Lattice study of the simplified model of M-theory for larger gauge groups

Lattice discretization of the supersymmetric Yang-Mills quantum mechanics is dis cussed. First results of the quenched Monte Carlo simulations, for D=4 and with higher g auge groups (3 <= N <= 8), are presented. We confirm an earlier (N=2) evidence tha t the system reveals different behaviours at low and high temperatures separated by a narrow transiti on region. These two regimes may correspond to a black hole and elementary excitations phases conjectured in the M-theory. Dependence of the "transition temperature" on N is consistent with 't Hooft scaling and shows a smooth saturation of lattice results towards the large N limit. Is not yet resolved if the observed change between the two regimes corresponds to a genuine phase transition or to a gentle crossover . A new, noncompact formulation of the lattice model is also proposed and its advantages are briefly discussed.Comment: 10 pages, 2 figures, Invited talk presented at the Sixth Workshop on Non-Perturbative QCD, American University of Paris, Paris, June, 200

Measuring an entropy in heavy ion collisions

We propose to use the coincidence method of Ma to measure an entropy of the system created in heavy ion collisions. Moreover we estimate, in a simple model, the values of parameters for which the thermodynamical behaviour sets in.Comment: LATTICE98(hightemp), 3 pages, LaTeX with two eps figure

Connected Correlators in Quantum Gravity

We discuss the concept of connected, reparameterization invariant matter correlators in quantum gravity. We analyze the effect of discretization in two solvable cases: branched polymers and two-dimensional simplicial gravity. In both cases the naively defined connected correlators for a fixed volume display an anomalous behavior, which could be interpreted as a long-range order. We suggest that this is in fact only a highly non-trivial finite-size effect and propose an improved definition of the connected correlator, which reduces the effect. Using this definition we illustrate the appearance of a long-range spin order in the Ising model on a two-dimensional random lattice in an external magnetic field $H$, when $H \to 0$ and $\beta=\beta_C$.Comment: 21 pages, 8 figure

RG flow in an exactly solvable model with fluctuating geometry

A recently proposed renormalization group technique, based on the hierarchical structures present in theories with fluctuating geometry, is implemented in the model of branched polymers. The renormalization group equations can be solved analytically, and the flow in coupling constant space can be determined.Comment: References updated, typos corrected and abstract sligtly changed. 10 pages. Pictex use

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

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

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

Density correlators in a self-similar cascade

Multivariate density moments (correlators) of arbitrary order are obtained for the multiplicative self-similar cascade. This result is based on the calculation by Greiner, Eggers and Lipa (reference [1]) where the correlators of the logarithms of the particle densities have been obtained. The density correlators, more suitable for comparison with multiparticle data, appear to have even simpler form than those obtained in [1].Comment: 9 pages, 3 figures, uses epsfig.st

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