Peculiar spectral statistics of ensembles of trees and starlike graphs
In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and pbranching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the \Lifshitz singularity" emerging in the onedimensional localization, while the spectral statistics of pbranching starlike graphs is less universal, being strongly dependent on p. The hierarchical structure of spectra of adjacency matrices is interpreted as sets of resonance frequencies, that emerge in ensembles of fully branched treelike systems, known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology, but has rather the numbertheoretic origin, re ecting the peculiarities of the rareevent statistics typical for onedimensional systems with a quenched structural disorder. The similarity of spectral densities of an individual dendrimer and of ensemble of linear chains with exponential distribution in lengths, demonstrates that dendrimers could be served as simple disorderless toy models of onedimensional systems with quenched disorder.
 Authors:

^{[1]};
^{[2]};
^{[3]};
^{[4]}
 Skolkovo Inst. of Science and Technology, Moscow (Russia)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Skolkovo Inst. of Science and Technology, Moscow (Russia)
 P.N. Lebedev Physical Inst. RAS, Moscow (Russia); Independent Univ. of Moscow (Russia). Poncelet Lab.
 National Research Univ. Higher School of Economics, Moscow (Russia); Russian Academy of Sciences (RAS), Moscow (Russian Federation)
 Publication Date:
 Report Number(s):
 LAUR1723905
Journal ID: ISSN 17425468
 Grant/Contract Number:
 AC5206NA25396
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Statistical Mechanics
 Additional Journal Information:
 Journal Volume: 2017; Journal Issue: 7; Journal ID: ISSN 17425468
 Publisher:
 IOP Publishing
 Research Org:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Mathematics; Graph spectra, rare event statistics
 OSTI Identifier:
 1375882
Kovaleva, V., Maximov, Yu, Nechaev, S., and Valba, O.. Peculiar spectral statistics of ensembles of trees and starlike graphs. United States: N. p.,
Web. doi:10.1088/17425468/aa7286.
Kovaleva, V., Maximov, Yu, Nechaev, S., & Valba, O.. Peculiar spectral statistics of ensembles of trees and starlike graphs. United States. doi:10.1088/17425468/aa7286.
Kovaleva, V., Maximov, Yu, Nechaev, S., and Valba, O.. 2017.
"Peculiar spectral statistics of ensembles of trees and starlike graphs". United States.
doi:10.1088/17425468/aa7286. https://www.osti.gov/servlets/purl/1375882.
@article{osti_1375882,
title = {Peculiar spectral statistics of ensembles of trees and starlike graphs},
author = {Kovaleva, V. and Maximov, Yu and Nechaev, S. and Valba, O.},
abstractNote = {In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and pbranching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the \Lifshitz singularity" emerging in the onedimensional localization, while the spectral statistics of pbranching starlike graphs is less universal, being strongly dependent on p. The hierarchical structure of spectra of adjacency matrices is interpreted as sets of resonance frequencies, that emerge in ensembles of fully branched treelike systems, known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology, but has rather the numbertheoretic origin, re ecting the peculiarities of the rareevent statistics typical for onedimensional systems with a quenched structural disorder. The similarity of spectral densities of an individual dendrimer and of ensemble of linear chains with exponential distribution in lengths, demonstrates that dendrimers could be served as simple disorderless toy models of onedimensional systems with quenched disorder.},
doi = {10.1088/17425468/aa7286},
journal = {Journal of Statistical Mechanics},
number = 7,
volume = 2017,
place = {United States},
year = {2017},
month = {7}
}