Some physical applications of random hierarchical matrices
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Some Physical Applications of Random Hierarchical Matrices V. A. Avetisova, A. Kh. Bikulova, O. A. Vasilyevb, c, S. K. Nechaevd, e, and A. V. Chertovichf a
Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, 119991 Russia b MaxPlanckInstitut für Metallforschung, D70569, Stuttgart, Germany c Institut für Theoretische und Angewandte Physik, Universität Stuttgart, D70569, Stuttgart, Germany d Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia e Université Paris Sud, 91405, Orsay Cedex, France f Moscow State University, Moscow, 119992 Russia email: [email protected] Received March 20, 2009
Abstract—The investigation of spectral properties of random blockhierarchical matrices as applied to dynamic and structural characteristics of complex hierarchical systems with disorder is proposed for the first time. Peculiarities of dynamics on random ultrametric energy landscapes are discussed and the statistical properties of scalefree and polyscale (depending on the topological characteristics under investigation) ran dom hierarchical networks (graphs) obtained by multiple mapping are considered. PACS numbers: 05.40.a, 87.15.hg DOI: 10.1134/S1063776109090155
1. INTRODUCTION The heightened interest in statistical properties of ensembles of random matrices that arose in the 1950s was primarily due to a number of problems in nuclear physics. In particular, it was found that random matri ces may serve as a simple model of neutron resonances of heavy nuclei, which correctly explains many of the observed statistical regularities in this field (the description of strongly excited states in terms of one particle models is an unsolvable problem). Soon it was found that the application of random matrices is not limited to only nuclear physics problems. Deep pene tration of random matrices in the physics of con densed media has led to a breakthrough in under standing of the behavior of conductivity in disordered mesoscopic systems. The applications of the theory of random matrices were subsequently extended to statis tical physics. Among the large number of publications in this field, we must specially mention recent works [1, 2] in which the statistics of limiting states of ensem bles of random matrices and their physical applica tions are considered. It should be recalled that a standard problem in the theory of random matrices is the calculation of the density of distribution of eigenvalues of random matri ces and the distribution of intervals between eigenval ues under the assumption that all matrix elements are independent random quantities assuming values in a certain preset ensemble (see, for example, [3]). In such a formulation, the theory of random matrices is used for describing a wide range of physical phenom ena; however, the theory in this form does not cover an
important class of complex systems that can be described using the concept of hierarchical (ultramet ric) organization [4, 5] of phase, dynamic, or
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