论文标题
图形和随机跳型模型的基态特征值的景观近似
Landscape approximation of the ground state eigenvalue for graphs and random hopping models
论文作者
论文摘要
我们考虑了用于图的运营商的本地化景观函数$ u $和基态特征值$λ$。我们首先表明，如果操作员满足某些半群核上限，则景观功能的最大值与基态特征值的倒数相当。 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.当带宽（跳距离）大于一个时，我们还会研究随机跳跃模型，并提供了强有力的数值证据，表明相似的近似值适用于光谱中的低洼能量。
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.
