# Eigenvalue statistics for random polymer models: Localization and delocalization

## TL;DR

This paper investigates the local eigenvalue statistics of one-dimensional random polymer models, revealing different statistical behaviors at critical energies versus other energies, and connecting these findings to localization and transport phenomena.

## Contribution

It proves that LES at critical energies follows a uniform clock process, while at other energies it is a Poisson process, advancing understanding of spectral statistics in random polymer models.

## Key findings

- LES at critical energies is a uniform clock process
- LES at non-critical energies is a Poisson point process
- Transition in LES is sharp at critical energies

## 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 [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.