We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice $Lambda$ by $ell$ bonds connecting the same adjacent vertices, thereby yieldin
g the lattice $Lambda_ell$. This relation is used to calculate the bond percolation threshold on $Lambda_ell$. We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality $d ge 2$ but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the $N to infty$ limits of several families of $N$-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as $N to infty$.
We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We exhibit a weigh
ted chromatic polynomial $Ph(G,q,w)$ associated with this problem that generalizes the chromatic polynomial $P(G,q)$. General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for $Ph(G,q,w)$ for lattice strip graphs $G$ with periodic longitudinal boundary conditions. The zeros of $Ph(G,q,w)$ in the $q$ and $w$ planes and their accumulation sets in the limit of infinitely many vertices of $G$ are analyzed. Finally, some related weighted graph coloring problems are mentioned.
We reconsider the two different facets of $pi$ and $K$ mesons as $q bar q$ bound states and approximate Nambu-Goldstone bosons. We address several topics, including masses, mass splittings between $pi$ and $rho$ and between $K$ and $K^*$, meson wavef
unctions, charge radii, and the $K-pi$ wavefunction overlap.
The successful description of current data provided by the Standard Model includes fundamental fermions that are color-singlets and electroweak-nonsinglets, but no fermions that are electroweak-singlets and color-nonsinglets. In an effort to understa
nd the absence of such fermions, we construct and study {it gedanken} models that do contain electroweak-singlet chiral quark fields. These models exhibit several distinctive properties, including the absence of any neutral lepton and the fact that both the $(uud)$ and $(ddu)$ nucleons are electrically charged. We also explore how such models could arise as low-energy limits of grand unified theories and, in this more restrictive context, we show that they exhibit further exotic properties.
Because quarks and gluons are confined within hadrons, they have a maximum wavelength of order the confinement scale. Propagators, normally calculated for free quarks and gluons using Dyson-Schwinger equations, are modified by bound-state effects in
close analogy to the calculation of the Lamb shift in atomic physics. Because of confinement, the effective quantum chromodynamic coupling stays finite in the infrared. The quark condensate which arises from spontaneous chiral symmetry breaking in the bound state Dyson-Schwinger equation is the expectation value of the operator $bar q q$ evaluated in the background of the fields of the other hadronic constituents, in contrast to a true vacuum expectation value. Thus quark and gluon condensates reside within hadrons. The effects of instantons are also modified. We discuss the implications of the maximum quark and gluon wavelength for phenomena such as deep inelastic scattering and annihilation, the decay of heavy quarkonia, jets, and dimensional counting rules for exclusive reactions. We also discuss implications for the zero-temperature phase structure of a vectorial SU($N$) gauge theory with a variable number $N_f$ of massless fermions.
We extend to larger unification groups an earlier study exploring the possibility of unification of gauge symmetries in theories with dynamical symmetry breaking. Based on our results, we comment on the outlook for models that seek to achieve this type of unification.
We suggest a solution to the problem of some apparently excessive contributions to the cosmological constant from Standard-Model condensates.
We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable $u=e^{-4K}$, where $K=J/(k_BT)$, for nonzero external magnetic field, $H$. Exact results are given for the phase diagram in the $u$ plane
for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real $h=H/(k_BT)$, these results provide new insights into features of our earlier study of this case. We also consider complex $h=H/(k_BT)$ and $mu=e^{-2h}$. Calculations of complex-$u$ zeros of the partition function on sections of the square lattice are presented. For the case of imaginary $h$, i.e., $mu=e^{itheta}$, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the $u$ plane. We find that in this case, the phase boundary ${cal B}_u$ contains a real line segment extending through part of the physical ferromagnetic interval $0 le u le 1$, with a right-hand endpoint $u_{rhe}$ at the temperature for which the Yang-Lee edge singularity occurs at $mu=e^{pm itheta}$. Conformal field theory arguments are used to relate the singularities at $u_{rhe}$ and the Yang-Lee edge.
We calculate Lorentz-invariant and gauge-invariant quantities characterizing the product $sum_a D_R(T^a) F^a_{mu u}$, where $D_R(T^a)$ denotes the matrix for the generator $T^a$ in the representation $R=$ fundamental and adjoint, for color SU(3). We
also present analogous results for an SU(2) gauge theory.
