Hierarchical pinning model with site disorder: Disorder is marginally relevant

TL;DR

This paper investigates a hierarchical pinning model with site disorder, demonstrating that disorder is marginally relevant and affects the critical point, with a novel proof leveraging the model's inhomogeneous Green function.

Contribution

It establishes the marginal relevance of disorder in a hierarchical pinning model with site disorder and introduces a new proof technique exploiting the Green function's inhomogeneity.

Findings

Disorder is marginally relevant at a specific parameter value b.

The critical point of the disordered system differs from the annealed case.

The proof utilizes the inhomogeneous Green function of the model.

Abstract

We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case [6, 9], there exists a value of a parameter b (enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model. The proof goes beyond the technique used in [9] and it takes explicitly advantage of the inhomogeneous character of the Green function of the model.