Optimizing at the Ergodic Edge

Using a simple, annealed model, some of the key features of the recently introduced extremal optimization heuristic are demonstrated. In particular, it is shown that the dynamics of local search possesses a generic critical point under the variation of its sole parameter, separating phases of too greedy (non-ergodic, jammed) and too random (ergodic) exploration. Comparison of various local search methods within this model suggests that the existence of the critical point is essential for the optimal performance of the heuristic.Comment: RevTex4, 17 pages, 3 ps-figures incl., for related information, see http://www.physics.emory.edu/faculty/boettcher/publications.htm

Extremal Optimization for Sherrington-Kirkpatrick Spin Glasses

Extremal Optimization (EO), a new local search heuristic, is used to approximate ground states of the mean-field spin glass model introduced by Sherrington and Kirkpatrick. The implementation extends the applicability of EO to systems with highly connected variables. Approximate ground states of sufficient accuracy and with statistical significance are obtained for systems with more than N=1000 variables using $\pm J$ bonds. The data reproduces the well-known Parisi solution for the average ground state energy of the model to about 0.01%, providing a high degree of confidence in the heuristic. The results support to less than 1% accuracy rational values of $\omega=2/3$ for the finite-size correction exponent, and of $\rho=3/4$ for the fluctuation exponent of the ground state energies, neither one of which has been obtained analytically yet. The probability density function for ground state energies is highly skewed and identical within numerical error to the one found for Gaussian bonds. But comparison with infinite-range models of finite connectivity shows that the skewness is connectivity-dependent.Comment: Substantially revised, several new results, 5 pages, 6 eps figures included, (see http://www.physics.emory.edu/faculty/boettcher/ for related information

Numerical Results for Ground States of Spin Glasses on Bethe Lattices

The average ground state energy and entropy for +/- J spin glasses on Bethe lattices of connectivities k+1=3...,26 at T=0 are approximated numerically. To obtain sufficient accuracy for large system sizes (up to n=2048), the Extremal Optimization heuristic is employed which provides high-quality results not only for the ground state energies per spin e_{k+1} but also for their entropies s_{k+1}. The results show considerable quantitative differences between lattices of even and odd connectivities. The results for the ground state energies compare very well with recent one-step replica symmetry breaking calculations. These energies can be scaled for all even connectivities k+1 to within a fraction of a percent onto a simple functional form, e_{k+1} = E_{SK} sqrt(k+1) - {2E_{SK}+sqrt(2)} / sqrt(k+1), where E_{SK} = -0.7633 is the ground state energy for the broken replica symmetry in the Sherrington-Kirkpatrick model. But this form is in conflict with perturbative calculations at large k+1, which do not distinguish between even and odd connectivities. We find non-zero entropies s_{k+1} at small connectivities. While s_{k+1} seems to vanish asymptotically with 1/(k+1) for even connectivities, it is indistinguishable from zero already for odd k+1 >= 9.Comment: 11 pages, RevTex4, 28 ps-figures included, related papers available at http://www.physics.emory.edu/faculty/boettcher

Coronal cooling and its signatures in the rapid aperiodic variability of Galactic black-hole candidates

The most popular models for the complex phase and time lags in the rapid aperiodic variability of Galactic X-ray binaries are based Comptonization of soft seed photons in a hot corona, where small-scale flares are induced by flares of the soft seed photon input (presumably from a cold accretion disc). However, in their original version, these models have neglected the additional cooling of the coronal plasma due to the increased soft seed photon input, and assumed a static coronal temperature structure. In this paper, our Monte-Carlo/Fokker-Planck code for time-dependent radiation transfer and electron energetics is used to simulate the self-consistent coronal response to the various flaring scenarios that have been suggested to explain phase and time lags observed in some Galactic X-ray binaries. It is found that the predictions of models involving slab-coronal geometries are drastically different from those deduced under the assumption of a static corona. However, with the inclusion of coronal cooling they may even be more successful than in their original version in explaining some of the observed phase and time lag features. The predictions of the model of inward-drifting density perturbations in an ADAF-like, two-temperature flow also differ from the static-corona case previously investigated, but may be consistent with the alternating phase lags seen in GRS 1915+105 and XTE J1550-564. Models based on flares of a cool disc around a hot, inner two-temperature flow may be ruled out for most objects where significant Fourier-frequency-dependent phase and time lags have been observed.Comment: 23 pages, including 8 figures and 2 tables; accepted for publication in ApJ; extended discussion w.r.t. original versio