Julien Poisat
email: poisat@math.univ-lyon1.fr

On quenched and annealed critical curves of random pinning model with finite range correlations

Keywords: AMS 2000 subject classifications: 82B44, 60K37, 60K05 Polymer models, Pinning, Annealed model, Disorder irrelevance, Correlated disorder, Renewal process, Markov renewal process, Intersection of renewal processes, Perron-Frobenius theory, subadditivity

This paper focuses on directed polymers pinned at a disordered and correlated interface. We assume that the disorder sequence is a q-order moving average and show that the critical curve of the annealed model can be expressed in terms of the Perron-Frobenius eigenvalue of an explicit transfer matrix, which generalizes the annealed bound of the critical curve for i.i.d. disorder. We provide explicit values of the annealed critical curve for q = 1,2 and a weak disorder asymptotic in the general case. Following the renewal theory approach of pinning, the processes arising in the study of the annealed model are particular Markov renewal processes. We consider the intersection of two replicas of this process to prove a result of disorder irrelevance (i.e. quenched and annealed critical curves as well as exponents coincide) via the method of second moment.

Introduction

Polymers are macromolecules which are modelized by self-avoiding or directed random walks. Take for instance S = (S n ) n≥0 a random walk on Z starting at 0 and such that |S n+1 -S n | ≤ 1. By polymer of dimension 1+1 and size N we will mean a realization of the directed random walk {(n, S n )} 0≤n≤N , where each segment [(n, S n ), (n + 1, S n+1 )] stands for a constitutive unit, called monomer.

Suppose now that a reward h is given to a configuration {(n, S n )} 0≤n≤N each time it touches the interface, i.e. each time S n = 0. One can then consider a distribution on polymers of size N whose density with respect to the initial distribution is equal, up to a renormalizing constant, to the Boltzmann factor exp (h × Card{n ∈ {1, . . . , N }|S n = 0}) .

Depending on the sign of h, this distribution favorizes or penalizes polymers pinned to the interface, and letting N go to infinity, the model, called homogeneous pinning model, undergoes a localization/delocalization transition.

Pinning models can also be used to study the interaction between two polymers, since the difference of two random walks is still a random walk. One can think for example of the two complementary strands of a DNA molecule: in this case, the values of n for which S n = 0 are the sites where the two strands are pinned, and the delocalization transition corresponds to DNA denaturation (or melting). One could argue that the binding strength between the two strands actually depends on the base pair, which is A-T or G-C. This corresponds to looking at a disordered model, i.e. a model in which the reward is n-dependent.

An assumption usually made is that the reward at site n writes

h n = h + βω n
where h ∈ R, β ≥ 0 and ω = (ω n ) n≥0 is a frozen realization of a sequence of independent standard gaussian random variables. The space of parameters is then partitioned in localized and delocalized phases, separated by a critical curve β → h c (β). The presence of disorder has important consequences on the model. For example, one can show that there is localization for h < 0 provided that disorder is strong enough (i.e. β large enough). If we consider the annealed model (i.e. the model in which the Boltzmann factor is averaged over disorder), we have the following lower bound:

h c (β) ≥ -log P (τ 1 < +∞) - β 2 2 (1)
where τ 1 is the first return time of S to 0. In the last few years, many rigorous results were given on relevance of disorder, which in particular answer the following question: when is (1) an equality? For these questions, as well as classical results on homogeneous and disordered pinning models, we refer to [START_REF] Giacomin | Random polymer models[END_REF], [START_REF] Giacomin | Renewal sequences, disordered potentials and pinning phenomena[END_REF], [START_REF] Toninelli | Localization transition in disordered pinning models[END_REF] and references therein.

In this paper we remove the independence assumption on ω and study the effect of correlations on the right-hand side of (1), i.e on the annealed critical curve. This is partly motivated by the long-range correlations in DNA sequence, see [START_REF] Chen | Characterizing long-range correlation properties in nucleotide sequences[END_REF] and [START_REF] Peng | Long-range correlations in nucleotide sequences[END_REF] on this topic. We also mention [START_REF] Allahverdyan | Unzipping of dna with correlated base sequence[END_REF] and [START_REF] Jeon | The effect of sequence correlation on bubble statistics in double-stranded dna[END_REF] where the effect of sequence correlation is investigated, in somewhat different contexts. In [START_REF] Allahverdyan | Unzipping of dna with correlated base sequence[END_REF], the authors study the effect of a pulling force applied to the extremity of a DNA strand on the number of broken base pairs (unzipping of DNA) in two correlated scenarii: integrable and nonintegrable correlations. In [START_REF] Jeon | The effect of sequence correlation on bubble statistics in double-stranded dna[END_REF], the authors consider the effect of sequence correlation on the bubble size distribution: by bubbles we mean broken base pairs, and if we keep in mind the analogy with pinning models, it corresponds to the excursions of the directed random walk between two visits at 0.

The disorder sequence in our model is a finite-order moving average of an i.i.d sequence, which is the simplest correlated sequence one can look at, and the reason for this choice will be clearer further in the text. This will be defined in Section 2, as well as the renewal sequence τ = (τ n ) n≥0 (the contact points) and the polymer measures. In Section 3, we introduce classical notions for these models: the free energy, the phase diagram and the (quenched and annealed) critical curve of the model. In the proof of Theorem 3.1, a new homogeneous model emerges, whose hamiltonian does not only depends on the number of renewal points but also on their mutual distances. In Section 4 we are interested in the annealed critical curve. The main results are Theorem 4.1, which states that the difference between the annealed critical curve in the correlated case and the annealed critical curve in the i.i.d. case can be expressed in terms of the Perron-Frobenius eigenvalue of an explicit transfer matrix, and Proposition 4.2, which gives a weak disorder asymptotic of the annealed critical curve. Note that the appearance of Perron-Frobenius eigenvalues is reminiscent of results on periodic copolymers, see [START_REF] Caravenna | A renewal theory approach to periodic copolymers with adsorption[END_REF]. In a second part of the paper (Section 5, Theorem 5.1), we show that under certain conditions (the same as i.i.d. disorder actually) quenched and critical curves (as well as exponents) coincide at high temperatures (small β). This is the regime of disorder irrelevance. We use the second moment method, which will lead us to study the exponential moments of two replicas of a certain Markov renewal process.

The model

Contact points between the polymer and the line

We follow the renewal theory approach of pinning. Let τ be a discrete renewal process such that τ 0 = 0 and τ n = n k=1 T k , where the inter-arrival times (or jumps) T k are i.i.d. random variables taking values in N * . Furthermore,

K(n) = P (T 1 = n) = L(n) n 1+α
where α ≥ 0 and L is a slowly varying function. Without losing in generality, we can assume that n≥1 K(n) = 1, i.e. τ is recurrent. We distinguish between positive recurrence (α > 1 or α = 1 and L is such that n≥1 L(n)/n < +∞) and null recurrence (α ∈ [0, 1) or α = 1 and L is such that n≥1 L(n)/n = +∞). We will denote by δ n the indicator of the event {n ∈ τ } = k≥0 {τ k = n} so that if ı N := sup{k ≥ 0|τ k ≤ N } is the number of renewal points before N , then ı N = N n=1 δ n . The letter E will denote expectation with respect to the renewal process.

We also suppose that for all n ≥ 1, K(n) > 0 (which implies aperiodicity). This assumption seems quite restrictive, but will be necessary in Section 4.2. If this condition on K were not fulfilled, we would simply have to reduce the state space of the matrices defined in Section 4 to {n ≥ 1|K(n) > 0} q and to assume that K is aperiodic.

Finite range correlations

Let (ε n ) n∈Z be a collection of independent standard gaussian random variables (independent from τ ), q ≥ 1 a fixed integer, and (a 0 , . . . , a q ) ∈ R q+1 such that a 2 0 + . . . + a 2 q = 1. Define the disorder sequence ω = (ω n ) n≥0 by the q-order moving average ω n = a 0 ε n + . . . + a q ε n-q . Then ω is a stationary centered gaussian process and its covariance function ρ n := Cov(ω 0 , ω n ) satisfies ρ 0 = 1 and n > q ⇒ ρ n = 0. From now, the notations P and E will be associated to disorder.

The quenched and annealed polymer measures

We define the (constraint) quenched polymer measures, which depend on two parameters, the averaged pinning reward h ∈ R and the amplitude of disorder β ≥ 0:

dP N,β,h,ω dP = 1 Z N,β,h,ω exp N n=1 (βω n + h)δ n δ N (2)
where

Z N,β,h,ω = E exp N n=1 (βω n + h)δ n δ N (3)
is the partition function. We also define its annealed counterpart:

d(P ⊗ P) N,β,h d(P ⊗ P) = 1 Z a N,β,h exp N n=1 (βω n + h)δ n δ N (4)
where Z a N,β,h = EZ N,β,h,ω .

Generalities

Free energy, phase diagram, and critical curve

We give some results which are well-known for i.i.d. disorder, and which can be generalized to ergodic disorder (see [START_REF] Giacomin | Random polymer models[END_REF]Thm 4.6,p.96]).

Proposition 3.1. For all h ∈ R and all β ≥ 0, there exists a nonnegative constant F (β, h) such that,

F (β, h) = lim N →+∞ 1 N log Z N,β,h,ω
P-almost surely and in L 1 (P).

Proof. We use the Markov property as in [10, Prop 4.2, p.91] or [9, (3.1), p.12] to write log

Z c N +M,β,h,ω ≥ log Z c M,β,h,ω + log Z c N,β,h,θ M ω
where θ is the shift operator. We then use Kingman's subadditive theorem (see [START_REF] Steele | Kingman's subadditive ergodic theorem[END_REF]). In our case, ω is ergodic because

ρ n n→∞ -→ 0 (see [6, Chp 14, §.2, Thm 2]). The phase diagram R + × R is then divided into a localized phase L = {(β, h)|F (β, h) > 0}
and a delocalized one

D = {(β, h)|F (β, h) = 0} .
For all β, define the critical point h c (β) := sup{h ∈ R|F (β, h) = 0}. By convexity of F (as the limit of convex functions), D is convex so the critical curve β → h c (β) is concave. Moreover, it is nonincreasing and h c (0) = 0. For detailed arguments, we refer to [START_REF] Giacomin | Random polymer models[END_REF].

Annealed free energy and annealed bound

The first difference that occurs when dealing with correlated disorder is that integrating on ω the Boltzmann factor does not yield a classical homogeneous model (see (6) below). As a consequence, we will need an additional argument to define the annealed free energy.

Lemma 3.1 (Hammersley's approximate subadditivity [START_REF] Hammersley | Generalization of the fundamental theorem on subadditive functions[END_REF]). Let h : N → R be such that for all n, m ≥ 1,

h(n + m) ≤ h(n) + h(m) + ∆(n + m),
with ∆ a non decreasing sequence satisfying:

∞ r=1 ∆(r) r(r + 1) < ∞.
Then, lim n→+∞ h(n) n exists and is finite.

Theorem 3.1. For all h ∈ R and all β ≥ 0, there exists a nonnegative constant F a (β, h) such that,

F a (β, h) = lim N →+∞ 1 N log Z a N,β,h . Moreover, if h a c (β) := sup{h ∈ R|F a (β, h) = 0} then h c (β) ≥ h a c (β). ( 5
)
Proof. First, we compute the variance (with respect to ω) of N n=1 ω n δ n . For every realization of τ , we have:

Var N n=1 ω n δ n = N i,j=1 Cov(ω i , ω j )δ i δ j = N n=1 δ n + 2 1≤i<j≤N ρ j-i δ i δ j . (6)
Then,

Z a N,β,h = E exp (h + β 2 2 ) N n=1 δ n + β 2 N -1 i=1 N -i k=1 ρ k δ i δ i+k δ N .
Now, we want some sort of superadditivity for the annealed partition function.

For a polymer of size N + M , observe that

1≤i<j≤N +M ρ j-i δ i δ j = 1≤i<j≤N ρ j-i δ i δ j + N +1≤i<j≤N +M ρ j-i δ i δ j + 1≤i≤N <j≤N +M ρ j-i δ i δ j .
Conditioned on the event {N ∈ τ }, the second term has the same law as 1≤i<j≤M ρ j-i δ i δ j . Moreover, the third term is greater than a constant C only depending only ρ and q. We can then write

Z a N +M,β,h ≥ E   exp   (h + β 2 2
)

N +M n=1 δ n + β 2 1≤i<j≤N +M ρ j-i δ i δ j   δ N δ N +M   ≥ e Cβ 2 Z a N,β,h Z a M,β,h
and we conclude by using Lemma 3.

1 to -log Z a N,β,h with ∆(n) = -Cβ 2 .
As in [10, Prop 5.1] we use Jensen's inequality to prove that

F (β, h) ≤ F a (β, h), (7)
which in turn yields the annealed bound [START_REF] Chen | Characterizing long-range correlation properties in nucleotide sequences[END_REF].

When disorder is i.i.d, (5) becomes h c (β) ≥ h c (0) -β 2 /2 := h a c (β) and the question of knowing whether this is an equality was studied in several papers and monographs (for example, [START_REF] Giacomin | Renewal sequences, disordered potentials and pinning phenomena[END_REF], [START_REF] Toninelli | Localization transition in disordered pinning models[END_REF], [START_REF] Giacomin | Random polymer models[END_REF] and references therein) where we learn that the answer depends on the values of α and β.

In the next subsection, the effect of correlations on h a c will be studied.

The annealed critical curve

4.1. The result for q = 1 and the reason why the technique used does not apply to q > 1 Proposition 4.1. If q = 1 then we have

h a c (β) = h c (0) - β 2 2 -log 1 + K(1) e ρ1β 2 -1
Proof. If q = 1, equality [START_REF] Cornfeld | Ergodic theory, Grundlehren der Mathematischen Wissenschaften[END_REF] gives:

Z a N,β,h = E exp (h + β 2 2 )ı N + ρ 1 β 2 N -1 n=1 δ n δ n+1 δ N .
The energetic contribution of a jump can only take two values: h+ (2ρ

K (q=1) (n) := K(n) if n > 1.
If q ≥ 2, the situation is more complicated because in this case we must consider the energetic contribution of a q-tuple of jumps instead of one of a single jump. For example, if q = 2, the energetic contribution of a jump of size 1 can be h+(1+2ρ 1 )β 2 /2 or h+(1+2ρ 1 +2ρ 2 )β 2 /2, depending on the value of the jump just before. This idea of looking at the sequence of q-tuples of consecutive inter-arrival times is developed in the next section.

An auxiliary Markov chain and the transfer matrix

From now we assume q ≥ 2. We will denote by t = (t 1 , . . . , t q ) a q-tuple in (N * ) q and if (t n ) n≥1 is a sequence of integers, then t n := (t n , . . . , t n+q-1 ). The projection on the first coordinate t → t 1 will be denoted by π 1 . Let G be a function defined on such q-tuples by G(t) = q k=1 ρ t1+...+t k , and which should be interpreted like this: if t is the q-tuple of the inter-arrival times of q + 1 consecutive renewal points on the interface, then G(t) gives the total contribution of correlations between disorder at theses points.

Notice that when we compute the value of G for some q-tuple of inter-arrival times, any inter arrival time strictly greater than q "does not count". To put it more precisely, we can consider a "cemetery state", denoted by ⋆, and define for all t ∈ N * and t ∈ (N * ) q , t * := t1 {t≤q} + ⋆1 {t>q} and t * = (t * 1 , . . . , t * q ). Then G can be considered as a function of t * instead of t, if we adopt the following natural conventions: ρ ⋆ = 0 and for all t ∈ {1, . . . , q, ⋆}, ⋆ + t = t + ⋆ = ⋆. From now we will use the following notations: E = {1, . . . , q, ⋆} and K(⋆) = n>q K(n).

In the following we will write s t (resp. s * t * ) if for all i ∈ {2, . . . , q},

s i = t i-1 (resp. s * i = t * i-1
). We now make the following remark: the sequence of q-tuples (T n ) n≥1 is a Markov chain on a countable state space, and its transition probability from state s = (s 1 , . . . , s q ) to state t = (t 1 , . . . , t q ) writes Q(s, t) := K(t q )1 {s t} .

Note that Q is irreducible because of the positiveness of the K(n)'s. We now define the nonnegative matrices Q β and Q * β ,which will play the role of transfer matrices, by

Q β (s, t) = e β 2 G(t) K(t q )1 {s t} and Q * β (s * , t * ) = e β 2 G(t * ) K(t * q )1 {s * t * } .
We will write Q * instead of Q * 0 . Since Q * β is an irreducible nonnegative matrix on the finite state space E q , we know by the Perron-Frobenius theorem that there exists a Perron-Frobenius eigenvalue λ(β) and an associated right eigenvector ν * β = (ν * β (x)) x∈E q with positive components (see [START_REF] Seneta | Non-negative matrices and Markov chains[END_REF]).

Statement of the results

We are now ready to state our main results. The first one expresses the annealed critical curve in terms of the Perron-Frobenius eigenvalue of the transfer matrix

Q * β . Theorem 4.1. For all α ≥ 0, for all β ≥ 0, h a c (β) = - β 2 2 -log λ(β).
It seems difficult to give a nice explicit expression of λ(β), since it is the Perron-Frobenius eigenvalue of a matrix of size (q + 1) q . For q = 2, we have computed

h a c (β) = - β 2 2 -log φ(β) -log   1 + 1 -ψ(β) φ(β) 2 2   where φ(β) = 1 + K(1)(e (ρ1+ρ2)β 2 -1) + K(2)(e ρ2β 2 -1) ψ(β) = 4K(1)(1 -K(1))e ρ1β 2 (e ρ2β 2 -1) 1 + K(2) 1 -K(1)
(e ρ2β 2 -1) .

In the general case, the asymptotic behaviour of the annealed critical curve for weak disorder can be explicited: Proposition 4.2. We have

h a c (β) β→0 ∼ -1 + 2 q n=1 ρ n P (n ∈ τ ) β 2 2 .
Before going into details, we outline the proof of Theorem 4.1. First, we introduce in Lemma 4.1 new Markov transition kernels built from the transfer matrices and an eigenvector associated to λ(β). From these we give a new law for the sequence of q-tuples of consecutive inter-arrival times, to which we associate what could be called a "q-correlated" renewal process. This process is in fact a particular Markov renewal process (these are processes in which the return times are not necessarily i.i.d., but driven by a Markov chain, see [START_REF] Asmussen | Applied probability and queues, Applications of Mathematics[END_REF] on this subject). With Lemma 4.4, we link the annealed free energy of our initial model to the homogeneous free energy of the new "q-correlated" renewal process. This will be the starting point of the proof of Theorem 4.1. Note that for positive recurrent renewal processes we give a shorter proof than in the general case.

A "q-correlated" renewal process related to the model

For all q-tuples t, define ν

β (t) = ν * β (t * ). Lemma 4.1. Qβ (s, t) := Q β (s,t)ν β (t) λ(β)ν β (s) and Q * β (s * , t * ) := Q * β (s * ,t * )ν * β (t * ) λ(β)ν * β (s * )
are Markov transition kernels.

Proof. For Q * β , the result is a direct consequence of the relation Q * β ν * β = λ(β)ν * β
and of the positiveness of λ(β) and ν β . For Qβ , we write for all s = (s 1 , . . . , s q ),

t Q β (s, t)ν β (t) = t≥1 e β 2 G(s2,...,sa,t) K(t)ν β (s 2 , . . . , s q , t) = t≥1 e β 2 G(s * 2 ,...,s * q ,t * ) K(t)ν * β (s * 2 , . . . , s * q , t * ) = t * ∈E e β 2 G(s * 2 ,...,s * q ,t * ) K(t * )ν * β (s * 2 , . . . , s * q , t * ) = λ(β)ν * β (s * ) = λ(β)ν β (s).
The result follows in the same way as for Q * β .

Since Q * β is a finite irreducible transition matrix (it has the same incidence matrix as Q * β , which is irreducible), it has a unique invariant probability measure that we denote by µ * β . If we define µ β a measure on (N * ) q by µ β (t) =

K(t0) K(t * 0 ) . . . K(tq-1) K(t * q-1 ) µ * β (t * ), then Lemma 4.2. µ β is the invariant probability of Qβ .
Proof. By a direct computation, µ β is a probability. Now we prove that it is invariant. For all t ∈ (N * ) q , we have

s µ β (s) Qβ (s, t) = λ(β) -1 e β 2 G(t) ν β (t)K(t q ) s≥1 µ β (s, t 1 , . . . , t q-1 ) ν β (s, t 1 , . . . , t q-1 ) = λ(β) -1 e β 2 G(t * ) ν * β (t * )K(t q ) × s≥1 K(s)K(t 1 ) . . . K(t q-1 ) K(s * )K(t * 1 ) . . . K(t * q-1 ) µ * β (s * , t * 1 , . . . , t * q-1 ) ν * β (s * , t * 1 , . . . , t * q-1 ) = λ(β) -1 e β 2 G(t * ) ν * β (t * )K(t * q ) µ β (t) µ * β (t * ) s≥1 K(s) K(s * ) µ * β (s * , t * 1 , . . . , t * q-1 ) ν * β (s * , t * 1 , . . . , t * q-1 ) = λ(β) -1 e β 2 G(t * ) ν * β (t * )K(t * q ) µ β (t) µ * β (t * ) s * ∈E q µ * β (s * , t * 1 , . . . , t * q-1 ) ν * β (s * , t * 1 , . . . , t * q-1 ) = µ β (t)
where for the last equality we use the fact that µ * β is the invariant probability of Q * β . We define a new law on the interarrival times (T n ) n≥1 , denoted by P β , by the following relations:

P β (T 1 = t 1 , . . . , T q = t q ) = q k=1 K(t k )
and for all k ≥ 0

P β (T k+q-1 = t q+1 |T k+1 = t 1 , . . . , T k+q = t q ) = Qβ ((t 1 , . . . , t q ), (t 2 , . . . , t q+1 ))
To determine T k+q+1 conditionnally to the past, only T * k+1 is relevant (and not T k+1 ) since it can be checked that

P β (T k+q-1 = t q+1 |T k+1 = t 1 , . . . , T k+q = t q ) = Qβ ((t * 1 , . . . , t * q ), (t * 2 , . . . , t * q+1 )) × K(t q+1 ) K(t * q+1 ) = P β (T k+q-1 = t q+1 |T * k+1 = t * 1 , . . . , T * k+q = t * q ).
Under P β , (τ n ) n≥0 is then a (delayed) Markov renewal process with markov modulating chain (T * k-q ) k≥q+1 , and semi-Markov kernel: for all n ≥ 1, x, y ∈ E q ,

P β (T k+q+1 = n, T * k+2 = y|T * k+1 = x) = Q * β (x, y) K(n) K(y q ) 1 {n * =yq} .
Lemma 4.3. For all h ∈ R and all β ≥ 0,

F a (β, h) = lim N →+∞ 1 N log E e (h+ β 2 2 )ıN +β 2 ı N n=1 G(T n) δ N .
Proof. On one hand, we have by integrating over disorder the partition function:

Z a N,β,h = E   exp   ı N (h + β 2 2 ) + β 2 1≤i<j≤N ρ j-i δ i δ j     = E exp ı N (h + β 2 2 ) + β 2 N -1 i=1 N -i k=1 ρ k δ i δ i+k .
On the other hand,

ıN n=1 G(T n ) = N i=1 q k=1 ρ k δ i δ i+k .
We prove the lemma by showing that there exists a constant C(ρ, q) such that

ıN n=1 G(T n ) - N -1 i=1 N -i k=1 ρ k δ i δ i+k ≤ C(ρ, q). Indeed, N i=1 q k=1 ρ k δ i δ i+k = N -1 i=1 q k=1 ρ k δ i δ i+k + q k=1 ρ k δ N δ N +k = N -1 i=1 N -i k=1 ρ k δ i δ i+k + q k=1 ρ k δ N δ N +k + N -1 i=N -q+1 q k=N -i+1 ρ k δ i δ i+k
where the second term is bounded in absolute value by q × max i=1...q |ρ i | and the third term by q(q+1) 2 × max i=1...q |ρ i |.

Lemma 4.4. For all h ∈ R and all β ≥ 0,

F a (β, h) = lim N →+∞ 1 N log E β e (h+ β 2 2 +log λ(β))ıN δ N .
Proof. By decomposing on the possible values of ı N , we have on one hand:

E e (h+ β 2 2 )ıN +β 2 ı N n=1 G(T n ) δ N = N n=1 e (h+ β 2 2 )n t1,...tn t1+...+tn=N e β 2 n k=1 G(t k ) Q(t 1 , t 2 ) . . . Q(t n-1 , t n )K ⊗q (t 1 ) = N n=1 e (h+ β 2 2 +log λ(β))n λ(β) n t1,...tn t1+...+tn=N Q β (t 1 , t 2 ) . . . Q β (t n-1 , t n )K ⊗q (t 1 ) = N n=1 e (h+ β 2 2 +log λ(β))n × t1,...tn t1+...+tn=N ν β (t 1 ) ν β (t n ) Qβ (t 1 , t 2 ) . . . Qβ (t n-1 , t n )K ⊗q (t 1 )
and on the other hand,

E β e (h+ β 2 2 +log λ(β))ıN δ N = N n=1 e (h+ β 2 2 +log λ(β))n t1,...tn t1+...+tn=N Qβ (t 1 , t 2 ) . . . Qβ (t n-1 , t n )K ⊗q (t 1 ) Since ν β (t) = ν * β (t *
) and ν * is a finite vector with positive components, there exists c and C two positive constants such that for all t 1 , t n , c ≤

ν β (t1)
ν β (tn) ≤ C. We conclude by using this remark and Lemma 4.3.

A short proof of Theorem 4.1 in the positive recurrent case

In accordance with Lemma 4.4, we will work on the homogeneous pinning model of the process τ under P β . In the positive recurrent case, a renewal-type lemma is obtained, which allows us to conclude. Lemma 4.5. If α > 1, or if α = 1 and L is such that n≥1 L(n)/n < ∞ then ıN N tends P β -almost surely and in L 1 (P β ) to a positive constant. Proof. From Lemma 4.2, under P β , the sequence of q-tuples (T k , . . . , T k+q-1 ) k≥0 is a positive recurrent Markov chain, with invariant probability measure µ β . If the previous conditions on α are satisfied, π 1 : t → t 1 (the projection on the first coordinate) is µ β -integrable. As a consequence,

τ N N = 1 N N k=1 π 1 (T k , . . . , T k+q-1 ) P β -a.s.
-→ c := t2,...,tq≥1

t 1 × µ β (t 1 , . . . , t q ) < ∞.
We deduce from this that ıN N P β -a.s.

→ 1 c > 0 by using the inequality τ ıN ≤ N ≤ τ ıN +1 . The convergence in L 1 follows from the Dominated Convergence Theorem.

From Lemma 4.4, h ≤ -β 2 2 -log λ(β) implies that F a (β, h) = 0. Suppose now that h = -β 2 2 -log λ(β) + ǫ with ǫ > 0. By Jensen's inequality, we have that 1 N log E β (e ǫıN ) ≥ ǫ E β (ı N ) N .
We conclude that F a (β, h) > 0 by using Lemma 4.5 and Lemma 4.4.

Proof of Theorem 4.1 in the general case

We now give a proof without any assumption on α. The starting point is Lemma 4.4 and we will actually identify the free energy of the pinning model associated to the law P β . Let's fix ǫ > 0. We introduce the matrices Qβ,F (s, t) = e -F tq Qβ (s, t)

and Q * β,F (s * , t * ) = e -F φF (t * q ) Q * β (s * , t * )
where φ F (s * ) = s * if s * ∈ {1, . . . , q} and

φ F (⋆) = - 1 F log t>q e -F t K(t) K(⋆) i.e. φ F (⋆) verifies e -F φF (⋆) K(⋆) = t>q e -F t K(t). (8)
We will denote by Λ(β, F ) the Perron-Frobenius eigenvalue of the irreducible matrix Q * β,F .

Lemma 4.6. There is a unique positive real denoted by F β (ǫ) such that

Λ(β, F β (ǫ)) = exp(-ǫ).
Proof. Componentwise, Q * β,F is smooth and strictly decreasing with respect to F . Since Λ(β, F ) is a simple root of the characteristic equation of Q * β,F (see [15, Thm 1.1]), Λ(β, F ) is also a smooth function of F by the Implicit Function Theorem. From the formula (see [START_REF] Seneta | Non-negative matrices and Markov chains[END_REF])

Λ(β, F ) = max v≥0 E q vi=1 min j:vj >0 ( Q * β,F v) j v j
one also obtains (see [START_REF] Giacomin | Random polymer models[END_REF]Appendix A.8]) that Λ(β, F ) is strictly decreasing in F and that Λ(β, F ) → 0 as F → ∞. Since Λ(β, 0) = 1 > exp(-ǫ), the result follows.

Let ν * be a Perron-Frobenius right eigenvector of Q * β,F β (ǫ) . We define ν by

ν(t) = ν * (t * ). ( 9
)
Lemma 4.7. The matrices

P F (s, t) := Qβ,F β (ǫ) (s, t) ν(t) ν(s) (10)
and

P * F (s * , t * ) := Q * β,F β (ǫ) (s * , t * ) ν * (t * ) ν * (s * ) ( 11
)
are stochastic and irreducible matrices. Furthermore, if we denote by l * the invariant probability measure of P * F , then l defined by

l(s) := l * (s * ) q j=1 K(s j )e -F β (ǫ)sj K(s * j )e -F β (ǫ)φ F β (ǫ) (s * j ) ( 12
)
is the invariant probability measure of P F .

Proof. The proof is left to the reader. It consists in straightforward computations very similar to Lemma 4.1 and Lemma 4.2. We use Lemma 4.6 to prove [START_REF] Hammersley | Generalization of the fundamental theorem on subadditive functions[END_REF],and ( 8), ( 9), [START_REF] Hammersley | Generalization of the fundamental theorem on subadditive functions[END_REF] to prove [START_REF] Giacomin | Random polymer models[END_REF].

Note that, like Q β and Qβ , P F satisfies the "consistancy" condition

P F (s, t) = 0 ⇔ s t.
This allows us to define a new law P (F ) on τ , the law for which (T n ) n≥1 is a Markov chain with transition kernel P F and initial distribution l.

Lemma 4.8. There exists two constants C ≥ c > 0 such that

ce F β (ǫ)N P (F ) (N ∈ τ ) ≤ E β (exp(ǫı N )δ N ) ≤ Ce F β (ǫ)N P (F ) (N ∈ τ ).
Proof. Decomposing the partition function and using [START_REF] Giacomin | Random polymer models[END_REF] we get

E β (exp(ǫı N )δ N ) = N n=1 t1,...,
P F (t 1 , t 2 ) . . . P F (t n-q , t n-q+1 )l(t 1 ) × ν(t 1 ) ν(t n-q+1 ) K ⊗q (t 1 )e -F (ǫ)(t1+...+tq) l(t 1 ) e ǫq
and, from ( 9), ( 12) and the finiteness of E q , the term in parenthesis is uniformly bounded by two positive constants C and c.

From this we deduce:

Lemma 4.9. For all ǫ > 0,

lim 1 N E β (exp(ǫı N )δ N ) = F β (ǫ) > 0.
This lemma (combined with Lemma 4.4) tells us that

F a (β, - β 2 2 -log λ(β) + ǫ) = F β (ǫ).
Proof. Since P (F ) (N ∈ τ ) ≤ 1, it will be sufficient to prove that lim inf

N →∞ P (F ) (N ∈ τ ) > 0.
We use an argument that has been already used in the study of Markov renewal processes arising in the study of periodic pinning (see [START_REF] Asmussen | Applied probability and queues, Applications of Mathematics[END_REF]Chp VII.4], [START_REF] Caravenna | A renewal theory approach to periodic copolymers with adsorption[END_REF] or [START_REF] Giacomin | Random polymer models[END_REF]Chp 3]). We choose arbitrarily the state 1 = (1, . . . , 1) ∈ (N * ) q . Consider (θ n ) n≥0 the following sequence of stopping times:

θ 0 = inf{n ≥ 1|T n = 1 } θ k+1 = inf{n > θ k |T n = 1 }
Since (T n ) n≥0 is positive recurrent under P (F ) , these stopping times are finite almost surely. If we now define the process τ θ by τ θ n := τ θn then it is clear that

P (F ) (N ∈ τ ) ≥ P (F ) (N ∈ τ θ )
By the strong Markov property, τ θ is a (delayed) renewal process whose interarrival times are on average equal to

m := E (F ) 1 (T 1 + . . . + T θ0-1 ) = E (F ) 1 ∞ n=1 π 1 (T n )1 {θ0>n} = t π 1 (t)E (F ) 1 ∞ n=1 1 {T n=t,θ0>n} = t π 1 (t) l(t) l(1 ) = k≥1 k(l • π -1 1 )(k) l(1 ) < ∞.
By the Renewal Theorem, we have

P (F ) (N ∈ τ θ ) N -→∞ → 1/m > 0
and the proof is complete.

Theorem 4.1 is now a direct consequence of Lemma 4.4 combined with Lemma 4.9.

The weak disorder asymptotic: proof of Proposition 4.2

We now give some lemmas which will be useful for the proof of Proposition 4.2. If I ⊂ E q and x, y ∈ E q then we will denote by Q * ,I the matrix with entries Q * ,I (x, y) = Q * (x, y)1 {y∈I} . If M is an n by n matrix then Com(M ) is the matrix of the cofactors of M , i.e. Com(M )(i, j) = (-1) i+j det M i,j where M i,j is the n -1 by n -1 matrix obtained by deleting the i-th line and the j-th column of M . Lemma 4.10. Q * is primitive and its invariant probability measure is K ⊗q (s * ) = K(s * 1 ) . . . K(s * q ).

Proof. For all t

* ∈ E q , (K ⊗q Q * 0 )(t * ) = s * ∈E q K ⊗q (s * )Q * 0 (s * , t * ) = s * ∈E K ⊗q (s * , t * 1 , . . . , t * q-1 )K(t * q ) = s * ∈E K(s * )K(t * 1 ) . . . K(t * q-1 )K(t * q ) = K ⊗q (t * ) so K ⊗q is the invariant probability measure of Q * 0 . Moreover, (Q * ) q (s * , t * ) = P T * q+1 = t * 1 , . . . , T * 2q = t * q |T * 1 = s * 1 , . . . , T * q = s * q = P T * q+1 = t * 1 , . . . , T * 2q = t * q = K ⊗q (t * )
which is positive under the assumptions of Section 2.1. Since (

Q * 0 ) q > 0, Q * 0 is primitive. Lemma 4.11. Tr (Com(Id -Q * )) = 0 and for all x ∈ E q Tr t Com(Id -Q * )Q * ,{x} Tr ( t Com(Id -Q * )) = K ⊗q (x).
Proof. In this proof we will use the properties of the Perron-Frobenius eigenvalue of a primitive matrix, that one can find for example in [START_REF] Seneta | Non-negative matrices and Markov chains[END_REF]. We define for all x ∈ E q :

p(x * ) := Tr t Com(Id -Q * )Q * ,{x} Tr ( t Com(Id -Q * ))
.

By Lemma 4.10, we only need to prove that p is the invariant probability measure of Q * . Since Q * is stochastic, 1 is clearly a right eigenvalue of Q * with associated eigenvector 1 (the vector with 1 on all its components). Moreover, Q * is primitive (Lemma (4.10)) so the Perron-Frobenius eigenvalue exists and all we have to prove is that |λ| ≤ 1 for every (possibly complex) eigenvalue of Q * . Indeed, if v is an eigenvector associated with such an eigenvalue, and x ∈ E q is such that v(x) = max y∈E q |v(y)| then

λv(x) = y∈E q Q * (x, y)v(y) so |λ||v(x)| ≤ |v(x)|, i.e |λ| ≤ 1.
This proves that 1 is the Perron-Frobenius eigenvalue of Q * , with associated eigenspace R1 . Now, from [15, Ch. 1, Corollary 2], we have that the rows of t Com(Id -Q * ) are all equal to the same left eigenvector (for the eigenvalue 1) of Q * , that we will denote by L. A first consequence is that Tr (Com(Id -Q * )) = 0 because the entries of L are either all positive or all negative. Another consequence is that if we define

m = (m(x)) x∈E q = Tr t Com(Id -Q * )Q * ,{x * } x∈E q then m(x) = L(x) for all x ∈ E q . Moreover, from the relation (Id -Q * ) t Com(Id -Q * ) = 0 we deduce that x∈E q m(x) = Tr t Com(Id -Q * ) . Since p is simply m renormal- ized by x∈E q m(x), it is the invariant probability of Q * .

Proof of Proposition 4.2.

In what follows, we will use the notations Q ′ 0 and Q ′′ 0 as shortcuts for ∂Q * β ∂β (0) and

∂ 2 Q * β ∂β 2 (0)
. First we will show that β → λ(β) is infinitely differentiable (C 2 would be enough). Let's define φ(β, λ) = det(λ Id -Q * β ) so that φ(β, X) is the characteristic polynomial of Q * β , and φ(β, λ(β)) = 0 for all β. The Perron-Frobenius eigenvalue of a nonnegative primitive matrix being a simple root of its characteristic equation, ∂φ ∂λ (β, λ(β)) = 0 for all β ≥ 0. Since φ is infinitely differentiable, the same holds for λ by the Implicit Function Theorem. Now, a straightforward computation shows that (we use that λ(0) = 1)

∂ ∂β log λ(β)| β=0 = λ ′ (0) ∂ 2 ∂β 2 log λ(β)| β=0 = λ ′′ (0) -λ ′ (0) 2 .
All we need to show then is

λ ′ (0) = 0 (13) λ ′′ (0) = 2 q n=1 ρ n P(n ∈ τ ). ( 14
)
By derivating the relation φ(β, λ(β)) = 0 we obtain

∂φ ∂β (0, 1) + λ ′ (0) ∂φ ∂λ (0, 1) = 0.
We already know that ∂φ ∂λ (0, 1) = 0 and since Q ′ 0 = 0 then ∂φ ∂β (0, 1) = 0, which leads to [START_REF] Lacoin | The martingale approach to disorder irrelevance for pinning models[END_REF].

All we have to do now is to prove [START_REF] Peng | Long-range correlations in nucleotide sequences[END_REF]. A Taylor expansion of det(λ(β) Id -Q * β ) gives:

det(λ(β) Id -Q * β ) = det Id -Q * + (λ ′′ (0) Id -Q ′′ 0 ) β 2 2 + o(β 2 ) = Tr t Com(Id -Q * )(λ ′′ (0) Id -Q ′′ 0 ) β 2 2 + o(β 2 )
where we have used the differential of the determinant: det

(A + H) = det(A) + Tr( t Com(A)H) + o( H ). But since det(Q * β -λ(β) Id) = 0 we have Tr t Com(Id -Q * )(λ ′′ (0) Id -Q ′′ 0 ) = 0 which yields λ ′′ (0) = Tr( t Com(Id -Q * )Q ′′ 0 ) Tr( t Com(Id -Q * ))
.

Note that Tr( t Com(Id -Q * )) = 0 (Lemma 4.10).

Let's now consider Q ′′ 0 as a function of (ρ n ) 1≤n≤q . Observe that

Q ′′ 0 (s * , t * ) = 2G(t * )Q * 0 (s * , t * )
so Q ′′ 0 linearly depends on (ρ n ) 1≤n≤q . We have then

Q ′′ 0 (ρ 1 , . . . , ρ q ) = Q ′′ 0 (0, . . . , 0, ρ q ) + Q ′′ 0 (ρ 1 , . . . , ρ q-1 , 0)
The result of the theorem is clearly true for q = 1 (remember that we have an explicit expression of h a c (β) in this case, see Proposition 4.1 ) so we can suppose that it is true for a (q -1)-order moving average and show that the result holds for q. The induction hypothesis then implies

Tr( t Com(Id -Q * )Q ′′ 0 (ρ 1 , . . . , ρ q-1 , 0)) Tr( t Com(Id -Q * )) = 2 q-1 n=1 ρ n P (n ∈ τ )
so the only thing left to prove is that

Tr( t Com(Id -Q * )Q ′′ 0 (0, . . . , 0, ρ q )) Tr( t Com(Id -Q * )) = 2ρ q P (q ∈ τ ). ( 15
)
Let's define I q = {s * ∈ E q s.t. ρ q appears in G(s * )} and notice that

Q ′′ 0 (0, . . . , 0, ρ q ) = 2ρ q Q * ,Iq .
We obtain from Lemma 4.11:

Tr( t Com(Id -Q * )Q * ,Iq ) Tr( t Com(Id -Q * )) = t * ∈Iq K ⊗q (t * ) = P (q ∈ τ )
which proves (5.6).

The irrelevance regime

In this section we will work with free partition functions (remove the δ N in definitions ( 2),( 3) and ( 4)). This has no incidence on the free energy.

Introduction and statement of the result

The following result states that under some assumptions on K and β, quenched and annealed critical curves and exponents are the same. This is the irrelevance regime.

Theorem 5.1. If ω is a gaussian moving average of finite order q and if α ∈ (0, 1/2)

or if α = 1/2 and L is such that ∞ n=1 1 nL(n) 2 < ∞ then there exists β 0 > 0 such that for β ≤ β 0 , h c (β) = h a c (β) and lim h→h a c (β) + log(F (β, h)) log(h -h a c (β)) = 1 α (16)
Disorder irrelevance has been proved by several authors, with different methods, in the case of i.i.d. disorder (see [START_REF] Lacoin | The martingale approach to disorder irrelevance for pinning models[END_REF], [START_REF] Toninelli | A replica-coupling approach to disordered pinning models[END_REF] and [START_REF] Alexander | The effect of disorder on polymer depinning transitions[END_REF]). A key element is the control of the second moment of the partition function at the annealed critical point, which is linked to the exponential moments of the number of intersections between two replicas of the initial renewal process τ . As in [START_REF] Lacoin | The martingale approach to disorder irrelevance for pinning models[END_REF], we will establish ( 16) by proving separately the lim inf and the lim sup parts. The lim inf part is just a consequence of Jensen's inequality F (β, h) ≤ F a (β, h) and of the behaviour of F a (β, h) near the annealed critical point. The lim sup part relies on the control of the second moment. In our case, additional difficulties arise from the presence of a Markov renewal process instead of a classical renewal process at the annealed critical point. Moreover the law of this Markov renewal process depends on β, so we will tackle a problem of continuity in β (see end of Section 5.3.1). Once the second moment is controlled, we use arguments from [START_REF] Lacoin | The martingale approach to disorder irrelevance for pinning models[END_REF] to conclude. Unlike what the title of [START_REF] Lacoin | The martingale approach to disorder irrelevance for pinning models[END_REF] suggests, there is no martingale involved in our problem.

Henceforth, we assume α satisfies the assumption of Theorem 5.1.

The lim inf part

The following proposition tells us that at the neighbourhood of the annealed critical point, the annealed free energy has the same behaviour as the homogeneous one.

Proposition 5.1. There exists a slowly varying function L ′ such that

F a (β, h a c (β) + ∆) ∆ց0 ∼ L ′ (∆)∆ 1/α .
Proof. The annealed free energy is defined by the implicit equation

Λ(β, F a (β, h a c (β) + ∆)) = e -∆
where Λ(β, F ) is the Perron-Frobenius eigenvalue of Q * β,F (see Lemma 4.6 and 4.9). This can be rewritten as:

1 -Λ(β, F a (β, h a c (β) + ∆)) = 1 -e -∆
and since the right-hand term is of the order of ∆ when ∆ goes to 0, it is enough to prove that the left-hand term is of the order of ∆ α .Indeed, if t * is such that

t * q ∈ {1, . . . , q} then Q * β,F (s * , t * ) -Q * β (s * , t * ) ∼ F ց0 -cste F t q but if t * q = ⋆, we have by Abelian arguments Q * β,F (s * , t * ) -Q * β (s * , t * ) = -L ⋆ (1/F )F α
where L ⋆ is a slowly varying function. We conclude the proof by writing

Λ(β, F ) -1 = Λ( Q * β,F ) -Λ( Q * β ) ∼ F ց0 DΛ Q * β ( Q * β,F -Q * β ) = -cste L ⋆ (1/F )F α .
where Λ is a differentiable function of the (q + 1) 2q entries of positive matrices.

The lim sup part

We adopt the following notations:

K β,x,y (n) = P β (T k = n, T * k-q+1 = y|T * k-q = x) and P β,x,y (n ∈ τ ) = k≥0 P β (τ k = n, T * k-q+1 = y|T * 1-q = x)
This section is organized as follows: in a first part we look at the intersection between two replicas of a Markov renewal process under the law P β . From this we control in a second part the second moment of the partition function at the annealed critical point. In a last part, we exploit this result to obtain the lim sup part of Theorem 5.1.

Intersection of Markov renewal processes

The main result of this part is:

Proposition 5.2. There exists β 0 > 0 such that for all β ≤ β 0 and for all l ∈ {0, . . . , q},

E ⊗2 β   exp(β 2 n≥1 δ (1) n δ (2) n+l )   < ∞.
As it will be explained further in the proof, it is enough to focus on the case l = 0, when the term inside the exponential is the number of intersections of two independent copies of a Markov renewal process with law P β . We begin with the following observation: Proposition 5.3. If τ (1) and τ (2) are two independent copies of a Markov renewal process with law P β , then τ (1) ∩ τ (2) is a (delayed) Markov renewal process.

The proof is left to the reader. It is a matter of writing that conditionally on the event that τ (1) and τ (2) meet at some point n, then the future, in particular the next intersection point, only depends on the states of the Markov modulating chains of τ (1) and τ (2) at n.

In the above proposition, the term Markov renewal process has to be understood in the large sense: it can happen (and actually it will be the case in the range of α's we consider) that (τ (1) ∩τ (2) ) n = +∞ for some n ≥ 1. We will denote by P ∩ β the law of this intersection Markov renewal process, with Markov modulating chain in (E q ) 2 , and (K ∩ β,x,y (n)) n≥1,x,y∈(E q ) 2 its semi-Markov (sub)kernel. Hence we have to prove that E ∩ β exp(β 2 n≥1 δ n ) < ∞ if β is small enough. We define the following matrices of Laplace transforms (for λ ≥ 0):

ϕ β,x,y (λ) := n≥1 e -λn P ∩ β,x,y (n ∈ τ ) φ β,x,y (λ) := n≥1 e -λn K ∩ β,x,y (n).
Notice that φ β (0) is the matrix of the P ∩ β,x,y (τ 1 < ∞)'s for x, y ∈ (E q ) 2 . Proposition 5.4. The matrix φ β (0) is irreducible and nonnegative. If we denote by θ(β) its Perron-Frobenius eigenvalue then

1. θ(0) = P ⊗2 ((τ (1) ∩ τ (2) ) 1 < ∞) = 1 -( n≥0 P (n ∈ τ ) 2 ) -1 < 1.
2. For all β, there exists a constant c such that

P ∩ β   n≥1 δ n ≥ N   ≤ c × θ(β) N .
Proof. First we prove the irreducibility. Let x = (x (1) , x (2) ) and y = (y (1) , y (2) ) be in (E q ) 2 . We want to prove that there exists a sequence x 0 := x, x 1 , x 2 , . . . , x i = y with i ≥ 1 such that

i k=0 P ∩ β,x k ,y k (τ 1 < ∞) > 0.
It is enough to show that

i k=0 K ∩ β,x k ,y k (n k ) > 0 ( 17
)
for some n k ≥ 1. One can find without much difficulty a path of positive probability on which τ (1) starts from x (1) , τ (2) starts from x (2) and they intersect at some point where respectively they are in states y (1) and y (2) . This path provides suitable i and (x k , n k ) 1≤k≤i .

For the first point, we will only prove the first part of the equality, that is θ(0) = P ⊗2 ((τ (1) ∩ τ (2) ) 1 < ∞). The other part has been stated several times in the literature (see [START_REF] Toninelli | A replica-coupling approach to disordered pinning models[END_REF] for instance). Remember that at β = 0 the Markov renewal process is in fact the initial (classical) renewal process, and so the quantities P ∩

x,y (n ∈ τ ), K ∩ x,y (n) and P ∩ x,y (τ 1 < ∞) do not depend on x. As a consequence, the quantity P ⊗2 ((τ (1) ∩ τ (2)

) 1 < ∞) = y∈(E q ) 2 P ∩ x,y (τ 1 < ∞)
is an eigenvalue of φ 0 (0) with positive right eigenvector 1 .

For the last point we have

P ∩ β   n≥1 δ n ≥ N   ≤ P ∩ β (τ 1 < ∞, . . . , τ N < ∞) ≤ x0,...,xN ∈(E q ) 2 N -1 i=0 P ∩ β,xi,xi+1 (τ 1 < ∞) ≤ x0,xN φ β (0) N x0,xN ≤ c × θ(β) N .
This proposition implies that if β is such that e β 2 θ(β) < 1 then

E ∩ β   exp(β 2 n≥1 δ n )   < ∞.
In other words, the only thing left to prove is that for β small enough, e β 2 θ(β) < 1. Actually we will prove that θ(β) is continuous at β = 0. Since we do not have direct access to P ∩ β , we first find a formula which is analogous to P ⊗2 ((τ (1) ∩ τ (2) )

1 < ∞) = 1 -( n≥0 P (n ∈ τ ) 2 ) -1 ,
i.e. which relates θ(β) to sums of Green functions of P ∩ β . Proposition 5.5. The matrix ϕ β (0) has finite components and is irreducible. If we denote by ϑ(β) the Perron-Frobenius eigenvalue of ϕ β (0) then

θ(β) = 1 -ϑ(β) -1 .
Before proving Proposition 5.5, we need a lemma, and for this lemma we need additional notations. We define E q ⋆ = {x ∈ E q : x q = ⋆} and for all x, y ∈ E q , Kx,y (n

) = K β,x,y (n) Q * β (x, y)
which is the probability under P β of making a jump of size n knowing departure state x and arrival state y. The letter β is omitted because the Kx,y (n)'s do not depend on it: actually, if y ∈ E q ⋆ , then Kx,y (n) = K(n) K(⋆) 1 n>q ; otherwise, Kx,y (n) = 1 n=yq . Lemma 5.1. For all x 0 , x 1 , . . . ,

x k ∈ E q , ( Kx0,x1 * Kx1,x2 * . . . * Kx k-1 ,x k )(n) n→∞ ∼ L(n) K(⋆)n 1+α ×|{1 ≤ l ≤ k, x l ∈ E q ⋆ }| (18)
and there exists c > 0 such that for all k, x 0 , x 1 , . . . , x k ∈ E q and n > kq

( Kx0,x1 * Kx1,x2 * . . . * Kx k-1 ,x k )(n) ≤ k c L(n) K(⋆)n 1+α (19)
Proof. Assertion (18) comes from the fact that if q(n) = L(n)/n 1+α is a probability kernel with L a slowly varying function, then q * k (n) ∼ kq(n) (see [START_REF] Giacomin | Random polymer models[END_REF]Lemma A.5]) and only the Kx,y 's for which y ∈ E q ⋆ contribute to the tail behaviour. For k = 1, ( 19) is clearly true. One can adapt the induction in the proof of [START_REF] Giacomin | Random polymer models[END_REF]Lemma A.5] to conclude.

Proof of Proposition 5.5. First we prove finiteness of the components. Let x = (x 1 , x 2 ), y = (y 1 , y 2 ) be in (E q ) 2 . We have

n≥0 P ∩ β,x,y (n ∈ τ ) = n≥0 P β,x1,y1 (n ∈ τ )P β,x2,y2 (n ∈ τ ) (20)
so we have to look at the tail behaviour of the P β,x1,y1 (n ∈ τ )'s. But for all x, y ∈ E q we can write

P β,x,y (n ∈ τ ) = P β,x (n ∈ τ y )
where τ y,n = τ θn(y) , and

θ 0 (y) = inf{k ≥ 0, T * k-q+1 = y} θ n+1 (y) = inf{k > θ n (y), T * k-q+1 = y}
By the markov renewal property, under P β , for all y ∈ E q , τ y is a (delayed, because we can start at x = y) classical renewal process. We are then left with proving that the interarrival distribution of τ y has (approximately) the same tail behaviour as the original renewal process (which satisfies the assumptions of Theorem 5.1). We then conclude with the result of [START_REF] Doney | One-sided local large deviation and renewal theorems in the case of infinite mean[END_REF] on renewal theorems with infinite mean to show that the series in (20) converges. We fix the state y ∈ E q , and write θ = θ 0 (y), which is finite almost surely ((T * n ) n≥1 is a recurrent Markov chain). We write J n = T * n-q the markov modulating chain. Then

P β,y (T 1 + . . . + T θ = n) = k≥1 E β,y 1 {θ=k} P z (T 0 + . . . + T θ = n|J 0 . . . J k ))
From our previous remark on the laws Ky,z and Lemma 5.1 we have

P β,y (T 0 + . . . + T θ = n|J 0 . . . J k ) = (K J0,J1 * K J1,J2 * . . . * K J k-1 ,J k )(n) n→∞ ∼ L(n) n 1+α × N ⋆ k K(⋆)
where

N ⋆ k := k i=1 1 {Ji∈E q ⋆ } .
From Markov chain theory (see [START_REF] Asmussen | Applied probability and queues, Applications of Mathematics[END_REF]Chap I.3] for example),

E β,y k≥1
1 {θ=k} N ⋆ k = E β,y N ⋆ θ = l * β (E q ⋆ ) l * β (y)
where l * β is the invariant probability measure of Q * β . Finally we have

P β,y (T 0 + . . . + T θ = n) ∼ l * β (E q ⋆ ) l * β (y)K(⋆) × L(n) n 1+α , (21)
but one has to justify the interchange of the integration and the asymptotic equivalent. Indeed, from the upper bound [START_REF] Toninelli | Localization transition in disordered pinning models[END_REF] in Lemma 5.1, one can apply the Dominated Convergence Theorem, because E(θ c ) < ∞ (it is not hard to see that the tail of θ decays exponentially fast). We prove the last point of the proposition. The following Markov renewal equation hold: for all x, y ∈ (E q ) 2 ,

P ∩ β,x,y (n ∈ τ ) = δ x,y 1 n=0 + z∈(E q ) 2 n k=1 P ∩ β,x,z (n -k ∈ τ )K ∩ β,z,y (k) (22) = δ x,y 1 n=0 + z∈(E q ) 2 n k=1 K ∩ β,x,z (k)P ∩ β,z,y (n -k ∈ τ ) (23)
Taking the Laplace transforms we get for λ > 0

ϕ β (λ) = Id +ϕ β (λ)φ β (λ) = Id +φ β (λ)ϕ β (λ).
Thanks to the first part of the proposition, that has been just proved, we can take the limit as λ goes to 0, which yields

ϕ β (0)(Id -φ β (0)) = (Id -φ β (0))ϕ β (0) = Id
from which we can conclude.

As a consequence, we will prove that ϑ(β) is continuous at β = 0, which is the same as proving that for all x, y in (E q ) 2 , the series n≥0 P ∩ β,x,y (n ∈ τ ) are continuous at β = 0. It is not difficult to see that for all n ≥ 0, x, y ∈ (E q ) 2 , the Green function P ∩ β,x,y (n ∈ τ ) is continuous in β but the continuity of the series

n≥0 P ∩ β,x,y (n ∈ τ )
is not immediate. We will see that the last quantity can be written as the L 2 norm of a certain function. The continuity will thus be proved on this L 2 norm via the Dominated Convergence Theorem. We define the following Fourier series:

φβ,x,y (θ) = n≥1 e iθn K β,x,y (n) φβ,x,y (θ) = n≥0 e iθn P β,x,y (n ∈ τ ) φsym β,x,y (θ) = n∈Z e iθn P β,x,y (|n| ∈ τ )
The matrix φ0 (θ) will be written φ(θ). The functions φβ,x,y are continuous whereas φβ,x,y (θ) and φsym β,x,y (θ) are in L 2 (-π, π) (the space of functions which are square integrable with respect to the Lebesgue measure on (-π, π)), because of our knowledge on the decay of the P β,x,y (n ∈ τ )'s (cf previous section).

Proposition 5.6. The matrix Id -φβ (θ) is θ-almost everywhere invertible, and

P β,x,y (n ∈ τ ) = 1 2π π -π e -inθ 2 Re((Id -φβ (θ)) -1 x,y ) -1 dθ.
Furthermore there exists a positive constant C such that for β small enough, for all x, y ∈

E q |[(Id -φβ (θ)) -1 ] x,y | ≤ C sup s,t∈E q |[(Id -φ(θ)) -1 ] s,t | (24
θ) and more generally, for all x, y ∈ E q , one can show by decomposing the probability (under P β ) starting from x that n is in τ y according to the value of τ the first time state y is reached, and by taking the Fourier transform, that φsym β,x,y (θ) = 1 + φ(x,y)

β (θ) 1 - φ(y) β (θ)
From the Markov renewal equations ( 22), written for P β instead of P ∩ β , one can deduce that almost everywhere, φβ (θ) = Id + φβ (θ) φβ (θ) = Id + φβ (θ) φβ (θ), which proves the first part of the proposition. Then we can write

P β,x,y (n ∈ τ ) = 1 2π π -π e -inθ φsym β,x,y (θ)dθ = 1 2π π -π e -inθ (2 Re( φβ,x,y (θ)) -1) dθ = 1 2π π -π e -inθ 2 Re([Id -φβ (θ)] -1 x,y ) -1 dθ,
which ends the proof of the first part. Now we prove the second part of the proposition (equation ( 24)). We recall that φβ,x,y (θ) = φx,y (θ) e β 2 G(y) ν * β (y) λ(β)ν * β (x) and φβ := φβ (θ = 0) is simply the matrix Q * , so

λ(β) = y∈E q e β 2 G(y) ν * β (y) ν * β (x) φx,y . (25)
We define the matrix R β (θ) = λ(β)(Id -φβ (θ)) -(Id -φ(θ)). It is enough to prove that the coefficients of R β (θ)(Id -φ(θ)) -1 decrease to 0 as β tends to 0, uniformly in θ. This is not immediate because there is a singularity at θ = 0. Recall that

(Id -φβ (θ)) -1 = t Com(Id -φ(θ)) det(Id -φ(θ)) .
We know that as θ goes to 0, det(Id -φ(θ)) is of the lowest order among the ( φx,yφx,y (θ))'s, for x, y ∈ E q , so we have to look at the elements of R β (θ) t Com(Id -φ(θ)). More precisely we have to check that

• as θ goes to 0 there are no terms of order 0, • all terms of higher order have coefficients which go to 0 as β goes to 0.

On one hand we have where ǫ β (x, y) := e β 2 G(y) ν ⋆ β (y) ν ⋆ β (x) -1 (which tends to 0 as β goes to 0). From these expressions, one can verify without much difficulty that the second point is satisfied. For the terms of order 0 in θ of R β (θ) t Com(Id -φ(θ)), it is equal to 0; it is shown by computation, using the fact that t Com(Id -φ) is constant on its columns (see Section 4.2).

We can now conclude this first part with Proof of Proposition 5.2. . We begin with l = 0, i.e. we show that

E ∩ β (exp(β 2 n≥1 δ n )) < ∞
for β small enough. From Proposition 5.4, it is enough to show that e β 2 θ(β) < 1 for β small and as θ(0) < 1, we will show that θ(β) → θ(0) as β → 0. From Proposition 5.5, this reduces to the continuity of ϑ(β) at β = 0 and so, to the continuity of the series n≥0 P ∩ β,x,y (n ∈ τ ) at β = 0, for all x, y ∈ (E q ) 2 . From (20) one can write (and so that h c (β) ≤ h a c (β)). Therefore we have all the ingredients to prove the theorem.

(

  Id -φ(θ)) x,x = ( φx,xφx,x (θ)) + y =x φx,y and for x = y, (Id -φ(θ)) x,y = ( φx,yφx,y (θ)) + φx,yOn the other hand, we easily compute, using (25),R β (θ) x,x = ǫ β (x, x)( φx,xφx,x (θ)) + y =x ǫ β (x, y) φx,y R β (θ)x,y = ǫ β (x, y)( φx,yφx,y (θ)) -ǫ β (x, y) φx,y

< 2 Lemma 5 . 4 .Lemma 5 . 5 . 2 ) δn+ β 2 2 Lemma 5 . 6 .

 254552256 x1,y1 (|n| ∈ τ )P β,x2,y2 (|n| ∈ τ ) where x = (x 1 , x 2 ) and y = (y 1 , y 2 ). From Proposition 5.6 and Parseval'Re((Id -φβ ) -1 x1,y1 ) -1, 2 Re((Id -φβ ) -1 x2,y2 ) -1 > L 2 (-π,π) . (27)Proof. In the first part of the paper it was proved that EZ N,β,h a c (β) ≥ c 1 (β)P β (n ∈ τ ) when considering the constraint partition function. With free partition function, it is not difficult to prove thatEZ N,β,h a c (β) ≥ c 1 (β). For all β ≤ β 0 (β 0 as in Proposition 5.7), there exists δ > 0, c ∈ (0, 1) such that inf N P(Z N,β,h a c ≥ δ) > c. Proof. From Lemma 5.2, we have for all u ∈ (0, 1),P(Z N,β,h a c ≥ uE(Z N,β,h a c )) ≥ (1 -u) 2 (E(Z N,β,h a c )) 2 E(Z 2 N,β,h a c )For N large enough and β ≤ β 0 , we have from Lemma 5.3 and Lemma 5.7,P(Z N,β,h a c ≥ uc(β) 2 ) ≥ P(Z N,β,h a c ≥ uE(Z N,β,h a c )) ≥ (1 -u) 2 (c(β)/2) 2 sup N E(Z 2 N,β,h a c ) ,which is positive thanks to Proposition 5.7. The result follows by choosing u close enough to 1. For all β > 0, there exists C β > 0 such thatEZ N,β,h a c P N,β,h a c (A γ N ) ≤ C β P β (A γ N ).Proof.EZ N,β,h a c P N,β,h a c (A γ N ) = EE 1 AN e (βωn+h a c (β))δn = E 1 AN e (-log λ(β)-β 2 Var ωnδn ≤ C(β)E 1 AN e -log λ(β) δn+ δn q k=1 ρ k β 2 δ n+k ≤ C ′ (β)P β (A N ). For all β > 0, γ < α, P β (A γ N )Let us choose arbitrarily z in E q . Then:P β (A γ N ) = P β (|τ ∩ [0, N ]| ≤ N γ ) ≤ P β (|τ z ∩ [0, N ]| ≤ N γ )which tends to 0 as N tends to +∞ because the tail exponent of the return times of τ z is α (see proof of Proposition 5.5, equation (21)), so|τ z ∩ [0, N ]| is of the order of N α .Proof of Proposition 5.8. We first prove that for all a ∈ (0, 1),P(P N,β,h a c (A N ) > a) ≥ EP N,β,h a c (A N ) -a.Indeed, this follows fromP(P N,β,h a c (A N ) > a) ≥ EP N,β,h a c (A N )1 {P N,β,h a c (AN )>a} and EP N,β,h a c (A N ) ≤ a + EP N,β,h a c (A N )1 {P N,β,h a c (AN )>a} Then, from Lemma 5.5,EP N,β,h a c (A N ) ≤ E P N,β,h a c (A N )1 {Z N,β,h a c ≥δ} + P(Z N,β,h a c < δ) ≤ δ -1 EZ N,β,h a c P N,β,h a c (A N ) + 1 -P(Z N,β,h a c ≥ δ) ≤ δ -1 C β P β (A N ) + 1 -P(Z N,β,h a c ≥ δ) . From Lemma 5.6 and Lemma 5.4, we have lim inf N EP N,β,h a c (A N ) ≥ c so lim inf N P(P N,β,h a c (A n ) > a) ≥ c -a and we conclude the proof by choosing a in (0, c).5.4. Conclusion: proof of Theorem 5.1We can now conclude: Proof of Theorem 5.1. The bound[START_REF] Doney | One-sided local large deviation and renewal theorems in the case of infinite mean[END_REF] and Proposition 5.1 tell us that h c (β) ≥ h a c (β) and lim inf h→h a c (β) + log F (β, h) log(h -h a c (β))

Acknowledgements

The author would like to thank Giambattista Giacomin for an enlightening discussion.

But for all s, t ∈ E q , (Id -φβ ) -1 s,t L 2 → (Id -φ0 ) -1 s,t because (24) in Proposition 5.6 allows the use of the Dominated Convergence Theorem. As a consequence the scalar product in the right-hand side of ( 26) is continuous at β = 0.

We now deal with the case 1 ≤ l ≤ q. Let us write τ (2) -l := {τ

and τ (1) ∩ (τ (2) -l) is a delayed renewal process with the same interarrival time distribution as τ (1) ∩ τ (2) , which is the case l = 0.

Control of the second moment

We will prove in this section Proposition 5.7. There exists β 0 > 0 such that for all β ≤ β 0 ,

Proof. Replica method gives:

n , ωnδ (2) n )

and

and take h = h a c (β). Then

Moreover, notice that C N (τ (1) , τ (2) )

where ρ + q = ρ q ∨ 0. By repeated use of Holder's inequality we prove that for all β > 0, there exists nonnegative constants C β , c 0 , c 1 , . . . , c q and e 0 , e 1 , . . . , e q such that

We conclude by using Proposition 5.2.

End of lim sup part

We define:

sometimes we will omit the superscript γ). We will prove Proposition 5.8. For all β ≤ β 0 (with β 0 as in Proposition 5.7), for all γ < α, there exists c > 0 such that

Once this is proved, the following proposition provides the limsup part of Theorem 5.1: Proposition 5.9. If for all γ < α, there exists some positive constant c such that lim inf

Proposition 5.9 is proved in [START_REF] Lacoin | The martingale approach to disorder irrelevance for pinning models[END_REF]. One can check that the independance assumption is not needed there.

To prove Proposition 5.8, we need the control of the second moment (see Proposition 5.7) and several lemmas, such as: Lemma 5.2 (Paley-Zygmund inequality). If Z is a nonnegative random variable with finite variance, and if 0 < u < 1, then

) .

Lemma 5.3. For all β > 0, lim inf N →+∞ EZ N,β,h a c ≥ c(β) > 0.


Olivier Glassey
email: olivier.glassey@idheap.unil.ch

Why TaxMe Makes Taxpayers Happy?

Keywords: Taxation, case study, usability, portal, data exchange, open source, user acceptance

TaxMe-Online is the online tax declaration system of the Canton of Bern in Switzerland, where each of the 26 Cantons has its own fiscal regime and taxation system. In 2008 it was used by almost 26 percent of the Canton of Bern taxpayers (42% used the TaxMe-CD or other software, and the last third chose the paper forms). The TaxMe portal furthermore gives taxpayer access to their fiscal data (taxation status, amounts paid, etc.) and allows them to send electronic vouchers. TaxMe-Online does not require any preliminary registration as the taxpayers receive their user ID at the same time as the tax declaration forms, and when they log in with their identification data, their identity data (name, address, etc.) are already available. Users do however have to sign a paper-based validation declaration: until they have done so, their online tax declaration is not considered as finalized. The tax administration does not have the right to access taxpayers' data until the receipt of this validation declaration. TaxMe-Online is built on open source components and solutions; data are coded before being sent electronically (Secure Socket Layer). 33% of the TaxMe-Online users say they are "very happy" with this way of filling in their tax declaration, but amongst citizens using a similar solution on CD-ROM or the paper-based declaration, only 18% say they are very happy. This paper tries to find out why the online solution scores much higher than other tax declaration systems. It comprises three main parts (i) the development of an assessment model; (ii) a description of the system and its functionalities; and (iii) an analysis of user acceptance. We investigated the point of view of the TaxMe-Online users on an empirical basis, most notably by analysing secondary sources such as surveys realized by the fiscal administration of the Canton of Bern and newspaper articles, and by conducting interviews with various stakeholders.

Introduction

Online taxation systems are becoming more and more common at the international levels: there are too many eTaxation projects to list them all, but let us mention a few such as eDavki in Slovenia, Revenue Online Service in Ireland, Taxis in Greece, FinanzOnline in Austria, e-Filing in Singapore, e-Revenue in Thailand, etc. In Switzerland we seem to be a bit behind with only one Canton offering a fully online tax declaration system. As each of the 26 Cantons has its own fiscal regime, there are very heterogeneous solutions. However most Cantons do offer a solution (downloadable or on CD-ROM) to support taxpayers in the process of filling their tax declaration. The Canton of St-Gall had the concept for such a product in 1998 already and nowadays this solution called eTaxes is used in 5 other Cantons (Zug, Schwytz, Tessin, Luzern and Basel-Stadt). eTaxes users fill their declaration on their computer and have the possibility of sending their data over the internet, as opposed to other Cantons such as Vaud, Valais or Geneva where users still have to print their tax declaration and send it back to their fiscal administration via postal mail. A barcode is printed on the declaration, allowing the administration to scan all data quickly and accurately. For the time being the only fully online solution is TaxMe-Online in the Canton of Bern: this system will be described more in details in section 3.

Assessment Model

In previous work, we described the reorganization of a fiscal administration through the introduction of computer-aided taxation (citation removed), and a private-public partnership where employers provide electronic fiscal data on their employees directly to the tax administration and pay taxes on their behalf (citation removed). In order to compare these two case studies and the TaxMe system described in this paper we wanted to develop a suitable analysis framework. Here we will only apply this framework to the TaxMe case as the first step of the implementation of this model and in future work we will do a full comparative analysis. First of all we needed to analyse level of services (a taxpayer seeking information on his situation or a taxpayer just willing to pay taxes online do not have the same needs), as well as communication channels and modes. In order to do so we used the work of [START_REF] Quirchmayr | An Evaluation Framework for e-Taxation Services[END_REF]) who define four levels of services in e-Taxation:

• Information and intention building phase: governments publish information on the Web and taxpayers can download forms. • Contracting phase: taxpayers can file their tax declaration (or returns) online.

• Service delivery phase: taxpayers can pay their taxes online.

• Aftercare phase: complains and appeals are managed online.

Regarding the channel types, [START_REF] Ebbers | Rethinking Service and Channel Strategies after the Hype[END_REF]) write that three of them stand out as being used the most by citizens: Web, phone and front desk. In order to study these channels they defined several modes based on who is the initiator of the interaction and on the level of interactivity:

• Allocution: an organization sends information to the citizens.

• Registration: a citizen sends information to the organisation.

• Consultation: a citizen uses an information source provided by the organisation in order to find the data he/she needs. • Conversation: a user requests information and the organization provides the information, tailored to the user's needs. [START_REF] Ebbers | Rethinking Service and Channel Strategies after the Hype[END_REF]) furthermore integrate the concept of (financial) transactions, where money (and not information) is exchanged between citizens and governmental organisations.

Table 1 combines these two approaches and can be used either to describe the current situation of eTaxation systems (as-is situation) or to identify user needs and requirements for a future system (to-be situation). In this example we have listed the existing communication modes corresponding to front desk/phone/online communication channels for each level of services. We consider the paper-based workflow for taxation to be integrated within the traditional front desk channel.

Contracting

Registration Registration

Service Delivery

Transaction Transaction

Aftercare

Conversation Conversation

Furthermore we needed dimensions and criteria in order to assess eTaxation systems. We combined two models that are not so different from one another and that cover similar grounds. The first one was developed by [START_REF] Quirchmayr | An Evaluation Framework for e-Taxation Services[END_REF] who propose a number of evaluation criteria for each of the following dimensions of e-Taxation services:

• Information quality: relevant, accurate, timely and context-pertinent information is provided.

• Usability: system is easy to use and provides support (search, download, help, FAQ, etc.)

• Multi-channel: information is provided on multiple channels and taxpayers can return their data likewise.

• Security and privacy: information is sent at the right person at the right place over secure communication channels. • Service quality: service should not only be faster but also better • Empowerment: instructions are available to help users master the different channels and services.

The second one is from [START_REF] Economides | Evaluating tax sites: An evaluation framework and its application[END_REF] who developed the Tax Site Evaluation Framework (TSEF) that is based on the following five categories:

• Content: quantity, quality and personalization.

• Presentation: appearance, format and multimedia.

• Usability: user interface, navigability, orientation, search, structure.

• Technical: security and privacy, performance, compatibility, reliability and maintenance.

• E-Services & Interactivity: quantity, quality, online payment, online learning, technical services, informational services.

However these two models only look at the supply-side (content and services provided by the tax administration) and do not allow us to analyse the demand-side or the user uptake of such systems. We used the work of (Gareis and Forte 2007) who based their assessment of online tax declaration use in nine European countries on the ACM model (Access to the internet, Competence in using the internet and Motivation for using the internet to communicate with tax authorities). We extended the motivation dimension by using the model of (Yung et al 2006) who applied the theory of planned behaviour to measure user acceptance of the e-Taxation system in Taiwan. They introduce concepts such as attitude towards perceived risk [START_REF] Stone | Perceived risk: Further considerations for the Marketing Discipline[END_REF], perceived usefulness [START_REF] Davis | Perceived Usefulness, Perceived Ease of Use, and User Acceptance of Information Technology[END_REF] or trust [START_REF] Gefen | E-government Adoption[END_REF], as well as subjective norms and intentions [START_REF] Taylor | Understanding Information Technology Usage: A test of Competing Models[END_REF].

Table 2 shows the two sides of our assessment model and lists the dimensions that we will analyse. We will not discuss in detail why we chose one indicator over the other from the literature mentioned above.

We will however point out that there is a correspondence between supply-side indicators and demandside indicators: content is linked to access, usability is related to competence, perceived risk is a function of the communication channel, trust is based on security and privacy guarantees, and so on.

Description of theTaxMe System

The tax administration of the Canton of Bern is responsible to levy Federal, Cantonal and municipal taxes for all natural and legal persons that are residents in the Canton of Bern. The tax administration is decentralized over five regions and has a central office in the City of Bern. It has the strategic objective of being one of the three best tax administrations in Switzerland and of being effective, efficient, and citizenoriented. In order to achieve this goal the tax administration relies (amongst others) on technology since 1999 when the fist experiences with a CD-based solution was conducted. In 2001 the concept of TaxMe-Online was introduced and it has since become a success story distinguished by a "Eugen e-Government Award" in 2008 and listed in 2009 as one of the 200 best Websites in Switzerland by the "anthrazit" magazine. In 2007 there were close to 100'000 users of TaxMe-Online, 125'000 in 2008 and an estimated 150'000 in 2009 (out of around 580'000 natural persons that pay taxes in the Canton of Bern). This is approximately a 20% annual growth rate.

The information system portfolio of the tax administration of the Canton of Bern comprises about 60 applications but we only describe several subsystems that are of interest for our case study:

• NESKO (Neues Steuerkonzept) is a tailor-made business application for tax assessors; it taxes automatically around 20% of tax declarations and it also routes tax declarations to specialist groups or to regions. • TaxInfo is an information portal where tax specialists explain the tax administration practices.

TaxInfo content is written in everyday language and complements the legal texts and regulations that are also available. • TaxMe-Online is the online tax declaration system (figure 1) where taxpayers enter data on their income, wealth, real estate properties, debts, deductions, etc. All taxpayers can access the system with identification data they received along with their tax declaration forms. They can furthermore chose not to receive any paper forms anymore, apart from a letter with this identification data. • TaxMe-Portal gives access to tax declarations and decisions of previous years, to tax provision, slips and deposits, to tax payment balance, etc (figure 2). It also allows users to upload digital versions of vouchers or additional documents such as medical bills. In order to use the TaxMe-Portal, taxpayers must first register and they then receive a one-time code card along with a personal identification number. Around 40'000 users are registered as of January 2010. The illustration of the TaxMe-Online work process (figure 3) is taken from a working document of the tax administration of the Canton of Bern; although it is in German, the various steps are described below: 1. Data is transmitted from the taxation system to the TaxMe-Online server; tax declaration is sent to the taxpayer, as well as the identification data to access the system. 2. Taxpayer fills in the tax declaration with TaxMe-Online; base data is already available from previous year's declaration and from other data sources within the administration of the Canton of Bern (such as residence or marital status). 3. Taxpayer sends a signed validation declaration (paper-based) to the fiscal administration; an employee enters the validation in the back-office taxation system. This business event launches the taxation process, as before it occurs the fiscal administration does not have the right to access data entered by taxpayers. 4. Data is transmitted from the TaxMe-Online server to the back-office taxation system. 5. Tax assessor handles the tax declaration and makes a taxation decision; the tax decision is available to the tax payer. • jTDS database connector.

• Java Barbecue barcode generator. All data exchange (both ways) are coded with a 128 bits key Secure Socket Layer (SSL); passwords and one time codes are stored in a database after a one-way coding (hash key 160 bits).

In order to complete the picture, we should also say that there are two software solutions developed by private companies that are also compliant with the tax administration standards: Dr. Tax and PEBE. Table 3

Assessment of the TaxMe System

As mentioned in section 2, we built our assessment model in order to compare TaxMe and two other case studies we developed previously. The model is thus applied ex-post to our analysis subject and this constitutes a limitation of our work. Indeed we did not formalize our research design ex-ante in order to study a given research question. We do however think that our exploratory approach can create some useful knowledge such as "good practices" or "lessons learned". In order to validate and refine our model we begin by applying it to the TaxMe-Online system only. Regarding the methodology, our survey is mostly based on secondary sources (documents and numbers provided by the fiscal administration of the Canton of Bern) and complemented by interviewing stakeholders of the Cantonal fiscal administration. We notably used the results of a survey conducted in November 2008 by the fiscal administration of the Canton of Bern: 4002 taxpayers were contacted and 1404 answered the survey, which is a return rate of 35.1%. They received a letter and a paper-based survey, as well as a link and a password to answer online. The main goal of this survey was to analyse the taxpayers' satisfaction regarding the services provided by the fiscal administration and to compare the three possible channels for tax declarations (paper, CD and software, online). The survey covered four dimensions: communication, contact, competencies, and cooperation. The notation was based on a Likert-scale from 1 to 6 (1-do not agree at all, 6-fully agree).

Table 3 shows the levels of services and the communication channels supported by the fiscal administration of the Canton of Bern. The traditional paper-based taxation process (tax declaration sent to taxpayers along with a paper guide on how to fill it, signed tax declaration sent back to the administration, tax decision made by tax assessors and notified to taxpayers, bills and payment via bank or post, letters of complaints and appeals) is compared to the online process enabled by TaxMe tools, whereas the phone support is only there for inquiries and questions. The possibility to pay online is quite new as was introduced in November 2009. The tax administration does not provide these tools directly; they have made conventions with eBilling providers such as banks or the Swiss Post. Without any communication or marketing around that service, there were already 8'000 registered users in January 2010. Our next step is the assessment of the supply-side services offered by the TaxMe system:

• Content: TaxInfo is very complete and combines the applicable legislation with a description of taxation practices in everyday language; TaxMe-Online is equivalent to the paper-based tax declarations with additional capabilities (such as automated calculations, suggestions on deductions that one might forget when using the paper forms, controls and estimation of the amounts of tax to be paid); TaxMe Portal also bring added value compared to the paper-based process, as it gives access to tax declarations and decisions of previous years, to tax provision, slips and deposits, to tax payment balance. • Usability: the system is quite intuitive with 92% of TaxMe-Online users rather or completely satisfied with tax declaration process (76% for the paper-based process), 91% rather of completely satisfied with the forms themselves (84% for the paper forms) and 95% rather of completely satisfied with the documentation (87% for the paper documentation). Considering that the system is rather new, these numbers show that those using TaxMe-Online have no problems regarding usability. • Multi-channel: users can choose between the paper-base tax declaration, the TaxMe CD-ROM, two compatible commercial applications, and TaxMe-Online. A proportion of taxpayers also use the services of professional tax accountants. • Security and Privacy: the TaxMe-Online solution has a page presenting very clearly the issues regarding privacy and security; however it has to be noted that some users prepare their declaration on the CD-ROM (feeling they have more privacy) and then enter data again using TaxMe-Online. • Technical: although we had no hard data on performance and reliability issues, let us mention that the system is managed on the basis of ITIL practices; furthermore it is completely standard and open-source based (see end of section 3). Both these elements do not prove per se that the system is technically up-to-date; they do however show that the system is developed and maintained according to state-of-the-art practices.

We will now comment user-uptake and user-acceptance, according to the dimensions defined in section 2:

• Access: the main reasons stated by respondents for not using TaxMe-Online are no internet connection, confidentiality issues, users being accustomed and satisfied with the paper-based process, and the fact of staying online which is perceived as a risk (see above, some people use the CD-ROM to prepare their tax declaration, thus being off-line, and then enter their data again using TaxMe-Online). We do not have a detailed break-down of these numbers and we cannot comment on how many taxpayers do not have internet access in the Canton of Bern. However we can suppose that it is similar to the internet access rate in Switzerland: according to the Swiss Statistical Office, 74% of Swiss households had access to internet in 2007. • Competence: it is really difficult to assess that dimension as there are two types of competencies to master here, the use of internet and the ability to fill in a tax declaration. It has to be noted that 69% of the taxpayers do their tax declaration by themselves, with another 31% getting help or using professional services. Tax professional do have access to the system on behalf of their customers, as long as the latter sign the validation declaration. However it seems that those using the internet do not have any problem with the system, with 96% percent of users rather or completely satisfied with the system. • Perceived Risk: we have already mentioned above that a proportion of the users do not use TaxMe-Online because of confidentiality issues and do not appreciate the fact of having to stay online during the whole time it takes to fill in the tax declaration. • Perceived Usefulness: with almost twice the proportion of very satisfied taxpayers amongst TaxMe-Online (33.1%) than amongst the paper forms users (17.7%), one can say that at least a proportion of users do perceive the usefulness of the online system. • Trust: all in all, taxpayers do trust their fiscal administration, with 80% of respondents stating that they are rather or very satisfied with how tax assessors handle their cases and with their "loyalty".

It is interesting to note that 85% of TaxMe-Online users say so (84% for the paper-based tax declaration), as opposed to 73% of the CD-Rom solution's users.

Conclusion and Future Work

We yet have to answer our initial question: why is it that almost twice as many TaxMe-Online users are very satisfied with the system, compared to the paper-based forms and the CD-ROM solution? Table 4 shows a summary of the results of the users satisfaction surveys (NB: numbers of rather satisfied users include very satisfied users). While there is no definite single answer, we believe that content and usability of the online system are key elements of satisfaction. On the other hand access and competence could possibly explain the differences between paper and digital solutions, but as figures are similar for the CD-ROM and paper channels, it probably does not have such an influence. Indeed, the numbers of people that have a computer in Switzerland is almost equivalent to the number of people with internet access, meaning that it is very likely that most users of the CD-ROM have access to the internet. Furthermore, if one is able to use the CD-ROM application, there is no reason that he or she could not use TaxMe-Online as they are very similar to operate. So if we rule out access and competence, that leaves us with perceived risk as an explanation. Although 80% of taxpayers globally trust their fiscal administration, it seems that a portion of users still want to be able to fill in their tax declaration (and to make simulations on how to optimize their fiscal situations) while being off-line and that they do not want to take any chances of the administration looking into their data and simulations (even if the system is built in order to make that impossible). This is corroborated by the fact that a number of CD-ROM users have indicated in the survey that they wished for an off-line version of the TaxMe-Online. This would indicate that the eTaxes project mentioned in the introduction has taken a smart option by allowing users to fill in their tax declaration off-line before sending it electronically to the fiscal administration. Last, there is a small monetary incentive in using TaxMe-Online: in order to postpone the return of the tax declaration (which is normally set March 15 th ) until November 15 th , there is a fee of CHF 20 if it is done by postal letter or email; when using the online system, it is free until September 15 th , and it costs CHF 10 until November 15 th . As we indicated in the opening of this paper, we are currently only testing and refining this assessment model by analysing the TaxMe tools (TaxMe-Online, TaxMe Portal and TaxInfo). Our next step will be a comparative study of three case studies by applying this analysis framework.

Figure 1 :

 1 Figure 1: TaxMe-Online user interface and tax declaration forms

Figure 2 :

 2 Figure 2: TaxMe-Online portal

Figure 3 :

 3 Figure 3: Taxation process using the TaxMe-Online system Without describing the technical architecture of TaxMe, let us state that the system was developed using open source components and technologies:• Several Apache projects, such as Axis (implementation of the SOAP protocol for Webservices), FOP (Formatting Objects Processor, a print formatter driven by XSL formatting objects), Struts for Web applications development), etc. • JBoss application server.• jTDS database connector.• Java Barbecue barcode generator. All data exchange (both ways) are coded with a 128 bits key Secure Socket Layer (SSL); passwords and one time codes are stored in a database after a one-way coding (hash key 160 bits).

Table 1 :

 1 Level of service and communication channels

		Front desk	Phone	Online
	Information and intention building	Allocution	Conversation Consultation

Table 2 :

 2 Evaluation of eTaxation services supply and user uptake

	Supply-side	User Uptake
	Content	Access
	Usability	Competence
	Multi-channel	Perceived Risk
	Security and Privacy	Perceived Usefulness
	Technical	Trust

  shows the breakdown of 2008 tax declarations turned in by December 31

st , 2009.

Table 3 :

 3 2008 tax declarations and channels used

		Number	%
	Paper	183'471	32
	TaxMe CD	160'779	28
	Dr. Tax	45'554	8
	PEBE	35'648	6
	TaxMe Online	149'761	26
	Other	1'011	<1
	Total	576'224	100

Table 4 :

 4 TaxMe levels of service and communication channels

		Front desk	Phone		Online
	Information and intention	Allocution:	paper	Conversation:	call	Consultation: TaxInfo
	building	guide to taxpayers	centres centre, regional tax	portal
	Contracting	Registration: full tax	n/a			Registration:	tax
		declaration	/				declaration sent over
		validation declaration				the internet
		sent by postal mail
	Service Delivery	Transaction:		n/a			Transaction:	online
		traditional payments				payments
	Aftercare	Conversation: paper-	Conversation:		Consultation: TaxInfo
		based workflow for	information on tax	and TaxMe portals
		complaint		decisions, negotiation	Conversation:
		management,		of	payment	possibility to ask for a
		appeals, etc.		arrangements		payment delay

Table 5 :

 5 TaxMe levels of service and communication channels

	TaxMe-	TaxMe	Paper
	Online	CD-ROM


3--Risk situa)ons as a snapshot AND as a pathway to development

• What are risks caused by?

• What do risks cause?

Poli%cs of risks

Environment and risks are part of a development process… How are they involved in jus+ce, governance or democracy issues?

5--Model and variables

Natural System Social System • Hegemonic and self--legi+mate GEC discourse

• Governmentality of risks, uncertain+es and more broadly insecurity in a post 9/11 world • Adap+ve strategies to achieve which socio-ecological system?

Conclusion: between renewed ques)ons and "more of the same" structural challenges

  System = to be situated in Time, across Space and among Sociesituated framing of human--environment interac+ons --The research ques+ons at stake Selected Variables depend on: