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

o m e g a

. The law of the simple symmetric random walk up to time

n

is modified by the exponential of the sum of

sitting on its range, with~

h

and

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} = p i^{2 / 3} h^{- 1 / 3}

. Here we study how disorder influences finer quantities. If the random variables

o m e g a_{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_{*} i n [0, c_{h}]

is a constant that only depends on the disorder. This contrast with the homogeneous model (i.e. when

), 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

m a t h c a l U n^{2 / 9}

from

- u_{*} n^{1 / 3}

and the right-most point at distance

m a t h c a l V n^{2 / 9}

from

(c_{h} - u_{*}) n^{1 / 3}

. Here again,

m a t h c a l U

and

m a t h c a l V

are constants that depend only on

o m e g a

.

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