Phase transitions for spatially extended pinning

We consider a directed polymer of length $N$ interacting with a linear interface. The monomers carry i.i.d. random charges $(\omega_i)_{i=1}^N$ taking values in $\mathbb{R}$ with mean zero and variance one. Each monomer $i$ contributes an energy $(\beta\omega_i-h)\varphi(S_i)$ to the interaction Hamiltonian, where $S_i \in \mathbb{Z}$ is the height of monomer $i$ with respect to the interface, $\varphi: \mathbb{Z} \to [0,\infty)$ is the interaction potential, $\beta \in [0,\infty)$ is the inverse temperature, and $h \in \mathbb{R}$ is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on $\mathbb{Z}$. We study both the quenched and the annealed free energy per monomer in the limit as $N\to\infty$. We show that each exhibits a phase transition along a critical curve in the $(\beta, h)$-plane, separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We derive variational formulas for the critical curves, and show that the quenched phase transition is at least of second order. We derive upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we derive the scaling limit of the annealed free energy as $\beta, h \downarrow 0$ in three different regimes for the tail exponent of $\varphi$.

1. Introduction 1.1. Motivation. Homogeneous pinning models, where a directed polymer receives a reward for every monomer that hits an interface, have been the object of intense study. Both discrete and continuous models have been analysed in detail, and a full understanding is available of the free energy, the phase diagram and the typical polymer configurations as a function of the underlying model parameters. Disordered pinning models, where the reward depends on random weights attached to the interface or where the shape of the interface is random itself, are much harder to analyse. Still, a lot of progress has been made in past years, in particular, the effect of the disorder on the scaling properties of the polymer has been elucidated to considerable depth. For an overview the reader is referred to the monographs by Giacomin [22], [23] and den Hollander [25], the review paper by Caravenna, Giacomin and Toninelli [14], and references therein.
Spatially extended pinning, where the interaction of the monomers depends on their distance to the interface, remains largely unexplored. For a discrete model with an interaction potential that decays sufficiently rapidly with the distance (at least polynomially fast with a sufficiently large exponent), pinning-like results have been obtained in Lacoin [26]. A continuum model for which the interaction potential is non-zero only in a finite window around the interface was analysed in Cranston, Koralov, Molchanov and Vainberg [19]. The goal of the present paper is to investigate what happens for more general interaction potentials, both for discrete and for continuous models with disorder.
The remainder of this section is organised as follows. In Section 1.2 we define our model, which consists of a directed polymer carrying random charges that interact with a linear interface at a strength that depends on their distance. In Sections 1.3 and 1.4 we look at the quenched, respectively, the annealed free energy, and discuss the qualitative properties of the phase diagram. In Section 1.5 we recall certain scaling properties of the Bessel random walk and its relation to the Bessel process, both of which play an important role in our analysis. In Section 1.6 we state three theorems when the underlying reference measure (describing the polymer without interaction) is a Markov chain. In Section 1.7 we state three theorems for the limit of weak interaction when the reference measure is the Bessel random walk, and show that this limit is related to the continuum version of our model when the reference measure is the Bessel process. In Section 1.8 we place the theorems in their proper perspective. In Section 1. 9 we list some open problems and explain how the proofs of the theorems are organised. (1) An irreducible nearest-neighbour recurrent Markov chain S := (S n ) n∈N 0 on Z starting at S 0 = 0, with law P = P 0 .
(2) An i.i.d. sequences of random charges ω := (ω n ) n∈N on R, with law P. which describes a directed polymer chain n → (n, S n ) of length N carrying charges n → ω n that interact with a linear interface according to the interaction potential x → ϕ(x) at inverse temperature β ∈ [0, ∞). Without loss of generality we may replace βω n by βω n − h, with h ∈ R the charge bias parameter, and assume that ω is standardized, i.e., Throughout the sequel we assume that (1.5) Moreover, defining τ 1 := inf{n ∈ N : S n = 0} to be the first return time of S to 0, we assume that there exists an α ∈ [0, ∞) such that n∈N P(τ 1 = n) = 1, P(τ 1 = n) = n −(1+α)+o (1) , n → ∞. (1.6) Note that E(τ 1 ) = ∞ for all α ∈ (0, 1). If S has period 2, then the last asymptotics is assumed to run along 2N.
Remark 1.1. [Bessel random walk] An example of a Markov chain S satisfying (1.6) that will receive special attention in this paper is the one with transition probabilities where for some α ∈ (0, 1) and ε > 0. This choice, which is referred to as the Bessel random walk, has a drift away from the origin (α < 1 2 ) or towards the origin (α > 1 2 ) that decays inversely proportional to the distance. The case d(x) ≡ 0 (α = 1 2 ) corresponds to simple random walk. The Bessel random walk was studied by Lamperti [27] and, more recently, by Alexander [2] (who actually considered the one-sided version (|S n |) n∈N 0 ). It is known that (1.6) holds in a sharp form [2, Theorem 2.1], namely, along 2N for some c ∈ (0, ∞). More refined asymptotics are available as well (see Section 1.5 below). See Giacomin [22], [23] and den Hollander [25] for details. Actually, in the copolymer model the interaction is via the bonds rather than the sites of the path, i.e., ϕ cop ((x, y)) = 1 {x+y≤0} , but we will ignore such refinements. Moreover, the standard parametrisation of the disorder in the copolymer model is −2β(ω n + h) rather than βω n − h. Again, this is the same after a change of parameters. Our choice has the advantage that the free energy is jointly convex in (β, h) and that the critical curve is non-negative (see Fig. 1). follows by standard super-additivity arguments. Since ϕ is bounded and P has finite exponential moments (recall (1.5)), the limit is finite. We will show in Appendix A.2 that = 0 P-a.s. and in L 1 (P), (1.14) so that (1.11) follows.
We will show in Appendix A.1 that, by (1.6), 16) and so it follows that f que (β, h) ≥ −ε(βE[|ω 1 |] + |h|). Since ε > 0 is arbitrary, we obtain the important inequality It is therefore natural to define the two phases which we refer to as the quenched localized phase, respectively, the quenched delocalized phase. From the monotonicity of h → f que (β, h) it follows that L que and D que are separated by a quenched critical curve h que c : [0, ∞) → [0, ∞) whose graph is ∂D que (see Fig. 1): From the convexity of (β, h) → f que (β, h) it follows that the lower level set Since D que is the upper graph of h que c , it follows that h que c is convex and hence continuous. In Section 1.6 we will see that h que c is finite everywhere. Since S is recurrent, it follows from the theory of the homogeneous pinning model (Giacomin [22], [23], den Hollander [25] for any x * ∈ Z with ϕ(x * ) > 0. Therefore we can dominate the quenched free energy for β = 0 by the free energy of the homogeneous pinning model with a strictly positive pinning reward.) Finally, from the monotonicity of The annealed free energy. The annealed partition function associated with (1.4) is This is the partition function of the homogeneous pinning model with potential ψ β,h . A delicate point is that ψ β,h does not have a sign: it may be a mixture of attractive and repulsive interactions. This comes from the fact that both the charge distribution and the interaction potential are general.
The annealed free energy is defined by For the constrained partition function again follows by standard super-additivity arguments. Since ψ β,h is bounded, the limit is finite. The analogue of (1.14), which will be proved in Appendix A.3, reads (1.26) As is clear from (1.21) and the fact that ϕ is non-negative, f ann (β, h) is non-increasing and convex as a function of h, and non-decreasing as a function of β but not necessarily convex. Later we will see that nonetheless β → h ann c (β) has a shape that is qualitatively similar to that of β → h que c (β) (see Fig. 2). An important property of the annealed free energy is that it provides an upper bound for the quenched free energy: by Jensen's inequality we have f que (β, h) ≤ f ann (β, h) for all β ∈ [0, ∞) and h ∈ R. Recalling (1.17), we therefore see that Unlike for the pinning model and the copolymer model, for general potentials ϕ the annealed free energy and the annealed critical curve are not known explicitly.
1.5. Scaling properties of the Bessel random walk. Part of our results below involve the annealed free energy and the annealed critical curve associated with a Brownian version of the model, where the reference measure is based on the Bessel process X := (X t ) t≥0 of dimension 2(1 − α) defined by where (B t ) t≥0 is standard Brownian motion on R. † We writeP =P 0 to denote its law when X 0 = 0. Informally, we may say that this process makes infrequent visits to 0 when α < 1 2 and frequent visits to 0 when α > 1 2 . The case α = 1 2 is standard Brownian motion.
, and so X t has density The relation with the Bessel random walk defined in Remark 1.1 is that the latter satisfies the invariance principle (see Lamperti [27]) Note that, for fixed t > 0, (1.29) gives The local time of X at 0 up to time T ≥ 0 is defined as the following limit in probability: The constant c α ensures thatÊ[L T (0)] = T 0 dt t 1−α = 1 α T α . We will informally writê L T (0) =: (1.33) Local limit theorems for the Bessel random walk have been established in [2, Theorem 2.4]. The following formulas hold in the limit as n → ∞, uniformly in a specified range of k ∈ Z. We assume that k − n is even, because otherwise P(S n = k) = 0.
1.6. General properties. Our first set of three theorems concerns the quenched and the annealed critical curve.
The left inequality in (1.39), which is known as the Monthus-Bodineau-Giacomin bound, was previously shown to hold for the copolymer model [10], [12]. We show that it holds for the general class of potentials satisfying (1.1). In Section 3.3 we will show that F ann (β, 0) > 0 for all β > 0. This implies that h ann c (β) > 0 for all β > 0, which via (1.39) settles the claim made at the end of Section 1.3 that h que c (β) > 0 for all β > 0.

1.7.
Scaling for weak interaction. Our second set of three theorems looks at the scaling of the annealed free energy in the limit of weak interaction, for the special case where S is the Bessel random walk with parameter α ∈ (0, 1) defined in Remark 1.1. We consider three different regimes for the tail behaviour of the interaction potential ϕ, namely, (1.42) Theorem 1.6. Suppose that α ∈ (0, 1) and ϑ ∈ (0, 1 − α). For everyβ ∈ (0, ∞) and h ∈ (0, ∞), lim with c the constant in (1.41). Note that, because of (1.35) and (1.41), c * [ϕ 2 ] < ∞ when ϑ > 1 − α and c * [ϕ] < ∞ when ϑ > 2(1 − α). In Appendix B.1 we will show that the annealed partition functions associated with the Bessel process appearing in Theorems 1.6-1.8 are finite, and so are the corresponding annealed free energies.
The annealed free energyF ann (β,ĥ) appearing in Theorems 1.6-1.8 has its own phase diagram, with phasesL ann := {(β,ĥ) :F ann (β,ĥ) > 0}, and with a critical curve that is a perfect power law (see Fig. 3), namely,  The constantĈ depends on α, ϕ and can be characterized as the unique solutionĈ ∈ (0, ∞) of the equationF ann (1,Ĉ) = 0. This constant is hard to identify in the first two regimes. In the third regime ϑ ∈ (2(1 − α), ∞) it is found by inserting (1.50) into the equation We show in Appendix B.2 that, for the the third regime ϑ ∈ (2(1 − α), ∞), the annealed free energyF ann (β,ĥ) can be computed explicitly, namely, (1.54) Remark 1.9. In view of the scaling limit for the annealed free energies described in Theorems 1.6-1.8, it is natural to expect a scaling limit for the corresponding annealed critical curves as well. Indeed, the continuum critical curve is the perfect power law in (1.50), where C and E depend on α and ϕ (and hence on ϑ). We conjecture that (1.50) captures the asymptotic behaviour for weak interaction of the discrete critical curve h ann c (β) as well, in the sense that in all three regimes we should have This scaling relation cannot be simply deduced from Theorems 1.6-1.8, because pointwise convergence of the free energies does not imply convergence of their zero-level sets, of which the critical curves are the boundaries. However, half of (1.55) follows because if the continuum free energy is strictly positive, then the rescaled discrete free energy eventually becomes strictly positive too in the weak interaction limit, which leads to In order to prove (1.55) extra work is needed: the scaling of the free energies in (1.43), (1.45) and (1.47) must be strengthened to a perturbative scaling, as shown in [11] and [13] for the copolymer model and in [17] for the pinning model.
1.8. Discussion. We comment on the results in Sections 1.6-1.7.

1.
The results in Theorems 1.3-1.5 are known for the special case where the interaction potential is that of the pinning model or the copolymer model defined in (1.10). However, the techniques used for these two cases do not carry over to the general class of potentials considered in (1.1). Intuitively, the reason why extension is possible is that the conditions stated in (1.1) say that, outside a large interval around the origin in Z, the interaction potential is controlled by a multiple of that of the copolymer model.

2.
As we will see in Section 2, the variational formula for h que c (β) mentioned in Theorem 1.3 involves a supremum over the space of all shift-invariant probability distributions on the set of infinite sequences of words of arbitrary length drawn from an infinite sequence of letters taking values in R × Z. The supremum involves a quenched rate function that captures the complexity of the interplay between the disorder of the charges and the excursions of the polymer away from the interface. This variational formula is hard to manipulate, but it is the starting point for the proofs of Theorems 1.4-1.5. The variational formula for h ann c (β) mentioned in Theorem 1.3 is simpler, but still not easy to manipulate (see (1.57) below).
3. Note that for h = 0 and β > 0 the annealed partition function Z ann N,β,h is bounded from below by the partition function of a homogenous pinning model with a strictly positive reward, which is localized. The lower bound in Theorem 1.4 therefore shows that h que c (β) > 0 for every β > 0. Since β → h que c (β) is convex, it must therefore be strictly increasing (see Fig. 1).
5. Theorems 1.6-1. 8 give detailed information about the scaling of the annealed free energy and the annealed critical curve in the limit of weak interaction. The scaling limits correspond to annealed free energies and annealed critical curves for Brownian versions of the model involving the Bessel process X α , which are interesting in their own right. The result is only valid for the Bessel random walk, and shows a trichotomy depending on the parameters α and ϑ.
• The regime ϑ ∈ (0, 1 − α) corresponds to a long-range interaction potential and is not pinning-like. When localized, the continuum polymer spends a positive fraction of the time near any height x ∈ R, and this fraction tends to zero as |x| ↓ 0 or |x| → ∞. Away from 0 it does not behave like the Bessel process conditioned to return to 0.
• The regime ϑ ∈ (1 − α, 2(1 − α)) corresponds to an intermediate-range interaction potential and exhibits some pinning-like features. When localized, the continuum polymer visits 0 a positive fraction of the time. Away from 0 it does not behave like the Bessel process conditioned to return to 0.
• The regime ϑ ∈ (2(1 − α), ∞) corresponds to a short-range interaction potential and is pinning-like. When localized, the continuum polymer visits 0 a positive fraction of the time. Away from 0 it behaves like the Bessel process conditioned to return to 0.
In the last regime the behaviour is similar to that of the homogeneous pinning model with Giacomin [22], [23], den Hollander [25]). In fact, the proof of Theorem 1.8 will show that the scaling in the last regime is valid for any ϕ such that c * [ϕ 2 ] and c * [ϕ] are finite, i.e., (1.41) may be replaced by the weaker condition ϕ( 6. The three regimes for ϑ represent three universality classes. The critical cases ϑ = 1 − α and ϑ = 2(1−α) are more delicate and we have skipped them. Also, we have not investigated what happens when the scaling of the interaction potential in (1.41) is modulated by a slowly varying function. For the same reason we have assumed that the error term in (1.8) is O(|x| −(1+ε) ) with ε > 0 rather than o(|x| −1 ), since the latter may give rise to modulation by slowly varying functions in (1.34) and (1.35) (see Alexander [2]). where we recall that τ 1 denotes the first return time of S to 0. Although the starting point 0 seems to play a special role in (1.57), it can be shown that the criterion in (1.57) is invariant under spatial shifts of ψ β,h (see Appendix C). A natural question is what happens when the random walk S is transient, i.e., P(τ 1 < ∞) = n∈N K(n) =: r < 1. For the constrained partition function Z ω,c N,β,h , working with a transient renewal process with law K is equivalent to working with a recurrent renewal process with law K/r and adding a depinning term N n=1 (log r)1 {Sn=0} in the exponential in (1.12). This amounts to replacing ψ β,h (x) by ψ β,h (x) + (log r)1 {x=0} , and so instead of (1.57) the localization condition for the annealed model becomes . Therefore h ann c (β) = 1 2 β 2 and, in fact, the left-hand side of (1.57) is ≤ 1 for h ≤ h ann c (β) and is = ∞ for h > h ann c (β). This means that the annealed critical curve h ann c does not depend on r, and hence neither do the bounds in (1.39). In other words, making the underlying renewal process transient or, equivalently, adding a homogeneous depinning term at zero, does not modify the annealed critical curve of the copolymer model. In essence this is due to the fact that the copolymer potential is long range (i.e., ψ β,h (x) does not vanish as x → −∞).
(II) For the pinning model, adding a depinning term at zero amounts to shifting h and this may have an effect. In essence this is due to the fact that the pinning potential ϕ(x) = ϕ pin (x) = 1 {x=0} is short range.
1.9. Open problems and outline.
For an overview, we refer the reader to Giacomin [23].
2. Determine whether the quenched phase transition is second order or higher order. For the copolymer model it is known that the phase transition is of infinite order when α = 0 (Berger, Giacomin and Lacoin [6]). The same is conjectured to be true for α ∈ (0, 1). 5. The qualitative shape of the critical curve in Fig. 1 depends on our assumption in (1.1) that ϕ ≥ 0. A reflected picture holds when ϕ ≤ 0. It appears that for ϕ with mixed signs there are two critical curves β → h que c,1 (β) and β → h que c,2 (β), separating a single quenched delocalized phase D que from two quenched localized phases L que 1 and L que 2 that lie above D que , respectively, below D que . What are the properties of these critical curves? 6. What happens when β = β N and h = h N with β N , h N ↓ 0 as N → ∞.

Is it possible to include non-nearest-neighbour random walks?
Outline. The remainder of this paper is organized as follows. Theorem 1.3 is proved in Section 2, Theorem 1.4 in Section 3 and Theorems 1.6-1.8 in Section 5. Appendices A and B collect a few technical facts that are needed along the way.

Proof of Theorem 1.3
In Section 2.1 we formulate annealed and quenched large deviation principles (LDPs) that are an adaptation to our model of the LDPs developed in Birkner [8] and Birkner, Greven and den Hollander [9]. The latter concern LDPs for random sequences of words cut out from random sequences of letters according to a renewal process. In Section 2.2 we formulate variational characterizations of the annealed and quenched critical curves that are an adaptation of the characterizations derived for the pinning model in Cheliotis and den Hollander [18] and for the copolymer model in Bolthausen, den Hollander and Opoku [12]. In Section 2.3 we explain how the variational characterizations follow from the LDPs via Varadhan's lemma.

Annealed and quenched LDP.
Our starting observation is that the partition function in (1.4) depends on the sequence of words Y = (Y i ) i∈N determined by the disorder and by the excursions of the polymer, namely, where τ = (τ i ) i∈N is the sequence of epochs of the successive visits of the polymer to zero (τ 0 = 0). Note that the random variables Y i take their values in the spaceΓ := n∈N Γ n with Γ := R × Z.
To capture the role of Y , we introduce its empirical process, where P inv (Γ N ) denotes the set of probability measures onΓ N that are invariant under the left-shiftθ acting onΓ N . The superscript ω reminds us that the random variables Y i are functions of ω. We must average over S while keeping ω fixed. Note that, under the annealed law P ⊗ P, Y is i.i.d. with the following marginal law q 0 onΓ: . . . , S n ) = (s 1 , . . . , s n ) τ 1 = n , n ∈ N, x 1 , . . . , x n ∈ R, s 1 , . . . , s n ∈ Z, where K(n) := P(τ 1 = n) and ν(dx) := P(ω 1 ∈ dx).
The specific relative entropy of Q w.r.t. q ⊗N 0 is defined by 4) whereπ N Q ∈ P(Γ N ) denotes the projection of Q onto the first N words, h( · | · ) denotes relative entropy, and the limit is non-decreasing. The following annealed LDP is standard (see Dembo and Zeitouni [20, Section 6.5]).

Proposition 2.1. [Annealed LDP]
The family (P ⊗ P)(R ω M ∈ · ), M ∈ N, satisfies the LDP on P inv (Γ N ) with rate M and with rate function I ann given by This rate function is lower semi-continuous, has compact level sets, has a unique zero at q ⊗N 0 , and is affine.
The quenched LDP is more delicate and requires extra notation. The reverse operation of cutting words out of a sequence of letters is glueing words together into a sequence of letters. Formally, this is done by defining a concatenation map κ fromΓ N to Γ N . This map induces in a natural way a map from P(Γ N ) to P(Γ N ), the sets of probability measures oñ Γ N and Γ N (endowed with the topology of weak convergence). The concatenation For Q ∈ P inv,fin (Γ N ), define Think of Ψ Q as the shift-invariant version of Q•κ −1 obtained after randomizing the location of the origin. This randomization is necessary because a shift-invariant Q in general does not give rise to a shift-invariant Q • κ −1 .
The following quenched LDP is a straight adaptation of the one derived in Birkner, Greven and den Hollander [9].

Proposition 2.2. [Quenched LDP]
For P-a.e. ω the family P(R ω M ∈ · ), M ∈ N, satisfies the LDP on P inv (Γ N ) with rate M and with rate function given by

8)
where α is the exponent in (1.6) and B δ (Q) is the δ-ball around Q (in any appropriate metric). This rate function is lower semi-continuous, has compact level sets, has a unique zero at q ⊗N 0 , and is affine. Remark 2.3. In [9] a formula was claimed for I que on P inv (Γ N ) \ P inv,fin (Γ N ) based on a truncation approximation for the average word length. As pointed out by Jean-Christophe Mourrat (private communication), the proof of this formula in [9] is flawed. The formula itself may still be correct, but no proof is currently available. In the present paper we will only need to know I que on P inv,fin (Γ N ). Then the constrained quenched partition function defined in (1.12) can be written as while the constrained annealed partition function defined in (1.23) can be written as (2.13) (2.14) As we will see below, the role of the conditions m Q < ∞ and I ann (Q) < ∞ under the two suprema is to ensure that Γ Φ β,h d(π 1 Q) < ∞, so that the suprema are well defined. The condition m Q < ∞ under the second supremum allows us to use the representation in (2.8).
We will see in Section 3 how the variational formulas in (2.13)-(2.14) can be exploited. . The only difficulty we need to deal with is the fact that both Q → m Q and Q → Φ * β,h (Q) = Γ Φ β,h d(π 1 Q) are neither bounded nor continuous in the weak topology. Therefore an approximation argument is required, which is worked out in detail in [12, Appendix A-D] for the case of the copolymer interaction potential in (1.10). This approximation argument shows why the restriction to m Q < ∞ and I ann (Q) < ∞ may be imposed, a key ingredient being that I ann (Q) < ∞ implies Φ * β,h (Q) < ∞. The proof in [12, Appendix A-D] readily carries over because our condition on the interaction potential in (1.1) reflects the properties of the copolymer interaction potential. We sketch the main line of thought. Throughout the sequel β, h > 0 are fixed.
Proof of Theorem 2.5. Following the argument in [12, Appendix A], we show that For every ̺ ∈ P(N), ν ∈ P(R) and p = (p n ) n∈N with p n ∈ P(Z n ), there exist γ > 0 and K = K(̺, ν, p; γ) > 0 such that Φ * β,h (Q) ≤ γh(π 1 Q | q ̺,ν,p ) + K for all Q ∈ P inv (Γ N ) with h(π 1 Q | q ̺,ν,p ) < ∞, where (compare with (2.3)) q ̺,ν,p (dx 1 , . . . , dx n ) × {(s 1 , . . . , s n )} = ̺(n) ν(dx 1 ) · · · ν(dx n ) p n (s 1 , . . . , s n ), n ∈ N, x 1 , . . . , x n ∈ R, s 1 , . . . , s n ∈ Z. (2.15) The proof uses the fact that the conditions in (1.1) allow us to approximate ϕ by a multiple of ϕ cop (recall (1.10)) uniformly on Z \ [−L, L] at arbitrary precision as L → ∞. The proof also uses a concentration of measure estimate for the disorder, which is proved in [12,Appendix D]. For g > 0, define the quenched free energy is the quenched partition function in which every letter gets an energetic penalty −g. Following the argument in [12, Appendix B], we use (1) and (2) to show that, for every g > 0, where R is the set of shift-invariant probability measures under which the concatenation of words produces a letter sequence that has the same asymptotic statistics as a typical realisation of Y , i.e.,  Here, in the passage from (2.18) to (2.21), the constraint in R disappears from the variational characterization, while I ann (Q) is replaced by I que (Q). The reason is that for every Q ∈ P inv (Γ N ) there exists a sequence (Q n ) n∈N in R such that Q = w − lim n→∞ Q n = Q and lim n→∞ I ann (Q n ) = I que (Q). The proof of (2.21) carries over verbatim. The supremum in the right-hand side of (2.21) is the same as the supremum in the right-hand side of (2.14). Again, this works because of the control enforced by (2). The supremum in the right-hand side of (2.24) is the same as the supremum in the right-hand side of (2.13).

Proof of Theorem 1.4
The upper bound in (1.39) is immediate from Theorems 2.4-2.5 and the inequality I que ≥ I ann (see also (1.27)). In Sections 3.1-3.3 we prove the lower bound in (1.39). This lower bound is the analogue of what for the copolymer model is called the Monthus-Bodineau-Giacomin lower bound (see Giacomin [22], den Hollander [25]).

3.1.
A sufficient criterion for quenched localization. The quenched rate function can be written as It can be shown that R(Q) ≥ 0 for all Q ∈ P inv (Γ N ): R(Q) has the meaning of a concatenation entropy (see Birkner, Greven and den Hollander [9]). Therefore, dropping this term in (2.14) we obtain the following sufficient criterion for quenched localization: The right-hand side resembles the necessary and sufficient criterion for annealed localization in (2.13), the only difference being the extra factor 1 + α.

3.2.
Reduction. Among the laws Q ∈ P inv (Γ N ) with a given marginal law q ∈ P(Γ), the product law Q = q ⊗N is the unique minimizer of the specific relative entropy H(Q | q ⊗N 0 ). Therefore the right-hand side of (3.3) reduces to where h(· | ·) denotes relative entropy. We next show that (3.3) reduces to an even simpler criterion. To that end, let C N := N n=1 Γ n be the subset of words of length at most N . Consider the lawq N ∈ P(Γ) defined by is the normalizing constant. The latter is finite because, by (2.10), Φ restricted to C N is the sum of at most N random variables with finite exponential moments. Note that also which yields h(q N | q 0 ) < ∞. Trivially, mq N ≤ N < ∞. Therefore we are allowed to pick q =q N in (3.4), so that (3.4) is satisfied when Since N is arbitrary, this in turn is satisfied when 9) where N α = ∞ is allowed. Conversely, if N α ≤ 1, then (3.4) is not satisfied. Indeed, as soon as N α < ∞ we may introduce the lawq ∈ P(Γ) defined by where the last inequality holds for any q because N α ≤ 1, and so (3.4) fails. Thus, (3.3) reduces to . Therefore, repeating the above steps and recalling (2.14), we conclude that It follows from (2.10), (2.12) and (3.9) that the condition N α > 1 is equivalent to f ann 1 1+α β, 1 1+α h > 0, (3.14) i.e., 1 1+α h < h ann c ( 1 1+α β), which by (3.12) implies f que (β, h) > 0, i.e., h < h que c (β). This completes the proof of the lower bound in (1.39).
Recalling ( By Jensen and the fact that ϕ ≡ 0, this sum is > 1 for all β > 0. Hence F ann (β, 0) > 0 for all β > 0, which settles the claim made at the end of Section 1.6.

Weak disorder and continuum limit
This section is a preparation for the proof of Theorems 1.6-1.8 in Section 5. We explain the main line of reasoning and highlight the key points, focusing on a slightly simpler setting.
We recall from (1.20)-(1.21) that the annealed partition function is defined by Note that (4.1) is the partition function of a homogeneous pinning model with potential ψ β,h and reference measure the Bessel random walk S defined in (1.7)-(1.8). Since we assume that (recall (1.41)) we see that for large x we have ψ β,h (x) ≈ 1 2 β 2 c 2 |x| −2ϑ − hc |x| −ϑ . Both terms will turn out to be relevant, because β and h will be scaled differently.
In this section we look at each term separately. Therefore we focus on the simplified model where ̺ δ : Z → R is a potential with the following properties: • There is a function ̺ : Z → R such that (4.5) • There are a ∈ R and γ > 0 such that We emphasize that these assumptions are satisfied for ̺ δ (x) = log M(βϕ(x)) (with δ = β 2 , ̺(x) = ϕ(x) 2 and a = c 2 /2) and for ̺ δ (x) = −hϕ(x) (with δ = h, ̺(x) = ϕ(x) and a = −c). We start from an expansion of the partition function. Namely, we set so that where we define C N,δ,k := In what follows we distinguish between two regimes for the exponent γ.
• 0 < γ < 2(1 − α): Recall from (1.30) that (|S N t |/ √ N ) t≥0 converges in distribution to the Bessel process (X t ) t≥0 . Fix δ = δ N ↓ 0, so that ̺ δ N ∼ δ N ̺(x) in (4.5) and χ δ N (x) ∼ δ N ̺(x) in (4.7). In view of (4.6), for fixed k ∈ N, 0 < t 1 < . . . < t k and N → ∞, we claim that because small values of X t ℓ , i.e., give a negligible contribution, as we show next. Indeed, by (4.5) and (4.7), we can bound χ δ (x) ≤ C 1 δ ̺(x) ≤ C 2 δ (1 + |x|) −γ for all x ∈ Z, for some constants C 1 , C 2 < ∞. Since P(S n ∈ ·|S 0 = m) stochastically dominates P(S n ∈ ·|S 0 = 0) for m ≥ 0 (as can be seen via a coupling argument), it follows from the uniform upper bound in (1.38) that where the last inequality holds by Riemann approximation with x ℓ = r ℓ / √ N . The last integral is finite because γ < 2(1 − α), and so we have shown that (4.12) Via the same argument, if in (4.12) we restrict the expectation in the left-hand side to the event k ℓ=1 {|S N t ℓ | ≤ ε N (t ℓ − t ℓ−1 )}, then we obtain a fraction of the right-hand side that vanishes as ε ↓ 0, uniformly in N, k ∈ N and 0 < t 1 < . . . < t k ≤ 1. This justifies (4.10). We next set so that the prefactor in (4.10) equals (δ N ) k , which is the right normalisation for the Riemann sum in (4.9) to converge to the corresponding integral. Indeed, (4.14) where the convergence is justified by (4.12), which shows that the values of t ℓ for which the gaps t ℓ − t ℓ−1 are small are negligible, and the second equality follows by integrating over unordered variables and applying Fubini's theorem. Looking back at (4.8), we obtain where the limit and the sum can be exchanged by (4.12).
• 2(1 − α) < γ < ∞: In this regime the last expected value in (4.10) will be seen to diverge, because the main contribution comes from values of S N t ℓ that are O(1), rather than O( √ N ). We again fix δ = δ N ↓ 0, so that ̺ δ N ∼ δ N ̺(x) and χ δn (x) ∼ δ N ̺(x). Then, as N → ∞, where we set n 0 := 0, x 0 := 0, and where q n (z, x) := P(S n = x|S 0 = z) is the transition kernel of the Bessel random walk. The asymptotic behaviour as n → ∞ of the latter is given by (1.34) also for any fixed z = 0, namely, q n (z, . This is proved in [2, Theorem 2.4] for z = 0, while for z = 0 it follows from the decomposition q n (z, then we claim that because the contribution of large values of S N t ℓ , i.e., S N t ℓ ≫ 1, is negligible, as we show next. Indeed, we argue as in (4.11) until the second last line, before the Riemann approximation. The integral in the last line of (4.11) diverges in the current regime γ > 2(1 − α). For this reason, we simply drop the exponential term and keep the simpler bound Since r∈Z (1 + |r|) 1−2α−γ < ∞ for γ > 2(1 − α), we have shown that, for some C < ∞, (4.20) Via the same argument, if in (4.20) we restrict the expectation in the left-hand side to the event k ℓ=1 {|S N t ℓ | ≥ M }, then we obtain a fraction of the right-hand side that vanishes as M → ∞ uniformly in N, k ∈ N and 0 < t 1 < . . . < t k ≤ 1. This justifies (4.18). We next set so that the prefactor in (4.18) equals (δ N ) k , which is the right normalisation for the Riemann sum in (4.9) to converge to the corresponding integral. Indeed, for any k ∈ N lim N →∞ where the convergence is justified by (4.20), which shows that the values of t ℓ for which the gaps t ℓ − t ℓ−1 are small are negligible. Looking back at (4.8), we obtain where the limit and the sum can be exchanged because of (4.20).
Recall the definition of the local timeL 1 (0) in (1.32). We show in Appendix B that where the limit is the Bessel process defined in (1.28). Recall (1.20)-(1.21). Since ψ β,h ∞ tends to zero as β, h ↓ 0, we can do a weak coupling expansion in the spirit of Caravenna, Sun and Zygouras [16]. Namely, fix T ∈ (0, ∞) and write (for easy of notation we pretend that T N is integer) Under the assumption in (1.41), we have Put n i = ⌊N t i ⌋, 1 ≤ i ≤ N , and approximate the sum in (5.3) by an integral with the help of (5.1) and (5.4), to get Inserting (5.6) into (5.2), we get This shows that, under the scaling in (5.5), the partition function converges to a continuum limit. A precise justification of the Riemann sum approximation and of the limits involved follows from the analysis performed in Section 4, where each term 1 2β 2 c 2 |x| −2ϑ andĥc |x| −ϑ in (5.7) was considered separately. More precisely, note that both the exponents γ = 2ϑ and γ = ϑ are such that γ < 2(1 − α), so that we can apply (4.14) and (4.15).
The annealed free energy is obtained aŝ where the interchange of the limits T → ∞ and N → ∞ is justified in Appendix A.4.

5.3.
The case ϑ ∈ (1 − α, 2(1 − α)). Pick A Riemann sum approximation in (5.3) now leads to where we recall that the "renormalized delta function"δ(·) is an informal notation hat we introduced in (1.32)-(1.33). To be more precise, note that the second termĥc |x| −ϑ is the same as in (5.7), but the first term is different and arises from the computation (recall (1.34)) which gives where we use that (recall (1.32) and the line following it) Note that c * [ϕ 2 ] < ∞ because of (1.35), (1.41) and the fact that ϑ > 1 − α. Again, the approximation can be justified by the arguments developed in Section 4. More precisely, the exponent γ = 2ϑ of the first term is such that γ > 2(1 − α), so that we can apply (4.14) and (4.15), while he exponent γ = ϑ of the second term is such that γ < 2(1 − α), so that we can apply (4.14) and (4.15), as in the previous case. Inserting (5.13)  This proves Theorem 1.7.
Any path that starts at x, ends at x and does not go below x can be cut into pieces that zig-zag between x and x + 1 and pieces that start at x + 1, end at x + 1 and do not go below x + 1. Hence we have where p x,x+1 := P(S 1 = x + 1 | S 0 = x). We know that F 0 (1 + ε) = ∞ ∀ ε > 0. We show that this implies F x (1 + ε) = ∞ ∀ ε > 0 for every x ∈ N. The proof is by induction on x. Fix x ∈ N 0 and suppose that F x (1 + ε) = ∞ ∀ ε > 0. We argue by contradiction. Suppose that F x+1 (1 + ε) < ∞ for ε small enough. Because r x > 0 and Q x (1 + ε) < ∞ for ε small enough, it follows that F x+1 (1 + ε)Q x (1 + ε) ≥ 1 ∀ ε > 0, and by continuity that F x+1 (1)Q x (1) ≥ 1. To get the contradiction it therefore suffices to show that where for both the last equality and the inequality we again use recurrence.
A.4. Interchange of limits. The following two lemmas, which give us a sandwich for the annealed free energies in the discrete and in the continuous model, are the key to showing that the limits in (5.11) may be interchanged. with  Proof. The proof is similar to that of Lemma A.1.
What makes Lemma A.1 useful is that N → log Z ann,− M,N,β,h is superadditive for every M ∈ N 0 , while N → log Z ann,+ N,β,h is subadditive. Consequently, Let A, B be the pair of exponents appearing in part (a) of Theorems 1.6-1.8, i.e., In this appendix we show that the annealed partition functions and the corresponding annealed free energies encountered in Theorems 1.6-1.8 are finite in each of the three regimes. We also give an explicit characterization of the annealed free energy and the annealed critical curve in the regime where ϑ ∈ (2(1 − α), ∞). We show that, for 0 < γ < 2(1 − α), these quantities grow at most exponentially as T → ∞: (B.2) By Cauchy-Schwarz, this implies that all the free energies in Theorems 1.6-1.8 are finite.
We first focus onẐ µ,T , which we rewrite aŝ We use the Markov property at times t 1 , . . . , t k to estimatê where the inequality holds becauseP 0 (X t ∈ ·) stochastically dominatesP x (X t ∈ ·) for any x ≥ 0 (by a standard coupling argument and the fact that X is a Markov process with continuous paths) and because x → x −γ is non-increasing. We thus obtain C k,T ≤ 0≤t 1 <···<t k ≤T dt 1 · · · dt k k ℓ=1Ê 0 X −γ t ℓ −t ℓ−1 (B.6) with t 0 = 0. By diffusive scaling we haveÊ 0 [X −γ t ] = t −γ/2 C, with C =Ê 0 [X −γ 1 ] < ∞ for 0 < γ < 2(1 − α) (recall (1.29)). The change of variables t k = T s k yieldŝ where, for ϑ ∈ (0, 1), with u 0 = 1 and u k = 1 is the normalization of the Dirichlet distribution, after setting s i = s k u i for i = 1, . . . , k − 1 in (B.8) we obtain Therefore the leading contribution to the sum in (B.12) is given by k ≈xT (more precisely, the sum restricted to 0 ≤ k < 2xT is at most 2x T exp(T e B/A ), while the contribution of the remaining terms with k ≥ 2xT is negligible). It follows that Let us focus onL ε T (0) for a moment. By explicit computation, for any k ∈ N 1 k!Ê L ε T (0) k = c k α ε 2(1−α)k 0≤t 1 <···<t k ≤T dt 1 · · · dt kP0 k ℓ=1 X t ℓ ∈ (0, ε) ≤ c k α ε 2(1−α)k 0≤t 1 <···<t k ≤T dt 1 · · · dt k k ℓ=1P 0 X t ℓ −t ℓ−1 ∈ (0, ε) , with t 0 := 0, where the inequality holds because, as we already remarked,P 0 (X t ∈ ·) stochastically dominatesP x (X t ∈ ·) for any x ≥ 0. We also remark that the inequality is asymptotically sharp as ε ↓ 0. Therefore it follows via (1.31) that, as ε ↓ 0, where I k (·) is defined in (B.8). This shows thatL ε T (0) is uniformly bounded in L k , for any k ∈ N. By uniform integrability, we can therefore exchange lim ε↓0 andÊ to get, recalling (B.16),C k,T = lim The steps in (B.10)-(B.14) (with ϑ = 1−α instead of ϑ = γ/2) show that not onlyZ µ,T < ∞ for all µ, T , but also lim T →∞ 1 T logZ µ,T < ∞ for all µ. B.2. Formula for the annealed free energy. To compute the annealed free energŷ F (β,ĥ) in the regime ϑ ∈ (2(1 − α), ∞), we use the first line in (B.21) to compute the Laplace transform ofZ µ,T . Writing e −λT = ( k ℓ=1 e −λ(t ℓ −t ℓ−1 ) ) e −λ(T −t k ) for λ ≥ 0, we get Appendix C. Localization criterion for the annealed model In this appendix we take a closer look at the criterion in (1.57) and show that it does not depend on the starting point of the walk. For the purpose of this appendix, let S = (S n ) n∈N 0 be any recurrent Markov chain on a countable space E, and let ψ : E → R be an arbitrary function. Denote by τ x := min{n ∈ N : S n = x} the first return time of S to x. Define (C.1) We will prove the following property: ∀x, y : It is convenient to introduce the following shorthand notation, for (possibly random) σ ∈ N: so that we may simply write A x = E x [e H(τ x ) ]. Given an arbitrary y, we can split this expected value according to the two complementary events {τ x < τ y } and {τ y < τ x }: The second term can be expanded by summing over all visits of S to y that precede the first return to x. If we define B xy := E x e H(τ x ) 1 {τ x <τ y } , C xy := E x e H(τ y ) 1 {τ y <τ x } , (C.5) then by the strong Markov property we get with the convention that A x = ∞ if B yx ≥ 1. Exchanging the roles of x and y, we get We are now ready to prove (C.2). Fix x, y. We show that if A x ≤ 1, then also A y ≤ 1. To simplify the notation, we abbreviate b := B xy , b ′ := B yx and c := C xy C yx , so that with the convention that the ratios equal ∞ if b ′ ≥ 1, respectively, b ≥ 1. Assume that A x ≤ 1. Then we must have b ′ < 1 and the formula A x = b + c 1−b ′ applies, which shows that also b < 1 (because c > 0). Hence we can write i.e., A y ≤ 1.