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We describe here a minimal theory of tight binding electrons moving on the square planar Cu lattice of the hole-doped cuprates and mixed quantum mechanically with pairs of them (Cooper pairs). Superconductivity occurring at the transition temperature T_c is the long-range, d-wave symmetry phase coherence of these Cooper pairs. Fluctuations necessarily associated with incipient long-range superconducting order have a generic large distance behaviour near T_c. We calculate the spectral density of electrons coupled to such Cooper pair fluctuations and show that features observed in Angle Resolved Photo Emission Spectroscopy (ARPES) experiments on different cuprates above T_c as a function of doping and temperature emerge naturally in this description. These include `Fermi arcs' with temperature-dependent length and an antinodal pseudogap which fills up linearly as the temperature increases towards the pseudogap temperature. Our results agree quantitatively with experiment. Below T_c, the effects of nonzero superfluid density and thermal fluctuations are calculated and compared successfully with some recent ARPES experiments, especially the observed `bending' or deviation of the superconducting gap from the canonical d-wave form., Comment: 14 pages, 8 figures (to appear in Phys. Rev. B)

Results of analytic and numerical investigations of first-passage properties of equilibrium fluctuations of monatomic steps on a vicinal surface are reviewed. Both temporal and spatial persistence and survival probabilities, as well as the probability of persistent large deviations are considered. Results of experiments in which dynamical scanning tunneling microscopy is used to evaluate these first-passage properties for steps with different microscopic mechanisms of mass transport are also presented and interpreted in terms of theoretical predictions for appropriate models. Effects of discrete sampling, finite system size and finite observation time, which are important in understanding the results of experiments and simulations, are discussed., Comment: 30 pages, 12 figures, review paper for a special issue of JSTAT

The effects of sampling rate and total measurement time have been determined for single-point measurements of step fluctuations within the context of first-passage properties. Time dependent STM has been used to evaluate step fluctuations on Ag(111) films grown on mica as a function of temperature (300-410 K), on screw dislocations on the facets of Pb crystallites at 320K, and on Al-terminated Si(111) over the temperature range 770K - 970K. Although the fundamental time constant for step fluctuations on Ag and Al/Si varies by orders of magnitude over the temperature ranges of measurement, no dependence of the persistence amplitude on temperature is observed. Instead, the persistence probability is found to scale directly with t/Dt where Dt is the time interval used for sampling. Survival probabilities show a more complex scaling dependence which includes both the sampling interval and the total measurement time tm. Scaling with t/Dt occurs only when Dt/tm is a constant. We show that this observation is equivalent to theoretical predictions that the survival probability will scale as Dt/L^z, where L is the effective length of a step. This implies that the survival probability for large systems, when measured with fixed values of tm or Dt should also show little or no temperature dependence., Comment: 27 pages, 10 figures

Time dependent STM has been used to evaluate step fluctuations as a function of temperature (300-450 K) on Ag(111) films grown on mica. The temporal correlation function scales as a power law in time, t^1/n with measured values of 1/n varying over a range of 0.19 pm 0.04 to 0.29 pm 0.04 with no dependence on temperature. The average value of 1/n = 0.24 pm 0.01 is consistent with step-edge diffusion limited fluctuations (n = z = 4, conserved noise). The magnitude of the time correlation function and the width of the fluctuations both scale with temperature with the same apparent activation energy of Eeff = 0.21 pm 0.02 eV, indicating that the correlation time is at most weakly temperature dependent. Direct analysis of the autocorrelation function confirms that the correlation time is at most weakly temperature dependent, and thus the apparent correlation length is strongly temperature dependent. This behavior can be reproduced by assuming that the apparent correlation length is governed by the longest wavelength of step fluctuations that can be sampled in the measurement time interval. Evaluation of the correlation time for previous measurements for Al/Si(111) (z =2) yields the same conclusion about measurement time interval. In both cases the ratio of the measurement time to the effective correlation time is on the order of 10.

Persistence probabilities of the interface height in (1+1)- and (2+1)-dimensional atomistic, solid-on-solid, stochastic models of surface growth are studied using kinetic Monte Carlo simulations, with emphasis on models that belong to the molecular beam epitaxy (MBE) universality class. Both the initial transient and the long-time steady-state regimes are investigated. We show that for growth models in the MBE universality class, the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady-state regimes. For the MBE universality class, the positive and negative persistence exponents in the steady-state are found to be $\theta^S_{+} = 0.66 \pm 0.02$ and $\theta^S_{-} = 0.78 \pm 0.02$, respectively, in (1+1) dimensions, and $\theta^S_{+} = 0.76 \pm 0.02$ and $\theta^S_{-} =0.85 \pm 0.02$, respectively, in (2+1) dimensions. The noise reduction technique is applied on some of the (1+1)-dimensional models in order to obtain accurate values of the persistence exponents. We show analytically that a relation between the steady-state persistence exponent and the dynamic growth exponent, found earlier to be valid for linear models, should be satisfied by the smaller of the two steady-state persistence exponents in the nonlinear models. Our numerical results for the persistence exponents are consistent with this prediction. We also find that the steady-state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state. The dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form., Comment: 28 pages, 16 figures

We report the results of numerical investigations of the steady-state (SS) and finite-initial-conditions (FIC) spatial persistence and survival probabilities for (1+1)--dimensional interfaces with dynamics governed by the nonlinear Kardar--Parisi--Zhang (KPZ) equation and the linear Edwards--Wilkinson (EW) equation with both white (uncorrelated) and colored (spatially correlated) noise. We study the effects of a finite sampling distance on the measured spatial persistence probability and show that both SS and FIC persistence probabilities exhibit simple scaling behavior as a function of the system size and the sampling distance. Analytical expressions for the exponents associated with the power-law decay of SS and FIC spatial persistence probabilities of the EW equation with power-law correlated noise are established and numerically verified., Comment: 11 pages, 5 figures

We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability $S(t)$ in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. $S(t)$ is shown to exhibit simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of experimental dynamical STM data on step fluctuations on Al/Si(111) and Ag(111) surfaces., Comment: RevTeX, 4 pages, 3 figures

We show experimentally and theoretically that the persistence of large deviations in equilibrium step fluctuations is characterized by an infinite family of independent exponents. These exponents are obtained by carefully analyzing dynamical experimental images of Al/Si(111) and Ag(111) equilibrium steps fluctuating at high (970K) and low (320K) temperatures respectively, and by quantitatively interpreting our observations on the basis of the corresponding coarse-grained discrete and continuum theoretical models for thermal surface step fluctuations under attachment/detachment (``high-temperature'') and edge-diffusion limited kinetics (``low-temperature'') respectively., Comment: LaTeX, 4 pages, 2 figures

The persistence behavior for fluctuating steps on the $Si(111)$ $(\sqrt3 \times \sqrt3)R30^{0} - Al$ surface was determined by analyzing time-dependent STM images for temperatures between 770 and 970K. The measured persistence probability follows a power law decay with an exponent of $\theta=0.77 \pm 0.03$. This is consistent with the value of $\theta= 3/4$ predicted for attachment/detachment limited step kinetics. If the persistence analysis is carried out in terms of return to a fixed reference position, the measured persistence probability decays exponentially. Numerical studies of the Langevin equation used to model step motion corroborate the experimental observations., Comment: LaTeX, 11 pages, 4 figures, minor changes in References section

A neural network model that exhibits stochastic population bursting is studied by simulation. First return maps of inter-burst intervals exhibit recurrent unstable periodic orbit (UPO)-like trajectories similar to those found in experiments on hippocampal slices. Applications of various control methods and surrogate analysis for UPO-detection also yield results similar to those of experiments. Our results question the interpretation of the experimental data as evidence for deterministic chaos and suggest caution in the use of UPO-based methods for detecting determinism in time-series data., Comment: 4 pages, 5 .eps figures (included), requires psfrag.sty (included)

Conserved growth models that exhibit a nonlinear instability in which the height (depth) of isolated pillars (grooves) grows in time are studied by numerical integration and stochastic simulation. When this instability is controlled by the introduction of an infinite series of higher-order nonlinear terms, these models exhibit, as function of a control parameter, a non-equilibrium phase transition between a kinetically rough phase with self-affine scaling and a phase that exhibits mound formation, slope selection and power-law coarsening., Comment: 7 pages, 4 .eps figures (Minor changes in text and references.)

We show by numerical simulations that discretized versions of commonly studied continuum nonlinear growth equations (such as the Kardar-Parisi-Zhang equation and the Lai-Das Sarma equation) and related atomistic models of epitaxial growth have a generic instability in which isolated pillars (or grooves) on an otherwise flat interface grow in time when their height (or depth) exceeds a critical value. Depending on the details of the model, the instability found in the discretized version may or may not be present in the truly continuum growth equation, indicating that the behavior of discretized nonlinear growth equations may be very different from that of their continuum counterparts. This instability can be controlled either by the introduction of higher-order nonlinear terms with appropriate coefficients or by restricting the growth of pillars (or grooves) by other means. A number of such ``controlled instability'' models are studied by simulation. For appropriate choice of the parameters used for controlling the instability, these models exhibit intermittent behavior, characterized by multiexponent scaling of height fluctuations, over the time interval during which the instability is active. The behavior found in this regime is very similar to the ``turbulent'' behavior observed in recent simulations of several one- and two-dimensional atomistic models of epitaxial growth. [pacs{61.50.Cj, 68.55.Bd, 05.70.Ln, 64.60.Ht}], Comment: 47 pages + 26 postscript figures, submitted to Phys. Rev. E