Bouchaud-M\'ezard model on a random network

We studied the Bouchaud-M\'ezard(BM) model, which was introduced to explain Pareto's law in a real economy, on a random network. Using "adiabatic and independent" assumptions, we analytically obtained the stationary probability distribution function of wealth. The results shows that wealth-condensation, indicated by the divergence of the variance of wealth, occurs at a larger $J$ than that obtained by the mean-field theory, where $J$ represents the strength of interaction between agents. We compared our results with numerical simulation results and found that they were in good agreement.Comment: to be published in Physical Review

Thermodynamics of the L\'evy spin glass

We investigate the L\'evy glass, a mean-field spin glass model with power-law distributed couplings characterized by a divergent second moment. By combining extensively many small couplings with a spare random backbone of strong bonds the model is intermediate between the Sherrington-Kirkpatrick and the Viana-Bray model. A truncated version where couplings smaller than some threshold \eps are neglected can be studied within the cavity method developed for spin glasses on locally tree-like random graphs. By performing the limit \eps\to 0 in a well-defined way we calculate the thermodynamic functions within replica symmetry and determine the de Almeida-Thouless line in the presence of an external magnetic field. Contrary to previous findings we show that there is no replica-symmetric spin glass phase. Moreover we determine the leading corrections to the ground-state energy within one-step replica symmetry breaking. The effects due to the breaking of replica symmetry appear to be small in accordance with the intuitive picture that a few strong bonds per spin reduce the degree of frustration in the system

Hidden Variables in Bipartite Networks

We introduce and study random bipartite networks with hidden variables. Nodes in these networks are characterized by hidden variables which control the appearance of links between node pairs. We derive analytic expressions for the degree distribution, degree correlations, the distribution of the number of common neighbors, and the bipartite clustering coefficient in these networks. We also establish the relationship between degrees of nodes in original bipartite networks and in their unipartite projections. We further demonstrate how hidden variable formalism can be applied to analyze topological properties of networks in certain bipartite network models, and verify our analytical results in numerical simulations

Simple observations concerning black holes and probability

It is argued that black holes and the limit distributions of probability theory share several properties when their entropy and information content are compared. In particular the no-hair theorem, the entropy maximization and holographic bound, and the quantization of entropy of black holes have their respective analogues for stable limit distributions. This observation suggests that the central limit theorem can play a fundamental role in black hole statistical mechanics and in a possibly emergent nature of gravity.Comment: 6 pages Latex, final version. Essay awarded "Honorable Mention" in the Gravity Research Foundation 2009 Essay Competitio

High-energy gluon bremsstrahlung in a finite medium: harmonic oscillator versus single scattering approximation

A particle produced in a hard collision can lose energy through bremsstrahlung. It has long been of interest to calculate the effect on bremsstrahlung if the particle is produced inside a finite-size QCD medium such as a quark-gluon plasma. For the case of very high-energy particles traveling through the background of a weakly-coupled quark-gluon plasma, it is known how to reduce this problem to an equivalent problem in non-relativistic two-dimensional quantum mechanics. Analytic solutions, however, have always resorted to further approximations. One is a harmonic oscillator approximation to the corresponding quantum mechanics problem, which is appropriate for sufficiently thick media. Another is to formally treat the particle as having only a single significant scattering from the plasma (known as the N=1 term of the opacity expansion), which is appropriate for sufficiently thin media. In a broad range of intermediate cases, these two very different approximations give surprisingly similar but slightly differing results if one works to leading logarithmic order in the particle energy, and there has been confusion about the range of validity of each approximation. In this paper, I sort out in detail the parametric range of validity of these two approximations at leading logarithmic order. For simplicity, I study the problem for small alpha_s and large logarithms but alpha_s log << 1.Comment: 40 pages, 23 figures [Primary change since v1: addition of new appendix reviewing transverse momentum distribution from multiple scattering

Anomalous biased diffusion in a randomly layered medium

We present analytical results for the biased diffusion of particles moving under a constant force in a randomly layered medium. The influence of this medium on the particle dynamics is modeled by a piecewise constant random force. The long-time behavior of the particle position is studied in the frame of a continuous-time random walk on a semi-infinite one-dimensional lattice. We formulate the conditions for anomalous diffusion, derive the diffusion laws and analyze their dependence on the particle mass and the distribution of the random force.Comment: 19 pages, 1 figur

Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels

To each hyperbolic Landau level of the Poincar\'e disc is attached a generalized negative binomial distribution. In this paper, we compute the moment generating function of this distribution and supply its decomposition as a perturbation of the negative binomial distribution by a finitely-supported measure. Using the Mandel parameter, we also discuss the nonclassical nature of the associated coherent states. Next, we determine the L\'evy-Kintchine decomposition its characteristic function when the latter does not vanish and deduce that it is quasi-infinitely divisible except for the lowest hyperbolic Landau level corresponding to the negative binomial distribution. By considering the total variation of the obtained quasi-L\'evy measure, we introduce a new infinitely-divisible distribution for which we derive the characteristic function

Central limit theorem for anomalous scaling due to correlations

We derive a central limit theorem for the probability distribution of the sum of many critically correlated random variables. The theorem characterizes a variety of different processes sharing the same asymptotic form of anomalous scaling and is based on a correspondence with the L\'evy-Gnedenko uncorrelated case. In particular, correlated anomalous diffusion is mapped onto L\'evy diffusion. Under suitable assumptions, the nonstandard multiplicative structure used for constructing the characteristic function of the total sum allows us to determine correlations of partial sums exclusively on the basis of the global anomalous scaling.Comment: The content of this manuscript was presented at the 3rd International Conference "Next Sigma-Phi", Kolymbari - Greece, 13-18 August 200

Alternative sampling for variational quantum Monte Carlo

Expectation values of physical quantities may accurately be obtained by the evaluation of integrals within Many-Body Quantum mechanics, and these multi-dimensional integrals may be estimated using Monte Carlo methods. In a previous publication it has been shown that for the simplest, most commonly applied strategy in continuum Quantum Monte Carlo, the random error in the resulting estimates is not well controlled. At best the Central Limit theorem is valid in its weakest form, and at worst it is invalid and replaced by an alternative Generalised Central Limit theorem and non-Normal random error. In both cases the random error is not controlled. Here we consider a new `residual sampling strategy' that reintroduces the Central Limit Theorem in its strongest form, and provides full control of the random error in estimates. Estimates of the total energy and the variance of the local energy within Variational Monte Carlo are considered in detail, and the approach presented may be generalised to expectation values of other operators, and to other variants of the Quantum Monte Carlo method.Comment: 14 pages, 9 figure