On scale-free and poly-scale behaviors of random hierarchical network

In this paper the question about statistical properties of block--hierarchical random matrices is raised for the first time in connection with structural characteristics of random hierarchical networks obtained by mipmapping procedure. In particular, we compute numerically the spectral density of large random adjacency matrices defined by a hierarchy of the Bernoulli distributions $\{q_1,q_2,...\}$ on matrix elements, where $q_{\gamma}$ depends on hierarchy level $\gamma$ as $q_{\gamma}=p^{-\mu \gamma}$ ($\mu>0$). For the spectral density we clearly see the free--scale behavior. We show also that for the Gaussian distributions on matrix elements with zero mean and variances $\sigma_{\gamma}=p^{-\nu \gamma}$, the tail of the spectral density, $\rho_G(\lambda)$, behaves as $\rho_G(\lambda) \sim |\lambda|^{-(2-\nu)/(1-\nu)}$ for $|\lambda|\to\infty$ and $0<\nu<1$, while for $\nu\ge 1$ the power--law behavior is terminated. We also find that the vertex degree distribution of such hierarchical networks has a poly--scale fractal behavior extended to a very broad range of scales.Comment: 11 pages, 6 figures (paper is substantially revised

First Passage Time Distribution and Number of Returns for Ultrametric Random Walk

In this paper, we consider a homogeneous Markov process \xi(t;\omega) on an ultrametric space Q_p, with distribution density f(x,t), x in Q_p, t in R_+, satisfying the ultrametric diffusion equation df(x,t)/dt =-Df(x,t). We construct and examine a random variable \tau (\omega) that has the meaning the first passage times. Also, we obtain a formula for the mean number of returns on the interval (0,t] and give its asymptotic estimates for large t.Comment: 20 page

A p-Adic Model of DNA Sequence and Genetic Code

Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and genetic code. Central role in our investigation plays an ultrametric p-adic information space which basic elements are nucleotides, codons and genes. We show that a 5-adic model is appropriate for DNA sequence. This 5-adic model, combined with 2-adic distance, is also suitable for genetic code and for a more advanced employment in genomics. We find that genetic code degeneracy is related to the p-adic distance between codons.Comment: 13 pages, 2 table

Non-Degenerate Ultrametric Diffusion

General non-degenerate p-adic operators of ultrametric diffusion are introduced. Bases of eigenvectors for the introduced operators are constructed and the corresponding eigenvalues are computed. Properties of the corresponding dynamics (i.e. of the ultrametric diffusion) are investigated.Comment: 19 pages, 3 figure

Random Hierarchical Matrices: Spectral Properties and Relation to Polymers on Disordered Trees

We study the statistical and dynamic properties of the systems characterized by an ultrametric space of states and translationary non-invariant symmetric transition matrices of the Parisi type subjected to "locally constant" randomization. Using the explicit expression for eigenvalues of such matrices, we compute the spectral density for the Gaussian distribution of matrix elements. We also compute the averaged "survival probability" (SP) having sense of the probability to find a system in the initial state by time $t$. Using the similarity between the averaged SP for locally constant randomized Parisi matrices and the partition function of directed polymers on disordered trees, we show that for times $t>t_{\rm cr}$ (where $t_{\rm cr}$ is some critical time) a "lacunary" structure of the ultrametric space occurs with the probability $1-{\rm const}/t$. This means that the escape from some bounded areas of the ultrametric space of states is locked and the kinetics is confined in these areas for infinitely long time.Comment: 7 pages, 2 figures (the paper is essentially reworked

Application of p-adic analysis to models of spontaneous breaking of the replica symmetry

Methods of p-adic analysis are applied to the investigation of the spontaneous symmetry breaking in the models of spin glasses. A p-adic expression for the replica matrix is given and moreover the replica matrix in the models of spontaneous breaking of the replica symmetry in the simplest case is expressed in the form of the Vladimirov operator of p-adic fractional differentiation. Also the model of hierarchical diffusion (that was proposed to describe relaxation of spin glasses) investigated using p-adic analysis.Comment: Latex, 8 page