The continuum disordered pinning model

Any renewal processes on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {N}}_0$$\end{document}N0 with a polynomial tail, with exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document}α∈(0,1), has a non-trivial scaling limit, known as the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}α-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i.i.d. random environment, called disordered pinning models. We show that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \left( \frac{1}{2}, 1\right) $$\end{document}α∈12,1 these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document}R in a white noise random environment, with subtle features: Any fixed a.s. property of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}α-stable regenerative set (e.g., its Hausdorff dimension) is also an a.s. property of the CDPM, for almost every realization of the environment. Nonetheless, the law of the CDPM is singular with respect to the law of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}α-stable regenerative set, for almost every realization of the environment. The existence of a disordered continuum model, such as the CDPM, is a manifestation of disorder relevance for pinning models with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \left( \frac{1}{2}, 1\right) $$\end{document}α∈12,1.


Introduction
We consider disordered pinning models, which are defined via a Gibbs change of measure of a renewal process, depending on an external i.i.d. random environment. First introduced in the physics and biology literature, these models have attracted much attention due to their rich structure, which is amenable to a rigorous investigation; see, e.g., the monographs of Giacomin [19,20] and den Hollander [13].
In this paper we define a continuum disordered pinning model (CDPM), inspired by recent work of Alberts et al. [4] on the directed polymer in random environment. The interest for such a continuum model is manifold: • It is a universal object, arising as the scaling limit of discrete disordered pinning models in a suitable continuum and weak disorder limit, Theorem 1.3. • It provides a tool to capture the emerging effect of disorder in pinning models, when disorder is relevant, Sect. 1.4 for a more detailed discussion. • It can be interpreted as an α-stable regenerative set in a white noise random environment, displaying subtle properties, Theorems 1.4, 1.5 and 1.6.

Renewal processes and regenerative sets
Let τ := (τ n ) n≥0 be a renewal process on N 0 , that is τ 0 = 0 and the increments (τ n −τ n−1 ) n∈N are i.i.d. N-valued random variables (so that 0 = τ 0 < τ 1 < τ 2 < · · · ). Probability and expectation for τ will be denoted respectively by P and E. We assume that τ is non-terminating, i.e., P(τ 1 < ∞) = 1, and K (n) := P(τ 1 = n) = L(n) n 1+α , as n → ∞, (1.1) where α ∈ (0, 1) and L(·) is a slowly varying function [8]. We assume for simplicity that K (n) > 0 for every n ∈ N (periodicity complicates notation, but can be easily incorporated). Let us denote by C the space of all closed subsets of R. There is a natural topology on C, called the Fell-Matheron topology [15,24,25], which turns C into a compact Polish space, i.e. a compact separable topological space which admits a complete metric. This can be taken as a version of the Hausdorff distance (see Appendix A for more details).
Identifying the renewal process τ = {τ n } n≥0 with its range, we may view τ as a random subset of N 0 , i.e. as a C-value random variable (hence we write {n ∈ τ } := k≥0 {τ k = n}). This viewpoint is very fruitful, because as N → ∞ the rescaled set τ N = τ n N n≥0 (1.2) converges in distribution on C to a universal random closed set τ of [0, ∞), called the α-stable regenerative set ([16], [19,Thm. A.8]). This coincides with the closure of the range of the α-stable subordinator or, equivalently, with the zero level set of a Bessel process of dimension δ = 2(1 − α) (see Appendix A), and we denote its law by P α .
Remark 1.1 Random sets have been studied extensively [24,25]. Here we focus on the special case of random closed subsets of R. The theory developed in [16] for regenerative sets cannot be applied in our context, because we modify renewal processes through inhomogeneous perturbations and conditioning (see (1.4)-(1.9) below). For this reason, in Appendix A we review and develop a general framework to study convergence of random closed sets of R, based on a natural notion of finite-dimensional distributions.

Disordered pinning models
Let ω := (ω n ) n∈N be i.i.d. random variables (independent of the renewal process τ ), which represent the disorder. Probability and expectation for ω will be denoted respectively by P and E. We assume that E[ω n ] = 0, Var(ω n ) = 1, ∃t 0 > 0 : (t) := log E[e tω n ] < ∞ for |t| ≤ t 0 . (1. 3) The disordered pinning model is a random probability law P ω N ,β,h on subsets of {0, . . . , N }, indexed by realizations ω of the disorder, the system size N ∈ N, the disorder strength β > 0 and bias h ∈ R, defined by the following Gibbs change of measure of the renewal process τ : where the normalizing constant The properties of the model P ω N ,β,h , especially in the limit N → ∞, have been studied in depth in the recent mathematical literature (see e.g. [13,19,20] for an overview). In this paper we focus on the problem of defining a continuum analogue of P ω N ,β,h . Since under the "free law" P the rescaled renewal process τ/N converges in distribution to the α-stable regenerative set τ , it is natural to ask what happens under the "interacting law" P ω N ,β,h . Heuristically, in the scaling limit the i.i.d. random variables (ω n ) n∈N should be replaced by a one-dimensional white noise (dW t ) t∈[0,∞) , where W = (W t ) t∈[0,∞) denotes a standard Brownian motion (independent of τ ). Looking at (1.4), a natural candidate for the scaling limit of τ/N under P ω N ,β,h would be the random measure P α;W T,β,h on C defined by where the continuum partition function Z α;W T,β,h would be defined in analogy to (1.5). The problem is that a.e. realization of the α-stable regenerative set τ has zero Lebesgue measure, hence the integral in (1.6) vanishes, yielding the "trivial" definition P α;W T,β,h = P α .
These difficulties turn out to be substantial and not just technical: as we shall see, a non-trivial scaling limit of P ω N ,β,h does exist, but, for α ∈ 1 2 , 1 , it is not absolutely continuous with respect to the law P α [hence no formula like (1.6) can hold]. Note that an analogous phenomenon happens for the directed polymer in random environment [4].
Recall that C denotes the compact Polish space of closed subsets of R. We denote by M 1 (C) the space of Borel probability measures on C, which is itself a compact Polish space, equipped with the topology of weak convergence. We will work with a conditioned version of the disordered pinning model (1.4), defined by (In order to lighten notation, when N / ∈ N we agree that P ω,c N ,β,h := P ω,c N ,β,h ). Recalling (1.2), let us introduce the notation Our first main result is the convergence in distribution of this random variable, provided α ∈ 1 2 , 1 and the coupling constants β = β N and h = h N are rescaled appropriately: (1.11) Theorem 1.3 (Existence and universality of the CDPM) Fix α ∈ ( 1 2 , 1), T > 0, β > 0,ĥ ∈ R. There exists a M 1 (C)-valued random variable P α;W,c T,β,ĥ , called the (conditioned) continuum disordered pinning model (CDPM), which is a function of the parameters (α, T,β,ĥ) and of a standard Brownian motion W = (W t ) t≥0 , with the following property: • for any renewal process τ satisfying (1. We refer to Sect. 1.4 for a discussion on the universality of the CDPM. We stress that the restriction α ∈ 1 2 , 1 is substantial and not technical, being linked with the issue of disorder relevance, as we explain in Sect. 1.4 (see also [10]).
Let us give a quick explanation of the choice of scalings (1.11). This is the canonical scaling under which the partition function Z ω N ,β N ,h N in (1.5) has a nontrivial continuum limit. To see this, write where ε β,h n := e βω n − (β)+h − 1. By Taylor expansion, as β, h tend to zero, one has the asymptotic behavior E[ε β,h n ] ≈ h and Var(ε β,h n ) ≈ β 2 . Using this fact, we see that the asymptotic mean and variance behavior of the first term (k = 1) in the above series is because P(n ∈ τ ) ≈ n α−1 /L(n), by (1.1) (see (2.10) below). Therefore, for these quantities to have a non-trivial limit as N tends to infinity, we are forced to scale β N and h N as in (1.11). Remarkably, this is also the correct scaling for higher order terms in the expansion for Z ω N ,β,h , as well as for the measure P ω N ,β N ,h N to converge to a non-trivial limit.
We now describe the continuum measure. For a fixed realization of the Brownian motion W = (W t ) t∈[0,∞) , which represents the "continuum disorder", we call P α;W,c T,β,ĥ the quenched law of the CDPM, while will be called the averaged law of the CDPM. We also introduce, for T > 0, the law P α;c T of the α-stable regenerative set τ restricted on [0, T ] and conditioned to visit T : The seeming contradiction between (1.14) and (1.15) is resolved noting that in (1.14) one cannot exchange "∀A ⊆ C" and "for P-a.e. W ", because there are uncountably many A ⊆ C (and, of course, the set A appearing in (1.15) depends on the realization of W ).
We conclude our main results with an explicit characterization of the CDPM. As we discuss in Appendix A, each closed subset C ⊆ R can be identified with two non-decreasing and right-continuous functions g t (C) and d t (C), defined for t ∈ R by As a consequence, the law of a random closed subset X ⊆ R is uniquely determined by the finite dimensional distributions of the random functions (g t (X )) t∈R and (d t (X )) t∈R , i.e. by the probability laws on R 2k given, for k ∈ N and −∞ < t 1 < t 2 < . . . < t k < ∞, by As a further simplification, it is enough to focus on the event that X ∩ [t i , t i+1 ] = ∅ for all i = 1, . . . , k, that is, one can restrict (x 1 , y 1 , . . . , x k , y k ) in (1.17) on the following set: where we restrict (x 1 , y 1 , . . . , x k , y k ) on the set (1.18), with t 0 = 0 and t k+1 := T . (2) A family of continuum partition functions for our model: These were constructed in [10] as the limit, in the sense of finite-dimensional distributions, of the following discrete family (under an appropriate rescaling): (ii) For all k ∈ N and 0 =: t 0 < t 1 < · · · < t k < t k+1 := T , and for (x 1 , y 1 , . . . , x k , y k ) restricted on the set R where we set y 0 := 0 and x k+1 := T , and where f α;c T ;t 1 ,...,t k (·) is defined in (1.19).

Discussion and perspectives
We conclude the introduction with some observations on the results stated so far, putting them in the context of the existing literature, stating some conjectures and outlining further directions of research.

(Disorder relevance)
The parameter β tunes the strength of the disorder in the model P ω,c N ,β,h (1.9), (1.4). When β = 0, the sequence ω disappears and we obtain the socalled homogeneous pinning model. Roughly speaking, the effect of disorder is said to be: • irrelevant if the disordered model (β > 0) has the same qualitative behavior as the homogeneous model (β = 0), provided the disorder is sufficiently weak (β 1); • relevant if, on the other hand, an arbitrarily small amount of disorder (any β > 0) alters the qualitative behavior of the homogeneous model (β = 0).
Recalling that α is the exponent appearing in (1.1), it is known that disorder is irrelevant for pinning models when α < 1 2 and relevant when α > 1 2 , while the case α = 1 2 is called marginal and is more delicate (see [20] and the references therein for an overview). It is natural to interpret our results from this perspective. For simplicity, in the sequel we set h N :=ĥ L(N )/N α , as in (1.11), and we use the notation P ω,c N T,β N ,h N (d(τ/N )) (1.10), for the law of the rescaled set τ/N under the pinning model.
In the homogeneous case (β = 0), it was shown in [31, Theorem 3.1] 1 that the weak limit of P α;c N T,0,h N (d(τ/N )) as N → ∞ is a probability law P α;c is absolutely continuous with respect to the reference law P α;c T (recall (1.13)): , (1.23) where L T (τ ) denotes the so-called local time associated to the regenerative set τ . We stress that this result holds with no restriction on α ∈ (0, 1).
Turning to the disordered model β > 0, what happens for α ∈ 0, 1 2 ? In analogy with [9,11], we conjecture that for fixed β > 0 small enough, the limit in distribution of P ω,c N T,β,h N (d(τ/N )) as N → ∞ is the same as for the homogeneous model (β = 0), i.e. the law P α;c T,0,ĥ defined in (1.23). Thus, for α ∈ 0, 1 2 , the continuum model is non-disordered (deterministic) and absolutely continuous with respect to the reference law.
This is in striking contrast with the case α ∈ 1 2 , 1 , where our results show that the continuum model P α;W,c T,β,ĥ is truly disordered and singular with respect to the reference law (Theorems 1.3, 1.4, 1.5). In other terms, for α ∈ 1 2 , 1 , disorder survives in the scaling limit (even though β N , h N → 0) and breaks down the absolute continuity with respect to the reference law, providing a clear manifestation of disorder relevance.
We refer to [10] for a general discussion on disorder relevance in our framework.
2. (Universality) The quenched law P α;W,c T,β,ĥ of the CDPM is a random probability law on C, i.e. a random variable taking values in M 1 (C). Its distribution is a probability law on the space M 1 (C)-i.e. an element of M 1 (M 1 (C))-which is universal: it depends on few macroscopic parameters (the time horizon T , the disorder strength and biasβ,ĥ and the exponent α) but not on finer details of the discrete model from which it arises, such as the distributions of ω 1 and of τ 1 : all these details disappear in the scaling limit.
Another important universal aspect of the CDPM is linked to phase transitions. We do not explore this issue here, referring to [10, §1.3] for a detailed discussion, but we mention that the CDPM leads to sharp predictions about the asymptotic behavior of the free energy and critical curve of discrete pinning models, in the weak disorder regime λ, h → 0.

(Bessel processes)
In this paper we consider pinning models built on top of general renewal processeses τ = (τ k ) k∈N 0 satisfying (1.1) and (1.7). In the special case when the renewal process is the zero level set of a Bessel-like random walk [1] (recall Remark 1.2), one can define the pinning model (1.4), (1.9) as a probability law on random walk paths (and not only on their zero level set).
Rescaling the paths diffusively, one has an analogue of Theorem 1.3, in which the CDPM is built as a random probability law on the space C([0, T ], R) of continuous functions from [0, T ] to R. Such an extended CDPM is a continuous process (X t ) t∈[0,T ] , that can be heuristically described as a Bessel process of dimension δ = 2(1 − α) interacting with an independent Brownian environment W each time X t = 0. The "original" CDPM of our Theorem 1.3 corresponds to the zero level set τ := {t ∈ [0, T ] : X t = 0}.
We stress that, starting from the zero level set τ , one can reconstruct the whole process (X t ) t∈[0,T ] by pasting independent Bessel excusions on top of τ (more precisely, since the open set [0, T ]\τ is a countable union of disjoint open intervals, one attaches a Bessel excursion to each of these intervals). 2 This provides a rigorous definition of (X t ) t∈ [0,T ] in terms of τ and shows that the zero level set is indeed the fundamental object. . Such a limit law would inherit scaling properties from the continuum partition functions, Theorem 2.4 (iii). (See also [29] for related work in the non-disordered caseβ = 0).

Organization of the paper
The rest of the paper is organized as follows.
• In Sect. 2, we study the properties of continuum partition functions.
• In Sect. 3, we prove Theorem 1.6 on the characterization of the CDPM, which also yields Theorem 1.3. • In Sect. 4, we prove Theorems 1.4 and 1.5 on the relations between the CDPM and the α-stable regenerative set. • In Appendix A, we describe the measure-theoretic background needed to study random closed subsets of R, which is of independent interest. • Lastly, in Appendices B and C we prove some auxiliary estimates.

Continuum partition functions as a process
In this section we focus on a family Z α;W,ĉ β,ĥ (s, t) 0≤s≤t<∞ of continuum partition functions for our model, which was recently introduced in [10] as the limit of the discrete family (1.21) in the sense of finite-dimensional distributions. We upgrade this convergence to the process level (Theorem 2.1), which allows us to deduce important properties (Theorem 2.4). Besides their own interest, these results are the key to the construction of the CDPM.

Fine properties of continuum partition functions
, with m ≤ n ∈ N, along the main diagonal and linearly interpolating Z ω,c β,h (·, ·) on each triangle. In this way, we can regard as random variables taking values in the space C([0, ∞) 2 ≤ , R), equipped with the topology of uniform convergence on compact sets and with the corresponding Borel σ -algebra. The randomness comes from the disorder sequence ω = (ω n ) n∈N . Even though our main interest in this paper is for α ∈ 1 2 , 1 , we also include the case α > 1 in the following key result, which is proved in Sect. 2.2 below.

Theorem 2.1 (Process level convergence of partition functions)
Let τ be a renewal process satisfying (1.1) and (1.7), and ω be an is defined as follows, with C α as in (1.20) and t 0 := s: (2.4)

Remark 2.2
The integral in (2.3) is defined by expanding formally the product of differentials and reducing to standard multiple Wiener and Lebesgue integrals. An alternative equivalent definition is to note that, by Girsanov's theorem, the law of By Theorem 2.1, we can fix a version of the continuum partition functions Z α;W,ĉ β,ĥ (s, t) which is continuous in (s, t). This will be implicitly done henceforth.
We can then state some fundamental properties, proved in Sect. 2.3.

Theorem 2.4 (Properties of continuum partition functions)
For all α ∈ 1 2 , 1 ,β > 0, h ∈ R the following properties hold: , and is independent of (iii) (Scaling Property) For any constant A > 0, one has the equality in distribution The rest of this section is devoted to the proof of Theorems 2.1 and 2.4. We recall that assumption (1.1) entails the following key renewal estimates, with C α as in (1.20): by the classical renewal theorem for α > 1 and by [12,18] for α ∈ (0, 1). Let us also note that the additional assumption (1.7) for α ∈ (0, 1) can be rephrased as follows: up to a possible change of the constants C, n 0 , ε.

Proof of Theorem 2.1
We is a tight family, because the finitedimensional distribution convergence was already obtained in [10] (see Theorem 3.1 and Remark 3.3 therein). We break down the proof into five steps.
Step 1. Moment criterion. We recall a moment criterion for the Hölder continuity of a family of multi-dimensional stochastic processes, which was also used in [3] to prove similar tightness results for the directed polymer model. Using Garsia's inequality [17,Lemma 2] with (x) = |x| p and ϕ(u) = u q for p ≥ 1 and pq > 2d, the modulus of continuity of a continuous function f : (2.12) which by triangle inequality, translation invariance and symmetry can be reduced to Step 2. Polynomial chaos expansion To simplify notation, let us denote 14) and rewrite N ,r as a polynomial chaos expansion:  17) where ψ N ,r (∅) := 1 and for I = {n 1 < n 2 < · · · < n k } ⊂ N, recalling (2.10), we can write 18) with n 0 := 0, n k+1 := r .
To prove (2.13), we write Z ω,c β N ,h N (0, s N ) = N ,q and Z ω,c β N ,h N (0, t N ) = N ,r , with q := s N and r := t N , so that 0 ≤ q < r ≤ N . For a given truncation level m = m(q, r, N ) ∈ (0, q), that we will later choose as (2.20) To establish (2.13) and hence tightness, it suffices to show that for each i = 1, 2, 3, (2.21) Step 3. Change of measure We now estimate the moments of ξ N ,i defined in (2.14).
It follows, in particular, that {ζ 2 N ,i } i,N ∈N are uniformly integrable. We can then apply a change of measure result established in [10, Lemma B.1], which asserts that we

Relation (2.21), and hence the tightness of
, and hence will be omitted. First we write 2 as  1 2 is finite and depends only on l, by (2.23).
(2.36) Let ε be as in condition (2.11). We first consider the case r − q ≥ ε 2 r , for which we bound (2.37) Applying the bound (2.32) with m = 0, since q < r , we obtain (2.38) By the same calculation as in (2.33), we then have, using r − q ≥ ε 2 r , which gives the desired bound (2.25). Now we consider the case r − q ≤ ε 2 r . Denote I := {n 1 , . . . , n k }. Recalling the definition of ψ N ,r in (2.18), we have (2.39) Since we assume m > 0, by (2.19) we have m = q − √ N (r − q) and q, r ≥ m + √ N . Recalling that u(n) = P(n ∈ τ ), we can bound the last factor in (2.39) as follows: We now apply (2.11), using the assumption r − q ≤ ε 2 r < εr and noting that Plugging this into (2.39) and recalling (2.18) then gives We can finally substitute this bound back into (2.36) and follow the same calculations as in (2.37)-(2.38), with an extra factor r −q r δ , to obtain .
By the same calculation as in (2.33), we then have Since δ > 0 and α > 1/2, this gives the desired bound (2.25) for E[| 1 | l ], provided l ∈ N is chosen large enough. This completes the proof.

Characterization and universality of the CDPM
In this section we prove Theorems 1.3 and 1.6. We recall that C is the space of all closed subsets of R, and refer to Appendix A for some key facts on C-valued random variables (in particular for the notion of restricted f.d.d., §A.3). Let us summarize our setting: • we have two independent sources of randomness: a renewal process τ = (τ n ) n≥0 satisfying ( Let us denote by X N the rescaled set τ/N ∩ [0, T ] (1.2), under the law P ω,c N T,β N ,h N . If we fix a realization of ω, then X N is a C-valued random variable (with respect to τ ).
Our strategy to prove Theorems 1.3 and 1.6 is based on two main steps: (1) first we define a suitable coupling of ω with a standard Brownian motion W ; (2) then we show that, for P-a.e. fixed realization of (ω, W ), the restricted f.d.d. of X N converge weakly as N → ∞ to those given in the right hand side of (1.22).
We can then apply Proposition A.6 (iii), which guarantees that the densities in (1.22) are the restricted f.d.d. of a C-valued random variable X ∞ , whose law on C we denote by P α;W,c T,β,ĥ ; furthermore, X N converges in distribution on C toward X ∞ as N → ∞, for Pa.e. fixed realization of (ω, W ). This is nothing but Theorem 1.3 in a strengthened form, with a.s. convergence instead of convergence in distribution (thanks to the coupling). Theorem 1.6 is also proved, once we note that P α;W,c T,β,ĥ is the unique probability law on C satisfying conditions (i) and (ii) therein, because restricted f.d.d. characterize laws on C, Proposition A.6 (i). It only remains to prove points (1)  Coming to point (2), we prove the convergence of the restricted f.d.d. of X N , i.e. the laws of the vectors (g t 1 (X N ), . . . , g t k (X N )) restricted on the event A X N t 1 ,...,t k defined in (4.16). Since X N = τ/N ∩ [0, T ] under the pinning law P ω,c T N,β N ,h N , we fix k ∈ N and 0 < t 1 < . . . < t k < T , as well as a continuous and bounded function F : R 2k → R, and we have to show that converges as N → ∞ to the integral of F with respect to the density in (1.22), i.e.
Denoting by P c N the law of the renewal process τ ∩ [0, N ] conditioned to visit N , In particular, the law P ω,c T N,β,h reduces to P c T N for β = h = 0. In this special case, the convergence I N → I is shown in the proof of Proposition A.8, (4.23) and the following lines, exploiting the renewal decomposition (4.25) for I N . In the general case, with P ω,c T N,β,h instead P c T N , we have a completely analogous decomposition, thanks to (3.4): We stress that the difference with respect to (4.25) is only given by the two terms in brackets appearing in the middle line. The first term in brackets converges to 1 as N → ∞, because max 0≤n≤N T |ω n | = O(log N ) (as we already remarked in §2.3). When we set a i = N x i and b i = N y i , the second term in brackets converges to its analogue in (3.2) involving the continuum partition functions, for P-a.e. fixed realization of (ω, W ) (thanks to our coupling), and is uniformly bounded by some (random) constant, because the continuum partition functions are a.s. continuous and strictly positive Theorem 2.4 (i). Since the convergence of (4.23)-(4.24) is shown by a Riemann sum approximation, the convergence of (3.1)-(3.2) follows immediately, completing the proof of point (2).

Key properties of the CDPM
In this section, we prove Theorems 1.4 and 1.5. The parameters α ∈ 1 2 , 1 , T > 0, β > 0 andĥ ∈ R are fixed throughout the section. We use in an essential way the continuum partition functions and P α;c T are probability measures on C 0,T , equipped with the Borel σ -algebra F. Recalling the definition (1.16) of the maps g t , d t , for n ∈ N let F n be the σ -algebra on C 0,T generated by g i 2 n T and d i 2 n T for 1 ≤ i ≤ 2 n − 1. Then (F n ) n∈N is a filtration on C 0,T that generates the Borel σ -algebra F on C 0,T , by Lemma A.2.
Let because we already know that the martingale limit lim n→∞ f W n (τ ) exists a.s.. We next identify f W n (τ ). Without loss of generality, assume T = 1. To remove duplicates among the random variables g i 2 n , d i 2 n , for 1 ≤ i ≤ 2 n − 1, let us set Then we claim that the following explicit expression for f W n (τ ) holds: In order to prove it, first note that f W n (τ ) must necessarily be a function of the vector   We can reduce to the caseĥ = 0 using Remark 2.2, in particular (2.5), writing Choosing p ∈ (1, ∞) close to 1, it suffices to prove (4.5) in the special caseĥ = 0. Henceforth we fixĥ = 0. For a given realization of τ , we note that the factors in the numerator of (4.3) are independent, by Theorem 2.4 (i). Therefore where we setβ n, j (τ ) :=β(b n, j (τ ) − a n, j (τ )) α−1/2 and we used the translation invariance and scaling property of Z α;W,ĉ β,0 (·, ·) established in Theorem 2.4 (ii)-(iii).
We claim that for any γ ∈ 0, 1 2 there exists c = c(γ ) > 0 such that forβ > 0 sufficiently small, Substituting this bound into (4.6) then gives The RHS diverges P α;c 1 (dτ )-a.s. as n → ∞, because {[a n, j (τ ), b n, j (τ )]} j∈{1}∪I n ∪{2 n } is a covering of τ with balls of diameter at most 2 −n , and τ a.s. has Hausdorff dimension α, which is strictly larger than 2α − 1 for α ∈ ( 1 2 , 1). The divergence follows from the definition of the Hausdorff dimension (see e.g. [5, Section III.5]. Lastly we prove (4.7). By (2.3), we have the representation where Y k is a random variable in the k-th order Wiener chaos expansion, and we recall that the series converges in L 2 for allβ > 0. By Taylor expansion, there exist ε, C > 0 such that For later convenience, let us define We then obtain having used the fact that E[Sβ ] = 0 in the last line. Observe that hence the first two terms in (4.8) give the correct asymptotic behavior (4.7). It remains to show that the three terms in brackets are o(β 2 ). Note that and moreover E[Y 4 1 ] ≤ (const.)E[Y 2 1 ] 2 < ∞, by the hyper-contractivity of Wiener chaos expansions (see e.g. [22,Thm. 3.50]). Writing Sβ =βY 1 + Tβ , we obtain and analogously The terms in bracket in Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Appendix A: Random closed subsets of R
In this section, we give a self-contained account of the theoretical background needed to study random closed sets of R.

A.1 Closed subsets of R
We denote by C the class of all closed subsets of R (including the empty set): We equip the set C with the so-called Fell-Matheron topology, built as follows. We first compactify R by defining R := R ∪ {±∞}, equipped with the metric (4.10) The Hausdorff distance of two compact non-empty subsets K , K ⊆ R is defined by where d(a, B) := inf b∈B d(a, b). (Note that d H (K , K ) ≤ ε if and only if for each x ∈ K there is x ∈ K with d(x, x ) ≤ ε, and vice versa switching the roles of K and K .) Coming back to C, one can identify a closed subset C ⊆ R with the compact non-empty subset C ∪ {±∞} ⊆ R. This allows to define a metric d FM on C: • for every open set G ⊆ R with G ∩ C = ∅, one has G ∩ C n = ∅ for large n; • for every compact set K ⊆ R with K ∩ C = ∅, one has K ∩ C n = ∅ for large n.
We also observe that the Fell-Matheron topology on closed subsets can be studied for more general topological space, together with the topology induced by the Hausdorff metric (4.11) on compact non-empty subsets (called myope topology): for more details, we refer to [24], [25,Appendixes B and C] and [30,Appendix B].

A.2 Finite-dimensional distributions
The space C is naturally equipped with the Borel σ -algebra B(C) generated by the open sets. By random closed subset of R we mean any C-valued random variable X . We are going to characterize the law of X , which is a probability measure on C, in terms of suitable finite-dimensional distributions, which provide useful criteria for convergence in distribution.
To every element C ∈ C we associate two non-decreasing and right-continuous functions t → g t (C) and t → d t (C), defined for t ∈ R with values in R as follows: It is therefore natural to describe a random closed set X in terms of the random functions t → g t (X ) and t → d t (X ). For convenience, we state results for both g and d, even if one could focus only on one of the two. We start with some basic properties of the maps g t (·) and d t (·).
Lemma A.2 For every t ∈ R, consider g t (·) and d t (·) as maps from C to R. Given a C-valued random variable X , we call g-finite-dimensional distributions (g-f.d.d.) of X the laws of the random vectors (g t 1 (X ), . . . , g t k (X )), for k ∈ N and t 1 , . . . , t k ∈ R. Analogously, we call d-f.d.d. the laws of the random vectors (d t 1 (X ), . . . , d t k (X )). We simply write f.d.d. to mean either g-f.d.d. or d-f.d.d., or both, when no confusion arises.
Since X is determined by the functions t → g t (X ) and t → d t (X ), it is not surprising that the law of X on C is uniquely determined by its f.d.d., and that criteria for convergence in distribution X n ⇒ X of C-valued random variables can be given in terms of f.d.d.. Some care is needed, however, because the maps g t (·) and d t (·) are not continuous on C. For this reason, given a C-valued random variable X , we denote by I g (X ) the subset of those t ∈ R for which the function s → g s (X ) is continuous at s = t with probability one: (4.14) One defines I d (X ) analogously. We then have the following result.

Proposition A.3 (Characterization and convergence via f.d.d.)
Let (X n ) n∈N , X be C-valued random variables.
(i) The set I g (X ) is cocountable, i.e. R\I g (X ) is at most countable.
(ii) The law of X is determined by its g-f.d.d. with indices t 1 , . . . , t k in a dense set T ⊆ R. (iii) Assume that X n ⇒ X . Then the g-f.d.d. of X n with indices in the cocountable set I g (X ) converge weakly to the g-f.d.d. of X : for all k ∈ N and t 1 , . . . , t k ∈ I g (X ), (iv) Assume that the g-f.d.d. of X n with indices in a set T ⊆ R with full Lebesgue measure converge weakly: for k ∈ N, t 1 , . . . , t k ∈ T there are measures μ t 1 ,...,t k on R k such that Then there is a C-valued random variable X such that X n ⇒ X . In particular, the g-f.d.d. of X with indices in the set T ∩ I g (X ) are given by μ t 1 ,...,t k .
The same conclusions hold replacing g by d.
Remark A.4 In Proposition A.3 (iv) it is sufficient that T has uncountably many points in every non-empty open interval (a, b) ⊆ R, as the proof shows. In fact, arguing as in [14,Th. 7.8 in Ch. 3], it is even enough that T is dense in R (in which case the f.d.d. of X must be recovered from μ t 1 ,...,t n by a limiting procedure, since T ∩ I g (X ) could be empty).
Remark A. 5 The map C → (g t (C)) t∈R allows one to identify C with a class of functions D 0 that can be explicitly described: The functions in D 0 are non-decreasing and right-continuous, hence càdlàg, and it turns out that the Fell-Matheron topology on C corresponds to the Skorokhod topology on D 0 . As a matter of fact, given the structure of D 0 , convergence f n → f in the Skorokhod topology is equivalent to pointwise convergence f n (x) → f (x) at all continuity points x of f . We do not prove these facts, because we do not use them directly. However, as the reader might have noticed, the key results of this section are translations of analogous results for the Skorokhod topology [6,14,21].

A.3 Restricted finite-dimensional distributions
It turns out that, in order to describe the f.d.d. of a C-valued random variable X , say the law of (g t 1 (X ), . . . , g t k (X )), with −∞ < t 1 < · · · < t k < +∞, it is sufficient to focus on the event We thus define the restricted g-f.d.d. of X as the laws of the vectors (g t 1 (X ), . . . , g t k (X )) restricted on the event A X t 1 ,...,t k . These are sub-probabilities, i.e. measures with total mass P(A X t 1 ,...,t k ) ≤ 1, and we equip them with the usual topology of weak convergence with respect to bounded and continuous functions. One defines analogously the restricted d-f.d.d. of X . We can then rephrase Proposition A.3 as follows. t 0 ,t 1 ,...,t k ,t k+1 , the event A X t 1 ,...,t k in (4.16) can be rewritten as
This can be shown using the general theory of regenerative sets [16], but it is instructive to prove it directly, as an application of Proposition A.6 (which shows, as a by-product, that the restricted f.d.d. (4.18) define indeed a probability law on C). We spell this out in the conditioned case, which is more directly linked to our main results, but the proof for the unconditioned case is analogous (and actually simpler).

Proofs
We now give the proofs of the results stated in the previous subsections.
Proof of Lemma A. 2 We start proving part (ii). Fix C ∈ C and t ∈ R and recall Remark A.1. If the function g · (C) is not continuous at t, i.e. g t− (C) < g t (C), defining C n := C + 1 n (i.e. translating C to the right by 1 n ) one has C n → C as n → ∞, is not continuous at C.
Assume now that g · (C) is continuous at t. We set s := g t (C) and distinguish two cases.
If C n → C, then (t − ε, t) ∩ C n = ∅ for large n, hence g t (C n ) ∈ (t − ε, t). This shows that g t (C n ) → g t (C), that is, g t (·) is continuous at C.
Thus g t (·) is continuous at C ∈ C if and only if g · (C) is continuous at t, proving part (ii). We now turn to part (i). Defining G t;m,ε (C) := 1 ε t+ε t max{g s (C), −m} ds, we can write g t (C) = lim m→∞ lim n→∞ G t;m,1/n (C) for all t ∈ R and C ∈ C. The measurability of g t (·) will follow if we show that G t;m,ε (·) is continuous, and hence measurable. If C n → C in C, we know that g s (C n ) → g s (C) at continuity points s of the non-decreasing function g · (C), hence for Lebesgue a.e. s ∈ R. Since g s (C) ≤ s, dominated convergence yields G t;m,ε (C n ) → G t;m,ε (C) as n → ∞, i.e. the function G t;m,ε (·) is continuous on C.
Finally, setting B := σ ((g t ) t∈T ), where T ⊆ R is a fixed dense set, the measurability of the maps g t (·) yields B ⊆ B(C). If we exhibit measurable maps ψ n : (C, B ) → (C, B(C)) such that C = lim n→∞ ψ n (C) in C, for all C ∈ C, it follows that the identity map ψ(C) := C is measurable from (C, B ) to (C, B(C)), as the pointwise limit of ψ n , hence B(C) ⊆ B .
Extracting a countable dense set {t i } i∈N ⊆ T , we define ψ n (C) := {g t 1 (C), . . . , g t n (C)} ∩ R, so that ψ n (C) is a finite subset of R and ψ n (C) → C in C. Since g t (·) is a measurable map from (C, B ) to R and (x 1 , . . . , x n ) → {x 1 , . . . , x k } ∩ R is a continuous, hence measurable, map from R k to (C, B(C)), it follows that ψ n : Proof of Proposition A.3. Since the path t → g t (X ) is increasing and rightcontinuous, its discontinuity points t, at which g t (X ) = g t− (X ), are at most countably many, a.s.. The corresponding fact that P(g t (X ) = g t− (X )) > 0 is possible for at most countably many t follows by a classilcal argument, see e.g. [6,Section 13]. This proves part (i). The proof of part (ii) is an easy consequence of Lemma A.2. A generator for the Borel σ -algebra B(C) is given by sets of the form {C ∈ C : g t 1 (C) ∈ A 1 , . . . , g t k (C) ∈ A k }, for k ∈ N, t 1 , . . . , t k ∈ T and A 1 , . . . , A k Borel subsets of B. Note that such sets are a π -system, i.e. they are closed under finite intersections. It is then a standard result that any probability on (C, B(C)) -in particular, the distribution of any C-valued random variable X -is characterized by its values on such sets, i.e. by its finite-dimensional distributions.
We now turn to part (iii). If X n ⇒ X on C, by Skorokhod's Representation Theorem [6, Th.6.7] we can couple X n and X so that a.s. X n → X in C. If t i , . . . , t k ∈ I g (X ), the maps g t i (·) are a.s. continuous at X , by Lemma A.2 (ii), hence one has the a.s. convergence (g t 1 (X n ), . . . , g t k (X n )) → (g t 1 (X ), . . . , g t k (X )), which implies weak convergence of the f.d.d..
We finally prove part (iv). Since C is a compact Polish space, every sequence (X n ) n∈N of C-valued random variables is tight, and hence relatively compact for the topology of convergence in distribution [6,Th.2.7]. We can then extract a subsequence X n k converging in distribution to some C-valued random variable X . To show that the whole sequence X n converges to X , by [6,Th.2.6] it is enough to show that for any other converging subsequence X n k ⇒ X , the random variables X and X have the same distribution.
By assumption, the f.d.d. of X n with indices t 1 , . . . , t k in a set T ⊆ R with full Lebesgue measure converge to μ t 1 ,...,t k . Since X n k ⇒ X , the f.d.d. of X are given by μ t 1 ,...,t k for indices in T ∩ I g (X ), by part (iii); analogously, the f.d.d. of X are given by μ t 1 ,...,t k for indices in T ∩ I g (X ). Thus X and X have the same f.d.d. with indices in T ∩ I g (X ) ∩ I g (X ). Since this set is dense, X and X have the same distribution, by Proposition A.3 (ii).
(4.21) It remains to express each term in the right hand side in terms of restricted f.d.d..
We first consider the case B = ∅. On the event {B(X ) = ∅} = {X ∩ (t 1 , t k ] = ∅} we have g t 1 (X ) = · · · = g t k (X ) ≤ t 1 . Recalling that μ t = μ rest t , we can write and this expression depends only on the restricted f.d.d..
If B = ∅, we can write B = { j 1 , . . . , j } with 1 ≤ ≤ k − 1 and 1 ≤ j 1 < . . . < j ≤ k − 1. Let us also set j 0 := 0 and j +1 := k. On the event {B = B}, we have g t j n−1 +1 (X ) = g t j n−1 +2 (X ) = . . . = g t jn (X ) ∈ (t j n−1 , t j n−1 +1 ], for every n = 1, . . . , + 1, therefore where we set (t 0 , · ] := [−∞, · ] and the product over m equals one when j n − j n−1 = 1. The first term in the right hand side of (4.22) is the f.d.d. μ t j 1 ,...,t j +1 (dx j 1 , . . . , dx j +1 ). However, by (4.20), this coincides with the restricted For part (ii), we proceed as in the proof of Proposition A.3 (iii). If X n ⇒ X on C, we couple X n and X so that a.s. X n → X in C, by Skorokhod's Representation Theorem. On the event that g · (X ) is continuous at t, if X ∩ (s, t] = ∅ then also X ∩ (s, t) = ∅, which implies X n ∩ (s, t) = ∅ for large n (Remark A.1). Analogously, on the event that d · (X ) is continuous at s, if X ∩ (s, t] = ∅ then also X ∩ [s, t] = ∅, which implies X n ∩ [s, t] = ∅ for large n. Therefore, if t 1 , . . . , t k ∈ I g (X ) ∩ I d (X ) and f : R k → R is bounded and continuous, Taking expectations of both sides, dominated convergence shows that the restricted f.d.d. of X n with indices in I g (X ) ∩ I d (X ) converge weakly toward the restricted f.d.d. of X . Finally, the proof of part (iii) is analogous to that of Proposition A.3 (iv). Any sequence X n of C-valued random variable is tight, hence it suffices to show that if X n k , X n k are subsequences converging in distribtion to X , X respectively, then X and X have the same law. Since the restricted g-f.d.d. of X n with indices in T converge, X and X have the same restricted g-f.d.d. with indexes in the dense set T ∩ I g (X ) ∩ I d (X ) ∩ I g (X ) ∩ I d (X ), by part (ii). It follows by part (i) that X and X have the same law.
Recall that u(i) := P(i ∈ τ ) and K (i) := P(τ 1 = i). A renewal decomposition yields where we write the factor 1 N 2k (which cancels with the product of the N 2 inside the square brackets) to make I N appear as a Riemann sum. Setting x i = a i /N , y i = b i /N for 1 ≤ i ≤ k, the summand converges pointwise to the integrand in (4.24), since by (1.1) and (2.10) To conclude that I N converges to the integral I in (4.24), we provide a suitable domination. By Potter bounds [8,Theorem1.5.6], for every ε > 0 there is a constant D ε such that L(m)/L( ) ≤ D ε max{(m/ ) ε , ( /m) ε } for all , m ∈ N. Since (y i − x i ) ≤ T , (x i − y i−1 ) ≤ T and max{α, β} ≤ αβ for α, β ≥ 1, we can write It follows that, for every ε > 0, the summand in I N is bounded uniformly in N by (4.26) It remains to show that, if we choose ε > 0 sufficiently small, this function has finite integral over the domain of integration in (4.24). Let us set η := min i=1,...,k+1 (t i −t i−1 ) and δ i := x i − y i−1 for i = 1, . . . , k + 1, δ i := y i − x i for i = 1, . . . , k, where t 0 = y 0 := 0 and x k+1 = t k+1 := T . Each of the quantities δ i , δ i can be smaller or larger than η/3, and we split the integral of (4.26) accordingly, as a sum of 2 2k+1 terms. Note that if δ i < η/3, either δ i−1 or δ i must exceed η/3 (because x i−1 < t i−1 < t i < y i and t i − t i−1 ≥ η). Whenever any δ i or δ i exceeds η/3, we replace them by η/3, getting an upper bound. This yields a factorization into a product of just four kind of basic integrals, i.e.

Lemma B.2
Let τ be a non-terminating renewal process satisfying (1.1), with α ∈ (0, 1), such that 2τ 1 has the same distribution as the first return to zero of a nearestneighbor Markov chain on N 0 with ±1 increments (Remark 1.2). Then (4.27) holds true.
Proof Let X and Y be two copies of such a Markov chain, starting at the origin at times −2 and 0 respectively, so that u(n + ) = P(X 2n = 0) and u(n) = P(Y 2n = 0).