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

Residual mean first-passage time for jump processes: theory and applications to L\'evy flights and fractional Brownian motion

We derive a functional equation for the mean first-passage time (MFPT) of a generic self-similar Markovian continuous process to a target in a one-dimensional domain and obtain its exact solution. We show that the obtained expression of the MFPT for continuous processes is actually different from the large system size limit of the MFPT for discrete jump processes allowing leapovers. In the case considered here, the asymptotic MFPT admits non-vanishing corrections, which we call residual MFPT. The case of L/'evy flights with diverging variance of jump lengths is investigated in detail, in particular, with respect to the associated leapover behaviour. We also show numerically that our results apply with good accuracy to fractional Brownian motion, despite its non-Markovian nature.Comment: 13 pages, 8 figure

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

Joint Probability Distributions for a Class of Non-Markovian Processes

We consider joint probability distributions for the class of coupled Langevin equations introduced by Fogedby [H.C. Fogedby, Phys. Rev. E 50, 1657 (1994)]. We generalize well-known results for the single time probability distributions to the case of N-time joint probability distributions. It is shown that these probability distribution functions can be obtained by an integral transform from distributions of a Markovian process. The integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non-Markovian character of the process.Comment: 13 pages, 1 figur

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

Functional Classical Mechanics and Rational Numbers

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility problem. A "functional" formulation of classical mechanics is suggested. The physical meaning is attached in this formulation not to an individual trajectory but only to a "beam" of trajectories, or the distribution function on phase space. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values and there are corrections to the Newton trajectories. We give a construction of probability density function starting from the directly observable quantities, i.e., the results of measurements, which are rational numbers.Comment: 8 page

The domain of partial attraction of a Poisson law

Groshev gave a characterization of the union of domains of partial attraction of all Poisson laws in 1941. His classical condition is expressed by the underlying distribution function and disguises the role of the mean λ of the attracting distribution. In the present paper we start out from results of the recent “probabilistic approach” and derive characterizations for any fixed λ>0 in terms of the underlying quantile function. The approach identifies the portion of the sample that contributes the limiting Poisson behavior of the sum, delineates the effect of extreme values, and leads to necessary and sufficient conditions all involving λ. It turns out that the limiting Poisson distributions arise in two qualitatively different ways depending upon whether λ>1 or λ<1. A concrete construction, illustrating all the results, also shows that in the boundary case when λ=1 both possibilities may occur.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45254/1/10959_2005_Article_BF01047000.pd