论文标题
在安德森过渡附近的非铁人对称随机矩阵的光谱刚度
Spectral rigidity of non-Hermitian symmetric random matrices near Anderson transition
论文作者
论文摘要
我们研究了Tzortzakis，Makris和Economou（TME）建议的Anderson模型的非热类似物的光谱刚度。这是一个$ l \ times l \ times l $紧密绑定的立方晶格，现场能量的真实和虚构部分都是独立的随机变量，均匀分布在$ -W/2 $和$ W/2 $之间。 TME模型可用于描述随机激光。在最近的一篇论文中，我们证明了该模型在$ W = w_c \ simeq 6 $中具有三维的安德森过渡。在这里，我们从数字上对tme $ l \ times l \ times l $立方晶格矩阵进行计算，并计算其复杂平面磁盘中特征值的数字差异。我们表明，在金属方面，安德森过渡的$ w <6 $，复杂的特征值彼此相互排斥，就像在复杂的ginibre合奏中一样，仅在包含$ n_c（l，w）$ eigenvalues的磁盘中。我们发现$ n_c（l，w）$与$ l $成比例，随着$ W $的减少，与能量水平的数量相似，$ n_c $ $ n_c $在Anderson型号的无能带中。
We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris and Economou (TME). This is a $L\times L \times L$ tightly bound cubic lattice, where both real and imaginary parts of on-site energies are independent random variables uniformly distributed between $-W/2$ and $W/2$. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at $W= W_c \simeq 6$ in three dimension. Here we numerically diagonalize TME $L \times L \times L$ cubic lattice matrices and calculate the number variance of eigenvalues in a disk of their complex plane. We show that on the metallic side $W < 6$ of the Anderson transition, complex eigenvalues repel each other as strongly as in the complex Ginibre ensemble only in a disk containing $N_c(L,W)$ eigenvalues. We find that $N_c(L,W)$ is proportional to $L$ and grows with decreasing $W$ similarly to the number of energy levels $N_c$ in the Thouless energy band of the Anderson model.
