Organic Halogen Group Parameters as Indicators of Ground Water Contamination[END_REF][START_REF] Xie | AOX contamination status and genotoxicity of AOX-bearing pharmaceutical wastewater[END_REF]. The analyses showed the initial presence of these compounds at low concentration in the feed water (Table 2) under the form of organochloride compound (AOCl). The results obtained for AOBr and AOI were below the limit of detection (0.5 µg L -1 ) in all the samples.

Table 2. Concentration of adsorbable organic chlorides (AOCl) in initial effluent and in the treated samples in dependence on the applied current density for experiments performed at a constant TOC flux of 3.

8 g h -1 m -2 . Samples AOCl (µg L -1 )
Initial effluent 77.8

Effluent treated (j = 73 A m -2 ) 1491

Effluent treated (j = 290 A m -2 ) 4270

Three samples of treated water at two different current densities were also analyzed. A significant increase in the concentration of AOCl was observed for all samples. For the lowest current density the concentration of AOCl was around 1.5 mg L -1 , though for the highest current it increased to a value almost three times higher. The formation of these compounds often comes from reactions between free chlorine (HOCl/ClO -) and dissolved organic compounds through various possible pathways such as oxidation, addition or electrophilic substitution. By increasing current density with a factor 3.9, the AOCl formation was multiplied by 3.0. These results are consistent with literature since it was reported that the formation of AOCl is tightly correlated with the electric charge without threshold effect related to the current density [START_REF] Schmalz | Electrochemical disinfection of biologically treated wastewater from small treatment systems by using boron-doped diamond (BDD) electrodes-Contribution for direct reuse of domestic wastewater[END_REF]. In a previous study on the tertiary treatment of wastewater (with similar TOC of 9.7 mg L -1 ) with a BDD anode, an electric charge of 0.13 A h L -1 resulted in the formation of 400 µg L -1 , while in this study 1413 µg L -1 was formed after passing the same electric charge (for j = 73 A m -2 and J = 540 L h -2 m -2 ) [START_REF] Schmalz | Electrochemical disinfection of biologically treated wastewater from small treatment systems by using boron-doped diamond (BDD) electrodes-Contribution for direct reuse of domestic wastewater[END_REF]. This higher value might be explained either by the higher chloride concentration (144 vs 500 mg L -1 ) or by the different electrode material used (further studies are required on the understanding of AOX formation with Ti4O7 REM). The results obtained in the literature and in this study show that it might be possible to control the formation of organo-halogenated compounds by optimizing the electric charge. Results reported in Table 2 have been obtained in a range of operating conditions for which >98% of CBZ degradation and >70% mineralization was achieved (mass transport limitation). Therefore, it might be possible to reduce the electric charge in order to further reduce the generation of AOCl. It is also important to take into consideration that the parameter AOCl is a global indicator of the presence of such chemicals; however it does not imply any information about the nature of their toxicological effects. It is therefore also necessary to conduct more detailed studies on the formation of specific most notorious representatives of organo-halogenated compounds and the mechanisms involved in these reactions in order to determine how to really avoid or control the formation of such toxic by-products [START_REF] Lin | Chlorinated Byproduct Formation during the Electrochemical Advanced Oxidation Process at Magne ĺi Phase Ti4O7 Electrodes[END_REF].

Energy consumption

Energy consumption is an important characteristic of the viability as well as a major drawback for electro-oxidation processes. Therefore, its proper analysis on a real effluent containing low concentration of pollutant was a crucial task of this study. The results are presented on fig. 5. One of the great advantages of REM is that it is possible to improve this energy consumption by simply increasing the TOC flux through the REM. Then, at high TOC flux, when the process is operated under current limitation, the energy consumption reached a plateau around 0.1 -0.2 kWh g -1 TOC removed. As explained previously, this current limitation was reached at different TOC flux depending on the current density applied. Besides, higher energy consumption was also always observed when using the highest current densities because of the higher cell potential.

For the optimization of the process, it is crucial to take into consideration all the parameters, including the energy consumption, the removal of target compounds and the formation of by-products. For determination of the optimal parameters, it was decided to choose the current density for which the formation of AOCl and perchlorate was minimized, i.e. the lowest current density (73 A m -2 ). Then, the optimal TOC flux through the membrane was selected as the maximal one allowing to achieve >98% removal of CBZ, i.e. 4.1 g h -1 m -2 (arrow in fig. 5B). With these operating conditions, the percent of mineralization was still excellent (70%). The energy consumption was 0.15 kWh g -1 TOC removed, which corresponds to 0.74 kWh m -3 . This high energy efficiency of electro-oxidation for the treatment of a real effluent with low concentration is clearly related to the mass transport enhancement allowed from the use of such REM. These promising results might promote the development of this process for such applications. For comparison, the French company SUEZ is currently estimating that a reasonable cost of a tertiary treatment (for capital and operational expenditures) would be between 7 and 14 euros / population equivalent / year, including an energy consumption of 1.45 kWh m -3 [START_REF] Penru | Journée de restitution des résultats du projet Micropolis[END_REF].

It is also of great interest to compare the results obtained here using Ti4O7 as REM for anodic oxidation to the use of other electrochemical systems, as reported in Table 3. Recently, Yang et al.

[40] combined the use of a BDD anode with the heterogeneous electro-Fenton process at the cathode for removal of the antibiotic ofloxacin in batch configuration. A mineralization rate of 80% was obtained after 8h of electrolysis from an effluent at 22 mgC L -1 . The energy consumption was about 3.5 kWh g -1 TOC (i.e. 77 kWh m -3 ). Besides, El Kateb et al. [START_REF] Kateb | Electrochemical advanced oxidation processes using novel electrode materials for mineralization and biodegradability enhancement of nanofiltration concentrate of landfill leachates[END_REF] used the same cathode material combined with plate Ti4O7 anode in batch configuration for TOC abatement of the concentrate of a landfill leachate pre-treated in an MBR. An energy consumption of 0.11 kWh g -1 TOC (49.5 kWh m -3 ) was mentioned for TOC removal. Those high values of EC could be mainly explained by the batch configuration of the system, which is dramatically penalizing for mineralization of low concentration of pollutants because of the limitation ascribed to the diffusion of pollutants from the bulk to the electrode surface [START_REF] Trellu | Electro-oxidation of organic pollutants by reactive electrochemical membranes[END_REF]. These much higher energy consumptions highlight the suitability of REM developed in this study for the treatment of low concentrations of pollutants in secondary effluents from municipal WWTP. The interest of flow-through reactors equipped with a Ti4O7 REM was also confirmed by the recent paper of Le et al. concerning PFAS degradation [START_REF] Le | Energy-Efficient Electrochemical Oxidation of Perfluoroalkyl Substances Using a Ti4O7 Reactive Electrochemical Membrane Anode[END_REF]. The energy consumption to remove PFOA (initial concentration 4.4 mg L -1 ) and PFOS (initial concentration 5 mg L -1 ) to below the detection limit (i.e. 86 and 35 ng L -1 ) were 5.1 and 6.7 kWh m -3 proving again the efficiency of Ti4O7 in flow-through configuration for highly stable micropollutants.

Conclusion

The main objective of this study was to analyze the viability of the anodic oxidation process using a substoichiometric titanium oxide reactive electrochemical membrane as anode. The laboratory pilot of this coupled process was operated to treat a secondary effluent from a municipal WWTP spiked with a pharmaceutical micropollutant (CBZ) at a concentration of 100 µg L -1 , which is close to an environmentally relevant value. The efficiency of this process was evaluated regarding to the removal of pollutants in terms of mineralization of the organic load as well as degradation of the micropollutant. Two major drawbacks of any electrochemical treatment processes as defined previously (toxic chlorinated by-product formation and energy consumption) were equally addressed by this study. Here are the main conclusions to draw from the results obtained.

• Mineralization and degradation demonstrated an efficacy above 70% removal for the low TOC flux, regardless to the current densities applied. Increasing the TOC flux revealed that the process effectiveness shifted from being limited by mass transport of pollutants to the surface of the anode to being controlled by the production of oxidant speices at the anode i.e. by the applied current density.

For each value of current density, a limiting value of TOC flux above which the effectiveness decreased was highlighted. The higher the current density, the higher this threshold value was observed.

• Formation of chlorates and perchlorates was strongly dependent on the applied current density.

Increasing the current density resulted in greater formation of chlorates, while perchlorates strongly raised in their concentration for the highest current density applied. A similar trend was observed for AOX. Even low current density (73 A m-2) led to the formation of AOX but their concentration was multiplied by 3 at high current density (290 A m -2 ). The main consequence is that only the use of low current density should be considered for such applications to Cl -containing effluents.

• It is possible to strongly decrease the energy consumption of the process by simply increasing the TOC flux in order to reduce mass transport limitations.

• Overall, by taking into consideration both advantages and drawbacks of this process, the optimal operating conditions were obtained at 73 A m -2 (in order to reduce the formation of toxic by-products) and a TOC flux through the membrane of 4.41 g h -1 m -2 , corresponding to 1.34 m 3 h -1 m -2 . These conditions allowed for >98% degradation of CBZ, 70% of mineralization with an energy consumption of 0.15 kWh g -1 of TOC removed (i.e. 0.74 kWh m -3 of treated wastewater). These results highlight that the development of REM based on Magnéli phases can be a cost effective electrochemical advanced oxidation process, even for the treatment of effluents with low concentrations of pollutants.

Fig. 1 .

 1 Fig. 1. Scheme of the pilot equipped with a Ti4O7 reactive electrochemical membrane (REM in green in the center) for electro-oxidation of secondary effluents doped with CBZ at 100 µg L -1

Fig. 2 .

 2 Fig. 2. Mineralization of the real effluent doped with 100 µg L -1 of CBZ in dependence on applied current density and TOC flux through the reactive electrochemical membrane.

  the REM (g h -1 m -2 ) j = 73 A / m2 j = 145 A / m2 j = 290 A / m2 j = 145 A m -2 j = 73 A m -2 j = 290 A m -2

Fig. 3 .

 3 Fig. 3. Degradation of 100 µg L -1 carbamazepine in the effluent in dependence of the TOC flux through the reactive membrane and the applied current density

Fig. 5 . 1 )

 51 Fig. 5. Energy consumption per meter cube of treated water as a function of volumetric flux. A. General view B. Increased view with a smaller scale of y axe.

Table 1 .

 1 Main characteristics of the secondary effluent.

	Parameter	Value	Parameter	Value mg.L -1
	COD	24.2 ± 2.7 mgO2 L -1	[NO2 -]	0.3 ± 0.1
	TOC	7 mg C L -1	[NO3 -]	13.8 ± 1.9
	Conductivity	2.5 ± 0.3 mS cm -1	[SO4 2-]	229 ± 4
	pH	7.2 -7.8	[Br -]	1.5 ± 0.2
	Abs 254 nm	0.128 -0.157	[Cl -]	556 ± 65
	[PO4 3-]/[NH4 + ]	not detected	[ClO3 -]	1.4 ± 0.2

Table 3 .

 3 Comparison of recently published results on the degradation of organic pollutants in aqueous media by electrochemical advanced oxidation processes.

	Authors	Ref.	System	Energy consumption	Mineralization	Comments
				(kWh.m -3 )	(%)
	This study	-	REM TiOx	0.74	70
	SUEZ	39	-	1.45	-	Recommendation
	Company
	Yang et al.	40	Batch BDD +	77	80
			Electro-Fenton		(8h)
	El Kateb et al.	29	Batch TiOx +	49.5	45
			Electro-Fenton		(8h)
	Lin et al.	41	Batch TiOx	14.2 -36.9	99
	Le et al.	27	REM TiOx	5.1 -6.7	-

Acknowledgements

This project is supported by Chimie Balard Cirimat Carnot Institute through the ANR program N°16 CARN 0008-01.The research group would like to thank the Carnot Institute for this financial support.

We are also grateful to Dr. Jelena Radjenovich ICREA Research Professor at Catalan Institute for Water Research, Gerone, Spain and her research team for the analysis of AOX of our samples.