510 research outputs found

    Collapse of 4D random geometries

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    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

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    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

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    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 p(q)=qβp(q) = q^{-\beta}, 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 β(2,)\beta\in (2,\infty) the transition between these two phases is first order, while for β(1,2]\beta \in (1,2] it is continuous. The transition becomes softer when β\beta approaches unity and eventually disappears at β=1\beta=1. 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

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    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

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    Based on the nuclear attenuation data obtained by the HERMES experiment on nitrogen and krypton nuclei, it is shown that the nuclear attenuation RMhR_M^{h} can be parametrised in a form of a linear polynomial P1=a11P_1=a_{11} + τa12\tau a_{12}, where τ\tau is the formation time, which depends on the energy of the virtual photon ν\nu and fraction of that energy zz carried by the final hadron. Three widely known parameterizations for τ\tau were used for the performed fit. The fit parameters a11a_{11} and a12a_{12} do not depend on ν\nu and zz

    Phase transition and topology in 4d simplicial gravity

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    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

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    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

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    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

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    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 m<1m<1. 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 (1m)N(1-m)N where NN 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 2β22\beta -2 if the weight wnw_n of a vertex of degree nn is asymptotic to nβn^{-\beta}.Comment: 57 pages, 14 figures. Minor change
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