Concentration and fluctuation phenomena in the localized phase of the pinning model

TL;DR

The work analyzes the localized phase of a pinning model with i.i.d. site disorder under mild mgf-boundedness assumptions. It proves that the free energy $f(h)$ is Gevrey-3 in $(h_c,\infty)$ and deduces quenched concentration bounds and CLTs for the contact number $L_n$ with disorder-dependent centering, together with a CLT for the centering itself with nondegenerate variance $w(h)$. A key device is a detailed network of correlation estimates yielding exponential decay of correlations, which in turn controls the maximal excursion $M_n$ (scaling as $\log n/\mu(h)$) and ensures almost-sure uniformity in $h$ across compact sets. The results extend Giacomin–Toninelli (2006) by weakening disorder assumptions and providing explicit Gevrey regularity, a full quenched CLT for the contact number, and a rigorous analysis of the centering fluctuations, with potential implications for disorder smoothing and big-jump phenomena. The work also introduces a Birkhoff-sum framework to handle almost-sure disorder-averaged quantities and discusses the relation to Griffiths-like phenomena, while leaving Griffiths-type singularities as an open problem for the true model.

Abstract

We focus on the localized phase of pinning models with i.i.d. site disorder on which we assume only that the moment generating function is bounded in a neighborhood of the origin. We develop quantitative correlation functions estimates for local observables that entail quantitative $C^\infty$ estimates on the free energy density, showing in particular that its regularity class is at least Gevrey-3 in the whole localized phase. We then explain how a quenched concentration bound and the quenched Central Limit Theorem (CLT) on the number of the pinned sites, i.e., the 'contact number', can be extracted from the regularity estimates on the free energy: this identifies the thermal fluctuations of the contact number. But the centering sequence in the quenched CLT is random in the sense that it is disorder dependent: we show that the (disorder induced) fluctuations of the centering are on the same scale of the thermal fluctuations by establishing a CLT, with a non degenerate variance, also for the centering. For what concerns the correlation and $C^\infty$ estimates, our work substantially generalizes and expands the analysis in [Giacomin and F. L. Toninelli, Lat. Am. J. Probab. 1 (2006), 149-180] that dealt with pinning models with restrictive conditions on the disorder distributions and in which less explicit, non uniform bounds were obtained.

Theorems & Definitions (45)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 1.6
• Proposition 1.7
• Proposition 1.8
• Remark 1.9
• Lemma 2.1