Eigenvalue Statistics for higher rank Anderson model over Canopy tree

TL;DR

This paper studies the local eigenvalue statistics of a higher rank Anderson model on a canopy tree at large disorder, showing convergence of the eigenvalue process to a compound Poisson process.

Contribution

It introduces a new analysis of eigenvalue statistics for a non-rank-one Anderson model on canopy trees, demonstrating convergence to a compound Poisson process.

Findings

Eigenvalue-counting process converges to a compound Poisson process.

Results hold for large disorder and non-rank-one perturbations.

Extends understanding of eigenvalue statistics in complex tree structures.

Abstract

This work is focused on the local eigenvalue statistics for the Anderson tight binding model with non-rank-one perturbations over the canopy tree, at large disorder. On the Hilbert space $\ell^2(\mathcal{C})$, where $ \mathcal{C} $ is the canopy tree, the random operator we consider is $\Delta_{\mathcal{C}}+\sum_{y\in J}\omega_y P_y$, where $\Delta_{\mathcal{C}}$ is the adjacency operator over the tree, $\{\omega_y\}_{y\in J}$ are i.i.d real random variables following some absolutely continuous distribution having a bounded density with compact support, and $P_y$ are projections on $\ell^2(\{x\in\mathcal{C}: dist(y,x)< m_0 \& y\prec x\})$. For this operator, we show that, the eigenvalue-counting point process converges to compound Poisson process.