论文标题
Lévy矩阵的移动性边缘
Mobility Edge for Lévy Matrices
论文作者
论文摘要
Lévy矩阵是对称的随机矩阵，其输入分布位于$α$稳定定律的吸引力领域。对于$α<1 $，物理学文献的预测表明，高维lévy矩阵应在$ e _ {\ mathrm {mob {mob}} $的点显示以下相变。应将与$ eigenvalues相对应的特征向量（ - e _ {\ mathrm {mob {mob}}，e _ {\ mathrm {mob {mob}}）$应被定位，而与本间隔之外的eigenvalues相对应的特征向下。此外，$ e _ {\ mathrm {mob}} $由$λ（e，α）= 1 $的（可能是独特的）积极解决方案给出，其中$λ$是$ e $和$ a $的明确函数。 我们证明了有关高维莱维矩阵的以下结果。 （1）如果$λ（e，α）> 1 $，则将其特征向量接近$ e $。 （2）如果$ e $位于集合$ \ big \ {x：λ（x，α）<1 \ big \} $的连接组件中，含有$ \ pm \ infty $，则本地化$ e $ eigenvalues的特征向量。 （3）对于$α$，足够接近$ 0 $或$ 1 $，有一个唯一的阳性解决方案$ e = e = e _ {\ mathrm {mob}} $ to $λ（e，α）= 1 $，表明存在（唯一的）相位过渡。 （a）如果$α$接近$ 0 $，则$ e _ {\ mathrm {mob}} $大约缩放为$ | | \logα|^{ - 2/α} $。 （b）如果$α$接近$ 1 $，则$ e _ {\ mathrm {mob}} $ scales作为$（1-α）^{ - 1} $。 我们的证明是通过分析Lévy基质的局部弱极限来进行的，lévy基质的局部弱极限是由泊松加权无限树上的某些无限尺寸，重尾的操作员给出的。
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.
