An Almost-Sure CLT for Stretched Polymers

We prove an almost sure CLT for spatial extension of stretched (meaning subject to a non-zero pulling force) polymers at very weak disorder in all dimensions d+1 larger than or equal to 4.


Introduction and Results
Directed polymers in random media were introduced in [7] as an effective model of Ising interfaces in systems with random impurities. The precise mathematical formulation appeared in the seminal paper [9], which triggered a wave of subsequent investigations. The model of directed polymers can be described as follows. Let η = (η k ) 0≤k≤n be a nearest-neighbour path on Z d starting at 0, and let γ = (γ k ) 0≤k≤n with γ k = (k, η k ) be the corresponding directed path in Z d+1 . Let also {V (x)} x∈Z d+1 be a collection of i.i.d. random variables with finite exponential moments, whose joint law is denoted by P. One is then interested in the behaviour of the path γ under the random probability measure µ ω n (γ) = (Z ω n;β ) −1 exp −β where β ≥ 0 is the inverse temperature. The behaviour of the path γ is closely related to the behaviour of the partition function Z ω n;β . Namely, one distinguishes between two regimes: the weak disorder regime, in which lim n→∞ Z ω n;β /E(Z ω n;β ) > 0, P-a.s., and the strong disorder regime, in which this limit is zero. It is known [2] that there is a sharp transition between these two regimes at an inverse temperature β c which is non-trivial when d ≥ 3. In the weak disorder regime (β < β c ), the path γ behaves diffusively, in that γ n satisfies a CLT. Diffusivity at sufficiently small values of β was first established in [9]; this was extended to an almost-sure CLT in [1]; a CLT (in probability) valid in the whole weak disorder regime was then obtained in [2].
In dimensions d ≥ 3 the sequence Z ω n;β /E(Z ω n;β ) is bounded in L 2 for all sufficiently small values of β. In such a situation local limit versions of the CLT, which hold in probability, were established in [16,18].
In the case of directed polymers the disorder is always strong in dimensions d = 1, 2 [3,14] and at sufficiently low temperatures. Concerning the (nondiffusive) behaviour in the strong disorder regime, we refer the reader to [4] and references therein.
In this work, we consider diffusive behaviour in dimensions d + 1 ≥ 4 for the related models of stretched polymers. The choice of notation d+1 indicates that stretched polymers on Z d+1 should be compared with directed polymers in d dimensions. However, a stretched path γ can be any nearest-neighbour path on Z d+1 , which is permitted to bend and to return to particular vertices an arbitrary number of times. The disorder is modelled by a collection {V (x)} x∈Z d+1 of i.i.d. non-negative random variables. Each visit of the path to a vertex x exerts the price e −βV (x) . The stretch is introduced in one of the following two natural ways: • The path γ starts at 0 and ends at a hyperplane at distance n from 0 and has arbitrary length. This is a model of crossing random walks in random potentials.
In dimension d + 1 = 2, it presumably provides a better approximation to Ising interfaces in the presence of random impurities.
• The path γ has a fixed length n, but it is subject to a drift, which can be interpreted physically as the effect of a force acting on the polymer's free end.
The precise model is described below. At this stage let us remark that models of stretched polymers have a richer morphology than models of directed polymers. Even the issue of ballistic behaviour for annealed models is non-trivial [10,8,13]. The issue of ballistic behaviour in the quenched case is still not resolved completely, and, in order to ensure ballisticity one needs to assume that the random potential V is strictly positive in the crossing case, and that the applied drift is sufficiently large in the fixed length case. Both conditions are designed to ensure a somewhat massive nature of the model.
As in the directed case, the disorder is always strong [21] in low dimensions d + 1 = 2, 3 or at sufficiently low temperatures.
In the case of higher dimensions d+1 ≥ 4, the existence of weak disorder on the level of equality between quenched and annealed free energies was established in [6,20]. The case of high temperature discrete Wiener sausage with drift was addressed in [17]. In the crossing case, a CLT in probability was established in [11] in all dimensions d + 1 ≥ 4 at sufficiently high temperatures.
The aim of the present paper is to establish an almost-sure CLT for the endpoint of the fixed-length version of the model of stretched polymers with non-zero drifts, also at sufficiently high temperatures and in all dimensions d + 1 ≥ 4.

Class of Models
Polymers. For the purpose of this paper, a polymer γ = (γ 0 , . . . , γ n ) is a nearestneighbour trajectory on the integer lattice Z d+1 . Unless stressed otherwise, γ 0 is always placed at the origin. The length of the polymer is |γ| ∆ = n and its spatial extension is X(γ) ∆ = γ n − γ 0 . In the most general case, neither the length nor the spatial extension are fixed.
Random Environment. The random environment is a collection {V (x)} x∈Z d+1 of nondegenerate non-negative i.i.d. random variables which are normalised by 0 ∈ supp(V ). There is no moment assumptions on V . The case of traps, p ∞ ∆ = P (V = ∞) > 0, is not EJP 0 (0), paper 0. Page 2/20 ejp.ejpecp.org excluded, but then we shall assume that p ∞ is small enough. In particular, we shall assume that P-a.s. there is an infinite connected cluster Cl ∞ (V ) of the set {x : V (x) < ∞} in Z d+1 . In fact, we shall assume more: Given R d+1 h = 0 and a number δ ∈ (0, 1 By construction, the cones Y h δ always contain at least one lattice direction ±e i , i = 1, . . . , d+1. We assume that it is possible to choose δ in such a fashion that, for any h, the intersection Cl h,δ For the rest of the paper, we fix such a δ ∈ (0, 1 √ d+1 ) and use the reduced notation Y h and Cl h ∞ (V ) for the corresponding cones (1.1) and percolation clusters. Weights and Path Measures. The reference measure p(γ) ∆ = (2(d + 1)) −|γ| is given by simple random walk weights. The polymer weights we are going to consider are quantified by two parameters: the inverse temperature β ≥ 0 and the external pulling The random quenched weights are given by (1. 2) The corresponding deterministic annealed weights are given by Note that the annealed potential is positive, non-decreasing and attractive, in the sense that (1.5) In the sequel, we shall drop the index β from the notation, and we shall drop the index h whenever it equals zero. With this convention, the quenched partition functions are defined by An almost sure CLT for stretched polymers

The Result
Fix h = 0. Then [19,5,10] λ = λ(β, h) for all sufficiently small β. The following two quantities play a central role in our limit If β is sufficiently small then v = 0 and the matrix Σ is positive definite and, moreover, v and Σ are the limiting spatial extension and, respectively, the diffusivity matrix for the annealed model. (Sections 4.1,4.2 in [10]). In Subsection 2.1 we recall further relevant facts about the annealed model.
Theorem A. Fix h = 0. Then, in the regime of very weak disorder, the following holds P-a.s. on the event {0 ∈ Cl ∞ (V )}: exists and is a strictly positive, square-integrable random variable.
• There exists a sequence { n } with lim n = 0, such that (1.12) We would like to stress that, in contrast to the case of directed polymers [2], our CLT does not pertain to the whole of the weak disorder region. The procedure of first fixing h = 0 and then going to β > 0 sufficiently small is essential. Furthermore, even in the regime we are working with, (1.12) falls short of the local CLT form of results as developed for directed polymers in [18]. These and related issues remain open in the context of stretched polymers.
Few remarks on the history of the problem: Flury [6] had established that under the conditions of Theorem A (and some additional moment assumptions of the potential V ) for on-axis exterior forces h. (1.13) was then extended to arbitrary directions h ∈ R d+1 by Zygouras [20]. In [6], the analysis was carried out directly in the canonical ensemble of polymers with fixed length n. In [20], the author derives results for the conjugate ensemble of the so-called crossing random walks.
Large deviations (LD) under both Q n,h and Q ω n,h were investigated in [19,5]. The results therein imply that, under the conditions of Theorem A, the model is ballistic in the sense that the value of the quenched rate function at zero is strictly positive. However, [19,5] do not imply a law of large numbers (LLN) even in the annealed case. In particular, these works do not contain information on the strict convexity of the corresponding rate functions. The issue of strict convexity for the annealed rate functions was settled in [10]. Therefore, (1.11) is a direct consequence of (1.13) and of the analysis of annealed canonical measures in [10].
The main new results of this work are (1.10) and (1.12). A version of Theorem A for the ensemble of crossing random walks appears in [11]. The length of crossing random walks is not fixed (only suppressed by an additional positive mass), and they are required to have their second endpoint on a distant hyperplane. In this way, crossing random walks in random potential are much more "martingale"-like than canonical random walks. Moreover, the canonical constraint of fixed length does not facilitate computations, to say the least. Finally, the CLT of [11] was only established in probability and not P-a.s. Thus, although the techniques developed in [11] are useful here, they certainly do not imply the claims of Theorem A, and an alternative approach was required.

Irreducible Decomposition, Basic Ensembles and Basic Partition Functions
(1.14) A cone-confined polymer which cannot be represented as the concatenation of two (nonsingleton) cone-confined polymers is said to be irreducible. We denote by T (x) the collection of all cone-confined paths leading from 0 to x, and by F(x) ⊂ T (x) the set of irreducible cone-confined paths. In the sequel we shall refer to F(x) and T (x) as to basic ensembles. The basic partition functions are defined by exists and is a strictly positive, square-integrable random variable.
• For every α ∈ R d+1 , For the rest of the paper, we shall focus on the proof of Theorem B.

Proof of Theorem B
To facilitate the exposition, we shall consider the case of on-axis external force h = he 1 . The proof, however, readily applies for any non-zero h ∈ R d+1 . By lattice symmetries, the mean displacement v = ∇λ(h) lies along the direction e 1 ; v = ve 1 . As it was already mentioned in the beginning of Subsection 1.2, v = 0 whenever β is small enough. We proceed assuming that both the drift and the speed are positive h, v > 0.

Three Main Inputs
The reduction to basic ensembles constitutes the central step of the Ornstein-Zernike theory. We rely on three facts: The first is the refined description of the annealed phase in the ballistic regime (which, in our regime, will always correspond to first fixing h = 0 and then choosing β > 0 small enough). Below, we shall summarize the required results from [10,12]. The second is an L 2 -type estimate on overlaps which holds for all β sufficiently small, and which could be understood as quantifying the notion of very weak disorder we employ here. The third is a maximal inequality for the so-called mixingales, due to McLeish. Unlike directed polymers, stretched polymers do not possess natural martingale structures, and McLeish's result happens to provide a convenient alternative framework.
Ornstein-Zernike theory of annealed models. Annealed asymptotics of t n in the ballistic regime are not related to the strength of disorder and hold for all values of β ≥ 0 and appropriately large drifts h . In particular, for each h = 0 fixed, the annealed model is ballistic for all sufficiently small β. We refer to [10, Sections 4.1 and 4.2] and to [12,Section 4.2] for the proof of the following: Fix h = 0; then, for all β > 0 small enough, λ(h) > 0, ∇λ(h) = 0 and Hess[λ](h) is positive definite. Furthermore, there exist a small complex neighbourhood U ⊂ C d+1 of the origin, an analytic function µ (with µ(0) = 0) on U and a non-vanishing analytic function κ = 0 on U such that: In view of Remark 1.2 the above bound is trivial whenever |x| > n.
Finally, it is a straightforward consequence of (2.1) that the following annealed CLT holds: with the second asymptotic equality holding uniformly in α on compact subsets of R d+1 .
An L 2 -estimate. Fix an external force h = 0. We continue to employ notation v = v(h, β). For a subset A ⊆ Z d+1 , let A be the σ-algebra generated by {V (x)} x∈A . We shall call such σ-algebras cylindrical.
Lemma 2.1. For any dimension d ≥ 3 there exist a positive non-decreasing function ζ d on (0, ∞) and a number ρ < 1/12 such that the following holds: If φ β (1) < ζ d (|h|), then there exist constants c 1 , c 2 < ∞ such that the random weights (1.15) satisfy: for all x, x , m, m , y, y , and all cylindrical σ-algebras A such that both t ω x, and t ω x , are A-measurable.
Remark 2.2. The above bound is non-trivial only if both |x| , |x | ≤ (Remark 1.2). Also, there is nothing sacred about the condition ρ < 1/12. We just need ρ to be sufficiently small. In fact, (2.4) holds with ρ = 0, although a proof of such statement would be a bit more involved.
In spite of its technical appearance, (2.4) has a transparent intuitive meaning: For ρ = 0, the expressions on the right-hand side are just local limit bounds for a couple of independent annealed polymers with exponential penalty for disagreement at their end-points. The irreducible terms have exponential decay. In the very weak disorder regime, the interaction between polymers does not destroy these asymptotics. The proof of Lemma 2.1 is relegated to the concluding Section 4.
In the sequel, we shall always work with the following filtration {A m }. Recall that we are discussing on-axis positive drifts h = he 1 which, for small β, give rise to on-axis limiting spatial extension v = ve 1 with v > 0. At this stage, define the hyperplanes H − m and the corresponding σ-algebras A m as Notation for asymptotic relations. The following notation is convenient, and we shall use it throughout the text: Given a (countable) set of indices I and two positive sequences {a α , b α } α∈I , we say that a α b α if there exists a constant c > 0 such that a α ≤ cb α for all α ∈ I . We shall use a α ∼ = b α if both a α b α and a α b α hold. For instance, for any > 0 fixed, e −c3k 2 / (1+ )/2 1 (1 + k) 1+ , (2.8) where the index set I is the set of pairs of integers (k, ) with k ≥ 0 and > 0.  (2.6). Recall that ρ < 1/12, and hence < 1/6.
In the sequel, we shall repeatedly derive variance bounds on quantities of the type ≤n Z (n) . The most general form of Z (n) we shall consider is where a (n) x, (y, m) are arrays of real or complex numbers. Assume that there exists another family of (non-negative) arrays â (n) x, and a number ν > 0 such that 2 . (2.12) Above we introduced a provisional notation d r (x) x, (y, m) E(f θxω y,m − f y,m | A −k ). (2.13) Taking the expectation of the square of the latter expression and, for each x, x , factorizing replicas using |ab| ≤ a 2 +b 2 2 , one derives the first inequality (2.11) directly from (2.4) and (2.10). Next, For any x ∈ H + +k , d +k (x) = 0, and the first term in (2.14) has exactly the same structure as the right-hand side of (2.13). On the other hand, if x ∈ H − +k and z ∈ H + +k , then, in view of Remark 1.2, f θxω z−x,m can be different from zero only if m ≥ d +k (x) and |z − x| ≤ m. Therefore, (2.12) is also a direct consequence of (2.4) and (2.10).
The following is a useful corollary: Proof. Consider first the right-hand side of (2.11). Since the non-trivial part is to check (2.15) for large values of k. In the latter case, we may assume that |x − v | > k|v| 2 for all x ∈ H − −k . Consequently, the sum on the right-hand side of (2.11) is bounded above by the last inequality being an application of (2.8). (2.15) follows.
Turning to the right-hand side of (2.12), we see that it remains to derive an upper bound on (2.17) The first sum above is treated as in (2.16). On the other hand, the second sum is bounded above as e −ν k , uniformly in all k sufficiently large. Since e −ν k (1+k) −1− , the bound (2.15) for E Z (n) − E(Z (n) | A +k ) 2 follows as well.
As an application of (2.15) we derive the following convergence result: Lemma 2.6. Assume that, for some ν > 0, the asymptotic bound (2.15) is, uniformly in n and ≤ n, satisfied withâ P-a.s. and in L 2 . In particular, assume that the asymptotic bound (2.10) is satisfied for an array b (n) x, (y, m) with some ν > 0 andb P-a.s. and in L 2 .
Proof. Part of the proof appeared in Subsection 5.3 of the review paper [12]. We rely on an expansion similar to the one employed by Sinai [16] and rewrite (2.21) as (see the beginning of Section 5.3 of [12] for details) In this way, t ω n ((56) in Section 5.3 of [12]) can be represented as Z . (2.28) The representation complies with (2.9) and (2.10) withâ Hence, by (2.15), The ω n term. Let us turn now to the correction term ω n in (2.26). The first summand to estimate is It tends to zero by Lemma 2.6. The second summand is Since t r − 1/κ is exponentially decaying in r, it is easy to see that (2.10) still holds with a applies.

Correction Terms
In this Section, we prove (2.35). The correction terms η ω n,i ; i = 1, 2, 3, will be treated separately. Recall that we are working with < 1/6 such that (2.4) holds with ρ = /2.  P-a.s. and in L 2 for each α ∈ R d fixed.
The η ω n,3 term. This is the most difficult term, and, at this stage, we need to rely on Lemma 2.4 rather than on Lemma 2.5. Recall that η ω n,3 By Lemma 2.6, we may remove the constraint m ≤ n − . Define, therefore, We need to prove: P-a.s. and in L 2 for each α ∈ R d fixed.
Proof of Lemma 3.3. For Z (n) defined in (3.19), the bound (2.10) is satisfied witĥ for any ν < c 2 . Applying (2.11), we infer that As in the derivation of (2.15), we may assume that k is sufficiently large, so that, in particular, |x − v| ≥ k|v| 2 for all x ∈ H − −k . In the latter case, the sum on the right-hand side of (3.22) is bounded above by r d e −c2r 2 / r 2 n ∧ 1 dr We shall repeatedly rely on (2.8). There are two cases to consider: Let us turn to the bound (2.12) on E Z (n) − E(Z (n) | A +k ) 2 . As before, we apply it withâ (n) x, = |x− v| √ n ∧ 1. We need to estimate In particular, by (2.6), Var η ω n,3 n −1/2+ and, as in the case ofη ω n,2 , we infer that there is P-a.s. and L 2 convergence to zero along lacunary subsequences n 2+δ , whenever δ satisfies (3.13). Hence, again as in the case ofη ω n,2 , we need to control the fluctuationsη ω N +r,3 −η ω N,3 over intervals of the form (3.14). As in (3.15), we make use of the decomposition We continue to work with d = 3. For each r = 1, . . . , R fixed the bound (2.10) is satisfied x, . (3.29) The expression forâ in (3.29) is similar to (3.21) with 1/ √ n being replaced by the higher order term R/N 3/2 . Literally repeating the derivation of (3.26) we infer that for each r = 1, . . . , R fixed random variables Z (N,r) in (3.28) satisfy: Applying (2.6) we conclude: For N and R in the range (3.14), and for any r = 1, . . . , R fixed, the variance of the first term on the right hand side of (3.28) is uniformly bounded As in the case of (3.18), the union bound suffices.
In particular, (3.11) carries over, and we infer convergence to zero along the lacunary sequence n 2+δ . In order to study the fluctuations of ≤n Z Since (3.11) holds, the second term on the right-hand side of (3.31) is worked out exactly as in (3.18). On the other hand,

Preliminaries
For u, v ∈ Z d+1 and m ∈ N, we set Moreover, we write T (u, v; n) for the set of all cone-confined paths γ ⊆ D(u, v) of length n leading from u to v, and F(u, v; n) for the corresponding subset of irreducible paths.  Indeed, in that case, either D(x, y) definiteness, let us assume the latter. We can then conclude that the random variable f ω x ,y ,m is independent of t ω x, t ω x , E(f ω x,y,m − f x,y,m | A). The same is thus also true of E(f ω x ,y ,m − f x ,y ,m | A), and the claim follows, since the latter has mean zero. A second observation is that, for any A ⊆ Z d+1 and the corresponding cylindrical t ω x , f ω x,y,m f ω x ,y ,m . Indeed, define g = (λ + log(2d + 2))(2 + m + m ) − h · (y + y ) and let Σ * be the sum over all the paths γ ∈ T (0, x; ), η ∈ F(x, y; m), γ ∈ T (0, x ; ) and η ∈ F(x , y ; m ). Then the attractivity property (1.5) implies that e −φ β ( γ (u)+ γ (u)+ η (u)+ η (u)) = e g E t ω x, t ω x , f ω x,y,m f ω x ,y ,m .
Note that (4.2) implies, in particular, that
In general, the left-hand side of (4.5) does not allow for a similar expression. Notice, however, that the attractivity property implies the lower bound ≥ t x, t x , = P ⊗ RW ∃k, k : (X k , L k ) = (x, ), (X k , L k ) = (x , ) , where P ⊗ RW denotes the law of a couple of independent random walks (X, L) and (X , L ) as above. It is important to observe that (4.5) would be an immediate consequence of the local limit theorem for random walks if its left-hand side was replaced by t x, t x , .
To prove (4.5), we thus have to prove that, in the very weak disorder regime, this local limit behaviour is not destroyed by the effective attractive interaction between the two paths resulting from averaging t ω x, t ω x , over the disorder.