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

Abstract

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.

Figures (9)

• Figure 2: We fix $N=10^4$, and run 100 random realizations with Bernoulli disorder in the 1D nearest neighbor hopping model \ref{['eq:HW1']}, in which each $a_j,(j=2,3,\cdots,N)$ is chosen as 1 with probability $1/2$ and 0 with probability $1/2$. Left: $\lambda_N$ and $\|u_N\|_\infty$ of $H_N$; Right: the associated $\lambda_N \|u_N\|_\infty$.
• Figure 3: The dependence of the product $\lambda_N \|u_N\|_\infty$ on $N$, for the 1D nearest neighbor hopping model \ref{['eq:HW1']} with Bernoulli disorder ($p=1/2$). To demonstrate the convergence to $\dfrac{\pi^2}{8}$ as $N\rightarrow\infty$, we plot the behavior of $\lambda_N \|u_N\|_\infty$ over a series of $N$. Specifically, we vary $N$ from 32 to $2^{24}=16\,777\,216$, and plot the landscape product for just a single random realization for each $N$.
• Figure 4: The behavior of $\lambda_N \|u_N\|_\infty$ by 500 random realizations, in which $N=10^4$, $a_j,(j=2,3,\cdots,N)$ are uniformly random numbers from [0,1].
• Figure 5: A graph $\Gamma=(\mathbb{Z},E)$ consists of vertices $\mathbb{Z}$ and edges $E=\{(i,j)\in \mathbb{Z}^2: 0<|i-j|\le 2\}$. We consider the subset $\{1,\cdots,7\}$ and the Dirichlet (sub-graph) Laplacian $-\Delta_{7,2}$ as in \ref{['eq:Lap72']}.
• Figure 6: An illustration showing the three subdivisions in \ref{['eqn:gaussian-poisson']}.

Theorems & Definitions (54)

• Remark 1.1
• Proposition 1
• Theorem 1: landscape product for graphs
• Remark 1.2
• Corollary 1: landscape product for band matrix model
• Remark 1.3
• Theorem 2: landscape product for 1D random hopping model
• Conjecture 1
• Theorem 3: landscape product for free band Laplacian
• Conjecture 2