Ruelle-Perron-Frobenius operator approach to the annealed pinning model with Gaussian long-range correlated disorder

TL;DR

This paper investigates the annealed pinning model with Gaussian long-range correlated disorder, demonstrating that fast decay of correlations preserves the critical behavior of the homogeneous model using spectral analysis of Ruelle-Perron-Frobenius operators.

Contribution

It establishes conditions under which the annealed critical behavior matches the homogeneous case, employing spectral methods from thermodynamic formalism.

Findings

Critical behavior matches homogeneous model with fast decay of correlations

Exponential decay of correlations yields sharper results

Provides large-temperature asymptotics of the critical curve

Abstract

In this paper we study the pinning model with correlated Gaussian disorder. The presence of correlations makes the annealed model more involved than the usual homogeneous model, which is fully solvable. We prove however that if the disorder correlations decay fast enough then the annealed critical behaviour is the same as the homogeneous one. Our result is sharper if the decay is exponential. The approach we propose relies on the spectral properties of a transfer or Ruelle-Perron Frobenius operator related to the model. We use results on these operators that were obtained in the framework of the thermodynamic formalism for countable Markov shifts. We also provide large-temperature asymptotics of the annealed critical curve under weaker assumptions.