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Variational representation and estimates for the free energy of a quenched charged polymer model

Julien Poisat

TL;DR

This work analyzes a one-dimensional directed charged polymer with quenched disorder, focusing on the quenched free energy and its variational representation. Using a quenched large deviation principle for words, the authors derive a variational formula $\bar{F}_{\rm que}(\beta)=\inf\{f: J(\beta,f)>0\}$ and establish existence and self-averaging of the free energy, linking the problem to renewal-process frameworks akin to pinning and copolymer models. They provide explicit high-temperature lower bounds and annealed upper bounds, and obtain low-temperature estimates for Gaussian and binary charge distributions, while also delivering corrections to results in the undirected model. The approach unifies disordered polymer analysis with renewal-LDP techniques, enabling sharper bounds and deeper insight into the freezing/collapse phenomena. These results advance rigorous understanding of how quenched disorder governs polymer conformations and transitions in one dimension, with potential implications for related disordered systems.

Abstract

Random walks with a disordered self-interaction potential may be used to model charged polymers. In this paper we consider a one-dimensional and directed version of the charged polymer model that was introduced by Derrida, Griffiths and Higgs. We prove new results for the associated quenched free energy, including a variational formula based on a quenched large deviation principle established by Birkner, Greven and den Hollander. We also take the occasion to (i) provide detailed proofs for state-of-the-art results pointing towards the existence of a freezing transition and (ii) proceed with minor corrections for two results previously obtained by the present author with Caravenna, den Hollander and P{é}tr{é}lis for the undirected model.

Variational representation and estimates for the free energy of a quenched charged polymer model

TL;DR

This work analyzes a one-dimensional directed charged polymer with quenched disorder, focusing on the quenched free energy and its variational representation. Using a quenched large deviation principle for words, the authors derive a variational formula and establish existence and self-averaging of the free energy, linking the problem to renewal-process frameworks akin to pinning and copolymer models. They provide explicit high-temperature lower bounds and annealed upper bounds, and obtain low-temperature estimates for Gaussian and binary charge distributions, while also delivering corrections to results in the undirected model. The approach unifies disordered polymer analysis with renewal-LDP techniques, enabling sharper bounds and deeper insight into the freezing/collapse phenomena. These results advance rigorous understanding of how quenched disorder governs polymer conformations and transitions in one dimension, with potential implications for related disordered systems.

Abstract

Random walks with a disordered self-interaction potential may be used to model charged polymers. In this paper we consider a one-dimensional and directed version of the charged polymer model that was introduced by Derrida, Griffiths and Higgs. We prove new results for the associated quenched free energy, including a variational formula based on a quenched large deviation principle established by Birkner, Greven and den Hollander. We also take the occasion to (i) provide detailed proofs for state-of-the-art results pointing towards the existence of a freezing transition and (ii) proceed with minor corrections for two results previously obtained by the present author with Caravenna, den Hollander and P{é}tr{é}lis for the undirected model.

Paper Structure

This paper contains 21 sections, 22 theorems, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Derrida-Griffiths-Higgs directed model
  3. Definition of the model
  4. Original convention
  5. Alternative convention
  6. Renewal times
  7. Predictions and state of the art
  8. Recursion relation
  9. The high temperature regime
  10. The low temperature regime
  11. Existence and variational representation of the free energy
  12. Existence and self-averaging
  13. The LDP for words in a quenched random letter sequence
  14. High-temperature estimate on the free energy
  15. A lower bound
  16. ...and 6 more sections

Key Result

Proposition 2.3

For every $1\le i< n$ and every charge sequence $\omega$, we have

Figures (2)

  • Figure 1: Quenched finite-volume free energy $(1/n) \log \bar{Z}^{\beta,\omega}_n$ versus inverse temperature $\beta$ for centered $\pm 1$ i.id. charges $\omega$ and polymer size $n=1000$ as defined in \ref{['eq:bar-pf']}. The high-temperature lower bound ($-\beta$) and the low-temperature lower bound from \ref{['eq:quenched-low-temp-LB']} respectively appear in blue and red.
  • Figure 2: Averaged quenched finite-volume free energy $(1/n) {\mathbb E} \log \bar{Z}^{\beta,\omega}_n$ versus inverse temperature $\beta$ for centered $\pm 1$ i.id. charges $\omega$ and polymer size $n=1000$ as defined in \ref{['eq:bar-pf']}. We have used $100$ samples to approximate the average over the charge distribution. The lower bound stated in Proposition \ref{['pr:prop-fe']} is shown in red.

Theorems & Definitions (48)

  • Conjecture 2.1
  • Remark 2.2
  • Proposition 2.3: DGH DeGrHi92
  • proof : Proof of Proposition \ref{['pr:DGH']}
  • Proposition 2.4: DGH DeGrHi92
  • Lemma 2.5: DGH DeGrHi92
  • proof : Proof of Lemma \ref{['lem:DGH']}
  • proof : Proof of Proposition \ref{['pr:DGH2']}
  • Remark 2.6
  • Proposition 2.7
  • ...and 38 more