A general smoothing inequality for disordered polymers

This note sharpens the smoothing inequality of Giacomin and Toninelli for disordered polymers. This inequality is shown to be valid for any disorder distribution with locally finite exponential moments, and to provide an asymptotically sharp constant for weak disorder. A key tool in the proof is an estimate that compares the effect on the free energy of tilting, respectively, shifting the disorder distribution. This estimate holds in large generality (way beyond disordered polymers) and is of independent interest.


Introduction and main results
Understanding the effect of disorder on phase transitions is a key topic in statistical physics. In a celebrated paper, Harris [9] proposed a criterion that predicts whether or not the addition of an arbitrarily small amount of quenched disorder is able to modify the critical behavior of a system close to a phase transition. The rigorous justification of this criterion for a class of pinning models has been an active direction of research in the mathematical literature (see Giacomin [6] for an overview). One of the key tools in this program is the smoothing inequality of Giacomin and Toninelli [7], [8]. It is the purpose of this note to generalize and sharpen this inequality. Section 1.1 provides motivation, Section 1.2 states the necessary model assumptions, Section 1.3 defines the free energy, Section 1.4 states our main theorems, while Section 1.5 discusses the context of these theorems. Proofs are given in Sections 2-4.
Given an R-valued sequence ω := (ω n ) n∈N (the disorder sequence), a function ϕ : E → R (the potential ), and parameters N ∈ N, β ≥ 0, h ∈ R (the system size, the disorder strength and the disorder shift), we define the partition function i.e., at each time n the Markov chain gets an exponential reward or penalty proportional to h + βω n , modulated by a factor ϕ(S n ). The sequence ω is to be thought of as a typical realization of a random process. Note that • the choice ϕ(x) := 1 {0} (x) corresponds to the pinning model (see Giacomin [5], [6], den Hollander [10]); • when E = Z and S is nearest-neighbor with symmetric excursions out of 0, the choice ϕ(x) := 1 (−∞,0] (x) corresponds to the copolymer model (see [5], [10]); † Thus, the modulating potential ϕ allows us to interpolate between different classes of models. When S is simple random walk on Z d and ϕ(x) ≈ |x| −ϑ as |x| → ∞ for some ϑ ∈ (0, ∞), the model displays interesting features that are currently under investigation (Caravenna and den Hollander [4]).

1.2.
Assumptions. Although our main focus will be on the model in (1.2), we list the assumptions that we actually need. We start with the disorder.
Assumption 1.1 (The disorder). The disorder ω = (ω n ) n∈N is an i.i.d. sequence of R-valued random variables, defined on a probability space (Ω , F , P), such that The crucial assumption is that the disorder distribution has locally finite exponential moments. The choice of zero mean and unit variance is a convenient normalization only (since we can play with the parameters β and h). (1.4) Next we state our assumptions on the partition function Z N,ω,β,h we will be able to handle, defined for N ∈ N, β ≥ 0, h ∈ R and P-a.e. ω ∈ R N (keeping in mind (1.2) as a special case). (1) Z N,ω,β,h is a function of N and of (h + βω n ) 1≤n≤N .
As a matter of fact, properties (1) and (2) are rather mild: they are satisfied for many (1 + d)-dimensional directed models (possibly after a minor modification of the partition function that does not change the free energy defined below). In contrast, property (3) is a more severe restriction. Roughly speaking, it says that the disorder can be avoided at a cost that is only polynomial in the system size.
1.3. Free energy. If Assumptions 1.1 and 1.2 are satisfied, then we can define the free energy A direct consequence of (1.5) is the inequality f(β, h; δ) ≥ 0, which is a crucial feature of the class of models we consider. In many interesting cases, like for pinning and copolymer models, the free energy is zero in some closed region of the parameter space and strictly positive in its complement, with both regions non-empty. When this happens, the free energy is not an analytic function and the model is said to undergo a phase transition. It is then of physical and mathematical interest to study the regularity of the free energy close to the critical curve separating the two regions.
More concretely, consider the case when h → Z N,ω,β,h is monotone (like for the model in (1.2) when ϕ has a sign), say non-decreasing, so that h → f(β, h; δ) is non-decreasing as well. Then for every β ≥ 0 there exists a critical value h c (β) ∈ R ∪ {±∞} such that f(β, h; 0) = 0 for h < h c (β) and f(β, h; 0) > 0 for h > h c (β) (we consider δ = 0 for simplicity). If h → f(β, h; 0) is continuous as well, as is typical, then f(β, h c (β); 0) = 0 and it is interesting to understand how the free energy vanishes as h ↓ h c (β). For homogeneous pinning models, i.e., when β = 0, it is known that (See [5, Theorem 2.1] for more precise estimates.) On the other hand, as soon as disorder is present, i.e., when β > 0, it was shown by Giacomin and Toninelli [7], [8] that, under some mild restrictions on the disorder distribution, (1.8) Comparing (1.7) and (1.8), we see that when α > 1 2 the addition of disorder has a smoothing effect on the way in which the free energy vanishes at the critical line.
1.4. Main results. The goal of this note is to generalize and sharpen (1.8), namely, to show that no assumption on the disorder distribution other than (1.3) is required, and to provide estimates on the constant c that are optimal in some sense (see below). We will stay in the general framework of Assumption 1.2, with no mention of "critical lines".
• Tilting. First we prove a smoothing inequality for f(β, h; δ) with respect to the tilt parameter δ rather than the shift parameter h. Although both tilting and shifting are natural ways to control the disorder bias, the latter is often preferred in the literature because the free energy typically is a convex function of the shift parameter h (like for the model in (1.2)). However, for the purpose of the smoothing inequality the tilt parameter δ turns out to be more natural.
Theorem 1.5 is proved in Section 2 through a direct translation of the argument developed in Giacomin and Toninelli [8]. The proof is based on the concept of rare stretch strategy, which has been a crucial tool in the study of disordered polymer models since the papers by Monthus [11], Bodineau and Giacomin [3].
• Shifting. Next we consider the effect of a disorder shift. In the Gaussian case, i.e., when P(ω 1 ∈ ·) = N (0, 1), tilting is the same as shifting: in fact P δ (ω 1 ∈ ·) = N (δ, 1) and so ω n under P δ is distributed like ω n + δ under P. Recalling property (1), we then get and, since M(δ) = e δ 2 /2 , it follows from (1.9) that if f(β,h; 0) = 0 withβ > 0, then This is precisely the smoothing inequality with respect to a disorder shift in (1.8), with an explicit constant (see also Giacomin [5, Theorem 5.6 and Remark 5.7]). For a general disorder distribution tilting is different from shifting. However, we may still hope that (1.11) holds approximately. This is what was shown in Giacomin and Toninelli [7], under additional restrictions on the disorder distribution and with non-optimal constants. The main result of this note, Theorem 1.8 below, shows that the effects on the free energy of tilting or shifting the disorder distribution are asymptotically equivalent, in large generality and with asymptotically optimal constants in the weak interaction limit. Since this result is unrelated to Theorem 1.5 and is of independent interest, we formulate it for a very general class of statistical physics models, way beyond disordered polymer models.

Assumption 1.7 (The partition function [II]). The partition function is defined as
, that are uniformly bounded, have a sign, say

14)
and satisfy −∞ < lim sup N →∞ We emphasize that the σ i 's need not be independent, nor exchangeable. A more detailed discussion on Assumption 1.7 is given below.
We can now state the approximate version of (1.11). The free energy f(β, h; δ) is again defined by (1.6).
Furthermore, δ → C ± β,δ δ is strictly increasing. The proof of Theorem 1.8 is given in Section 3. The general strategy and consists in showing that the derivatives of f(β, h; δ) with respect to δ and h are comparable. Compared to Giacomin and Toninelli [7], several estimates need to be sharpened considerably.
• Smoothing. Combining Theorems 1.5 and 1.8, we finally obtain our smoothing inequality with respect to a shift, with explicit control on the constant. Theorem 1.9 (Smoothing inequality with respect to a disorder shift). Subject to Assumptions 1.1, 1.2 and 1.7, there is an ε 0 > 0 with the following property: if f(β,h; 0) = 0 for someβ ∈ (0, ε 0 ) andh ∈ R, then for t ∈ (−βε 0 ,βε 0 ), , and is such that (1.18) 1.5. Discussion. We comment on the results obtained in Section 1.4.
1. The version of the smoothing inequality in Theorem 1.9, with the precision on the constant, is picked up and used in Berger, Caravenna, Poisat, Sun and Zygouras [2] to obtain the sharp asymptotics of the critical curve β → h c (β) for pinning and copolymer models in the weak disorder regime β ↓ 0, for the case α ∈ (1, ∞) (recall (1.1)).
2. The smoothing inequality in (1.17), at the level of generality at which it is stated, is optimal in the following sense.
• We cannot hope for an exponent strictly larger than 2 in the right-hand side of (1.17), because pinning models with P(τ 1 = n) ∼ (log n)/n 3/2 are in the "irrelevant disorder regime", and it is known that • We cannot hope for an asymptotically smaller constant, i.e., lim (β,δ)→(0,0) A β,δ < 1, because the proof in Berger, Caravenna, Poisat, Sun and Zygouras [2] would yield a contradiction (the lower bound would be strictly larger than the upper bound).
Of course, for specific models the inequality (1.17) can sometimes be strengthened. For instance, pinning models satisfying (1. 3. Compared with Assumption 1.2, Assumption 1.7 prescribes a specific form for the partition function Z N,ω,β,h and therefore is more restrictive. On the other hand, in view of the minor constraints put on the σ i 's, (1.13) is so general that the absence of any restrictive conditions like (2) or (3) makes Assumption 1.7 effectively much weaker than Assumption 1.2. For instance, since (1.2) is a special case of (1.13), with P N (·) = P( · ∩ {S N = 0}) (which, incidentally, explains why P N is allowed to be a finite measure, and not necessarily a probability), the model in (1.2) satisfies Assumption 1.7 as soon as the function ϕ is bounded and has a sign, without the need for any requirement like (1.1).
We emphasize that many other (also non-directed) disordered models fall into Assumption 1.  .6) and (1.13) that (with obvious notation) Therefore, when the σ n 's are uniformly bounded but not necessarily non-negative, we can first perform a uniform translation to transform them into non-negative random variables, next apply (1.15), and finally use (1.19) to come back to the original σ n 's. Still, the non-negativity assumption on the σ n 's in (1.14) cannot be dropped from Theorem 1.8. In fact, if f(β, h; δ) is differentiable in h and δ, then (1.15) implies that This relation, which is a necessary condition for (1.15) when the free energy is differentiable, may be violated when the σ n 's take both signs. For instance, let (σ n ) n∈N under P N := P be i.i.d. with P(σ n = −1) = P(σ n = +1) = 1 2 , and let the marginal distribution of the disorder be P(ω n = −a −1 ) = a 2 /(a 2 + 1), P(ω n = a) = 1/(a 2 + 1) with a > 0 (note that E(ω 1 ) = 0 and Var(ω 1 ) = 1, so that (1.3) is satisfied). The free energy is easily computed: In particular, and hence (1.20) does not hold for a = 1 (the left-hand side is ≈ β 2 , while the right-hand side is ≈ β 4 ). Intuitively, such a discrepancy arises for values of h at which ∂f ∂h (0, h; 0) = 0, which means that the average E N,ω,0,h ( 1 N N n=1 σ n ) tends to zero as N → ∞, where P N,ω,β,h is the Gibbs law associated to the partition function Z N,ω,β,h (see (3.2) below). When the σ n 's are non-negative, their individual variances under P N,ω,0,h must be small, but this is no longer true when the σ n 's can also take negative values. This is why one might have
Lemma 2.1. The following implication holds: Proof. Fix ∈ N large enough so that the above conditions hold, and for ω ∈ R N denote by T 1 (ω), T 2 (ω), . . . the distances between the endpoints of the stretches in A : Note that {T k } k∈N is i.i.d. with marginal law given by GEO(P(A )). In particular, Henceforth we suppress the subscripts β, h. Since (ϑ (T 1 +...+T i )− ω) (0, ] ∈ A by construction, applying properties (2)-(3) in Assumption 1.2 and the definition of G, we get (2.4) where we set Z 0 := 1 for convenience. Recalling (1.6) and Remark 1.4, for P-a.e. ω we can write, by the strong law of large numbers and Jensen's inequality, (2.5) If G − γC > 0, then we can choose ∈ N large enough (but finite!) such that the right-hand side is strictly positive. This proves (2.1).

2.2.
Proof of Theorem 1.5. We use Lemma 2.1. Fix β > 0, h ∈ R, δ ∈ (−t 0 , t 0 ) and ε > 0, and define the set of good atypical stretches as so that G = f(β, h; δ) − ε by construction. It remains to determine C, for which we need to estimate the probability of P(A ) from below.
3. Asymptotic equivalence of tilting and shifting: proof of Theorem 1.8 Throughout this section, we work under Assumptions 1.1 and 1.7.
3.1. Notation. Denote the empirical average of the variables σ i 's by The finite-volume Gibbs measure associated with the partition function in (1.13) is the probability on Ω N defined, for N ∈ N, ω ∈ R N , β ≥ 0 and h ∈ R, by   Note that, by (1.14), 3.2. Preparation. Before proving Theorem 1.8, we need some preparation. Recalling (3.1), we define for [a, b] ⊆ R with a < b a restricted version of the partition function and the free energy, in which the empirical average σ N is constrained to lie in [a, b]: N,ω,β,h . (3.5) The corresponding restricted Gibbs measure is the probability defined by (recall (3.2)) In particular, for x ∈ R we may define where a n ↑ x and b n ↓ x are arbitrary strictly monotone sequences (it is easily seen that the limit does not depend on the choice of these sequences). Note that f {x} (β, h; δ) = −∞ when x ∈ [0, s 0 ], by (1.14). The following result is standard: Recalling (3.5), we see that By compactness, there exists an x ∈ [0, s 0 ] such that I n ↓ {x}, i.e., n∈N I n = {x}. If I n = [a n , b n ], then we set J n := [a n − 1 n , b n + 1 n ], so that we still have J n ↓ {x}, and x lies in the interior of each J n . Since f In (β, h; δ) ≤ f Jn (β, h; δ), recalling (3.8) we obtain (3.13) and the proof of (3.9) is complete.
3.3. Proof of Theorem 1.8. By (3.9), it suffices to show that (1.15) is satisfied with f {x} instead of f, for every fixed x ∈ [0, s 0 ]. It is of course important that the constants C ± β,δ do not depend on x.
1. First we consider the case x = 0. We claim that (3.14) Since the right-hand side of (3.14) is a constant that does not depend on β ≥ 0, δ ∈ (−t 0 , t 0 ) and h ∈ R, (1.15) is trivially satisfied with f {0} instead of f, whatever the definition of C ± β,δ is. To prove (3.14) note that, by Cauchy-Schwarz, because 0 ≤ σ n = |σ n | ≤ s 0 by (1.14). Recalling (3.5), for every N ∈ N we get where we use Jensen. Note that the right-hand side is a finite constant.
2. Next we consider the case x ∈ (0, s 0 ]. Roughly speaking, the strategy of the proof is to show that the derivatives of the free energy with respect to δ and to h are comparable. Unless otherwise specified, we work with generic values of the parameters in the admissible range β ≥ 0, h ∈ R and δ ∈ (−t 0 , t 0 ). Henceforth we fix 0 < a < b < ∞. Recalling (3.5) and ( where m δ := E δ (ω n ) = (log M) (δ) by (2.9). Subtracting a centering term with zero mean, we get where the second equality follows easily from (3.5) via (3.6). Note that f n (ω, y) depends on the ω i 's for i = n, not on ω n . Therefore (3.20) can be rewritten as , we see that this is precisely what we want, because Var δ (ω 1 ) ≈ 1 for δ small. In order to turn these arguments into a proof, we need to estimate the dependence of f n (ω, y) on y. To that end we note that because 0 ≤ σ n ≤ s 0 , by (1.14). Therefore ∂ ∂y f n (ω, y) ≥ 0, ∂ ∂y e −s 0 β y f n (ω, y) ≤ 0, (3.24) and integrating these relations we get Introducing the function taking y = m δ in (3.25) and integrating over y, we easily obtain the bounds ωn m δ f n (ω, y) dy ≤ g βs 0 (ω n − m δ ) + f n (ω, m δ ). (3.27)

6.
Let us now substitute the estimate (3.27) into (3.22). Since f n (ω, m δ ) does not depend on ω n , the expectation over E δ factorizes and we obtain (3.29) We next want to replace f n (ω, m δ ) by f n (ω, ω n ) = E [a,b] N,ω,β,h σ n (recall (3.21)). To this end, we again apply (3.25), this time with y = ω n and y = m δ . Since f n (ω, m δ ) does not depend on ω n , we have