We consider the localization landscape function u and ground state eigenvalue λ 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 $\lambda {\Vert u\Vert }_{\infty }$ 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 $\lambda {\Vert u\Vert }_{\infty }$ approaches ${\pi }^2/8$ for Bernoulli off-diagonal disorder, and has the same upper bound of ${\pi }^2/8$ for $\mathrm{Uniform}([0,1])$ 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.

我们考虑图上算子的定位景观函数u和基态特征值λ 。我们首先证明，如果算子满足某些半群核上限，则景观函数的最大值与基态特征值的倒数相当。这意味着景观产品的一般上限和下限$\lambda {\Vert 你\Vert }_{无穷大}$对于多种模型，包括安德森模型和随机跳跃（键无序）模型，在大致等距的图上${\mathbb{Z}}^d$，以及一些类似分形的图，例如谢尔宾斯基垫片图。接下来，我们专门研究随机跳跃模型$\mathbb{Z}$，并表明随着链条规模的增长，景观产品$\lambda {\Vert 你\Vert }_{无穷大}$方法${\pi }^2/8$对于伯努利非对角无序，并且具有相同的上限${\pi }^2/8$为了$\mathrm{制服}（[0,1]）$非对角线紊乱。我们还对带宽（跳跃距离）大于 1 时的随机跳跃模型进行了数值研究，并提供了强有力的数值证据，证明类似的近似适用于光谱中的低位能量。