图与随机跳跃模型的基态特征值景观近似
Landscape approximation of the ground state eigenvalue for graphs and random hopping models
AI总结：
本文研究了图上算子的局域化景观函数与基态特征值的关系，证明了在满足半群核上界时景观函数最大值与基态特征值倒数可比，并在多种无序模型上推导了景观乘积的上下界，同时针对一维随机跳跃模型给出了精确的渐近极限与数值证据。
摘要：
我们考虑图上算子的局域化景观函数 $u$ 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界，则景观函数的最大值与基态特征值的倒数是可比的。这为景观乘积 $λ\\|u\\|_\infty$ 提供了广义的上下界，适用于多个模型，包括 Anderson 模型和随机跳跃（键无序）模型，这些模型位于与 $\mathbb{Z}^d$ 粗糙等距的图上，以及一些类分形图（如 Sierpinski gasket 图）上。接着，我们专门研究 $\mathbb{Z}$ 上的随机跳跃模型，并证明随着链尺寸的增长，对于 Bernoulli 非对角无序，景观乘积 $λ\\|u\\|_\infty$ 趋近于 $π^2/8$；对于 Uniform([0,1]) 非对角无序，具有相同的 $π^2/8$ 上界。我们还数值研究了带宽（跳跃距离）大于 1 的随机跳跃模型，并提供了强有力的数值证据，表明类似的近似对谱中的低能态成立。
英文摘要：
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 $λ\|u\|_\infty$ 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 $λ\|u\|_\infty$ approaches $π^2/8$ for Bernoulli off-diagonal disorder, and has the same upper bound of $π^2/8$ 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.
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