Moments of the inverse participation ratio for the Laplacian on finite regular graphs
Moments of the inverse participation ratio for the Laplacian on finite regular graphs
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DOI：
10.1088/1751-8121/aaebb2
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发表时间：
2015-06
期刊：
影响因子：
--
通讯作者：
Timothy B. P. Clark;A. Del Maestro
中科院分区：
文献类型：
--
作者：
Timothy B. P. Clark;A. Del Maestro
本文研究了z度n阶有限随机正则图的Laplacian特征向量的逆参与比的一阶矩和二阶矩。通过精确对角化一个大的z-正则图集，我们发现，当n变大时，每个图上的逆参与率的平均值，当在一个大的图系综上平均时，接近数值3。这个普适数被理解为对应于IPR的四次多项式在适当维超球面上的平均值的大n极限。对于一个大的，但不是详尽的合奏图，平均方差的所有图形拉普拉斯特征向量的逆参与比偏离其连续超球平均值，由于大的图形到图形的波动，所产生的高度本地化的模式的存在。
We investigate the first and second moments of the inverse participation ratio (IPR) for all eigenvectors of the Laplacian on finite random regular graphs with n vertices and degree z. By exactly diagonalizing a large set of z-regular graphs, we find that as n becomes large, the mean of the inverse participation ratio on each graph, when averaged over a large ensemble of graphs, approaches the numerical value 3. This universal number is understood as the large-n limit of the average of the quartic polynomial corresponding to the IPR over an appropriate -dimensional hypersphere of . For a large, but not exhaustive ensemble of graphs, the mean variance of the inverse participation ratio for all graph Laplacian eigenvectors deviates from its continuous hypersphere average due to large graph-to-graph fluctuations that arise from the existence of highly localized modes.