Topological jamming and the glass transition in a frustrated system.

The relationship between extended structures, glassy dynamics and an underlying critical point is examined in the context of a lattice model of fluctuating lines. Monte Carlo simulations are used to construct an effective, coarse-grained dynamics for the "order parameter" near the critical point. Analysis of the effective dynamics reveals that the critical point is associated with diverging barriers leading to the observed Vogel-Fulcher divergence of the relaxation times. A direct connection is established between the presence of extended structures and the activated dynamics.

Critical geometry of two-dimensional passive scalar turbulence.

Passive scalars advected by a magnetically driven two-dimensional turbulent flow are analyzed using methods of statistical topography. The passive tracer concentration is interpreted as the height of a random surface, and the scaling properties of its contour loops are analyzed. Various exponents that describe the loop ensemble are measured and compared to a scaling theory. This leads to a geometrical criterion for the intermittency of scalar fluctuations.

Nonlinear measures for characterizing rough surface morphologies

We develop an approach for characterizing the morphology of rough surfaces based on the analysis of the scaling properties of contour loops, i.e., loops of constant height. Given a height profile of the surface we perform independent measurements of the fractal dimension of contour loops, and the exponent that characterizes their size distribution. Scaling formulas are derived, and used to relate these two geometrical exponents to the roughness exponent of a self-affine surface, thus providing independent measurements of this important quantity. Furthermore, we define the scale-dependent curvature, and demonstrate that by measuring its third moment departures of the height fluctuations from Gaussian behavior can be ascertained. These nonlinear measures are used to characterize the morphology of computer generated Gaussian rough surfaces, surfaces obtained in numerical simulations of a simple growth model, and surfaces observed by scanning-tunneling microscopes. For experimentally realized surfaces the self-affine scaling is cut off by a correlation length, and we generalize our theory of contour loops to take this into account.