One dimensional pinning models have been widely studied in the physical and mathematical literature, also in presence of disorder. Roughly speaking, they undergo a transition between a delocalized phase and a localized one. In mathematical terms these models are obtained by modifying the distribution of a discrete renewal process via a Boltzmann factor with an energy that contains only one body potentials. For some more complex models, notably pinning models based on higher dimensional renewals, it has been shown that other phases may be present. We study a generalization of the one dimensional pinning model in which the energy may depend in a nonlinear way on the contact fraction: this class of models contains the circular DNA case considered in the bio-physics literature. We give a full solution of this generalized pinning model in absence of disorder and show that another transition appears. In fact the systems may display up to three different regimes: delocalization, partial localization and full localization. What happens in the partially localized regime can be explained in terms of the "big-jump" phenomenon for sums of heavy tail random variables under conditioning. We then show that disorder completely smears this second transition and we are back to the delocalization versus localization scenario. In fact we show that the disorder, even if arbitrarily weak, is incompatible with the presence of a big-jump.

中文翻译：

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一类广义钉扎模型的定位、大跳跃机制和无序的影响

一维钉扎模型在物理和数学文献中得到了广泛的研究，也存在无序的情况。粗略地说，它们经历了离域阶段和局部阶段之间的过渡。在数学方面，这些模型是通过使用仅包含一个身体电位的能量的 Boltzmann 因子修改离散更新过程的分布来获得的。对于一些更复杂的模型，特别是基于更高维度更新的固定模型，已经表明可能存在其他阶段。我们研究了一维钉扎模型的推广，其中能量可能以非线性方式取决于接触分数：此类模型包含生物物理学文献中考虑的圆形 DNA 情况。我们在没有无序的情况下给出了这个广义钉扎模型的完整解决方案，并表明出现了另一个转变。事实上，系统最多可以显示三种不同的状态：离域、部分本地化和完全本地化。在部分局部化的情况下发生的事情可以用条件下重尾随机变量总和的“大跳跃”现象来解释。然后我们证明无序完全抹黑了第二次转变，我们又回到了离域与定位的场景。事实上，我们证明了这种无序，即使是任意弱的，也与大跳跃的存在不相容。在部分局部化的情况下发生的事情可以用条件下重尾随机变量总和的“大跳跃”现象来解释。然后我们证明无序完全抹黑了第二次转变，我们又回到了离域与定位的场景。事实上，我们证明了这种无序，即使是任意弱的，也与大跳跃的存在不相容。在部分局部化的情况下发生的事情可以用条件下重尾随机变量总和的“大跳跃”现象来解释。然后我们证明无序完全抹黑了第二次转变，我们又回到了离域与定位的场景。事实上，我们证明了这种无序，即使是任意弱的，也与大跳跃的存在不相容。