Abstract:
Contents: A. Introduction B. High Temperature Expansions for the Ising Model C. Characteristic Functions and Cumulants D. The One Dimensional Chain E. Directed Paths and the Transfer Matrix F. Moments of the Correlation Function G. The Probability Distribution in Two Dimensions H. Higher Dimensions I. Random Signs J. Other Realizations of DPRM K. Quantum Interference of Strongly Localized Electrons L. The Locator Expansion and Forward Scattering Paths M. Magnetic Field Response N. Unitary Propagation O. Unitary Averages P. Summing all Paths in High Dimensions Q. The Ising Model on a Square Lattice R. Singular Behavior S. The Two Dimensional Spin Glass T. Results for the Two Dimensional Spin Glass

Abstract:
Inhomogeneities in deposition may lead to formation of rough surfaces, whose height fluctuations can be probed directly by scanning microscopy, or indirectly by scattering. Analytical or numerical treatments of simple growth models suggest that, quite generally, the height fluctuations have a self-similar character. The roughness and dynamic exponents are expected to be universal; depending only on the underlying mechanism that generates self-similar roughness. Despite its ubiquitous occurrence in theory and simulations, experimental confirmations of dynamic scaling have been rare. I shall briefly review the theoretical foundations of dynamic scaling, and suggest possible reasons for discrepancies with experimental results.

Abstract:
In many growth processes particles are highly mobile in an active layer at the surface, but are relatively immobile once incorporated in the bulk. We study models in which atoms are allowed to interact, equilibrate, and order on the surface, but are frozen in the bulk. Order parameter correlations in the resulting bulk material are highly anisotropic, reflecting its growth history. In a flat (layer by layer) growth mode, correlations perpendicular to the growth direction are similar to a two dimensional system in equilibrium, while parallel correlations reflect the dynamics of such a system. When the growing film is rough, various couplings between height and order parameter fluctuations are possible. Such couplings modify the dynamic scaling properties of surface roughness, and may also change the critical behavior of the order parameter. Even the deterministic growth of the surface profile can result in interesting textures for the order parameter.

Abstract:
The lectures examine several problems related to non-equilibrium fluctuations of interfaces and flux lines. The first two introduce the phenomenology of depinning, with particular emphasis on interfaces and contact lines. The role of the anisotropy of the medium in producing different universality classes is elucidated. The last two lectures focus on the dynamics of lines, where transverse fluctuations are also important. We shall demonstrate how various non-linearities appear in the dynamics of driven flux lines. The universality classes of depinning, and also dynamic roughening, are illustrated in the contexts of moving flux lines, advancing crack fronts, and drifting polymers.

Abstract:
We consider a model of a reconstructed crystal surface, first considered by Villain and Vilfan (Europhys. Lett. 12, p. 523 (1990) and Surf. Sci. 257, p. 368 (1991)) for the gold (110) surface, in which roughening occurs via the formation of anisotropic steps traversing the entire length of the crystal. The model is studied by a mapping to a spin--1/2 Fermion system in 1+1 dimensions, which, in the absence of islands, is precisely the Hubbard model. We consider a general $\pbyo$ reconstruction, in the presence of inter--step interactions and closed islands. Our analysis predicts the existence of a new type of rough phase, with incommensurate correlations in the reconstruction order parameter and unusual momentum space singularities at a characteristic ``Fermi momentum'' and its harmonics, analagous to the Luttinger liquid of one--dimensional Fermions. The general phase structure for $p>1$ is as follows: for $p>2$, there is a flat ordered (FO), a rough incommensurate (RI), and a flat incommensurate phase (FI). The FO--RI and FO--FI transitions are of the commensurate to incommensurate type, and the FI--RI transition is in the Kosterlitz--Thouless (KT) universality class. For $p=2$, the FI phase is replaced by a flat disordered phase (FD), and there may be a new rough disordered phase (RD). The FO--FD transition is now of Ising type, and the FD--RD and RI--RD transitions are in the KT universality class.

Abstract:
Spatial configurations of randomly charged polymers, known as polyampholytes (PAs), are very sensitive to the overall excess charge Q. Analytical arguments, supported by Monte Carlo simulations and exact enumeration studies, lead to the following picture: For Q < Q_c = q_0 N^2 (q_0 is the elementary charge, N is the number of monomers in the polymer), the radius of gyration R_g of the polymer decreases with decreasing temperature T and the polymer becomes compact, while for Q>Q_c the polymer stretches with decreasing T. At low T, the dense states are described by Debye-Huckel theory, while the expanded states resemble a necklace of globules connected by strings. At such temperatures, the transition between the dense and the expanded states with increasing Q, is reminiscent of the breakup of a charged drop.

Abstract:
We consider polymers in which M randomly selected pairs of monomers are restricted to be in contact. Analytical arguments and numerical simulations show that an ideal (Gaussian) chain of N monomers remains expanded as long as M< (N/M)^(nu). The number of links needed to collapse a polymer in three dimensions thus scales as N^(phi), with (phi) > 0.43.

Abstract:
Using a combination of analytic arguments and numerical simulations, we determine lower and upper bounds for the energy barriers to the motion of a defect line in a random potential at low temperatures. We study the cases of magnetic flux lines in high-$T_{c}$ superconductors in 2 and 3 dimensions, and of domain walls in 2 dimensional random-field Ising models. The results show that, under fairly general conditions, energy barriers have the same scaling as the fluctuations in free energy, except for possible logarithmic factors. This holds not only for barriers between optimal configurations of the line, but also for barriers separating any metastable configuration from a configuration of minimal energy. Similar arguments may be applicable to other elastic media with impurities, such as bunches of flux lines.

Abstract:
We study analytically and numerically the winding of directed polymers of length $t$ around each other or around a rod. Unconfined polymers in pure media have exponentially decaying winding angle distributions, the decay constant depending on whether the interaction is repulsive or neutral, but not on microscopic details. In the presence of a chiral asymmetry, the exponential tails become non universal. In all these cases the mean winding angle is proportional to $\ln t$. When the polymer is confined to a finite region around the winding center, e.g. due to an attractive interaction, the winding angle distribution is Gaussian, with a variance proportional to $t$. We also examine the windings of polymers in random systems. Our results suggest that randomness reduces entanglements, leading to a narrow (Gaussian) distribution with a mean winding angle of the order of $\sqrt{\ln t}$.

Abstract:
We study a simple growth model for (d+1)-dimensional films of binary alloys in which atoms are allowed to interact and equilibrate at the surface, but are frozen in the bulk. The resulting crystal is highly anisotropic: Correlations perpendicular to the growth direction are identical to a d-dimensional two-layer system in equilibrium, while parallel correlations generally reflect the (Glauber) dynamics of such a system. For stronger in-plane interactions, the correlation volumes change from oblate to highly prolate shapes near a critical demixing or ordering transition. In d=1, the critical exponent z relating the scaling of the two correlation lengths varies continuously with the chemical interactions.