Hierarchical pinning model: low disorder relevance in the $b=s$ case

TL;DR

This paper investigates a hierarchical pinning model and finds that when parameters b and s are equal, disorder has a negligible effect on the phase transition, unlike in other parameter cases.

Contribution

It demonstrates that in the case b=s, disorder is weakly relevant, showing quenched and annealed critical points coincide, which contrasts with previous cases.

Findings

Disorder is weakly relevant when b=s.

Quenched and annealed critical points coincide in this case.

Contrasts with the case where b ≠ s.

Abstract

We consider a hierarchical pinning model introduced by B.Derrida, V.Hakim and J.Vannimenus which undergoes a localization/delocalization phase transition. This model depends on two parameters $b$ and $s$. We show that in the particular case where $b=s$, the disorder is weakly relevant, in the sense that at any given temperature, the quenched and the annealed critical points coincide. This is in contrast with the case where $b \neq s$.