Two Pinning Models with Markov disorder

TL;DR

This paper investigates two disordered pinning models with Markovian inhomogeneities, deriving critical curves and phase diagrams, and highlights their relevance to DNA denaturation.

Contribution

It introduces novel analytical results for pinning models with Markov disorder, including explicit formulas for critical curves and phase diagrams.

Findings

Derived annealed critical curve using Perron-Frobenius eigenvalue.

Established the limit of quenched free energy for correlated disorder.

Connected the number of critical points to Markov chain states.

Abstract

Disordered pinning models deal with the (de)localization tran- sition of a polymer in interaction with a heterogeneous interface. In this paper, we focus on two models where the inhomogeneities at the interface are not independent but given by an irreducible Markov chain on a finite state space. In the first model, using Markov renewal tools, we give an expression for the annealed critical curve in terms of a Perron-Frobenius eigenvalue, and provide examples where exact computations are possible. In the second model, the transition matrix vary with the size of the system so that, roughly speaking, disorder is more and more correlated. In this case we are able to give the limit of the averaged quenched free energy, therefore providing the full phase diagram picture, and the number of critical points is related to the number of states of the Markov chain. We also mention that the question of pinning in correlated disorder appears in the context of DNA denaturation.