Eigenvalue statistics for random polymer models: Localization and delocalization

Abstract

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 (D. Damanik, R. Sims, and G. Stolz (2004) and S. De Bièvre and F. Germinet (2000)), and non-trivial transport for wave packets with initial states supported at an integer point (S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz (2003)). 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 non-trivial transport, refining the connection between non-trivial transport and the critical energies. Finally, we also prove that the transition in the unfolded LES is sharp at the critical energies.