Landscape approximation of the ground state eigenvalue for graphs and random hopping models

We consider the localization landscape function $u$ and ground state eigenvalue $\lambda$ for operators on graphs. We first show that the maximum of the landscape function is comparable to the reciprocal of the ground state eigenvalue if the operator satisfies certain semigroup kernel upper bounds. This implies general upper and lower bounds on the landscape product $\lambda\|u\|_\infty$ for several models, including the Anderson model and random hopping (bond-disordered) models, on graphs that are roughly isometric to $\mathbb{Z}^d$, as well as on some fractal-like graphs such as the Sierpinski gasket graph. Next, we specialize to a random hopping model on $\mathbb{Z}$, and show that as the size of the chain grows, the landscape product $\lambda\|u\|_\infty$ approaches $\pi^2/8$ for Bernoulli off-diagonal disorder, and has the same upper bound of $\pi^2/8$ for Uniform([0,1]) off-diagonal disorder. We also numerically study the random hopping model when the band width (hopping distance) is greater than one, and provide strong numerical evidence that a similar approximation holds for low-lying energies in the spectrum.