Variational characterization of the critical curve for pinning of random polymers

Abstract

In this paper we look at the pinning of a directed polymer by a onedimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process [Birkner, Greven and den Hollander, Probab. Theory Related Fields 148 (2010) 403-456], we derive variational formulas for the quenched, respectively, annealed critical curve separating the two phases. These variational formulas are used to obtain a necessary and sufficient criterion, stated in terms of relative entropies, for the two critical curves to be different at a given inverse temperature, a property referred to as relevance of the disorder. This criterion in turn is used to show that the regimes of relevant and irrelevant disorder are separated by a unique inverse critical temperature. Subsequently, upper and lower bounds are derived for the inverse critical temperature, from which sufficient conditions under which it is strictly positive, respectively, finite are obtained. The former condition is believed to be necessary as well, a problem that we will address in a forthcoming paper.Random pinning has been studied extensively in the literature. The present paper opens up a window with a variational view. Our variational formulas for the quenched and the annealed critical curve are new and provide valuable insight into the nature of the phase transition. Our results on the inverse critical temperature drawn from these variational formulas are not new, but they offer an alternative approach, that is, flexible enough to be extended to other models of random polymers with disorder. quenched vs. annealed large deviation principle, quenched vs. annealed critical curve, relevant vs. irrelevant disorder, critical temperature. que c (0) = 0; see Appendix A. It was shown in Alexander and Sidoravicius [2] that h que c (β) > 0 for β ∈ (0, ∞). Therefore we have the qualitative picture drawn in Figure 2. We further remark that lim β→∞ h que c (β)/β is finite if and only if supp(μ 0 ) is bounded from above.The mean value of the disorder is E(βω 0 −h) = −h. Thus, we see from Figure 2 that for the random pinning model localization may even occur for moderately negative mean values of the disorder, contrary to what happens for the homogeneous pinning model, where localization occurs only for a strictly positive parameter; see Appendix A. In other words, even a globally repulsive random interface can pin the polymer: all that the polymer needs to do is to hit some positive values of the disorder and avoid the negative values of the disorder.

Influence of Disorder For the Polymer Pinning Model

“…Theorem 2.2 collects several results proven in a series of papers [4,10,16,17,19,28,29,31] in several manners, the full necessary and sufficient condition for disorder relevance (together with the sharp critical point shift when α = 1/2) being given only recently in [12]. In [12,19,28,29], the authors estimate the fractional moment of the partition function up to the correlation length by a change of measure argument, and then use a coarse-graining procedure to glue these estimates together.…”

“…There exist several proofs of Theorem 2.1, with very different techniques: a second moment method in [1,39], martingale techniques in [34], and a large deviation/variational formula approach in [17]. The case α = 0 is studied in depth in [5]: the annealed and quenched critical points are equal, whatever β is.…”

“…In [12,19,28,29], the authors estimate the fractional moment of the partition function up to the correlation length by a change of measure argument, and then use a coarse-graining procedure to glue these estimates together. A coarse-graining argument is also in order in [4], whereas in [17], a large deviation approach is used, enabling the authors to give a variational formula for the quenched critical point (and as a matter of fact, also for the annealed one). In [15,31], the authors prove the smoothing of the free energy (12) via a rare-stretch strategy, that we outline in Section 2.2: it stresses the influence of rare regions in disorder relevance.…”

“…Also, h que c (β) ≥ h a c (β), see Section 1.4 (note that under some condition h c (0) − h a c (β) ∼ cst. β 2 , see (17) Another important feature of the model is the order of the phase transition: one desires to know how fast the free energy vanishes when h approaches h c (β). Formally, one defines…”

“…It has been proven in [17] and in [29, § 6] that the critical point shift β → h que c (β) − h a c (β) is non-decreasing, so that disorder relevance is monotone in β. In particular, there exists a critical value β c ≥ 0 such that h que…”