510 research outputs found
Collapse of 4D random geometries
We extend the analysis of the Backgammon model to an ensemble with a fixed
number of balls and a fluctuating number of boxes. In this ensemble the model
exhibits a first order phase transition analogous to the one in higher
dimensional simplicial gravity. The transition relies on a kinematic
condensation and reflects a crisis of the integration measure which is probably
a part of the more general problem with the measure for functional integration
over higher (d>2) dimensional Riemannian structures.Comment: 7 pages, Latex2e, 2 figures (.eps
Correlation functions and critical behaviour on fluctuating geometries
We study the two-point correlation function in the model of branched polymers
and its relation to the critical behaviour of the model. We show that the
correlation function has a universal scaling form in the generic phase with the
only scale given by the size of the polymer. We show that the origin of the
singularity of the free energy at the critical point is different from that in
the standard statistical models. The transition is related to the change of the
dimensionality of the system.Comment: 10 Pages, Latex2e, uses elsart.cls, 1 figure include
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
Causal and homogeneous networks
Growing networks have a causal structure. We show that the causality strongly
influences the scaling and geometrical properties of the network. In particular
the average distance between nodes is smaller for causal networks than for
corresponding homogeneous networks. We explain the origin of this effect and
illustrate it using as an example a solvable model of random trees. We also
discuss the issue of stability of the scale-free node degree distribution. We
show that a surplus of links may lead to the emergence of a singular node with
the degree proportional to the total number of links. This effect is closely
related to the backgammon condensation known from the balls-in-boxes model.Comment: short review submitted to AIP proceedings, CNET2004 conference;
changes in the discussion of the distance distribution for growing trees,
Fig. 6-right change
Simple parameterization of nuclear attenuation data
Based on the nuclear attenuation data obtained by the HERMES experiment on
nitrogen and krypton nuclei, it is shown that the nuclear attenuation
can be parametrised in a form of a linear polynomial + , where is the formation time, which depends on the energy of the
virtual photon and fraction of that energy carried by the final
hadron. Three widely known parameterizations for were used for the
performed fit. The fit parameters and do not depend on
and
Phase transition and topology in 4d simplicial gravity
We present data indicating that the recent evidence for the phase transition
being of first order does not result from a breakdown of the ergodicity of the
algorithm. We also present data showing that the thermodynamical limit of the
model is independent of topology.Comment: 3 latex pages + 4 ps fig. + espcrc2.sty. Talk presented at
LATTICE(gravity
Replica analysis of a preferential urn model
We analyse a preferential urn model with randomness using the replica method.
The preferential urn model is a stochastic model based on the concept "the rich
get richer." The replica analysis clarifies that the preferential urn model
with randomness shows a fat-tailed occupation distribution. The analytical
treatments and results would be useful for various research fields such as
complex networks, stochastic models, and econophysics.Comment: 6 pages, 2 figure
Exotic trees
We discuss the scaling properties of free branched polymers. The scaling
behaviour of the model is classified by the Hausdorff dimensions for the
internal geometry: d_L and d_H, and for the external one: D_L and D_H. The
dimensions d_H and D_H characterize the behaviour for long distances while d_L
and D_L for short distances. We show that the internal Hausdorff dimension is
d_L=2 for generic and scale-free trees, contrary to d_H which is known be equal
two for generic trees and to vary between two and infinity for scale-free
trees. We show that the external Hausdorff dimension D_H is directly related to
the internal one as D_H = \alpha d_H, where \alpha is the stability index of
the embedding weights for the nearest-vertex interactions. The index is
\alpha=2 for weights from the gaussian domain of attraction and 0<\alpha <2 for
those from the L\'evy domain of attraction. If the dimension D of the target
space is larger than D_H one finds D_L=D_H, or otherwise D_L=D. The latter
result means that the fractal structure cannot develop in a target space which
has too low dimension.Comment: 33 pages, 6 eps figure
Condensation in nongeneric trees
We study nongeneric planar trees and prove the existence of a Gibbs measure
on infinite trees obtained as a weak limit of the finite volume measures. It is
shown that in the infinite volume limit there arises exactly one vertex of
infinite degree and the rest of the tree is distributed like a subcritical
Galton-Watson tree with mean offspring probability . We calculate the rate
of divergence of the degree of the highest order vertex of finite trees in the
thermodynamic limit and show it goes like where is the size of the
tree. These trees have infinite spectral dimension with probability one but the
spectral dimension calculated from the ensemble average of the generating
function for return probabilities is given by if the weight
of a vertex of degree is asymptotic to .Comment: 57 pages, 14 figures. Minor change
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