Coarse graining, fractional moments and the critical slope of random copolymers

For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function.


Introduction
We consider a model of copolymer at a selective interface introduced in [10], which has attracted much attention among both theoretical physicists and probabilists (we refer to [11] for general references and motivations).
Let S := {S n } n≥0 be the symmetric Simple Random Walk on Z started at S 0 = 0, with law P SRW such that the increments {S i − S i−1 } i≥0 are IID and P SRW (S 1 = ±1) = 1/2. The partition function of the model of size N is given by where h ≥ 0, λ ≥ 0 and {ω n } n≥1 is a sequence of IID standard Gaussian random variables (the quenched disorder). We adopt the convention that, if S n = 0, then sign(S n ) := sign(S n−1 ). One interprets λ as the inverse temperature (or coupling strength) and h as an "asymmetry parameter": if h > 0, since the ω n 's are centered, the random walk overall prefers to be in the upper half-plane (S ≥ 0). It is known that the model undergoes a delocalization transition: if the asymmetry parameter exceeds a critical value h c (λ) then the fraction of "monomers" S n , n ≤ N , which are in the upper half-plane tends to 1 in the thermodynamic limit N → ∞ (delocalized phase), while if h < h c (λ) then a nonzero fraction of them is in the lower half-plane (localized phase). What mostly attracts attention is the slope, call it m c , of the curve λ → h c (λ) in the limit λ ց 0: m c is expected to be a universal quantity, i.e., independent of the details of the law P SRW and of the disorder distribution (see next section for a more extended discussion on this point). Already the fact that the limit slope is well-defined and positive is highly non-trivial [5]. Until now, all what was known rigorously about m c is that 2/3 ≤ m c ≤ 1, but numerically the true value seems to be rather around 0.83 [6]. The upper bound comes simply from annealing, i.e., from Jensen's inequality, as explained in next section. Our main new result is that m c is strictly smaller than 1. The proof works through a coarse-graining procedure in which one looks at the system on the length-scale k(λ, h), given by the inverse of the annealed free energy. The other essential ingredient is a change-of-measure idea to estimate fractional moments of the partition function (this idea was developed in [12] and [7], and used in the context of copolymers in [3]). Coarse-graining schemes, implemented in a way very different from ours, have already played an important role in this and related polymer models; we mention in particular [5], [4] and [1].

The general copolymer model
As in [3], we consider a more general copolymer model which includes (1.1) as a particular case. Since the critical slope is not proven to exist in this general setting, Theorem 2.1 will involve a lim sup instead of a limit.
Consider a renewal process τ := {τ 0 , τ 1 , . . .} of law P, where τ 0 := 0 and {τ i − τ i−1 } i∈N is an IID sequence of integer-valued random variables. We call K(n) := P(τ 1 = n) and we assume that n∈N K(n) = 1 (the renewal is recurrent) and that K(·) has a power-law tail: with α > 0 and C K > 0. As usual, the notation A(x) The copolymer model we are going to define depends on two parameters λ ≥ 0 and h ≥ 0, and on a sequence ω := {ω 1 , ω 2 , . . .} of IID standard Gaussian random variables (the quenched disorder), whose law is denoted by P. For a given system size N ∈ N and disorder realization ω, we define the partition function Z N,ω := Z N,ω (λ, h) as where 1 {A} is the indicator function of the event A.
To see that the "standard copolymer model" (1.1) is a particular case of (2.2), let τ := {n ≥ 0 ∈ N : S n = 0} and as a consequence K(n) := P SRW (inf{k > 0 : S k = 0} = n). It is known that in this case K(·) satisfies (2.1) with α = 1/2, see [9, Ch. III] (the fact that in this case (2.1) holds only for n ∈ 2N, while K(n) = 0 for n ∈ 2N + 1 due to the periodicity of the simple random walk, entails only elementary modifications in the arguments below). Next, observe that if s i ∈ {−1, +1} denotes the sign of the excursion of S between the successive returns to zero τ i−1 and τ i , under P SRW the sequence {s i } i∈N is IID and symmetric (and independent of the sequence τ ). Therefore, performing the average on the {s i } i in (1.1) one immediately gets (2.2).
The infinite-volume free energy is defined as where existence of the limit is a consequence of superadditivity of the sequence {E log Z N,ω } N and the inequality f ≥ 0 is immediate from which is easily seen inserting in the expectation in right-hand side of (2. Indeed, one has from which it is not difficult to prove that and therefore the claim (2.6). For h ≥ λ, (2.8) follows from the fact that the right-hand side of (2.7) is bounded above by 1, while the left-hand side of (2.8) is always non-negative. For h < λ, just observe that and that (2.10) The limit (2.8) is called annealed free energy.
What is known about the critical line and its slope at the origin. The critical point is known to satisfy the bounds The upper bound, proven recently in [3, Th. 2.10], says that the annealed inequality (2.6) is strict for every λ. The lower bound was proven in [2] for the model (1.1) and in the general situation (2.2) in [11], and is based on an idea by C. Monthus [14]. We mention that (the analog of) the lower bound in (2.11) was recently proven in [15] and [4] to become optimal in the limit λ → ∞ for the "reduced copolymer model" introduced in [2, Sec. 4] (this is a copolymer model where the disorder law P depends on the coupling parameter λ).
As already mentioned, much attention has been devoted to the slope of the critical curve at the origin, in short the "critical slope", (2.12) Existence of such limit is not at all obvious (and indeed was proven [5] only in the case of the "standard copolymer model" (1.1)), but is expected to hold in general. While the proof in [5] was given in the case P(ω 1 = ±1) = 1/2, it was shown in [13] (by a much softer argument) that the results of [5] imply (always for the model (1.1)) that the slope exists and is the same in the Gaussian case we are considering here. Moreover, the critical slope is expected to be a function only of α and not of the full K(·), at least for 0 < α < 1, and to be independent of the choice of the disorder law P, as long as the ω n 's are IID, with finite exponential moments, centered and of variance 1. In contrast, it is known that the critical curve λ → h c (λ) does in general depend on the details of K(·) (this follows from [3, Prop. 2.11]) and of course on the disorder law P. The belief in the universality of the critical slope is supported by the result of [5] which, beyond proving that the limit (2.12) exists, identifies it with the critical slope of a continuous copolymer model, where the simple random walk S is replaced by a Brownian motion, and the ω n 's by a white noise. Until recently, nothing was known about the value of the critical slope, except for which follows from (2.6) and from the lower bound in (2.11) (note that the strict upper bound (2.11) does not imply a strict upper bound on the slope). None of these bounds is believed to be optimal. In particular, as we mentioned in the introduction, for the standard copolymer model (1.1) numerical simulations [6] suggest a value around 0.83 for the slope. This situation was much improved in [3]: if α > 1, then [3, Ths. 2.9 and 2.10] (2.14) Note that α > 1 and α ≤ 1 are profoundly different situations: the inter-arrival times of the renewal process have finite mean in the former case and infinite mean in the latter. Moreover, it was proven in [3, Th. 2.10] that there exists α 0 < 1 (which can be estimated to be around 0. Note that this does not cover the case of the standard copolymer model (1.1), for which α = 1/2.
A new upper bound on the critical slope. Our main result is that the upper bound in (2.13) is always strict: It is interesting to note that the upper bound (2.16) depends only on the exponent α and not the details of K(·). This is coherent with the mentioned belief in universality of the slope.
The new idea which allows to go beyond the results of [3, Th. 2.10] is to bound above the fractional moments of Z N,ω in two steps: (1) first we chop the system into blocks of size k, the correlation length of the annealed model, and we decompose Z N,ω according to which of the blocks contain points of τ (2) only at that point we apply the inequality (3.12), where each of the a i corresponds to one of the pieces into which the partition function has been decomposed.
Remark 2.2. Theorem 2.1 holds in the more general situation where ω is a sequence of IID random variables with finite exponential moments and normalized so that E ω 1 = 0, E ω 2 1 = 1. We state the result and give the proof only in the Gaussian case simply to keep technicalities at a minimum. The extension to the general disorder law can be obtained following the lines of [3,Sec. 4.4].

Proof of Theorem 2.1.
Fix α > 0, 1/(1 + α) < γ < 1 and define The reason why we restrict to γ > 1/(1 + α) will be clear after (3.34). From now on we take h = ρλ, where the value of will be chosen close to 1 later. Let and note that, irrespective of how ρ is chosen, k can be made arbitrarily large choosing λ small (which is no restriction since in Theorem 2.1 we are interested in the limit λ ց 0).
One sees from (2.8) that, apart from an inessential factor 2, k is just the inverse of the annealed free energy, i.e., We will show that, if ρ(α, γ) is sufficiently close to 1, there exists λ 0 := λ 0 (γ, K(·)) > 0 such that for 0 < λ < λ 0 there exists c := c(γ, λ, K(·)) < ∞ such that for every N ∈ kN. In particular, by Jensen's inequality and the fact that the sequence From now on we assume that (N/k) is integer and we divide the interval {1, . . . , N } into blocks We have then the identity (see Fig. 1) (3.10). In this example we have a number N/k = 4 of blocks, ℓ = 2 and i1 = 2, while i ℓ = N/k = 4 by definition (cf. (3.9)). Big black dots denote the ni's, white dots denote the ji's, while small dots are all the other points of τ . Note that j1 − n1 < k, as it should, and that there is no point of τ between a white dot and the next big black dot. In this example, the set M of (3.14) is {2, 3, 4}, and as a consequence W defined in (3.15) is B2 ∪ B3 ∪ B4. In words: n1 is the first point of τ after 0, j1 is the last point of τ which does not exceed n1 + k − 1, n2 is the first point after j1, and so on. The index of the block containing nr defines ir.
n 2 ≥n 1 +k and, for I ⊂ N, We have then, using the inequality which holds for 0 ≤ γ ≤ 1 and a i ≥ 0, and note that 1 ≤ |M | < 2ℓ. With the conventions of Fig. 1, W is the union of the blocks B i which either contain a big black dot or such that B i−1 contains a big black dot. Note also that, for every r, the interval [n r , j r ] is a subset of W . We want first of all to show that the ϕ's can be effectively replaced by constants. To this purpose, we use the inequality ϕ((j r , n r+1 ]) ≤ 2ϕ((j r , n r+1 ] \ W ) ϕ ((j r , n r+1 ] ∩ W ) (3.16) (with the convention that ϕ(∅) = 1 and j 0 := 0), where W was defined in (3.15). This is simply due to the fact that, if I 1 and I 2 are two disjoint subsets of N, one has ϕ(I 1 ∪ I 2 ) ≤ 2ϕ(I 1 )ϕ(I 2 ). We note that the two factors in the right-hand side of (3.16) are independent random variables. Moreover, since we observe that the law of depends only on (i 1 , . . . , i ℓ ) and not on the n r 's and j r 's, and that, once (i 1 , . . . , i ℓ ) is fixed, (3.18) is the product of ℓ independent random variables. As a consequence, Thanks to (3.12) and to the choice h = ρλ, for every I ⊂ N where the second inequality is implied by our assumption γ < ρ, cf. (3.2). Then, In order to estimate the remaining average, we use Hölder's inequality with p = 1/γ and q = 1/(1 − γ): where, under the modified law E := E (i 1 ,...,i ℓ ) , the {ω i } i∈N are still Gaussian, independent and of variance 1, but ω i has average 1/ √ k if i ∈ W (i 1 , . . . , i ℓ ), while ω i has average 0, as under E, if i / ∈ W (i 1 , . . . , i ℓ ). Since E is still a product measure, it is immediate to check that where we used the fact that |W | = k|M | and |M | ≤ 2ℓ. Next, we observe that with I = (j r , n r+1 ] ∩ W . Thanks to the definition of k and to h = ρλ, (3.25) equals In conclusion, we proved with the convention that U (0) := 1. In (3.28) we used independence of Z nr,jr for different r's (recall that E is a product measure) to factorize the expectation. The heart of the proof of Theorem 2.1 is the following: Lemma 3.1. There exists λ 0 (γ, K(·)) > 0 such that the following holds for λ < λ 0 . If, for some ε > 0, then the quantity in square brackets in (3.28) is bounded above by where C 1 := C 1 (ε, k, K(·)) < ∞.
Here and in the following, the positive and finite constants C i , i ≥ 1 depend only on the arguments which are explicitly indicated, while C K is the same constant which appears in (2.1).
Assume that Lemma 3.1 is true and that (3.30)-(3.31) are satisfied. Then, If moreover ε satisfies then it follows from [11,Th. A.4] that for every N ∈ kN. Indeed, the sum in the right-hand side of (3.33) is nothing but the partition function of a homogeneous pinning model [11,Ch. 2] of length N/k with pinning parameter ξ such that the system is in the delocalized phase (this is encoded in (3.34)). More precisely: to obtain (3.35) it is sufficient to apply Proposition 3.2 below, with α replaced by (1 + α)γ − 1 > 0 and K(·) replaced by (3.36) Proposition 3.2. [11,Th. A.4] If K(·) is a probability on N which satisfies (2.1) for some α > 0, then for every ξ < 1 there exists c = c(K(·), ξ) such that for every Inequality (3.5) is then proven; note that C 2 depends on λ through k. The condition γ > 1/(1 + α) which we required since the beginning guarantees that the sum in (3.34) converges. Note that (3.34) depends on K(·) only through α; this is important since we want ρ in (2.16) to depend only on α and not on the whole K(·).
Proof of Lemma 3.1. First of all, we get rid of U (N − n ℓ ) and effectively we replace n ℓ by N : the quantity in square brackets in (3.28) is upper bounded by Explicitly, one may take (recall (2.9) and N − n ℓ ≤ k), while the supremum in (3.39) is easily seen from (2.1) to depend only on k and K(·).
Recall that by convention i 0 = j 0 = 0, and let also from now on n ℓ := N and n 0 := 0. We do the following: • for every 1 ≤ r ≤ ℓ such that i r > i r−1 + 2 (which guarantees that between j r−1 and n r there is at least one full block), we use This is true under the assumption that k ≥ k 0 with k 0 (γ, K(·)) large enough, i.e., λ ≤ λ 0 (γ, K(·)) with λ 0 small, since n r − j r−1 ≥ k(i r − i r−1 − 2) and sup n>2 (n/(n − 2)) 1+α = 3 1+α . • for every 1 ≤ r ≤ ℓ such that i r ≤ i r−1 + 2, we leave K(n r − j r−1 ) as it is. Then, (3.38) is bounded above by Now we can sum over j r , n r , 1 ≤ r < ℓ, using the two assumptions (3.30)-(3.31). We do this in three steps: • First, for every r ∈ J we sum over the allowed values of j r−1 (provided that r > 1, otherwise there is no j r−1 to sum over) using (3.30) and the constraint 0 ≤ j r−1 − n r−1 < k. The sum over all such j r−1 gives at most where C 4 takes care of the fact that possibly 1 ∈ J. At this point we are left with the task of summing r∈{1,...,ℓ}\J (observe that U (j 0 − n 0 ) = 1) over all the allowed values of n r , 1 ≤ r < ℓ and of j r−1 , r ∈ {1, . . . , ℓ} \ J. • Secondly, using (3.31), we see that if r ∈ {1, . . . , ℓ} \ J then the sum of K(n r − j r−1 )U (j r−1 − n r−1 ) over the allowed values of n r and j r−1 gives at most ε (or 1 if r = 1). The contribution from all the n r , j r−1 with r ∈ {1, . . . , ℓ} \ J is therefore at most This is best seen if one starts to sum on n r , j r−1 for the largest value of r ∈ {1, . . . , ℓ} \ J and then proceeds to the second largest, and so on. • Finally, the sum over all the n r 's with r ∈ J is trivial (the summand does not depend on the n r 's) and gives at most k |J| . In conclusion, we have upper bounded (3.41) by (3.46) If 0 < α < 1 it is clear from the definition of Y (k, C K , α) that the last factor equals 1 and (3.32) is proven. For α ≥ 1, C K Y (k, C K , α) k α can be made as small as wished with k large (i.e. choosing λ 0 small), so that we can again assume that the last factor in (3.46) does not exceed 1. Lemma 3.1 is proven.
Proof of (3.31). If we choose b := b(ε, α) small, j ≤ bk implies k − j ≥ k/2. Therefore, bk j=0 n≥k (if k is sufficiently large and b(ε, α) is suitably small). As for the rest of the sum: again from Lemma 3.4 and (3.48), one has U (j) ≤ (ε/2)P(j ∈ τ ) for every bk ≤ j < k. Then, In the last equality, we used the fact that is the P-probability that the first point of τ which does not precede k equals n. The sum over n ≥ k of (3.62) then clearly equals 1, since τ is recurrent. We detail below the proof of this inequality in order to leave no doubts on the fact that the constant C 9 depends only on α. This was not emphasized in the proof of [3, Lemma 4.3] since it was not needed there. For α ≥ 1, it follows from [3, Eqs. (4.23) and (4.49)] that the lim sup in the right-hand side of (A.2) is actually a limit, and equals exp(−q/2) (the q/8 which appears in [3, Eq. (4.49)] can be immediately improved into q/2). As a side remark, the expectation in (A.2), irrespective of the value of α and N , is not smaller than exp(−q/2); this just follows from the convexity of the exponential function: 1 + e −(q/N )x 2 ≥ e −q/(2N ) x .