Scaling limit of a one-dimensional polymer in a repulsive i.i.d. environment

Abstract

The purpose of this paper is to study a one-dimensional polymer penalized by its range and placed in a random environment ω. The law of the simple symmetric random walk up to time n is modified by the exponential of the sum of βωz−h sitting on its range, with h and β positive parameters. It is known that, at first order, the polymer folds itself to a segment of optimal size c_h n^{1 ∕ 3} with c_h = π^{2 ∕ 3} h^{−1 ∕ 3}. Here we study how disorder influences finer quantities. If the random variables ω_z are i.i.d. with a finite second moment, we prove that the left-most point of the range is located near − u_∗ n^{1 ∕ 3}, where u∗∈[0,c_h] is a constant that only depends on the disorder. This contrasts with the homogeneous model (i.e. when β=0), where the left-most point has a random location between −c_h n^{1 ∕ 3} and 0. With an additional moment assumption, we are able to show that the left-most point of the range is at distance U n^{2 ∕ 9} from −u_∗ n^{1 ∕ 3} and the right-most point at distance V n^{2 ∕ 9} from (c_h−u_∗)n^{1 ∕ 3}. Here again, U and V are constants that depend only on ω.