# Estimates on path delocalization for copolymers at selective interfaces

Giacomin, G; Toninelli, F (2005). Estimates on path delocalization for copolymers at selective interfaces. Probability Theory and Related Fields, 133(4):464-482.

## Abstract

Starting from the simple symmetric random walk {Sn}n, we introduce a new process whose path measure is weighted by a factor exp factor $\exp (\lambda\sum^N_{n=1}(\omega_n+h){\rm sign}\,(S_n))$, with α,h≥0, {Wn}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 Wn+h>0 and Sn<0 if Wn+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 α 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].

## Abstract

Starting from the simple symmetric random walk {Sn}n, we introduce a new process whose path measure is weighted by a factor exp factor $\exp (\lambda\sum^N_{n=1}(\omega_n+h){\rm sign}\,(S_n))$, with α,h≥0, {Wn}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 Wn+h>0 and Sn<0 if Wn+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 α 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].

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