Jak\v{s}i\'{c}-Last Theorem for Higher Rank Perturbations

TL;DR

This paper extends the Jakšić-Last theorem to higher rank perturbations in the generalized Anderson model, demonstrating the equivalence of certain trace measures across models including dimers and polymers.

Contribution

It proves a Jakšić-Last type theorem for higher rank perturbations in the Anderson model, covering models like dimers and polymers.

Findings

Trace measures are equivalent for almost every realization.

The theorem applies to models with finite-rank projections.

Includes applications to dimer and polymer models.

Abstract

We consider the generalized Anderson Model $\Delta+\sum_{n\in\mathcal{N}}\omega_n P_n$, where $\mathcal{N}$ is a countable set, $\{\omega_n\}_{n\in\mathcal{N}}$ are i.i.d random variables and $P_n$ are rank $N<\infty$ projections. For these models we prove theorem analogous to that of Jak\v{s}i\'{c}-Last on the equivalence of the trace measure $\sigma_n(\cdot)=tr(P_nE_{H^\omega}(\cdot)P_n)$ for $n\in\mathcal{N}$ a.e $\omega$. Our model covers the dimer and polymer models.