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

Julien Poisat|arXiv (Cornell University)|Nov 20, 2012

Theoretical and Computational Physics27 references4 citations

TL;DR

This paper studies the annealed pinning model with Gaussian long-range correlated disorder using the Ruelle-Perron-Frobenius (RPF) operator framework. By analyzing the spectral properties of the transfer operator on countable Markov shifts, the authors prove that if disorder correlations decay sufficiently fast—exponentially or polynomially with exponent α > 0—the annealed critical behavior matches that of the homogeneous model, including the same critical exponent. The result establishes the robustness of the homogeneous phase transition under weakly correlated disorder.

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.

Motivation & Objective

• To understand the critical behavior of the annealed pinning model when disorder exhibits long-range Gaussian correlations.
• To determine under what conditions the annealed critical exponent matches that of the homogeneous (non-random) model.
• To extend the Ruelle-Perron-Frobenius operator approach beyond finite-range correlations to infinite-range, long-range correlated disorder.
• To establish large-temperature asymptotics of the annealed critical curve under weaker assumptions on correlation decay.

Proposed method

• Formalizes the annealed pinning model with Gaussian long-range correlated disorder using a renewal process and Hamiltonian with i.i.d. interarrival times.
• Derives the annealed partition function and free energy by averaging over disorder, leveraging Gaussian moment generating functions.
• Applies the Ruelle-Perron-Frobenius (RPF) operator to infinite symbolic dynamics on countable Markov shifts to analyze spectral properties.
• Uses results from thermodynamic formalism for countable Markov shifts to control the growth of transfer operator iterates and derive critical behavior.
• Establishes subadditivity and spectral gap estimates via the RPF operator to bound the free energy and critical curve.
• Applies perturbation theory to the RPF operator under small disorder strength, linking spectral data to critical exponent behavior.

Experimental results

Research questions

• RQ1 Does the annealed critical exponent of the pinning model with long-range correlated Gaussian disorder match that of the homogeneous model?
• RQ2 Under what decay conditions on the correlation structure of the disorder does the annealed critical behavior remain unchanged compared to the homogeneous case?
• RQ3 Can the Ruelle-Perron-Frobenius operator framework be extended from finite-range to infinite-range correlated disorder in pinning models?
• RQ4 What is the large-temperature (small β) asymptotic behavior of the annealed critical curve under weak correlation decay?
• RQ5 How do the spectral properties of the transfer operator relate to the phase transition in the annealed model?

Key findings

• If the Gaussian disorder correlations decay fast enough—specifically, with polynomial decay rate α > 0 or exponential decay—the annealed critical exponent matches that of the homogeneous model.
• For exponential decay, the result is sharper: the critical behavior is identical to the homogeneous case, with no correction in the critical exponent.
• The RPF operator approach allows rigorous control of the annealed free energy via spectral analysis on countable Markov shifts, extending prior results limited to finite-range correlations.
• Large-temperature asymptotics of the annealed critical curve are derived under weaker assumptions, including polynomial decay with α > 0.
• The critical curve satisfies h_c^a(β) ∼ −(β²/2)(1 + 2∑ₙ ρₙP(n∈τ)) as β↓0, under the moment condition ∑nK(n) < ∞.
• The upper bound on the critical curve holds for all β ≥ 0 when ∑n nK(n) < ∞, derived via Jensen’s inequality on the annealed partition function.