双曲方格上的无序 I：安德森退局域化与绝对连续谱

中文摘要 AI 辅助

我们研究双曲格上的安德森模型。该图是欧几里得格与每个顶点具有五个正方形的正则双曲方格的笛卡尔积。我们证明了在弱无序下，正测度的确定性集合上绝对连续谱的存在性和奇异谱的缺失，并且当单点密度在[-1,1]上有下界时，对于0<λ≤2也成立。截断柯西分布和受控扰动在区间上给出纯绝对连续谱。

英文摘要

We study the Anderson model on hyperbolic lattices. The graph is the Cartesian product of the Euclidean lattice with the regular hyperbolic square lattice having five squares at each vertex. We prove the existence of absolutely continuous spectrum and the absence of singular spectrum on a deterministic set of positive measure at weak disorder, and for $0<λ\le2$ when the single-site density is bounded below on $[-1,1]$. Truncated Cauchy distributions and controlled perturbations give purely absolutely continuous spectrum on intervals.

发表机构

• Bocconi University（博科尼大学）
• Northwestern University（西北大学）

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