A copolymer near a selective interface:
variational characterization of the free energy
In this paper we consider a random copolymer near a selective interface separating two solvents. The configurations of the copolymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The excursion length distribution is assumed to have a tail that is logarithmically equivalent to a power law with exponent . The monomers carry i.i.d. real-valued types whose distribution is assumed to have zero mean, unit variance, and a finite moment generating function. The interaction Hamiltonian rewards matches and penalizes mismatches of the monomer types and the solvents, and depends on two parameters: the interaction strength and the interaction bias . We are interested in the behavior of the copolymer in the limit as its length tends to infinity.
The quenched free energy per monomer has a phase transition along a quenched critical curve separating a localized phase, where the copolymer stays close to the interface, from a delocalized phase, where the copolymer wanders away from the interface. We derive variational formulas for both these quantities. We compare these variational formulas with their analogues for the annealed free energy per monomer and the annealed critical curve , both of which are explicitly computable. This comparison leads to:
A proof that for all and in the annealed localized phase.
A proof that for all and .
A proof that with for and for with .
An estimate of the total number of times the copolymer visits the interface in the interior of the quenched delocalized phase.
An identification of the asymptotic frequency at which the copolymer visits the interface in the quenched localized phase.
The copolymer model has been studied extensively in the literature. The goal of the present paper is to open up a window with a variational view and to address a number of open problems.
AMS 2000 subject classifications. 60F10, 60K37, 82B27.
Key words and phrases. Copolymer, selective interface, free energy, critical curve, localization vs. delocalization, large deviation principle, variational formula, specific relative entropy.
Acknowledgment. FdH thanks M. Birkner and F. Redig for fruitful discussions. EB was supported by SNSF-grant 20-100536/1, FdH by ERC Advanced Grant VARIS 267356, and AO by NWO-grant 613.000.913.
1 Introduction and main results
In Section 1.1 we define the model. In Sections 1.2 and 1.3 we define the quenched and the annealed free energy and critical curve. In Section 1.4 we state our main results, while in Section 1.5 we place these results in the context of earlier work. For more background and key results in the literature, we refer the reader to Giacomin , Chapters 6–8, and den Hollander , Chapter 9.
1.1 A copolymer near a selective interface
Let be i.i.d. random variables with a probability distribution on having zero mean and unit variance:
and a finite cumulant generating function:
Write to denote the distribution of . Let
denote the set of infinite directed paths on (with ). Fix and . For given , let
be the -step Hamiltonian on , and let
be the -step path measure on , where is any probability distribution on under which the excursions away from the interface are i.i.d., lie with equal probability above and below the interface, and have a length whose probability distribution on has infinite support and a polynomial tail
Note that the Hamiltonian in (1.4) only depends on the signs of the excursions and on their starting and ending points in , not on their shape.
Example. For the special case where is the binary distribution and is simple random walk on , the above definitions have the following interpretation (see Fig. 1). Think of in (1.3) as the path of a directed copolymer on , consisting of monomers represented by the edges , , pointing either north-east of south-east. Think of the lower half-plane as water and the upper half-plane as oil. The monomers are labeled by , with indicating that monomer is hydrophilic and that it is hydrophobic. Both types occur with density . The factor in (1.4) equals or depending on whether monomer lies in the water or in the oil. The interaction Hamiltonian in (1.4) therefore rewards matches and penalizes mismatches of the monomer types and the solvents. The parameter is the interaction strength (or inverse temperature), the parameter plays the role of the interaction bias: corresponds to the hydrophobic and hydrophilic monomers interacting equally strongly, while corresponds to the hydrophilic monomers not interacting at all. The probability distribution of the copolymer given is the quenched Gibbs distribution in (1.5). For simple random walk the support of is and the exponent is : as (Feller , Chapter III).
1.2 Quenched free energy and critical curve
It was further noted that
This lower bound comes from the strategy where the path spends all of its time above the interface, i.e., for . Indeed, in that case for , resulting in -a.s. as by the strong law of large numbers for (recall (1.1)). Since as by (1.6), the cost of this strategy under is negligible on an exponential scale.
In view of (1.8), it is natural to introduce the quenched excess free energy
to define the two phases
and to refer to as the quenched delocalized phase, where the strategy of staying above the interface is optimal, and to as the quenched localized phase, where this strategy is not optimal. The presence of these two phases is the result of a competition between entropy and energy: by staying close to the interface the copolymer looses entropy, but it gains energy because it can more easily switch between the two sides of the interface in an attempt to place as many monomers as possible in their preferred solvent.
General convexity arguments show that and are separated by a quenched critical curve given by
with the property that , is strictly increasing and finite on , and is strictly convex on . Moreover, it is easy to check that , the supremum of the support of (see Fig. 2).
The following bounds are known for the quenched critical curve:
The upper bound was proved in Bolthausen and den Hollander , and comes from an annealed estimate on . The lower bound was proved in Bodineau and Giacomin , and comes from strategies where the copolymer dips below the interface during rare stretches in where the empirical density is sufficiently biased downwards.
Remark: In the literature is typically assumed to be regularly varying at infinity, i.e.,
However, the proof of (1.12) in  and  can be extended to satisfying the much weaker assumption in (1.6). In the literature is sometimes assumed to have Gaussian or sub-Gaussian tails, which is stronger than (1.2). Also this is not necessary for (1.12). Throughout our paper, (1.2) and (1.6) are the only conditions in force (with a sole exception indicated later on).
1.3 Annealed free energy and critical curve
In this partition sum only the excursions of the copolymer below the interface contribute. The annealed version of the model has partition sum
where is expectation w.r.t. . The annealed excess free energy is therefore given by
which are referred to as the annealed delocalized phase, respectively, the annealed localized phase, and are separated by an annealed critical curve given by
1.4 Main results
Our variational characterization of the excess free energies and the critical curves are contained in the following theorem.
given by explicit variational formulas, such that
(ii) For every , and are the unique solutions of the equations
(iii) For every , and are the unique solutions of the equations
Next we state six corollaries that are consequences of the variational formulas. The first three corollaries are strict inequalities for the excess free energies and the critical curves.
for all .
If , then for all .
If , then for all .
The fourth corollary concerns the slope of the quenched critical curve at . For , let
and let be the unique solution of the equation . We say that is asymptotically periodic when there exists a such that if and only if for large enough.
The last two corollaries concern the typical path behavior. Let denote the path measure associated with the constrained partition sum defined in (1.15). Write to denote the number of times returns to the interface up to time .
For every and ,
For every ,
provided this derivative exists. (By convexity, at least the left-derivative and the right-derivative exist.)
1. The main importance of our results in Section 1.4 is that they open up a window on the copolymer model with a variational view. Whereas the results in the literature were obtained with the help of a variety of estimation techniques, Theorem 1.1 provides variational formulas that are new and explicit. As we will see in Section 3, these variational formulas are not easy to manipulate. However, they provide a natural setting, and are robust in the sense that the large deviation principles on which they are based (see Section 2) can be applied to other polymer models as well, e.g. the pinning model with disorder (Cheliotis and den Hollander ). Still other applications involve certain classes of interacting stochastic systems (Birkner, Greven and den Hollander ). For an overview, see den Hollander .
2. The gap between the excess free energies stated in Corollary 1.2 has never been claimed in the literature, but follows from known results. Fix . We know that is strictly positive, strictly decreasing and linear on , and zero on (see Fig. 3). We also know that is strictly positive, strictly decreasing and convex on , and zero on . It was shown in Giacomin and Toninelli [18, 19] that drops below a quadratic as , i.e., the phase transition is “at least of second order” (see Fig. 2). Hence, the gap is present in a left-neighborhood of . Combining this observation with the fact that and , it follows that the gap is present for all . Note: The above argument crucially relies on the linearity of on . However, we will see in Section 3 that our proof of Corollary 1.2 is robust and does not depend on this linearity.
3. For a number of years, all attempts in the literature to improve (1.12) had failed. As explained in Orlandini, Rechnitzer and Whittington  and Caravenna and Giacomin , the reason behind this failure is that any improvement of (1.12) necessarily requires a deep understanding of the global behavior of the copolymer when the parameters are close to the quenched critical curve. Toninelli  proved the strict upper bound in Corollary 1.3 with the help of fractional moment estimates for unbounded disorder and large subject to (1.2) and (1.13), and this result was later extended by Bodineau, Giacomin, Lacoin and Toninelli  to arbitrary disorder and arbitrary , again subject to (1.2) and (1.13). The latter paper also proved the strict lower bound in Corollary 1.4 with the help of appropriate localization strategies for small and , where (theoretical bound) and (numerical bound), which unfortunately excludes the simple random walk example in Section 1.1 for which . Corollaries 1.3 and 1.4 settle the strict inequalities in full generality subject to (1.2) and (1.6).
4. A point of heated debate has been the value of
which is believed to be universal, i.e., to depend on alone and to be robust under changes of the fine details of the interaction Hamiltonian. The existence of was proved in Bolthausen and den Hollander  for associated with simple random walk () and binary disorder. The proof uses a Brownian approximation of the copolymer model. This result was extended in Caravenna and Giacomin  to satisfying (1.13) with and disorder with a moment generating function that is finite in a neighborhood of the origin. The proof uses a Lévy approximation of the copolymer model. The Lévy copolymer serves as the attractor of a universality class, indexed by the exponent . For , the existence of the limit has remained open. The bounds in (1.12) imply that , and various claims were made in the literature arguing in favor of , respectively, . However, in Bodineau, Giacomin, Lacoin and Toninelli  it was shown that for and for . Corollary 1.5 improves these two lower bounds. We do not have an upper bound. In  it was shown that for , which was later extended to in Toninelli . For an overview, see Caravenna, Giacomin and Toninelli .
5. A numerical analysis for simple random walk () and binary disorder carried out in Caravenna, Giacomin and Gubinelli  (see also Giacomin , Chapter 9) showed that . Since , the following conjecture is natural.
for all .
In  is was also shown that
Thus, the quenched critical curve lies “somewhere halfway” between the two bounds in (1.12), and so it remains a challenge to quantify the strict inequalities in Corollaries 1.3 and 1.4. Some quantification for the upper bound was offered in Bodineau, Giacomin, Lacoin and Toninelli , and for the lower bound in Toninelli . Our proofs of Corollaries 1.3 and 1.4 sharpen these quantifications.
6. Because of (1.12), it was suggested that the quenched critical curve possibly depends on the exponent of alone and not on the fine details of . However, it was shown in Bodineau, Giacomin, Lacoin and Toninelli  that, subject to (1.2), for every , and there exists a satisfying (1.13) such that is -close to the upper bound, which rules out such a scenario. Our variational characterization in Section 3 confirms this observation, and makes it quite evident that the fine details of do indeed matter.
7. Special cases of Corollaries 1.6 and 1.7 were proved in Biskup and den Hollander  (for simple random walk and binary disorder) and in Giacomin and Toninelli [17, 20] (subject to (1.13) and for disorder satisfying a Gaussian concentration of measure bound). However, no formulas were obtained for the relevant constants.The latter two papers prove the bound under the average quenched measure, i.e., under . For the pinning model with disorder, the same result as in Corollary 1.6 was derived in Mourrat  with the help of the variational characterization obtained in Cheliotis and den Hollander .
In Section 2 we recall two large deviation principles (LDP’s) derived in Birkner  and Birkner, Greven and den Hollander , which describe the large deviation behavior of the empirical process of words cut out from a random letter sequence according to a random renewal process with exponentially bounded, respectively, polynomial tails. In Section 3 we use these LDP’s to prove Theorem 1.1. In Sections 4–8 we prove Corollaries 1.2–1.7. Appendices A–D contain a number of technical estimates that are needed in Section 3.
In Cheliotis and den Hollander , the LDP’s in  were applied to the pinning model with disorder, and variational formulas were derived for the critical curves (not the free energies). The Hamiltonian is similar in spirit to (1.4), except that the disorder is felt only at the interface, which makes the pinning model easier than the copolymer model. The present paper borrows ideas from . However, the new challenges that come up are considerable.
2 Large deviation principles: intermezzo
In this section we recall the LDP’s from Birkner  and Birkner, Greven and den Hollander , which are the key tools in the present paper. Section 2.1 introduces the relevant notation, while Sections 2.2 and 2.3 state the annealed, respectively, quenched version of the LDP. Apart from minor modifications, this section is copied from . We repeat it here in order to set the notation and to keep the paper self-contained.
Let be a Polish space, playing the role of an alphabet, i.e., a set of letters. Let be the set of finite words drawn from , which can be metrized to become a Polish space. Write and to denote the set of probability measures on and .
Fix , and satisfying (1.6). Let be i.i.d. -valued random variables with marginal law , and i.i.d. -valued random variables with marginal law . Assume that and are independent, and write to denote their joint law. Cut words out of the letter sequence according to (see Fig. 5), i.e., put
Under the law , is an i.i.d. sequence of words with marginal law on given by
We define as the tilted version of given by
Note that if , then has an exponentially bounded tail. For we write instead of . We write and for the analogues of and when is replaced by defined in (2.4).
The reverse operation of cutting words out of a sequence of letters is glueing words together into a sequence of letters. Formally, this is done by defining a concatenation map from to . This map induces in a natural way a map fsrom to , the sets of probability measures on and (endowed with the topology of weak convergence). The concatenation of equals , as is evident from (2.3).
Let be the set of probability measures on that are invariant under the left-shift acting on . For , let be the specific relative entropy of w.r.t. defined by
where denotes the projection of onto the first words, denotes relative entropy, and the limit is non-decreasing. The following lemma relates the specific relative entropies of w.r.t. and .
For and ,
with defined in (2.4) and the average word length under ( denotes expectation under the law and is the length of the first word).
Proof. Observe from (2.4) that
where is the length of the -th word and the second equality uses that . Let and use (2.5), to get the claim.
2.2 Annealed LDP
For , let be the periodic extension of the -tuple to an element of , and define
This is the empirical process of -tuples of words. The superscript indicates that the words are cut from the letter sequence . The following annealed LDP is standard (see e.g. Dembo and Zeitouni , Section 6.5).
For every , the family , , satisfies the LDP on with rate and with rate function given by
This rate function is lower semi-continuous, has compact level sets, has a unique zero at , and is affine.
It follows from Lemma 2.1 that
where , the annealed rate function for .
2.3 Quenched LDP
To formulate the quenched analogue of Theorem 2.2, we need some more notation. Let be the set of probability measures on that are invariant under the left-shift acting on . For such that , define
Think of as the shift-invariant version of obtained after randomizing the location of the origin. This randomization is necessary because a shift-invariant in general does not give rise to a shift-invariant .
For , let denote the truncation map on words defined by
i.e., is the word of length obtained from the word by dropping all the letters with label . This map induces in a natural way a map from to , and from to . Note that if , then is an element of the set
Define (w-lim means weak limit)
i.e., the set of probability measures in under which the concatenation of words almost surely has the same asymptotic statistics as a typical realization of .
(Birkner ; Birkner, Greven and den Hollander ) Assume (1.2) and (1.6). Then, for –a.s. all and all , the family of (regular) conditional probability distributions , , satisfies the LDP on with rate and with deterministic rate function given by
This rate function is lower semi-continuous, has compact level sets, has a unique zero at , and is affine.
The difference between (2.15) for and (2.16–2.17) for can be explained as follows. For , the word length distribution has a polynomial tail. It therefore is only exponentially costly to cut out a few words of an exponentially large length in order to move to stretches in that are suitable to build a large deviation with words whose length is of order 1. This is precisely where the second term in (2.17) comes from: this term is the extra cost to find these stretches under the quenched law rather than to create them “on the spot” under the annealed law. For , on the other hand, the word length distribution has an exponentially bounded tail, and hence exponentially long words are too costly, so that suitable stretches far away cannot be reached. Phrased differently, and is qualitatively similar to and , for which we see that the expression in (2.17) is finite if and only . It was shown in , Lemma 2, that