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Title: Peculiar spectral statistics of ensembles of trees and star-like graphs

In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and p-branching 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 p-branching star-like 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 tree-like systems, known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, re ecting the peculiarities of the rare-event statistics typical for one-dimensional 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 disorder-less toy models of one-dimensional systems with quenched disorder.
Authors:
 [1] ;  [2] ;  [3] ;  [4]
  1. Skolkovo Inst. of Science and Technology, Moscow (Russia)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Skolkovo Inst. of Science and Technology, Moscow (Russia)
  3. P.N. Lebedev Physical Inst. RAS, Moscow (Russia); Independent Univ. of Moscow (Russia). Poncelet Lab.
  4. National Research Univ. Higher School of Economics, Moscow (Russia); Russian Academy of Sciences (RAS), Moscow (Russian Federation)
Publication Date:
Report Number(s):
LA-UR-17-23905
Journal ID: ISSN 1742-5468
Grant/Contract Number:
AC52-06NA25396
Type:
Accepted Manuscript
Journal Name:
Journal of Statistical Mechanics
Additional Journal Information:
Journal Volume: 2017; Journal Issue: 7; Journal ID: ISSN 1742-5468
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 star-like graphs. United States: N. p., Web. doi:10.1088/1742-5468/aa7286.
Kovaleva, V., Maximov, Yu, Nechaev, S., & Valba, O.. Peculiar spectral statistics of ensembles of trees and star-like graphs. United States. doi:10.1088/1742-5468/aa7286.
Kovaleva, V., Maximov, Yu, Nechaev, S., and Valba, O.. 2017. "Peculiar spectral statistics of ensembles of trees and star-like graphs". United States. doi:10.1088/1742-5468/aa7286. https://www.osti.gov/servlets/purl/1375882.
@article{osti_1375882,
title = {Peculiar spectral statistics of ensembles of trees and star-like 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 p-branching 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 p-branching star-like 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 tree-like systems, known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, re ecting the peculiarities of the rare-event statistics typical for one-dimensional 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 disorder-less toy models of one-dimensional systems with quenched disorder.},
doi = {10.1088/1742-5468/aa7286},
journal = {Journal of Statistical Mechanics},
number = 7,
volume = 2017,
place = {United States},
year = {2017},
month = {7}
}