1,956 research outputs found
Phase diagram of the mean field model of simplicial gravity
We discuss the phase diagram of the balls in boxes model, with a varying
number of boxes. The model can be regarded as a mean-field model of simplicial
gravity. We analyse in detail the case of weights of the form , which correspond to the measure term introduced in the simplicial
quantum gravity simulations. The system has two phases~: {\em elongated} ({\em
fluid}) and {\em crumpled}. For the transition between
these two phases is first order, while for it is continuous.
The transition becomes softer when approaches unity and eventually
disappears at . We then generalise the discussion to an arbitrary set
of weights. Finally, we show that if one introduces an additional kinematic
bound on the average density of balls per box then a new {\em condensed} phase
appears in the phase diagram. It bears some similarity to the {\em crinkled}
phase of simplicial gravity discussed recently in models of gravity interacting
with matter fields.Comment: 15 pages, 5 figure
Solution of the Dyson--Schwinger equation on de Sitter background in IR limit
We propose an ansatz which solves the Dyson-Schwinger equation for the real
scalar fields in Poincare patch of de Sitter space in the IR limit. The
Dyson-Schwinger equation for this ansatz reduces to the kinetic equation, if
one considers scalar fields from the principal series. Solving the latter
equation we show that under the adiabatic switching on and then off the
coupling constant the Bunch-Davies vacuum relaxes in the future infinity to the
state with the flat Gibbons-Hawking density of out-Jost harmonics on top of the
corresponding de Sitter invariant out-vacuum.Comment: 20 pages, including 4 pages of Appendix. Acknowledgements correcte
A Random Matrix Approach to VARMA Processes
We apply random matrix theory to derive spectral density of large sample
covariance matrices generated by multivariate VMA(q), VAR(q) and VARMA(q1,q2)
processes. In particular, we consider a limit where the number of random
variables N and the number of consecutive time measurements T are large but the
ratio N/T is fixed. In this regime the underlying random matrices are
asymptotically equivalent to Free Random Variables (FRV). We apply the FRV
calculus to calculate the eigenvalue density of the sample covariance for
several VARMA-type processes. We explicitly solve the VARMA(1,1) case and
demonstrate a perfect agreement between the analytical result and the spectra
obtained by Monte Carlo simulations. The proposed method is purely algebraic
and can be easily generalized to q1>1 and q2>1.Comment: 16 pages, 6 figures, submitted to New Journal of Physic
Network Transitivity and Matrix Models
This paper is a step towards a systematic theory of the transitivity
(clustering) phenomenon in random networks. A static framework is used, with
adjacency matrix playing the role of the dynamical variable. Hence, our model
is a matrix model, where matrices are random, but their elements take values 0
and 1 only. Confusion present in some papers where earlier attempts to
incorporate transitivity in a similar framework have been made is hopefully
dissipated. Inspired by more conventional matrix models, new analytic
techniques to develop a static model with non-trivial clustering are
introduced. Computer simulations complete the analytic discussion.Comment: 11 pages, 7 eps figures, 2-column revtex format, print bug correcte
Network of inherent structures in spin glasses: scaling and scale-free distributions
The local minima (inherent structures) of a system and their associated
transition links give rise to a network. Here we consider the topological and
distance properties of such a network in the context of spin glasses. We use
steepest descent dynamics, determining for each disorder sample the transition
links appearing within a given barrier height. We find that differences between
linked inherent structures are typically associated with local clusters of
spins; we interpret this within a framework based on droplets in which the
characteristic ``length scale'' grows with the barrier height. We also consider
the network connectivity and the degrees of its nodes. Interestingly, for spin
glasses based on random graphs, the degree distribution of the network of
inherent structures exhibits a non-trivial scale-free tail.Comment: minor changes and references adde
Spectrum of the Product of Independent Random Gaussian Matrices
We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M
independent NxN Gaussian random matrices in the large-N limit is rotationally
symmetric in the complex plane and is given by a simple expression
rho(z,\bar{z}) = 1/(M\pi\sigma^2} |z|^{-2+2/M} for |z|<\sigma, and is zero for
|z|> \sigma. The parameter \sigma corresponds to the radius of the circular
support and is related to the amplitude of the Gaussian fluctuations. This form
of the eigenvalue density is highly universal. It is identical for products of
Gaussian Hermitian, non-Hermitian, real or complex random matrices. It does not
change even if the matrices in the product are taken from different Gaussian
ensembles. We present a self-contained derivation of this result using a planar
diagrammatic technique for Gaussian matrices. We also give a numerical evidence
suggesting that this result applies also to matrices whose elements are
independent, centered random variables with a finite variance.Comment: 16 pages, 6 figures, minor changes, some references adde
Maximal-entropy random walk unifies centrality measures
In this paper analogies between different (dis)similarity matrices are
derived. These matrices, which are connected to path enumeration and random
walks, are used in community detection methods or in computation of centrality
measures for complex networks. The focus is on a number of known centrality
measures, which inherit the connections established for similarity matrices.
These measures are based on the principal eigenvector of the adjacency matrix,
path enumeration, as well as on the stationary state, stochastic matrix or mean
first-passage times of a random walk. Particular attention is paid to the
maximal-entropy random walk, which serves as a very distinct alternative to the
ordinary random walk used in network analysis.
The various importance measures, defined both with the use of ordinary random
walk and the maximal-entropy random walk, are compared numerically on a set of
benchmark graphs. It is shown that groups of centrality measures defined with
the two random walks cluster into two separate families. In particular, the
group of centralities for the maximal-entropy random walk, connected to the
eigenvector centrality and path enumeration, is strongly distinct from all the
other measures and produces largely equivalent results.Comment: 7 pages, 2 figure
Evolution of scale-free random graphs: Potts model formulation
We study the bond percolation problem in random graphs of weighted
vertices, where each vertex has a prescribed weight and an edge can
connect vertices and with rate . The problem is solved by the
limit of the -state Potts model with inhomogeneous interactions for
all pairs of spins. We apply this approach to the static model having
so that the resulting graph is scale-free with
the degree exponent . The number of loops as well as the giant
cluster size and the mean cluster size are obtained in the thermodynamic limit
as a function of the edge density, and their associated critical exponents are
also obtained. Finite-size scaling behaviors are derived using the largest
cluster size in the critical regime, which is calculated from the cluster size
distribution, and checked against numerical simulation results. We find that
the process of forming the giant cluster is qualitatively different between the
cases of and . While for the former, the giant
cluster forms abruptly at the percolation transition, for the latter, however,
the formation of the giant cluster is gradual and the mean cluster size for
finite shows double peaks.Comment: 34 pages, 9 figures, elsart.cls, final version appeared in NP
Levy targeting and the principle of detailed balance
We investigate confined L\'{e}vy flights under premises of the principle of
detailed balance. The master equation admits a transformation to L\'{e}vy -
Schr\"{o}dinger semigroup dynamics (akin to a mapping of the Fokker-Planck
equation into the generalized diffusion equation). We solve a stochastic
targeting problem for arbitrary stability index of L\'{e}vy drivers:
given an invariant probability density function (pdf), specify the jump - type
dynamics for which this pdf is a long-time asymptotic target. Our
("-targeting") method is exemplified by Cauchy family and Gaussian target
pdfs. We solve the reverse engineering problem for so-called L\'{e}vy
oscillators: given a quadratic semigroup potential, find an asymptotic pdf for
the associated master equation for arbitrary
On the top eigenvalue of heavy-tailed random matrices
We study the statistics of the largest eigenvalue lambda_max of N x N random
matrices with unit variance, but power-law distributed entries, P(M_{ij})~
|M_{ij}|^{-1-mu}. When mu > 4, lambda_max converges to 2 with Tracy-Widom
fluctuations of order N^{-2/3}. When mu < 4, lambda_max is of order
N^{2/mu-1/2} and is governed by Fr\'echet statistics. The marginal case mu=4
provides a new class of limiting distribution that we compute explicitely. We
extend these results to sample covariance matrices, and show that extreme
events may cause the largest eigenvalue to significantly exceed the
Marcenko-Pastur edge. Connections with Directed Polymers are briefly discussed.Comment: 4 pages, 2 figure
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