227 research outputs found
Native ultrametricity of sparse random ensembles
We investigate the eigenvalue density in ensembles of large sparse Bernoulli
random matrices. We demonstrate that the fraction of linear subgraphs just
below the percolation threshold is about 95\% of all finite subgraphs, and the
distribution of linear chains is purely exponential. We analyze in detail the
spectral density of ensembles of linear subgraphs, discuss its ultrametric
nature and show that near the spectrum boundary, the tail of the spectral
density exhibits a Lifshitz singularity typical for Anderson localization. We
also discuss an intriguing connection of the spectral density to the Dedekind
-function. We conjecture that ultrametricity is inherit to complex
systems with extremal sparse statistics and argue that a number-theoretic
ultrametricity emerges in any rare-event statistics.Comment: 24 pages, 9 figure
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 on matrix elements, where
depends on hierarchy level as (). 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 , the tail of the
spectral density, , behaves as for and , while for
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
Spontaneous Symmetry Breaking and Phase Coexistence in Two-Color Networks
We have considered an equilibrium ensemble of large Erd\H{o}s-Renyi
topological random networks with fixed vertex degree and two types of vertices,
black and white, prepared randomly with the bond connection probability, .
The network energy is a sum of all unicolor triples (either black or white),
weighted with chemical potential of triples, . Minimizing the system
energy, we see for some positive formation of two predominantly unicolor
clusters, linked by a "string" of black-white bonds. We have
demonstrated that the system exhibits critical behavior manifested in emergence
of a wide plateau on the -curve, which is relevant to a spinodal
decomposition in 1st order phase transitions. In terms of a string theory, the
plateau formation can be interpreted as an entanglement between baby-universes
in 2D gravity. We have conjectured that observed classical phenomenon can be
considered as a toy model for the chiral condensate formation in quantum
chromodynamics.Comment: 9 pages, 4 figure
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
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
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 . 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 (where is some critical
time) a "lacunary" structure of the ultrametric space occurs with the
probability . 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
- …