d_c=4 is the upper critical dimension for the Bak-Sneppen model

Numerical results are presented indicating d_c=4 as the upper critical dimension for the Bak-Sneppen evolution model. This finding agrees with previous theoretical arguments, but contradicts a recent Letter [Phys. Rev. Lett. 80, 5746-5749 (1998)] that placed d_c as high as d=8. In particular, we find that avalanches are compact for all dimensions d<=4, and are fractal for d>4. Under those conditions, scaling arguments predict a d_c=4, where hyperscaling relations hold for d<=4. Other properties of avalanches, studied for 1<=d<=6, corroborate this result. To this end, an improved numerical algorithm is presented that is based on the equivalent branching process.Comment: 4 pages, RevTex4, as to appear in Phys. Rev. Lett., related papers available at http://userwww.service.emory.edu/~sboettc

Gamma-Ray Bursts from Up-Scattered Self-Absorbed Synchrotron Emission

We calculate the synchrotron self-Compton emission from internal shocks occurring in relativistic winds as a source of gamma-ray bursts, with allowance for self-absorption. For plausible model parameters most pulses within a Gamma-Ray Burst (GRB) are optically thick to synchrotron self-absorption at the frequency at which most electrons radiate. Up-scattering of photon number spectra harder than $\nu^0$ (such as the self-absorbed emission) yields inverse Compton photon number spectra that are flat, therefore our model has the potential of explaining the low-energy indices harder than $\nu^{-2/3}$ (the optically thin synchrotron limit) that have been observed in some bursts. The optical counterparts of the model bursts are sufficiently bright to be detected by such experiments as LOTIS, unless the magnetic field is well below equipartition.Comment: to be published in ApJL, 5 pages, 3 color figure

Extremal Optimization at the Phase Transition of the 3-Coloring Problem

We investigate the phase transition of the 3-coloring problem on random graphs, using the extremal optimization heuristic. 3-coloring is among the hardest combinatorial optimization problems and is closely related to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the ``backbone'', an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size $n$ up to 512. For graphs up to this size, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about $O(n^{3.5})$ update steps. Finite size scaling gives a critical mean degree value $\alpha_{\rm c}=4.703(28)$. Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available at http://www.physics.emory.edu/faculty/boettcher

GRB990123: The Case for Saturated Comptonization

The recent simultaneous detection of optical, X-ray and gamma-ray photons from GRB990123 during the burst provides the first broadband multi-wavelength characterization of the burst spectrum and evolution. Here we show that a direct correlation exists between the time-varying gamma-ray spectral shape and the prompt optical emission. This combined with the unique signatures of the time-resolved spectra of GRB990123 convincingly supports earlier predictions of the saturated Comptonization model. Contrary to other suggestions, we find that the entire continuum from optical to gamma-rays can be generated from a single source of leptons (electrons and pairs). The optical flux only appears to lag the gamma-ray flux due to the high initial Thomson depth of the plasma. Once the plasma has completely thinned out, the late time afterglow behavior of our model is the same as in standard models based on the Blandford-McKee (1976) solution.Comment: 10 pages, including 3 figures and 1 table, submitted to The Astrophysical Journal Letter

Aging in Dense Colloids as Diffusion in the Logarithm of Time

The far-from-equilibrium dynamics of glassy systems share important phenomenological traits. A transition is generally observed from a time-homogeneous dynamical regime to an aging regime where physical changes occur intermittently and, on average, at a decreasing rate. It has been suggested that a global change of the independent time variable to its logarithm may render the aging dynamics homogeneous: for colloids, this entails diffusion but on a logarithmic time scale. Our novel analysis of experimental colloid data confirms that the mean square displacement grows linearly in time at low densities and shows that it grows linearly in the logarithm of time at high densities. Correspondingly, pairs of particles initially in close contact survive as pairs with a probability which decays exponentially in either time or its logarithm. The form of the Probability Density Function of the displacements shows that long-ranged spatial correlations are very long-lived in dense colloids. A phenomenological stochastic model is then introduced which relies on the growth and collapse of strongly correlated clusters ("dynamic heterogeneity"), and which reproduces the full spectrum of observed colloidal behaviors depending on the form assumed for the probability that a cluster collapses during a Monte Carlo update. In the limit where large clusters dominate, the collapse rate is ~1/t, implying a homogeneous, log-Poissonian process that qualitatively reproduces the experimental results for dense colloids. Finally an analytical toy-model is discussed to elucidate the strong dependence of the simulation results on the integrability (or lack thereof) of the cluster collapse probability function.Comment: 6 pages, extensively revised, final version; for related work, see http://www.physics.emory.edu/faculty/boettcher/ or http://www.fysik.sdu.dk/staff/staff-vip/pas-personal.htm