Random walks and polymers in the presence of quenched disorder

TL;DR

This paper reviews recent advances in understanding how quenched disorder affects random walk and polymer models, highlighting phenomena like anomalous diffusion, localization, and critical behavior at phase transitions.

Contribution

It provides a comprehensive overview of disorder effects on random walks and polymers, including new insights into anomalous diffusion, localization, and non-self-averaging at critical points.

Findings

Disorder induces anomalous diffusion and localization in random walk models.

Disorder causes non-self-averaging at critical points in polymer models.

Analysis of wetting and DNA denaturation models reveals critical disorder effects.

Abstract

After a general introduction to the field, we describe some recent results concerning disorder effects on both `random walk models', where the random walk is a dynamical process generated by local transition rules, and on `polymer models', where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points : thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. We describe the results of this analysis for the bidimensional wetting and for the Poland-Scheraga model of DNA denaturation.