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.
From: Marco Zamparo [view email]
[v1]
Mon, 27 May 2024 09:38:50 UTC (48 KB)
[v2]
Fri, 4 Apr 2025 10:00:20 UTC (50 KB)
[v3]
Wed, 26 Aug 2026 10:54:01 UTC (82 KB)
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