Smoothing Effect of Quenched Disorder on Polymer Depinning Transitions

Abstract

Abstract. We consider general disordered models of pinning of directed polymers on a defect line. This class contains in particular the (1 + 1)-dimensional interface wetting model, the disordered Poland-Scheraga model of DNA denaturation and other (1 + d)-dimensional polymers in interaction with flat interfaces. We consider also the case of copolymers with adsorption at a selective interface. Under quite general conditions, these models are known to have a (de)localization transition at some critical line in the phase diagram. In this work we prove in particular that, as soon as disorder is present, the transition is at least of second order, in the sense that the free energy is differentiable at the critical line, so that the order parameter vanishes continuously at the transition. On the other hand, it is known that the corresponding non-disordered models can have a first order (de)localization transition, with a discontinuous first derivative. Our result shows therefore that the presence of the disorder has really a smoothing effect on the transition. The relation with the predictions based on the Harris criterion is discussed.2000 Mathematics Subject Classification: 60K35, 82B41, 82B44

The Effect of Disorder on Polymer Depinning Transitions

“…The P V -a.s. existence and the nonrandomness of the last limit are proved in [12], for the cases we consider here. Clearly f d (β, u) depends only on βu, and f a (β, u) depends only on βu + log M V (β).…”

“…For recurrent chains satisfying (1.4) it is easily seen (see [1]) that u d c (β) = 0 for all β > 0, and hence (1.5) u a c (β) = −β −1 log M V (β), and, again from [1], for the deterministic or annealed model the transition is first order if and only if E 1 X < ∞; in particular it is first order for c > 2 but not for c < 2. It is proved in [12] that (again as known nonrigorously from the physics literature-see e.g. [6]) the annealed specific heat exponent is (2c − 3)/(c − 1).…”

“…This incorporates both the directed case in which we view the object as existing in Z × Σ with configuration given by the space-time trajectory (0, 0), (1, X 1 ), .., (N, X N ), and the undirected case in which the object exists in Σ with configuration given by the trajectory 0, X 1 , .., X N , with i merely an index. A number of physically disparate phenomena are subsumed in this formulation, which was studied in [1], [10] and [12] in the mathematics literature; closely related models have been studied extensively in the physics literature, for example in [5], [8], [16], [17] and [18]. Taking Σ = Z d we obtain a directed quenched random copolymer in d + 1 dimensions interacting, with strength that depends on the random monomer types, with a potential located in the first coordinate axis.…”

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.…”

“…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. For a more complete overview of the results and techniques employed, we refer the reader to [27].…”