Absolutely continuous spectrum for random Schrödinger operators on the Bethe strip

The Bethe strip of width m is the cartesian product $mathbb {B}times lbrace 1,ldots ,mrbrace$, where $mathbb {B}$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have “extended states” for small disorder. More precisely, we consider Anderson‐like Hamiltonians $H_lambda =frac{1}{2} Delta otimes 1 + 1 otimes A,+,lambda mathcal {V}$ on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, $mathcal {V}$ is a random matrix potential, and λ is the disorder parameter. Given any closed interval $Isubset big (!-!sqrt{K}+a_{{rm max}},sqrt{K}+a_{rm {min}}big )$, where a_min and a_max are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator H_λ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.