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

Nicolas Bouchot

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 $βω_z - h$ 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 $c_h = π^{2/3} h^{-1/3}$. 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 $u_* \in [0,c_h]$ is a constant that only depends on the disorder. This contrast 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 $\mathcal U n^{2/9}$ from $-u_* n^{1/3}$ and the right-most point at distance $\mathcal V n^{2/9}$ from $(c_h-u_*) n^{1/3}$. Here again, $\mathcal{U}$ and $\mathcal{V}$ are constants that depend only on $ω$.

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 . The law of the simple symmetric random walk up to time is modified by the exponential of the sum of sitting on its range, with~ and positive parameters. It is known that, at first order, the polymer folds itself to a segment of optimal size with . Here we study how disorder influences finer quantities. If the random variables are i.i.d.\ with a finite second moment, we prove that the left-most point of the range is located near , where 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 and . With an additional moment assumption, we are able to show that the left-most point of the range is at distance from and the right-most point at distance from . Here again, and are constants that depend only on .
Paper Structure (31 sections, 34 theorems, 251 equations, 2 figures)

This paper contains 31 sections, 34 theorems, 251 equations, 2 figures.

Table of Contents

  1. Introduction
  2. About the homogeneous setting
  3. First convergence result
  4. Second order convergence result
  5. Comments on the results, outline of the paper
  6. Expansion of the log-partition function.
  7. Coupling and almost sure results
  8. Local limit conjecture
  9. Related works
  10. The case of varying parameters $\beta,h$.
  11. Link with the random walk among Bernoulli obstacles.
  12. Link with the directed polymer model.
  13. Outline of the paper.
  14. Second order expansion and optimal position
  15. A rewriting of the partition function
  16. ...and 16 more sections

Key Result

Theorem 1.1

For all $h > 0$, define $c_h \vcentcolon= (\pi^2 h^{-1})^{1/3}$. Then, for any $h,\beta>0$, $\mathbb{P}$-almost surely we have the following convergence

Figures (2)

  • Figure 1: A typical trajectory under the polymer measure for a given $u_*$ and large $n$
  • Figure 2: Reflection of the trajectory $b \to (0,b]$ with respect to the horizontal line at $b$

Theorems & Definitions (64)

  • Theorem 1.1: berger2020one
  • Theorem 1.2: bouchot1
  • Theorem 1.3
  • Proposition 1.4
  • proof
  • Definition 1.1
  • Proposition 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • proof : Proof of Theorem \ref{['th-1/6']}-\ref{['eq:th-1/6-part']}
  • ...and 54 more