Lévy矩阵的迁移率边缘
Mobility Edge for Lévy Matrices
AI总结：
本文证明了高维Lévy矩阵在迁移率边缘$E_{\mathrm{mob}}$处的相变现象，确立了特征向量的局域化与退局域化条件，并给出了$E_{\mathrm{mob}}$在$α$接近0或1时的渐近标度。
摘要：
Lévy矩阵是其元素分布处于$α$-稳定律吸引域内的对称随机矩阵。对于$α< 1$，物理学文献的预测表明，高维Lévy矩阵应在点$E_{\mathrm{mob}}$处表现出以下相变。对应于$(-E_{\mathrm{mob}},E_{\mathrm{mob}})$内特征值的特征向量应是退局域化的，而对应于该区间外特征值的特征向量应是局域化的。此外，$E_{\mathrm{mob}}$由方程$λ(E,α) =1$的（推测唯一的）正解给出，其中$λ$是关于$E$和$α$的显式函数。我们证明了关于高维Lévy矩阵的以下结论。(1) 若$λ(E,α) > 1$，则特征值接近$E$的特征向量是退局域化的。(2) 若$E$位于集合$\big\{ x : λ(x,α) < 1 \big\}$包含$\pm \infty$的连通分量中，则特征值接近$E$的特征向量是局域化的。(3) 对于$α$充分接近$0$或$1$的情况，方程$λ(E,α) = 1$存在唯一的正解$E = E_{\mathrm{mob}}$，证明了（唯一）相变的存在性。(a) 若$α$接近$0$，则$E_{\mathrm{mob}}$的标度近似为$|\log α|^{-2/α}$。(b) 若$α$接近$1$，则$E_{\mathrm{mob}}$的标度为$(1-α)^{-1}$。我们的证明通过分析Lévy矩阵的局部弱极限来完成，该极限由Poisson加权无穷树上的某个无穷维重尾算子给出。
英文摘要：
Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an $α$-stable law. For $α< 1$, predictions from the physics literature suggest that high-dimensional Lévy matrices should display the following phase transition at a point $E_{\mathrm{mob}}$. Eigenvectors corresponding to eigenvalues in $(-E_{\mathrm{mob}},E_{\mathrm{mob}})$ should be delocalized, while eigenvectors corresponding to eigenvalues outside of this interval should be localized. Further, $E_{\mathrm{mob}}$ is given by the (presumably unique) positive solution to $λ(E,α) =1$, where $λ$ is an explicit function of $E$ and $α$. We prove the following results about high-dimensional Lévy matrices. (1) If $λ(E,α) > 1$ then eigenvectors with eigenvalues near $E$ are delocalized. (2) If $E$ is in the connected components of the set $\big\{ x : λ(x,α) < 1 \big\}$ containing $\pm \infty$, then eigenvectors with eigenvalues near $E$ are localized. (3) For $α$ sufficiently near $0$ or $1$, there is a unique positive solution $E = E_{\mathrm{mob}}$ to $λ(E,α) = 1$, demonstrating the existence of a (unique) phase transition. (a) If $α$ is close to $0$, then $E_{\mathrm{mob}}$ scales approximately as $|\log α|^{-2/α}$. (b) If $α$ is close to $1$, then $E_{\mathrm{mob}}$ scales as $(1-α)^{-1}$. Our proofs proceed through an analysis of the local weak limit of a Lévy matrix, given by a certain infinite-dimensional, heavy-tailed operator on the Poisson weighted infinite tree.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
