Spectral statistics for unitary transfer matrices of binary graphs
Spectral statistics for unitary transfer matrices of binary graphs
复制标题
二元图的酉传递矩阵的谱统计
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
G. Tanner
中科院分区：
文献类型：
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作者：
G. Tanner
量子图最近被引入作为模型系统来研究具有混沌经典极限的线性波问题的谱统计。本文通过考虑具有酉转移矩阵的任意有向图来推广这一方法。当对周期轨道简并类进行对角线求和时，发现对形状因子的贡献呈指数增长。更详细地研究了一类特殊的图，即所谓的二值图。对于这些，周期轨道对相互关联的条件（包括由于传递矩阵的一致性而产生的关联）可以明确地给出。在一些低维情况下，利用组合技术可以对相关周期轨道对对形状因子的贡献进行求和。随着二值图顶点数的增加，其结果逐渐收敛于随机矩阵。
Quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits. It is proposed here to generalize this approach by considering arbitrary, directed graphs with unitary transfer matrices. An exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes. A special class of graphs, so-called binary graphs, is studied in more detail. For these, the conditions for periodic orbit pairs to be correlated (including correlations due to the unitarity of the transfer matrix) can be given explicitly. Using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low-dimensional cases. Gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs.