The analogy between self-avoiding walks and the zero-component magnetic system on a lattice is considered. The magnetic system possesses a natural boundary in the ($T-H$) plane below which we cannot continue it analytically. It is found, however, that we do not have to cross this boundary to obtain the semidilute regime. The scaling relation of des Cloizeaux is confirmed. There is also discovered a new phase below the boundary which is identified as a collapsed phase. This is quite unexpected because no attractive interactions are allowed.

We show how a generalization of continued fractions can be used to develop a renormalization-group formalism to study the behavior of maps of the two-torus. Such maps may mimic the universal behavior of dynamical systems with three mutually incommensurate frequencies. Numerical evidence indicates that "chaos" may occur in maps which are invertible. While we do not see scaling at chaos onset, subcritical scaling is observed and explained by the number theory.

Any attempt to describe bound states of three or more Dirac particles, or of two or more such particles in the presence of an external potential, must take into account the problem of continuum dissolution: A Hamiltonian which involves only the sum of free Dirac Hamiltonians for the particles plus local interactions will not have normalizable eigenfunctions because of the mixing of positive- and negative-energy states. A simple model is constructed to illustrate the validity of this "folk theorem," first noted by Brown and Ravenhall in 1951.

As the low-frequency power input to a nonlinear medium is increased, the power spectrum of wave motion converges to a universal distribution proportional to the inverse of the frequency. Experiments which can probe the transition to $\frac{1}{f}$ noise are proposed.

Canonical discrete-time operator difference equations are introduced as an alternative approach to the numerical solution of a quantum field theory. We apply these techniques to the solution of the operator Heisenberg equations of motion describing the problem of quantum-mechanical tunneling. Our numerical solutions accurately depict the time evolution of $〈q^n〉$ and of a local probability measure.

With reliance on Regge calculus and the Regge-Einstein action, models of four-dimensional Euclidean gravity are simulated numerically. The scale $a=l_0$ is set by fixing the expectation value of a length. Monte Carlo calculations support the following results: In the infinite-volume limit, canonical dimensions are realized and a finite action density is obtained.

When near-integrable Hamiltonians systems are perturbed by dissipation, then the stable orbits become simple attracting sinks, the Kolmogorov-Arnol'd-Moser tori are destroyed, and persistent chaotic motion disappears. We determine analytically the mean lifetime, the quasistatic distribution, and the fraction trapped into the various sinks for a dissipatively perturbed area-preserving twist map.

The path-integral method is used for determination of the quantum corrections to the free energy of nonlinear systems. All quantum effects of the harmonic part of the potential are considered and a variational principle is used to account for the quantum corrections due to the anharmonic part. Correct renormalized frequencies are obtained at any temperature and an effective potential to be inserted in the configurational integral is found. A new general expression for the partition function at any temperature in the low-coupling limit is obtained.

The structure of constrained Hamiltonian systems is investigated. The structure functions are shown to obey remarkable identities which ensure the existence of the Becchi-Rouet-Stora symmetry for any gauge theory.

We have studied the shape of large clusters in the lattice-animal, percolation, and growing-percolation models. By calculating the radius of gyration tensor we find that in these models the clusters have an anisotropic shape. The results suggest that the critical droplets in related isotropic equilibrium models, such as the Ising model, may also be anisotropic. We have also determined the leading nonanalytic correction-to-scaling exponent by analyzing the anisotropy data and find that for percolation in two dimensions $\theta \simeq 0.47$.

We study the correspondence between quantum and classical dynamics with Wigner and Husimi distribution functions, respectively. The nature of quantum-mechanical regular and irregular behaviors is discussed with the use of Wigner and Husimi distribution functions. By numerical studies of the chaotic state, the Husimi distribution function is found to be a better representation than the Wigner distribution function, because coarse graining is usually involved implicitly in an observational process using the Wigner distribution function.

A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of $s$ spins is shown to scale as $\ln s$. This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.

A new and quite tractable model for spreading phenomena is proposed, which contains as a special case the Eden model and a model for epidemics. Two exponents are defined, one static and one kinetic ("growth"). The surprising feature is that the kinetic exponent can be continuously tuned while the static one does not change. Thus the dynamic universality classes are quite independent of the static one. This is the first one-cluster growth model showing dynamic universality classes unrelated to static ones and thereby yields insight into a generic feature for growth models.

Irreversibility stabilizes certain locally interacting discrete systems against the nucleation and growth of a most-stable phase, thereby enabling them to behave in a computationally complex and nonergodic manner over a set of positive measure in the parameter space of their local transition probabilities, unlike analogous reversible systems.

For a closed system of two coupled nonlinear oscillators, chaotic orbits are punctuated by holes associated with stable periodic orbits. For the corresponding class of Hamiltonian maps we demonstrate that the combined area for all holes of size $\epsilon$ or greater scales as a power law with exponent $\beta$ and asymptotic area $0<\mathrm{}\mu <1\mathrm{}$. In contrast to previous results, this is a global scaling property, valid for a set of positive Lebesgue measure. It suggests that these chaotic orbits are fat fractals, i.e., Cantor-set-like objects of positive area. We numerically compute lower bounds on their area.

We study an extension of Lévy flight to include self-repulsion in the path of the walk. We call the extension node-avoiding Lévy flight and we show its equivalence to the $n\to 0$ limit of a statistical mechanical model for a magnetic system with long-range interactions between the spins. By use of this equivalence we are able to make a detailed comparison between the results of the $\epsilon$ expansion for the magnetic model, a Monte Carlo simulation of the Lévy flight model, and the results of a Flory-type argument. This is the first comparison of the $\epsilon$ expansion for $\epsilon \ll 1$ with a numerical simulation for any model. Some speculations are made on applications of the model of node-avoiding Lévy flight.

The asymptotic expansions derived for high-frequency scattering by cylinders and spheres are reexamined in detail. It is found that earlier results require small but significant corrections. The accuracy of the new analytical results is confirmed by comparison with numerical evaluations.

For field equations of Hamiltonian form the relation between constants of motion and solutions of the linearized equation is discussed. A known result is that the Poisson bracket of a constant of motion with the field variable solves the linearized equation. Here the following converse result is obtained: If a δu which satisfies the linear equation is of the form δu=[u,T], then T is a constant of motion. Further, ∂T/∂t is also a constant. The sequence [T,∂T/∂t], [[T,∂T/∂t],T],. . . is shown to produce other nontrivial constants.

Randomness and chaos in physical systems are usually ultimately attributed to external noise. But it is argued here that even without such random input, the intrinsic behavior of many nonlinear systems can be computationally so complicated as to seem random in all practical experiments. This effect is suggested as the basic origin of such phenomena as fluid turbulence.

It is shown that (i) the N-component Ashkin-Teller model in d=2 has continuous O(N) symmetry, (ii) its critical properties near the decoupled Ising transition are determined by the massive Gross-Neveu [(ψ¯ψ${)}^2$] model in general and the (integrable) massless version along a line, (iii) the known results for the massless Gross-Neveu model imply that the first-order transition found as N→∞ persists down to N=3 along this line, (iv) the supersymmetry of the N=3 model implies that the leading singularity of the free energy is zero along the first-order line, and (v) the N=4 model along this line decouples into two identical N=2 models up to subleading corrections.