We argue that microcanonical Monte Carlo techniques can provide direct information on the values of renormalized coupling constants in numerical renormalization-group studies. The method is tested on SU(2) lattice gauge theory with fundamental and adjoint couplings and on the two-dimensional O(3) Heisenberg model.

We present a reanalysis of the renormalization-group calculation to first order in $\epsilon =6-d$, where $d$ is the spatial dimensionality, of the exponent, $t$, which describes the behavior of the conductivity of a percolating network at the percolation threshold. If we set $t=(d-2\mathrm{})\mathrm{}{\nu }_p+\zeta$, where ${\nu }_p$ is the correlation-length exponent, then our result is $\zeta =1+\left(\frac{\epsilon }{42}\right)$. This result clarifies several previously paradoxical results concerning resistor networks and shows that the Alexander-Orbach relation breaks down at order $\epsilon$.

The Suzuki-Trotter formula has been used to get the $m$ th approximant to the classical representation of the partition function of the $N$-spin $S=\frac{1}{2}$ $\mathrm{XY}$ chain. The equivalent two-dimensional ($N\times 2m$) Ising model with four-spin interactions has been studied in detail by use of the numerically exact transfer-matrix method for $T>~0.05$ and $m<~8$. The limiting $m\to \infty$ behavior has been estimated from the extrapolation formula of the form $E\left(m\right)=E\left(\infty \right)+\frac{a}{m^2}$. The limiting value of the energy agrees with the exact solution of Katsura.

The velocity autocorrelation of the one-dimensional stochastic Lorentz model is analyzed both numerically and theoretically. It is found that the tail becomes proportional to $-t^{-[\frac{\mathrm{}(1-D)\mathrm{}}{2}+1]}$ in the case that impurities distribute fractally with the dimension $D$. The usual analyses are included for the special case $D=0\mathrm{}$.

Some solvable fractal models of the percolating backbone are used to investigate the critical behavior of a random, $d$-dimensional, isotropic, elastic medium. The critical exponent $T$ for the elastic moduli is found to be appreciably greater than the conductivity exponent $t$, and the ratio of bulk to shear modulus is found to have the universal value $\frac{4}{d}$. A comparison with the effective-medium and the Clausius-Mossotti-type approximations leads to the conjecture that the result $\frac{4}{d}$ is in fact exact.

The vacuum expectation value of the stress-energy tensor for the Hartle-Hawking state in Schwarzschild space-time has been calculated for the conformal scalar field. $〈T_{\mu }^{\nu }〉$ separates naturally into the sum of two terms. The first coincides with an approximate expression suggested by Page. The second term is a "remainder" that may be evaluated numerically. The total expression is in good qualitative agreement with Page's approximation. These results are at variance with earlier results by Fawcett whose error is explained.

We introduce an exact enumeration method for calculating the survival probability $P\left(n\right)\mathrm{}$ for the $n$-step random walker on a lattice with randomly distributed high-concentration traps, $c$. Using it we show that our data scale as $\ln P\left(n\right)\mathrm{}\simeq a\rho$, where $\rho ={[-\ln \mathrm{}(1-c\left)\right]}^{\frac{2}{\mathrm{}(D+2\mathrm{})\mathrm{}}}{n}^{\frac{D}{\mathrm{}(D+2\mathrm{})\mathrm{}}}$, when $\rho \gtrsim 10$ in $D=2\mathrm{}$ and $D=3\mathrm{}$ dimensions. This value of $\rho$ corresponds to $P\left(n\right)\mathrm{}\lesssim {10}^{-13\mathrm{}}$.

In order to solve multidimensional tunneling problems that cannot be treated by the normal instanton techniques, we introduce the path decomposition expansion formalism, and show its usefulness by solving three generic examples: the symmetric and asymmetric double well, and the decay problem. The technique allows us to handle excited states and backscattering effects in nonseparable potentials.

Growth is described in terms of an active zone defined as the region where new particles join the existing cluster. This zone is characterized by the probability $P(r,N)\mathrm{}\mathrm{dr}$ that the $N\mathrm{th}$ particle is deposited a distance $r$ from the center of mass. Our Monte Carlo simulations for two-dimensional diffusion-limited aggregation and the Eden process show that, for large $N$, $P(r,N)={\left[{\left(2\mathrm{}\pi \right)}^{\frac{1}{2}}{\xi }_N\right]}^{-1\mathrm{}}\exp [-\frac{{(r-{\bar{r}}_N)}^2}{2{\xi }_N^2}]$, where ${\bar{r}}_N\sim N^{\nu }$ and ${\xi }_N\sim N^{\bar{\nu }}$. We find $\bar{\nu }<\nu$ indicating the presence of two distinctly diverging lengths in these processes.

Schrödinger's equation and the heat equation are connected to one another by analytic continuation, as are the Feynman integral and the Wiener integral. We extend this form of correspondence to the relativistic domain by identifying Poisson processes whose real version gives rise to the telegrapher's equation and which when analytically continued produce the Dirac equation.

Evidence is presented for oscillatory and chaotic behavior of the electrical conduction measured in barium sodium niobate single crystals at elevated temperatures. Analysis of the chaotic state verifies the existence of a strange attractor, whose dimension and entropy are determined. The observations indicate a Ruelle-Takens-Newhouse scenario.

Consideration is given to equations of the form $\frac{\partial n}{\partial t}=D{\nabla }^{2}n+Q\left(n\right)\mathrm{}$; the reaction-rate $Q\left(n\right)\mathrm{}$ is such that $\frac{d^{2}Q}{d{n}^2}<~0$ for all values of $n>~0$. A lower bound $n_{-}=n_{-}(\overrightarrow{\mathrm{x}},t)\equiv 〈\gamma 〉 (<~n)$ is obtained as the expectation value of the solution to the associated Brownian-motion equation $\frac{\partial \gamma }{\partial t}=-\overrightarrow{\mathrm{u}}\mathrm{}\left(t\right)\mathrm{}·\nabla \gamma +Q\left(\gamma \right)$, where $\gamma =\gamma (\overrightarrow{\mathrm{x}},t)$ is an auxiliary stochastic variable and the random velocity $\overrightarrow{\mathrm{u}}\mathrm{}\left(t\right)\mathrm{}$ is Gaussian. An upper bound $n_{+} (>~n)$ emerges as a concomitant of $n_{-}$, and an approximate analytical solution is shown to be given by $\tilde{n}\equiv \frac{1}{2}(n_{-}+n_{+})$ with the accuracy bound $|n-\tilde{n}|<~\frac{1}{2}(n_{+}-n_{-})$ expressed analytically.

Time development of cellular automata in $d$ dimensions is mapped onto equilibrium statistical mechanics of Ising models in $d+1\mathrm{}$ dimensions. Directed percolation is equivalent to a cellular automaton, and thus to an Ising model. For a particular case of directed percolation we find ${\nu }_{\parallel }=2\mathrm{}$, ${\nu }_{\perp }=1\mathrm{}$, ${\eta }_{\perp }=0\mathrm{}$.

We prove Birkhoff's theorem in conformal gravity: The most general spherically symmetric, electrovac solution is static, and hence absolutely stable against spherically symmetric perturbations. This result goes beyond the zero-energy theorem since it applies even when the solution is not asymptotically flat. On the basis of the known examples, we conjecture that any field theory lacking linearized spin-0 modes possesses a Birkhoff theorem.

We derive the effective Lagrangian which describes the interaction between a charged particle and a magnetic moment in the nonrelativistic limit. It is shown that neutral particles with a magnetic moment will exhibit the Aharonov-Bohm effect in certain circumstances. We suggest several types of experiments.

We compute numerically the eigenvalues of a family of two-dimensional Hamiltonians which give rise to regular or chaotic classical motion depending on the particular choice of parameters. A close relationship is found between the spectral statistics and the fraction of classical phase space covered by chaotic trajectories. In the extreme regular and chaotic cases the system displays Poisson and Gaussian-orthogonal-ensemble statistics, respectively. A one-parameter random-matrix model is proposed to describe the spectral statistics in intermediate situations.

A simple and direct method is described for constructing exact quasiperiodic solutions of the focusing nonlinear Schrödinger equation. An algebraic constraint is imposed on the initial values of a set of auxiliary variables. The solution, expressed in terms of these auxiliary variables, is quasiperiodic in general, and represents a wider class of solutions than those obtained from the inverse-scattering transform. As an example of our method, we present solutions corresponding to a modulated plane wave.

An exact path-integral solution of the Dirac equation with a Coulomb potential is found. The energy-dependent Green's function of the second-order Dirac-Coulomb equation, expressed as a polar-coordinate path integral, is reduced to the exactly solvable path integral for an isotropic harmonic oscillator by a coordinate transformation combined with a local time rescaling. The Green's function of the linear Dirac equation is evaluated with the help of the Biedenharn transformation. The energy spectrum for the bound states is also obtained.

We obtain the rule governing many-body wave functions for particles obeying fractional statistics in two (space) dimensions. It generalizes and continuously interpolates the usual symmetrization and antisymmetrization. Quantum mechanics of more than two particles is discussed and some new features are found.

We show that the collision time may be interpreted as the time average of a flux-flux correlation function. This interpretation leads quantum mechanically to a complex time whose real part is identical to the usual definition as provided by F. T. Smith. The imaginary part is identical, in the semiclassical limit, to the imaginary time associated with tunneling.