Pinning of a renewal on a quenched renewal

We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $\sigma$, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $\sigma$ have infinite mean. The"polymer"-- of length $\sigma_N$ -- is given by another renewal $\tau$, whose law is modified by the Boltzmann weight $\exp(\beta\sum_{n=1}^N \mathbf{1}_{\{\sigma_n\in\tau\}})$. Our assumption is that $\tau$ and $\sigma$ have gap distributions with power-law-decay exponents $1+\alpha$ and $1+\tilde \alpha$ respectively, with $\alpha\geq 0,\tilde \alpha>0$. There is a localization phase transition: above a critical value $\beta_c$ the free energy is positive, meaning that $\tau$ is \emph{pinned} on the quenched renewal $\sigma$. We consider the question of relevance of the disorder, that is to know when $\beta_c$ differs from its annealed counterpart $\beta_c^{\rm ann}$. We show that $\beta_c=\beta_c^{\rm ann}$ whenever $ \alpha+\tilde \alpha \geq 1$, and $\beta_c=0$ if and only if the renewal $\tau\cap\sigma$ is recurrent. On the other hand, we show $\beta_c>\beta_c^{\rm ann}$ when $ \alpha+\frac32\, \tilde \alpha<1$. We give evidence that this should in fact be true whenever $ \alpha+\tilde \alpha<1$, providing examples for all such $ \alpha,\tilde \alpha$ of distributions of $\tau,\sigma$ for which $\beta_c>\beta_c^{\rm ann}$. We additionally consider two natural variants of the model: one in which the polymer and disorder are constrained to have equal numbers of renewals ($\sigma_N=\tau_N$), and one in which the polymer length is $\tau_N$ rather than $\sigma_N$. In both cases we show the critical point is the same as in the original model, at least when $ \alpha>0$.


Motivations
A variety of polymer pinning models have been studied in theoretical and mathematical physics in the past decades, see [15,19,20] for reviews. We introduce in this paper a new type of disordered pinning model that we call pinning on a quenched renewal. Before giving its definition, we recall two well-studied related models that motivate the introduction and the study of this new model.

Pinning on an inhomogeneous defect line
The disordered pinning model was introduced by Poland and Scheraga [29] to model DNA denaturation, and it has recently been the subject of extensive rigorous mathematical studies, cf. [19,20]. We recall its definition.
Let τ = {τ 0 = 0, τ 1 , . . .} be a discrete recurrent renewal process of law P, and let ω = (ω n ) n∈N be a sequence of IID random variables of law denoted P, with finite exponential moment M (λ) = E[e λω ] < +∞, for λ > 0 small enough. Then, for β > 0 and h ∈ R, and a fixed realization of ω (quenched disorder), the Gibbs measure is defined by dP β,ω where Z β,ω N,h is the partition function, which normalizes P β,ω N,h to a probability measure.
This Gibbs measure corresponds to giving a (possibly negative) reward βω n + h to the occurrence of a renewal point at n. In this context it is natural to think of the polymer configuration as the space-time trajectory of some Markov chain, with the polymer interacting with a potential whenever it returns to some particular site, for example the origin; then τ represents the times of these returns. This system is known to undergo a localization phase transition when h varies, and much attention has been given to the question of disorder relevance, that is, whether the quenched and annealed systems (with respective partition functions Z β,ω N,h and EZ β,ω N,h ) have different critical behaviors. The so-called Harris criterion [24] is used to predict disorder relevance, and its characterization in terms of the distribution of the renewal time τ 1 has now been mathematically settled completely [1,4,6,14,16,22,27,30], the complete necessary and sufficient condition for critical point shift being given only recently, in [6].
An interesting approach to this problem is based on a large deviation principle for cutting words out of a sequence of letters [9], see [14]: quantities of the model, such as critical points, are expressed as variational formulas involving (quenched and annealed) rate functions I que and I ann . In [14], the authors consider a version of τ truncated at a level tr (whose law we denote P (tr) ), and it is implicitly shown that a lower bound on the critical point shift is, for all β > 0, Pinning of a renewal on a quenched renewal where Supp χ denotes the support of the distribution of a random variable χ. At times we will add the assumption d τ ≤ d σ , (1.7) which ensures that certain partition functions carrying the restriction σ N = τ N cannot be 0, for sufficiently large N .
Proof. We have where θ is the "shift operator applied to increments": Therefore the sequence (log Z σ N,β ) n∈N is superadditive, and using Kingman's subadditive Theorem [26], one gets the P σ -a.s. existence of the limit in (1.1), and moreover F(β) = F(β, P σ ) = sup (1.10) The non-negativity trivially follows from the fact that β ≥ 0, and the convexity is classical and comes from a straightforward computation.

The annealed model
We will compare the quenched-renewal pinning model to its annealed counterpart, with partition function E σ Z σ N,β = E τ,σ e β|τ ∩σ|σ N 1 {σ N ∈τ } . The annealed free energy is defined by (1.14) The existence of the annealed free energy is straightforward, using the superadditivity Note that this model does not treat τ and σ symmetrically. Closely related is what we will call the homogeneous doubled pinning model, in which the length of the polymer is fixed, rather than the number of renewals in σ: the partition function and free energy are Z hom n,β := E τ,σ e β|τ ∩σ|n 1 {n∈τ ∩σ} , F hom (β) = lim n→∞ 1 n log Z hom n,β , and the corresponding critical point is denoted β hom c . This model is exactly solvable, see [19,Ch. 2], and in particular, its critical point is β hom c = − log P τ,σ (τ ∩ σ) 1 < ∞) = log 1 + E σ,τ [|τ ∩ σ|] −1 , with the convention that 1 ∞ = 0. Proposition 1.3. The annealed system is solvable, in the sense that for β > 0, F ann (β) is the solution F of n≥1 e −Fn P σ,τ (σ n ∈ τ ) = 1 e β − 1 The proof of this proposition uses standard techniques, and is postponed to Appendix A.2. It relies on a rewriting of the partition function, together with an estimate on the probability P σ,τ (σ n ∈ τ ) that we now present (its proof is also postponed to Appendix A.2). Lemma 1.4. Under (1.5), with α > 0, there exists a slowly varying function ϕ * (·) such that, for α * from (1.16), P σ,τ (σ n ∈ τ ) = ϕ * (n) n −(1+α * ) . (1.17) From Jensen's inequality we have F(β) ≤ F ann (β), so that β c ≥ β ann c . When τ ∩ σ is recurrent, β ann c = 0, and our main theorem will say that β c = 0 as well. In the transient case we have β ann c > 0 and we can ask whether β c = β ann c or not.

Results and comments
Our main result says that in the case where τ ∩ σ is recurrent, or transient with α + α = 1, the quenched and annealed critical points are equal (so both are 0 in the recurrent case). When α+ α < 1, a sufficient condition for unequal critical points involves the exponent α * appearing in Lemma 1.4.

Theorem 2.2.
For any α, α > 0 with α + α < 1 and any slowly varying functions ϕ(·), ϕ(·), there exist distributions for τ and σ satisfying P τ ( We expect that, following the same scheme of proof, one can extend Theorem 2.2 to the case α = 0, α ∈ (0, 1). However, in the interest of brevity we do not prove it here, since it would require separate estimates from [3].
Let us now make some comments about these results.

1.
Since the renewal τ ∩ σ can be recurrent only if α + α ≥ 1, one has from (2.1) that β c = 0 if and only if β ann c = 0, that is if and only if τ ∩ σ is recurrent. This is notable in that |τ ∩ σ| = +∞ P σ,τ −a.s. is enough for τ to be pinned on a quenched trajectory of σ for all β > 0, even though a typical σ-trajectory is very sparse.
Pinning of a renewal on a quenched renewal Moreover, Theorem 2.2 supports the conjecture that the quenched and annealed critical points differ whenever ρ > 1, see Conjecture 1.8 in [10].

6.
One may argue that, in view of the annealed critical exponent 1/α * , the condition α * > 1/2 is somehow reminiscent of the Harris criterion for disorder relevance/irrelevance. Applying this criterion without further precaution tells that disorder should be irrelevant if 1/α * > 2 and relevant if 1/α * < 2. However, in the Harris criterion formulation, disorder is irrelevant if it does not change the critical behavior provided that the disorder strength is sufficiently small. Compared to the pinning model (where Harris' criterion has been confirmed, as mentioned in Introduction), the difference here -as well as for the Random Walk Pinning Model -is that one cannot tune the stength of the disorder: a quenched realization of σ is given, and there is no extra parameter to play with to get an arbitrarily small disorder strength. This is for example what makes it easier for us to prove a critical point shift in the case α * > 1/2, since according to Harris' prediction, disorder should be relevant whatever the disorder strength is -which is here though to be fixed, positive. The case α * ∈ (0, 1/2] is therefore more subtle, however our Theorem 2.2 provides examples where 0 < α * ≤ 1/2 but where a shift in the critical points still occur (so to speak, disorder is strong enough to shift the critical point).

Variations of the main model: balanced and elastic polymers
Pinning, in the quenched or annealed model, means that τ hits a positive fraction of the sites in the quenched renewal σ, up to some σ N . The number of renewals in τ is unlimited. We can alternatively consider the balanced polymer with τ constrained to satisfy τ N = σ N .
A second alternative pinning notion asks whether a positive fraction of τ renewals hit sites of σ, by considering a polymer of length τ N instead of σ N . Physically this may be viewed as an elastic polymer, since τ has a fixed number N of monomers and needs to stretch or contract to place them on renewals of σ.
First variation: balanced polymer, τ N = σ N We first consider τ constrained to have τ N = σ N , since in that case, both pinning notions are equivalent.
We introduceẐ The proof of Proposition 1.1 applies here, establishing the existence and non-randomness of this limit. For the balanced polymer we need the condition (1.7), for otherwise the partition function is 0 with positive probability (whenever σ N = N d σ , see (1.6)), and therefore It is easy to see that thereforeF (β) < 0 for small β. In this sense the constraint τ N = σ N dominates the partition function, which is unphysical, so we assume E σ [σ 1 ] ≥ E τ [τ 1 ], which ensuresF (β) ≥ 0 for all β > 0. There then exists a critical pointβ c := inf{β :F(β) > 0} such thatF (β) > 0 if β >β c andF (β) = 0 if β ≤β c . The positivity ofF (β) implies that τ visits a positive proportion of σ-renewal points, and also that a positive proportion of τ -renewals sit on a σ-renewal point. Clearly, one has that Z σ N,β ≥Ẑ σ N,β , so thatβ c ≥ β c . The following proposition establishes equality, and is proved in Section 6.
Pinning of a renewal on a quenched renewal Hence, here again,β c = 0 if and only if τ ∩ σ is recurrent.
Second variation: elastic polymer of length τ N In the elastic polymer, pinning essentially means that a positive fraction of τ -renewals fall on σ-renewals. The partition function is Standard subadditivity methods to establish existence of lim N →∞ N −1 logZ σ N,β do not work here, but we can consider the lim inf instead, since its positivity implies pinning in the above sense. We therefore definē which is non-decreasing in β, and the critical pointβ c := inf{β :F(β) > 0}. Compared to the balanced polymer, here it is much less apparent a priori that we should haveβ c ≥ β c . It could be favorable for τ to jump far beyond σ N to reach an exceptionally favorable stretch of σ, before τ N . The original model (1.8) does not permit this type of strategy, so the question is whether this allows pinning in the elastic polymer with β < β c . The answer is no, at least if α > 0, or if α = 0 and α ≥ 1, as the following shows; the proof is in Section 6. We do not know whetherβ c = β c when α = 0 and α < 1.

Organization of the rest of the paper
We now present a brief outline of the proofs, and how they are organized.
In Section 3, we prove the first part (2.1) of Theorem 2.1, to establish pinning of the quenched system when β > β ann c , with α + α ≥ 1. We use a rare-stretch strategy to get a lower bound on the partition function: the idea is to consider trajectories τ which visit exponentially rare stretches where the renewals of σ have some (not-well-specified) special structure which reduces the entropy cost for τ to hit a large number of σ sites. In Section 3.3, we classify trajectories of σ according to the size of this entropy cost reduction, then select a class which makes a large contribution to the annealed free energy F ann (β) > 0 (we will fix β > β ann c ). We call trajectories in this class accepting, and the localization strategy of Section 3.4 consists in visiting all accepting segments of σ.
More detailed heuristics are presented in Section 3.1. We also prove (2.3) in Section 3.2, thanks to a finite volume criterion.
In Section 4, we prove the second part (2.2) of Theorem 2.1. First, we rewrite the partition function as that of another type of pinning model -see Section 4.1; this reduces the problem to a system at the annealed critical point β ann c > 0. We then employ a fractional moment method, combined with coarse-graining, similar to one developed for the disordered pinning model [6,16,21,22], and later used for the random walk pinning model [7,11,12]. In Sections 4.2-4.3, we show how one can reduce to proving a finite-size estimate of the fractional moment of the partition function. Then to show that the fractional moment of the partition function is small, we describe a general change of measure argument in Section 4.4. The change of measure is based on a set J of trajectories σ, defined in Section 4.5, which has high probability, but becomes rare under certain conditioning which, roughly speaking, makes the partition function Z σ N,β large. Identification of such an event is the key step; our choice of J only works in full generality for α * > 1/2, necessitating that hypothesis.
In Section 5, we give another sufficient condition for differing critical points, see (5.2).
Then, we construct examples of τ, σ for which this condition holds, to prove Theorem 2.2.
In Section 6, we study the variants of the model introduced in Section 2.2, and prove Propositions 2.3-2.3. We mainly use some particular localization strategies to obtain lower bounds on the free energy of these two models.
The Appendix is devoted to the proof of technical lemmas on renewal processes.

Notations
Throughout the paper, c i are constants which depend only on the distributions of τ and σ, unless otherwise specified.

Sketch of the proof
In the recurrent case, we may use super-additivity argument to obtain directly a lower bound on the quenched free energy via a finite-volume criterion, as presented in Section 3.2. This gives us that β c = 0 when α * < 0, together with a lower bound on the order of the quenched phase transition. However, it is not helpful when α * = 0 and in particular in the transient case, when we try to prove that β c = β ann c > 0, and we need a novel idea.
We use a rare-stretch localization strategy to show that, when α + α ≥ 1, we have F(β) > 0 for all β > β ann c . The idea is to identify "accepting" stretches of σ that we then require τ to visit; the contribution to the free energy from such trajectories τ is shown sufficient to make the free energy positive. Here, the definition of an accepting stretch is made without describing these stretches explicitly, in the following sense: for fixed large , different configurations {σ 1 , . . . , σ } are conducive to large values of |τ ∩ σ| σ to different degrees, as measured by the probability P τ |τ ∩ σ| σ ≥ δ ) for some appropriate δ. We divide (most of) the possible values of this probability into a finite number of intervals, which divides configurations σ into corresponding classes; we select the class responsible for the largest contribution to the annealed partition function, and call the corresponding trajectories σ accepting. This is done in Section 3.3. We then get in Section 3.4 a lower bound for the quenched partition function by considering trajectories of length N which visit all accepting stretches of σ of length .

Finite-volume criterion and proof of (2.3)
The finite-volume criterion comes directly from (1.10): if there exists some N = N β such that E σ [log Z σ N β ,β ] ≥ 1, then we have that F(β) ≥ N −1 β > 0. We will use this simple observation to obtain a lower bound on the free energy in the case α * < 0, showing both β c = 0 and (2.3).
We write Z σ N,β = P τ (σ N ∈ τ )E τ e β|τ ∩σ|σ N |σ N ∈ τ , so by Jensen's inequality we get Let us now estimate the terms on the right. For the second term, we get thanks to (1.19) that for large N (hence large σ N ), log P τ (σ N ∈ τ ) ≥ −2 log σ N . Then, we have E σ [log σ N ] ≤ 2 α −1 log N for N large enough, as established below in (3.28) -in fact the factor 2 can be replaced by a constant arbitrarily close to 1.
For the case α * = 0, it is more delicate since E σ E τ |τ ∩ σ| σ N |σ N ∈ τ grows only as a slowly varying function, and may not be enough to compensate the logarithmic term in (3.1). We develop another approach in the following Sections, to prove that β c = β ann c also when α * = 0. Once we have proven that β c = β ann c , we get directly from F(β) ≤ F ann (β) and Proposition 1.3 that thus proving (2.3) also when α * = 0.
Note that f n is actually a function of the finite renewal sequence B := {0, σ 1 , . . . , σ n }, so we may write it as f n (B). We decompose the probability that appears in f n according to what portion of the cost I( δ)n is borne by σ versus by τ : we fix ε > 0 and write Then by Lemma 3.1, there exists some n ε large enough so that for n ≥ n ε , e −(1+ε)I( δ)n ≤ 1 2 P σ,τ |τ ∩ σ| n ≥ δn, σ n ∈ τ, σ n ≤ 2b n .
Then, (3.12) gives that where the last inequality holds provided that n is large enough (n ≥ n ε ). This leads us to define, for n ≥ n ε the event A n for σ (or more precisely for {0, σ 1 , . . . , σ n }) of being accepting, by (3.14) Then (3.13) gives the lower bound

Localization strategy
Let us fix β > β ann c = β hom c , δ as in (3.10) and large (in particular ≥ n ε , so that (3.15) holds for n = .) We divide σ into segments of jumps which we denote Q 1 , Q 2 , . . ., that is, We now write A for A , and a for a( ). Let (3.16) where the inequality is from (3.13). Informally, the strategy for τ is then to visit all accepting segments, with no other restrictions on other regions. On each such segment one has an energetic gain of at least e (β δn−(a+ε)I( δ))n , thanks to (3.15).
We define M 0 := 0, and iteratively M i := inf{j > M i−1 ; Q j ∈ A}; then M i − M i−1 are independent geometric random variables of parameter p A . Imposing visits to all the accepting segments Q Mi (that is imposing σ Mi , σ (Mi+1) ∈ τ ), one has So, using (3.15), and with the convention that P τ (0 ∈ τ ) = 1, we have Letting k go to infinity, and using the Law of Large Numbers twice, we get We are left with estimating 1 E σ log P τ σ (M1−1) ∈ τ . For any α ≥ 0 and non-random times n, (1.19) ensures that, for any η > 0, one can find some n η such that, for all n ≥ n η , Therefore if η is chosen large enough, and ≥ η , one has The following is proved below.
Combining (3.21) with Lemma 3.2, one gets for ≥ η , and provided that η is large so that 1 log ≤ η, From (3.19) and (3.22), and assuming that η is sufficiently small (depending on ε), we where we used (3.16) in the second inequality, and (3.10) in the third one. If we choose ε small enough, we therefore have that F(β) ≥ 1 4 p A F hom (β) > 0, as soon as β > β ann c = β hom c . This completes the proof of (2.2) Theorem 2.1.

Coarse-graining and change of measure: proof of the second part of Theorem 2.1
In this section and the following ones, we deal with the case α + α < 1. In particular, one has 0 ≤ α < 1, 0 < α < 1 and α * > 0. Moreover, the renewal τ ∩ σ is transient, so E σ,τ [|τ ∩ σ|] < +∞, and the annealed critical point is β ann The proof is decomposed into several parts: first, we use an alternative representation of the partition function in order to obtain a new model which naturally incorporates the quantities E τ,σ [1 {σn∈τ } ]. This model has an annealed critical point u ann c = 0; we apply a fractional moment/coarse-graining/change of measure argument to show the quenched critical point is strictly positive. The core of the proof is Section 4.4 where our choice of change of measure for σ (on a block of L renewals) is specified. The change has a simple form, given by a density η1 J L + 1 J c L for some small η and event J L , which reduces the measure of J L . The difficulty is in selecting J L to have small probabilty, but also so that on J c L the size of the partition function is substantially reduced, in the sense of Lemma 4.1. In other words, we need to identify a small set J L of configurations σ which make the main contribution to the annealed partition function.

Alternative representation of the partition function
We use a standard alternative representation of the partition function, used for example in the Random Walk Pinning Model, see [10] and [7,11,12], and in other various context: it is the so-called polynomial chaos expansion, which is the cornerstone of [13].
We write e β = 1 + z (called Mayer expansion), and expand (1 + z) where i 0 := 0. We want to interpret the expansion (4.1) as the partition function of a new pinning model, with a renewal ν representing the new polymer. To that end, let us define so that n∈N K * (n) = 1, and we denote by ν a renewal process with law P ν and interarrival distribution P ν (ν 1 = n) = K * (n). In particular, ν is recurrent. Lemma 1.4 (proven in Appendix A.2) gives the asymptotic behavior of K * (n): We define z ann Then we write z = z ann c e u : thanks ThusŽ σ N,u does correspond to a pinning model: an excursion (ν i , ν i+1 ] of ν is weighted by e u w(σ, ν i , ν i+1 ). Note that, since E σ [w(σ, a, b)] = 1, the annealed partition function, is the partition function of a homogeneous pinning model, with an underlying recurrent renewal ν: the annealed critical point is therefore u ann c = 0.

Fractional moment method
We are left with studying the new representation of the partition function,Ž σ N,u , and in particular, we want to show that its (quenched) critical point is positive. For that purpose, it is enough to show that there exists some u > 0 and some ζ > 0 such that Indeed, using Jensen's inequality, one has that so that (4.7) implies F(β) = 0.

The coarse-graining procedure
Let us fix the coarse-graining length L = 1/u, and decompose the system into blocks of size L: B i := {(i − 1)L + 1, . . . , iL}, i ≥ 1. Considering a system of length nL, for n ∈ N, we decompose the partition function according to which blocks are visited by ν, and where each visit begins and ends: Then, we denote by Z I the partition function with u = 0, and where ν is restricted to visit only blocks B i for i ∈ I: if I = {1 ≤ i 1 < · · · < i m = n − 1}, (4.10) and we are left with estimating E σ [(Z I ) ζ ]. We choose ζ given by (1 + α * /2)ζ = 1 + α * /4. We will show in the next two sections that for every δ > 0, there exists L 0 such that for any L ≥ L 0 With (4.11), this shows that We choose δ so thatK(n) = e ζ δn −(1+α * /4) sums to 1, making it the inter-arrival probability of a recurrent renewal processτ . We then have E σ (Ž σ nL,u ) ζ ≤ Pτ (n ∈τ ) ≤ 1, which yields (4.7) and concludes the proof of the second part of Theorem 2.1.

Change of measure argument
To estimate Z I and prove (4.12), we use a change of measure, but only on the blocks B i , for i ∈ I. The idea is to choose an event J L depending on {0, σ 1 , . . . , σ L } which has a small probability under P σ , but large probability under the modified measure, see With the event J L to be specified, we define, for some (small) η > 0, Using Hölder's inequality, we have The first term on the right in (4.15) is easily computed: assuming we choose J L with We are left to estimate E σ g I Z I . For this it is useful to The following lemma fills in the missing pieces of the preceding, so we can conclude that and moreover, for 0 ≤ a < b ≤ L with b − a ≥ εL, Additionally, for all L, Pinning of a renewal on a quenched renewal Together, (4.19) and (4.20) say that J L is a small set of configurations σ which make the main contribution to the annealed partition function. As we will see, changing the measure as in (4.14) for a given choice of the set I of blocks in (4.9) shrinks the analog of J L for each block in I, and thereby reduces the size of the partition function contribution due to I, by reducing the "reward" for visiting each block, to the point that pinning does not occur in the quenched system.
To bound E σ g I Z I we need the following extension of Lemma 4.1, which concerns a single block, to cover all blocks. The proof is in section 4.5.
For fixed δ > 0, we claim that, by taking ε, η small enough in Lemma 4.2, if L is large enough, This, together with (4.15) and (4.16), enables us to conclude that (4.12) holds, with δ replaced by δ = 2δ 2ζ , since ζ(1 + α * /2) where we set We next show that given δ > 0, for ε, η sufficiently small, for any given I and 1 ≤ k ≤ m where we used the convention that i 0 = 0, f 0 = 0 and B i0 = {0}. Then, we easily get where we used the existence of a constant c 3 such that, for all a ≥ 1 and L large, ϕ * (aL)/ϕ * (L) ≤ c 3 a α * /2 . In the end, we have It remains to bound the rest of (4.24). We decompose the expectation according to whether the interval [D i k , F i k ] is far from the upper end of the block B i k , that is, Then, using (4.26) if i k − i k−1 ≥ 2, or bounding the probability by 1 if i k − i k−1 = 1, we bound this from above for large L by For the last inequality, we used (1.19) We can now repeat the argument of (4.24). Combining the bounds (4.28) and (4.30), one gets that where we used the fact that ε α * ∧1 γ −α * = γ. Combining (4.27) and (4.31), we obtain (4.24) with δ = c 4 η + c 9 γ, completing the proof of (4.22) and thus of (4.12).

Proof of Lemma 4.1: choice of the change of measure,
We rewrite the partition function in (4.20) as follows: Pinning of a renewal on a quenched renewal where, for any renewal trajectory ν, we defined the event A L (ν) = {(σ, τ ) : σ ν k ∈ τ for all k such that ν k ∈ [0, L]} . (4.33) Observe that if we find a set G of (good) trajectories of ν satisfying To get (4.20), it is therefore enough to show that provided L is large and b − a ≥ εL, we have that for all ν ∈ G P σ,τ J c L | A L (ν) ≤ η/2.
It is standard that S L,1 /B k L has a nontrivial limiting distribution if k L → ∞ (see Section 2.4), and since 0 < α < 1 we also have b n = B n . Define where r L L→∞ → ∞ slowly, to be specified. Here our use of block size k L 1 when E ν [ν 1 ] = ∞ is purely a technical device to deal with the fact we only know n → P(n ∈ τ ) is approximately monotone, not exactly; the spirit of our argument is captured in the k L ≡ 1 case. See Remark 4.3 for further comments.
The heuristic is that large gaps σ i − σ i−1 lower the probability of A L (ν), since they make τ less likely to "'find" the locations σ ν k ; equivalently, A L (ν) reduces the occurrence of large gaps, and in particular reduces the probability that S L,j is a large multiple of its typical size B k L . More precisely, our goal is to show that the conditioning by A L (ν) (for ν ∈ G with appropriate G) induces a reduction in E σ [Y L ] of a size y L which is much larger than Var(Y L ) and Var(Y L |A L (ν)); then, one can take J L of the form Y L − E σ [Y L ] ≤ −y L /2 , and obtain P σ (J L ) → 0, and P σ,τ (J L |A L (ν)) → 1 as L → ∞.
Step 1. We first estimate E[Y L ] and Var(Y L ), without the conditioning on A L (ν). Suppose first that k L → ∞. Since P σ (σ 1 > n) n→∞ ∼ 1 α ϕ(n)n − α , one easily has that, for fixed r > 0, as L → ∞, where we used (2.9) for the last equivalence. Therefore when r L L→∞ → ∞ slowly enough we have Pinning of a renewal on a quenched renewal and hence In the alternative case k L ≡ 1, we have B k L ≡ d σ (see (1.6), (2.8)) so whenever r L → ∞ we have Step 2. We now study the influence of the conditioning by A L (ν) on the events {σ i − σ i−1 > r B k L } and {S L,j > r L B k L }. As we have noted, heuristically one expects the probabilities to decrease, and this is readily shown to be true if P τ (n ∈ τ ) is decreasing in n, but in general such monotonicity only holds asymptotically. So instead we show that, for L large enough, where L → 0 is defined below, see (4.47).
We need to quantify what this conclusion says about the variables S L,j . The following is easily established: there exists δ 0 > 0 such that for all p ∈ (0, 1), ∈ (0, 1) and k ≥ 1 with kp < δ 0 , we have Taking p = P σ (σ 1 > r L B k L ), k = k L so that pk ≤ δ 0 provided r L is large (see (4.38)), taking = L and using the stochastic domination, we obtain that for L large, for all ν, Step 3. We next want to show that for certain ν and j we can make a much stronger statement than (4.41). Specifically, with L fixed, we say an interval I is visited (in ν) if I ∩ ν = φ, and (for 3 ≤ j < γ L − 2) we say the sub-block Q j = ((j − 1)k L , jk L ] is capped (in ν) if Q j−2 ∪ Q j−1 and Q j+1 ∪ Q j+2 are both visited.

Pinning of a renewal on a quenched renewal
We now prove that, if j is such that Q j is capped in ν, we have P σ,τ S L,j > r L B k L A L (ν) ≤ 2(r L ) −(1−α)/2 P σ,τ S L,j > r L B k L . (4.50) Suppose that Q j is capped in ν , and that s is an index such that (ν s−1 , ν s ] ∩ Q j = ∅: we write s ≺ Q j . Note that the events H s,j = σ : max i∈(νs−1,νs]∩Qj are conditionally independent given A L (ν), and (by exchangeability) satisfy where k := ν s − ν s−1 ≤ 5k L (since Q j is capped), and := |(ν s−1 , ν s ] ∩ Q j | ≤ k. Furthermore, since α ∈ [0, 1), by (1.19) we have P(m ∈ τ ) =φ(m)m −(1−α) for some slowly varyingφ(·). Therefore On the other hand, since 0 < α < 1, σ n /B n has a non-degenerate limiting distribution with positive density on (0, ∞) (see Section 2.4). Using again that P(m ∈ τ ) = ϕ(m)m −(1−α) , we therefore find that there exist a constant c 11 such that, for any k ≤ 5k L ,  (4.54) or equivalently, using (4.51), Since the {H s,j : s ≺ Q j } are independent for fixed j (even when conditioned on A L (ν)), with s≺Qj H s,j = {S L,j > r L B k L } and P σ,τ (S L,j > r L B k L ) → 0 as L → ∞, we have from (4.55) that for large L Step 4. We now control the number of capped blocks Q j . Let and define what is heuristically a lower bound for the typical size of |C(ν)| (see (4.59) and its proof): We now restrict our choice of γ L , k L as follows (with further restriction to come): k L L such that γ L = L/k L is slowly varying, and as L → ∞ we have γ L → ∞ slowly enough so ϕ * (k L ) ∼ ϕ * (L) if α * < 1 or m * (k L ) ∼ m * (L) if α * = 1, which is possible since for α * = 1, m * (n) is slowly varying. We obtain D L ∼ γ α * L 1.
We then fix κ > 0 and define the set G of good trajectories by G = {ν : |C(ν)| ≥ κD L }. We wish to choose κ = κ(η, ε) sufficiently small so that, provided that L is large enough and b − a ≥ εL, one has For a fixed ε, for large L we have D M ≥ εD L for all M ≥ εL, so it suffices to prove this for ε = 1, that is, to consider only a = 0, b = L. We now consider two cases.
In the third inequality, we used Lemma A.1 (Lemma A.2 in [22], that we recall in the Appendix since we use it several times) to remove the conditioning at the expense of the constant c 16 . In the last inequality, we used (4.61), taking κ sufficiently small.
• Case of α * = 1, E ν [ν 1 ] = +∞. First, we claim that with our choice of k L , we have P ν (Q j is capped) L→∞ → 1. Indeed, summing over possible locations of a last visit to Q j gives that for 1 ≤ j ≤ γ L and i ≤ (j − 1)k L , where we used k L −1 m=0 P(ν 1 > m) = m * (k L ) in the second inequality, and (1.19) in the last line. Note that the lower bound in (4.62) is uniform in the specified i, j. Therefore, applying (4.62) three times, we obtain uniformly in j ≤ γ L where in the second inequality, we used Lemma A.1 to remove the conditioning at the expense of the constant c 16 .
Step 5. We now have the ingredients to control how the conditioning by We wish to choose γ L , k L so that, for L sufficiently large, L . Since ε n 0 and L/k L is slowly varying, we have B k L L so ε L ≤ ε L so a sufficient condition is γ 1−α * L ε −1 L . But our only restriction so far is from (ii) after (4.57), that γ L → ∞ slowly enough, so we may choose γ L to satisfy this sufficient condition also.
Thanks to (4.41), (4.55) and (4.65), we have for ν ∈ G and large L (4.66) Step 6. We now define Let us compare P(J L ) and P(J c L |A L (ν)) for ν ∈ G. Since α * > 1/2, γ L D −2 L is of order L −(2α * −1)∧1 → 0, so we can choose r L to satisfy γ L D −2 L r α L → 0, which is compatible with our previous requirement on r L involving (4.54). Using (4.39) and Chebyshev's inequality we then get that On the other hand, by (4.41), conditionally on A(ν) with ν ∈ G, the variables 1 {S L,j >r L b k L } , 1 ≤ j ≤ γ L , are (jointly) stochastically dominated by a collection of independent Bernoulli variables with parameter (1 + 2 L )P σ (S L,1 > r L B k L ). Hence, Var(Y L |A L (ν)) ≤ 2γ L P σ (S L,1 > r L B k L ) and as in (4.68) we obtain We have thus proved (4.19) and (4.34), and hence also (4.20).

Remark 4.3.
If n → P(n ∈ τ ) is non-increasing, then we have that k = 0 for all k, and we can replace (4.41) with The term γ L L does not appear in the computation in (4.66), and we can drop the condition (4.65). We can therefore choose γ L = L, k L = 1 in all cases, not just when Then for α * < 1, we have D L of order L α * ϕ * (L) −1 , and the condition γ L D −2 L → 0 in Step 6 becomes L 2α * −1 ϕ * (L) −2 1.
Proof of Lemma 4.2. The proof of Lemma 4.1 is for a single interval [0, L], and we now adapt it to the whole system: we take the same definition for J L . Then, recalling g I from (4.14), we have similarly to (4.32) Pinning of a renewal on a quenched renewal By expanding the product over k ∈ {1, . . . , m}, we obtain (1 S L,j >r L B k L ) in some B i k by independent Bernoullis, remains valid if we also condition on any information about the other B i , i = i k . This means we can ignore the dependencies between different i k ∈ I, and if L is sufficiently large we get that, since ν Bi k ∈ G for all k ∈ K 4 where the bound comes from (4.69).

Then, by independence of the ν Bi
where we used (4.59) in the last inequality. Therefore, plugging (4.73)-(4.74) in (4.72), we get that

Proof of Theorem 2.2: examples with unequal critical points
In this Section, our goal is to exhibit an example of renewal processes τ, σ in which the critical points are unequal. As suggested by comment 6 below Theorem 2.2, in order to do so, one must somehow increase the strength of disorder: this is the idea of our example, where we construct a distribution of τ and σ so that P τ (σ j ∈ τ ) is typically much smaller than E σ P τ (σ j ∈ τ ) -in other words, in the representation (5.1), the weigths w(σ, a, b) (which have mean 1) will be typically very small but with a large variance. Our technique is based on a fractional moment estimate, similar in spirit to that of [31] (which estimates critical point shifts in presence of strong disorder), compare in particular (5.1) below with [31, Eq. (2.15)-(2. 16)].

Pinning of a renewal on a quenched renewal
We use the representation (4.4) which yields that for ζ ∈ (0, 1), then for sufficiently small u > 0 we have (4.7), which establishes inequality of critical points since u ann c = 0. Fix α, α, ϕ, ϕ as in the theorem, and for ε ∈ (0, 1/2) and even N (large enough) define where p N,ε is chosen so that n≥1 K N,ε (n) = 1. Define K N ,ε (n) and p N ,ε similarly with α, ϕ, N in place of α, ϕ.N , and suppose τ, σ have distributions K N,ε , K N ,ε respectively. Note that p N,ε , p N ,ε are in (1/2, 1) provided N is sufficiently large. We use superscripts N, N , ε to designate these parameters in the corresponding probability measures and expectations, for example P N, N ,ε σ,τ (·) and E N,ε σ [·]. We always take N ≥ N . It suffices to show that (5.2) holds, for some N, N , ε, ζ: we prove below that the main contribution to the numerator and denominator of (5.2) comes from jumps of size 1. For the denominator in τ , we observe that For the numerator in (5.2), we write For m < N/2 we have P N ,ε τ (m ∈ τ ) = ε m (and similarly for σ for m < N /2), so it follows that if ε is small enough (and N large) Second, there exists c 18 (ε) and slowly varying ψ 0 , ψ 0 , ψ, ψ such that for all sufficiently large N, N , Note that by (1.19) the inequalities in (5.6) and (5.7) hold for sufficiently large m, so the aspect to be proved is that the given ranges of m are sufficient in the case of distributions K N,ε , K N ,ε . We henceforth take N ≥ 2 N 1/(1−α) ψ( N ) and ζ ∈ ( α/(1 − α), 1). Applying (5.5) and (5.7) to part of the sum on the right in (5.4) then yields that for some slowly varying ψ 1 , provided N is large, Then taking N large and applying (5.6) and (5.7) to the rest of that same sum yields that for some slowly varying ψ 2 ,   (5.11) and for 0 ≤ k < N/2 Now (5.5) follows from (5.11) when N/2 ≤ m < N , so we consider m ≥ N . It is enough to show that for all m ≥ N and N ≤ j ≤ m, Pinning of a renewal on a quenched renewal For any N ≤ j ≤ m, we have that where we used (5.12) for the terms k < N/2, and (5.11) for the terms N/2 ≤ k < N . We now show that there is a constant c 18 such that uniformly for k < N/2, j ≥ N , The probability on the left is equal to P τ (τ 1 = j − k)/P τ (τ 1 > j − N ). Therefore for each j ≥ 3N/2, the probabilities P τ (G m = m − k|F m = m − j), k < N/2 are all of the same order so are bounded by c 20 /N for some c 20 .
For the other j values, that is, N ≤ j < 3N/2, the probabilities P τ (G m = m − k | F m = m − j), j − N < k ≤ j − N/2, are all equal so have common value at most 2/N . For k ≤ j − N the probabilities P τ (G m = m − k|F m = m − j) are uniformly smaller than this common value provided N is large, because n −(1+α) ϕ(n) ≤ 2p N,ε /N for all n ≥ N . Therefore the bound of 2/N applies for all k ≤ j − N/2, which includes all k < N/2. This proves (5.15), which with (5.14) proves (5.13) and thus (5.5).
It remains to prove (5.6); (5.7) is equivalent. Let δ ∈ (0, 1) to be specified, and define , n 0 (m) = max{n : a n ≤ m(log m) β } , where a n is as in (2.9). If τ n = m for some n > n 0 (m), it means that τ is "very compressed" in the sense that τ n a n . It is easily checked that there exist slowly varying ψ 3 , ψ 4 such that n 0 (m) ∼ m α ψ 3 (m) as m → ∞ (5.16) and n ≥ k N := N α/(1−α) ψ 4 (N ) =⇒ 2nN ≤ a n (log a n ) β . (5.17) It follows from (5.16) that we can choose the slowly varying function ψ such that for large N , We first handle the "very compressed" case, that is n > n 0 (m): using Lemma A.3, provided N is large, P N,ε τ (τ n = m for some n > n 0 (m)) ≤ P N,ε τ (τ n0(m) < m) Next, we consider the "not very compressed" case, with a large gap, that is, n ≤ n 0 (m) and M n ≥ m . Let m ≥ N 1/(1−α) ψ(N ), 1 ≤ n ≤ n 0 (m) and 1 ≤ j < m; note m ≥ N provided N is large. For vectors (t 1 , . . . , t n ) ∈ Z n , consider the mapping which adds j to the first coordinate t i satsifying t i > m , when one exists. This map is one-to-one, and provided m is large, it decreases the corresponding probability P N,ε τ (∆ 1 = t 1 , . . . , ∆ n = t n ) by at most a factor of (δ/ log m) 2(1+α) . It follows that Summing over 1 ≤ j < m we obtain ∈ [m, 2m)) .

Variations of the model: proofs of Proposition 2.3 and Proposition 2.4
We start with the proof of Proposition 2.4, since we adapt it for the proof of Proposition 2.3, in which we need to control additionally |τ | σ N .
As in Section 3, we divide the system into blocks B i := {0, σ (i−1)L+1 −σ (i−1)L , . . . , σ iL − σ (i−1)L }, with L to be specified. For b N from (2.9), there exists v 0 > 0 such that P σ (σ N > v 0 b N ) ≤ 1/4 for N large. Define the event of being good by G L := (σ 1 , . . . , σ L ) : Since β > β c , there exists L 0 such that, for L ≥ L 0 We now set I = I(σ) = {i : B i ∈ G L } = {i 1 , i 2 , . . .}, the set of indices of the good blocks, and set i 0 = 0. There can be at most v 0 b L τ -renewals per block, so restricting trajectories to visit only blocks with index in I, we get that for all m ∈ N Pinning of a renewal on a quenched renewal with the convention that P τ (τ 1 = 0) means 1. Then, letting m → ∞, we get s. (6.5) Let us estimate the last term. Thanks to (1.5), we have that provided L is large. Then, Lemma 3.2 applies: for L large, since P σ ( Hence for some L 0 , for L ≥ L 0 , (6.6) and provided that L is large enough, we have thatF (β) > 0 for any β > β c , meaning β c ≥β c .

Proof of Proposition 2.4(ii)
Observe that the annealed systems for the original and elastic polymers have the same critical point, by Remark 1.2. Therefore β ann c ≤β c . When α = 0, α ≥ 1, it then follows from Theorem 2.1 that β c ≤β c , so equality holds.
Hence it remains to prove β c ≤β c assuming α > 0, by showing that pinning in the elastic polymer (length-τ N system) implies pinning in the original polymer (length-σ N system.) In the recurrent case, β c = 0 so there is nothing to prove. So we assume transience, which here implies α, α ∈ (0, 1). Let v N = e 2 αβN/α and τ , so is nonrandom, up to a null set) and assume thatF (β) > 0. We define the truncated Then it is not hard to show that P σ -a.s., for large N , It is essential here that the truncation in the partition function be at mv N , not at the much larger value v mN , as we want the allowed length of trajectories to grow essentially only linearly in m. But we need to know that with this length restriction, the log of the partition function is still of order mN .
With this lemma in hand, we easily have that Then thanks to Remark 1.2 (and because 1 which gives that F(β) > 0 for any β >β c , that is β c ≤β c . This concludes the proof of Proposition 2.4(i).
For any fixed j 1 < · · · < j m with |j i − j i−1 | ≤ v N (representing possible values of the J i ), we can decompose a product of X variables as follows: Hence, summing over all such j 1 < · · · < j m , we get the bound Here in a mild abuse of notation we write E J [· | σ] for the expectation over U , and we will write E σ,J for the expectation over (σ, U ). Since J 1 is a stopping time, the summands on the right side of (6.10) are i.i.d. functions of (σ, U ). By (6.10) and Jensen's inequality, and hence, writing x − for the negative part of x and using (3.27), It then follows from (6.10) that (6.14) We will show that there is a choice of N, The lemma will then follow from (6.14) and (6.15).
Fix K to be specified and define From (6.9), we then have N,β,0 < K e NF (β)/2 → 0 as N → ∞. (6.16) We define the marking probabilities which only depends on σ 1 , . . . , σ j , and We may view this as follows: for each j < R N we mark σ j with probability Q N,j , independently, and we mark σ R N with probability 1; σ J1 is the first σ j to be marked. As a result we have P J (J 1 = j | σ) =Q N,j (σ), and J 1 ≤ v N . Note that this weights J 1 toward values j for which X N,j is large, which, heuristically, occurs when σ j follows a favorable stretch of the disorder σ.
We now consider (6.15), and write For the sum in (6.17), using 1 − e −x ≤ x we get X N,j ≥ e NF (β)/2 Q N,j ≥ e NF (β)/2Q N,j for all j < R N , and therefore For the last term in (6.17), from (6.12) we have Combining these bounds and averaging over σ, we get from (6.17) that For the first term on the right side of (6.18), when R N (σ) < v N we have from the definition of R N that (6.19) We therefore get using (6.16) that 20) provided that K and then N are chosen large enough.
For the last term in (6.18), we have from (6.19) and (6.13) that In addition, it is routine that there exists a constant c 39 such that E σ log σ n log n 2 ≤ c 39 for all n ≥ 2.
Hence using (6.16) and the Cauchy-Schwarz inequality Combining (6.21) and (6.22) we get that if we take K and then N is large enough, Plugging (6.20) and (6.23) into (6.18), we finally get (6.15).

Proof of Proposition 2.3
As noted in Section 2.2, we only need prove thatβ c ≤ β c , so let us fix β > β c , and show thatF (β) > 0. We write Z σ,f N,β (H) for E τ [e β|τ ∩σ|σ N 1 H ], for an event H. (Note this partition function may involve trajectories not tied down, that is, with σ N / ∈ τ .) A first observation is that for fixed q ≥ 1, By (1.7), if we fix q large enough and then take N large, the minimum here is achieved We continue notations from Section 6.1: we take v 0 such that P σ (σ n ≥ v 0 b n ) ≤ 1/4, use G L from (6.1), take L ≥ L 0 large so that (6.3) holds, and consider the set of good blocks I = {i : B i ∈ G L }. The idea of the proof is similar to that of Proposition 2.4, but in addition, we need to control the size of τ L on good blocks.
On the event E 1 ∩ E 2 , as in the proof of Proposition 2.4 we restrict to τ visiting only good blocks B i , including visits to the endpoints σ (i−1)L , σ iL . Since (by definition of G L ) on each good block B i there are at most v 0 b L τ -renewals, up to τ m such τ visit at least m := m/(v 0 b L ) good blocks. We also choose L large so that m ≤ 1 8 m: on the event E 1 , it ensures that i m ≤ 7m/8. We denote k 0 ≤ mL the index such that τ k0 = σ i m L : since τ mL − τ k0 ≤ τ mL , we have on E 1 , restricting to τ visiting the first m good blocks, where we used that for L large enough, P σ (E 1 ∩ E 2 ) ≥ 1/2. Therefore dividing by mL and letting m → ∞ gives where we used Lemma 6.2 to get that lim m→∞ 1 mL E σ 1 {σ mL/8 ≥dτ mL} log P τ (τ mL ≤ σ mL/8 ) = 0.

Case 2.
We now deal with the case when µ σ := E σ [σ 1 ] < +∞ and E τ [τ 1 ] ≤ E σ [σ 1 ]. Here b n = µ σ n, see (2.9). Let us fix m ∈ N large, and consider a system consisting in m blocks of length L. Decomposing according to whether the first block is good or not we have, recalling G L from (6.1), Recalling the indicator 1 {σ L ∈τ } in the definition of Z σ L,β , if τ k0 = σ L for some k 0 , then the relation τ mL − τ k0 ≤ τ mL guarantees that Z σ,f mL,β (τ mL ≤ σ mL ) ≥ Z σ L,β × P τ (τ mL ≤ σ mL − σ L ). Provided m exceeds some m 0 we have d τ m/(m − 1) < E[τ 1 ] ≤ µ σ , and therefore for sufficiently large L we have P σ (σ (m−1)L ≥ d τ mL) ≥ 1/2. Since B 1 is independent of all other blocks and P σ (G L ) ≥ 1/2, it then follows from (6.31) that for σ ∈ G L , We choose ε = 1 40 F(β), and fix m 1 ≥ m 0 such that one can apply Lemma 6.3. Then, we take L large enough such that 1 Combining this with (6.30) and (6.32), we obtain for sufficiently large L To conclude the proof of Proposition 2.3, we use that (logẐ σ N,β ) N ∈N is an ergodic super-additive sequence, so thatF (β) ≥ sup N ∈N In view of (6.25), we therefore have that, if L is large enough, for q as specified after (6.24), Then we can use (6.34) to obtain, by taking L large enough, Proof of Lemma 6.2. We have the following crude bound: there exists a constant c 40 > 0 such that, for every k ≥ d τ n, where the last inequality follows from the fact that P τ τ 1 > k n is bounded away from 1, for all k ≥ d τ n. Since E σ [σ 1 ] = +∞, we may choose a sequence α n with α n /n → +∞, and uniformly in x ≥ 1/10, P σ (σ xn ≤ α n ) n→∞ → 0. We get where we used (6.37) in the second inequality. Letting n → ∞, we see that the limit is 0 thanks to our choice of α n .
Hence, observing that (m − 2)µ τ ≥ d τ m provided that m has been fixed large enough, we can get Hence we obtain, using (6.39) which in turn also gives that lim inf n→∞

A.1 Some estimates on renewal processes
First of all, we state a result that we use throughout the paper, which is Lemma A.2 in [22] (that was slightly generalized in [2] to cover the case α = 0).
As noted in Section 2.4, for a n as in (2.10), τ n /a n converges to an α-stable distribution with some density h, which is bounded and satisfies Further, by the local limit theorem for such convergence, see [23, §50], and the fact that a α k ∼ kϕ(a k ), for any given 0 < θ < K < ∞ we have as k → ∞ uniformly over m ∈ [θa k , Ka k ]. Together with (A.2) and Lemma A.3 below, which deals with the case m a k , we thereby obtain the following uniform bound.
Lemma A.2. Assume α ∈ (0, 1). There exists a constant c 42 > 0 such that, for k large enough and all m ≥ k, P τ (τ k = m) ≤ c 42 k P τ (τ 1 = m) , For the lower tail we have the following.
Lemma A.3. Assume α ∈ (0, 1). There exist a constant c 43 such that, for all < 1/2 and n ≥ 1 P τ (τ n ≤ a n ) ≤ exp −c 43 In particular, for any γ ∈ (0, 1), for all n ≥ 1, . Note that this lemma implies that the density h also satisfies for some constant c 45 > 0 and sufficiently small x.
We can approximately optimize (A.6) by taking t = t n = t n ( ) given by Λ(−t n ) t n = −2 a n n .
Such a solution exists since t −1 Λ(−t) → −∞ as t → 0, t −1 Λ(−t) → −d τ as t → ∞, and 2 a n /n ≥ 2d τ . We therefore end up with P τ (τ n ≤ a n ) ≤ exp(− a n t n ), so we need a lower bound on a n t n .
We emphasize that the constants c i and n 0 do not depend on . It follows that λ a n ≥ c 49 /t n , or equivalently a n t n ≥ c 49 −(λ−1) , which with (A.9) completes the proof for n ≥ n 0 . We finish by observing that for q = max n<n0 a n , for all n < n 0 and ≤ 1/2 we have P τ (τ n ≤ a n ) ≤ P τ (τ 1 ≤ q) < 1, and P τ (τ n ≤ a n ) = 0 if < 1/q. Therefore after reducing c 43 if necessary, (A.4) also holds for n < n 0 .
Let us finally prove some analogue of Lemma A.3 in the case α = 1 with E[τ 1 ] = +∞, that will be needed below. Recall in that case the definition (2.9) of a n , and that τ n /a n → 1 in probability.
Of course, this result is not optimal when is not close to 0, but it is sufficient for our purpose.
Proof. We use the same notations as for the proof of Lemma A.3. The difference here is that, as t ↓ 0, Λ(−t) ∼ −tm(1/t), where we recall thatm(x) = E τ [τ 1 ∧ x] diverges as a slowly varying function as x → ∞. Hence, we still have that t −1 Λ(−t) → −∞ as t ↓ 0, and as in (A.8) we may define t n = t n ( ) as the solution of Λ(−t n )t −1 n = −2 a n /n . Then we estimate t n : using that a n ∼ nm(a n ), and that Λ(−t) ∼ −tµ(1/t), we get that there is a constant c 50 such that for any n ≥ 1, m(1/t n ) ≤ c 50 m(a n ) .