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

TL;DR

This paper develops a variational framework for analyzing the quenched free energy of a one-dimensional directed charged polymer model, providing new formulas, detailed proofs, and corrections for prior results.

Contribution

It introduces a variational formula for the quenched free energy based on large deviation principles and refines existing results for the charged polymer model.

Findings

Derived a variational representation for the quenched free energy.

Provided detailed proofs supporting the existence of a freezing transition.

Made minor corrections to previous results on the undirected model.

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{\'e}tr{\'e}lis for the undirected model.