Multiplicity bound of Singular Spectrum for higher rank Anderson models

In this work, we prove a bound on multiplicity of the singular spectrum for certain class of Anderson Hamiltonians. The class of operator is $H^\omega=\Delta+\sum_{n\in\mathbb{Z}^d}\omega_n P_n$ on the Hilbert space $\ell^2(\mathbb{Z}^d)$, where $\Delta$ is discrete laplacian, $P_n$ are projection onto $\ell^2(\{x\in\mathbb{Z}^d:n_il_i<x_i\leq (n_i+1)l_i\})$ for some $l_1,\cdots,l_d\in\mathbb{N}$ and $\{\omega_n\}_n$ are i.i.d real bounded random variables following absolutely continuous distribution. We prove that the multiplicity of singular spectrum is bounded above by $2^d-d$ independent of $\{l_i\}_{i=1}^d$. When $l_i+1\not\in 2\mathbb{N}\cup3\mathbb{N}$ for all $i$ and $gcd(l_i+1,l_j+1)=1$ for $i\neq j$, we also prove that the singular spectrum is simple.