Estimates on path delocalization for copolymers at selective interfaces

Abstract

Abstract. Starting from the simple symmetric random walk {Sn}n, we introduce a new process whose path measure is weighted by a factor exp λ N n=1 (ωn + h) sign (Sn) , with λ, h ≥ 0, {ωn}n a typical realization of an IID process and N a positive integer. We are looking for results in the large N limit. This factor favors Sn > 0 if ωn + h > 0 and Sn < 0 if ωn + h < 0. The process can be interpreted as a model for a random heterogeneous polymer in the proximity of an interface separating two selective solvents. It has been shown [6] that this model undergoes a (de)localization transition: more precisely there exists a continuous increasing function λ −→ hc(λ) such that if h < hc(λ) then the model is localized while it is delocalized if h ≥ hc(λ). However, localization and delocalization were not given in terms of path properties, but in a free energy sense. Later on it has been shown that free energy localization does indeed correspond to a (strong) form of path localization [3]. On the other hand, only weak results on the delocalized regime have been known so far. We present a method, based on concentration bounds on suitably restricted partition functions, that yields much stronger results on the path behavior in the interior of the delocalized region, that is for h > hc(λ). In particular we prove that, in a suitable sense, one cannot expect more than O(log N ) visits of the walk to the lower half plane. The previously known bound was o(N ). Stronger O(1)-type results are obtained deep inside the delocalized region.The same approach is also helpful for a different type of question: we prove in fact that the limit as λ tends to zero of hc(λ)/λ exists and it is independent of the law of ω1, at least when the random variable ω1 is bounded or it is Gaussian. This is achieved by interpolating between this class of variables and the particular case of ω1 taking values ±1 with probability 1/2, treated in [6].

A Numerical Approach to Copolymers at Selective Interfaces

“…(4) Finally, in Section 5, we report the results of a numerical attempt to determine the critical curve. While this issue has to be treated with care, mostly for the reasons raised in point 4 above, we observe a surprising phenomenon: the critical curve appears to be very close to h (m) (·) for a suitable value of m. By the universality result proven in [15], building on the free energy Brownian scaling result proven in [5], the slope at the origin of h c (·) does not depend on the law of ω. Therefore if really h (m) (·) = h c (·), since the slope at the origin of h (m) (·) is m, m is the universal constant we are looking for.…”

“…In non disordered models one can in fact prove that the polymer becomes transient and that it visits the unfavorable solvent, or any point below a fixed level, only a finite number of times. Recently it has been shown [15] that such a result cannot hold as stated, at least for h < h(λ), for the disordered copolymer. However the results in [15] leave open the possibility of Brownian scaling in the whole delocalized region.…”

“…• For (λ, h) in the interior of D one can prove by large deviation arguments that there are o(N ) visits to the unfavorable solvent and by more sophisticate arguments that these visits are actually O(log N ) [15]. These results are in sharp contrast with what happens in L and in this sense they are satisfactory.…”

“…The details of the proof are in Appendix B and we point out in particular Proposition B.2, that gives a necessary and sufficient condition for localization. This viewpoint on the transition, derived from [15,Section 4], helps substantially in interpreting the irregularities in the behavior of {Z N,ω } N as N ր ∞. (3) In Section 4 we pick up the conjecture of Brownian scaling in the delocalized regime both in the intent of testing it and in trying to asses with reasonable confidence that (λ, h) is in the interior of D. In particular, we present quantitative evidences in favor of the fact that the upper bound h = h(λ) defined in (1.7) is strictly greater than the critical line.…”

Hierarchical pinning models, quadratic maps and quenched disorder

“…The idea of considering non-integer moments (this time, of R n − 1) plays an important role also in the present paper. (3) A number of results on the behavior of the paths of the model have been proven addressing the question of what can be said about the trajectories of the system once we know that the free energy is zero (or positive) [17,18]. One can in fact prove that if f(β, h) > 0 then the process sticks close to the origin (in a strong sense) and it is therefore in a localized (L) regime.…”