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 $\mathit{\beta }{\mathit{\omega }}_z-h$ 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 $c_h={\mathrm{\pi }}^{2∕3}{h}^{-1∕3}$. Here we study how disorder influences finer quantities. If the random variables ${\mathit{\omega }}_z$ are i.i.d. with a finite second moment, we prove that the left-most point of the range is located near $-u_{\ast }{n}^{1∕3}$, where $u_{\ast }\in [0,c_h]$ is a constant that only depends on the disorder. This contrasts with the homogeneous model (i.e. when $\mathit{\beta }=0$), 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 $\mathcal{U}{n}^{2∕9}$ from $-u_{\ast }{n}^{1∕3}$ and the right-most point at distance $\mathcal{V}{n}^{2∕9}$ from $(c_h-u_{\ast })n^{1∕3}$. Here again, $\mathcal{U}$ and $\mathcal{V}$ are constants that depend only on ω.