标题： 随机聚合物模型的特征值统计：局域化与非局域化 显示英文标题

标题： Eigenvalue statistics for random polymer models: Localization and delocalization

摘要： 我们研究与一维格子随机聚合物模型相关的局部特征值统计（LES）。 我们考虑由两种聚合物构成的模型。 每种聚合物是一个有限区间的格点，具有有限势。 这些聚合物沿$\mathbb{Z}$按照伯努利分布分布。 这些模型的确定性谱是密集的纯点谱，并且已知包含有限多个临界能量。 在本文中，我们证明在这些临界能量处的 LES 由均匀钟过程描述，而在确定性谱中任何其他能量处的展开特征值的 LES 是泊松点过程。 这些结果增强了我们对这些模型的理解，这些模型在避开临界能量的任何能量区间内表现出动力学局域化 [Damanik, Sims, Stolz] [De Bievre, Germinet]，以及在初始状态支持于整数点的波包中表现出非平凡输运 [Jitomirskaya, Schultz-Baldes, Stolz]。 我们表明，这些初始状态在包含所有临界能量的能量区间的谱子空间上的投影表现出非平凡输运，从而细化了非平凡输运与临界能量之间的联系。 最后，我们还证明在展开的 LES 中，临界能量处的转变是尖锐的。 显示英文摘要

摘要： We study the local eigenvalue statistics (LES) associated with one-dimensional lattice models of random polymers. We consider models constructed from two polymers. Each polymer is a finite interval of lattice points with a finite potential. These polymers are distributed along $\mathbb{Z}$ according to a Bernoulli distribution. The deterministic spectrum for these models is dense pure point, and is known to contain finitely-many critical energies. In this paper, we prove that the LES centered at these critical energies is described by a uniform clock process, and that the LES for the unfolded eigenvalues, centered at any other energy in the deterministic spectrum, is a Poisson point process. These results add to our understanding of these models that exhibit dynamical localization in any energy interval avoiding the critical energies [Damanik, Sims, Stolz] [De Bievre, Germinet], and nontrivial transport for wave packets with initial states supported at an integer point [Jitomirskaya, Schultz-Baldes, Stolz]. We show that the projection of these initial states onto spectral subspaces associated with any energy interval that contains all of the critical energies exhibit nontrivial transport, refining the connection between nontrivial transport and the critical energies. Finally, we also prove that the transition in the unfolded LES is sharp at the critical energies.