Eigenvalue Statistics for higher rank Anderson model over Canopy tree

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 ${\mathrm{\ell }}^2(\mathcal{C})$, where $\mathcal{C}$ is the canopy tree, the random operator we consider is ${\mathrm{\Delta }}_{\mathcal{C}}+{\sum }_{y\in J}{\omega }_y{P}_y$, where ${\mathrm{\Delta }}_{\mathcal{C}}$ 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 ${\mathrm{\ell }}^2(\{x\in \mathcal{C}:dist(y,x)<m_0\mathrm{\&}y\prec x\})$. For this operator, we show that, the eigenvalue-counting point process converges to compound Poisson process.