Exact results for average cluster numbers in bond percolation on infinite-length lattice strips.

We calculate exact analytic expressions for the average cluster numbers 〈k〉_{Λ_{s}} on infinite-length strips Λ_{s}, with various widths, of several different lattices, as functions of the bond occupation probability p. It is proved that these expressions are rational functions of p. As special cases of our results, we obtain exact values of 〈k〉_{Λ_{s}} and derivatives of 〈k〉_{Λ_{s}} with respect to p, evaluated at the critical percolation probabilities p_{c,Λ} for the corresponding infinite two-dimensional lattices Λ. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in 〈k〉_{Λ_{s}} determine the radii of convergence of series expansions for small p and for p near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.

Improved lower bounds on the ground-state entropy of the antiferromagnetic Potts model.

We present generalized methods for calculating lower bounds on the ground-state entropy per site, S(0), or equivalently, the ground-state degeneracy per site, W=e(S(0)/k(B)), of the antiferromagnetic Potts model. We use these methods to derive improved lower bounds on W for several lattices.

Ground state entropy of the Potts antiferromagnet on homeomorphic expansions of kagomé lattice strips.

We present exact calculations of the chromatic polynomial and resultant ground state entropy of the q-state Potts antiferromagnet on lattice strips that are homeomorphic expansions of a strip of the kagomé lattice. The dependence of the ground state entropy on the form of homeomorphic expansion is elucidated.

Lower bounds on the ground-state entropy of the Potts antiferromagnet on slabs of the simple cubic lattice.

We calculate rigorous lower bounds for the ground-state degeneracy per site, W, of the q-state Potts antiferromagnet on slabs of the simple cubic lattice that are infinite in two directions and finite in the third and that thus interpolate between the square (sq) and simple cubic (sc) lattices. We give a comparison with large-q series expansions for the sq and sc lattices and also present numerical comparisons.

Exact results for average cluster numbers in bond percolation on lattice strips.

We present exact calculations of the average number of connected clusters per site, , as a function of bond occupation probability p, for the bond percolation problem on infinite-length strips of finite width Ly, of the square, triangular, honeycomb, and kagomé lattices Lambda with various boundary conditions. These are used to study the approach of , for a given p and Lambda, to its value on the two-dimensional lattice as the strip width increases. We investigate the singularities of in the complex p plane and their influence on the radii of convergence of the Taylor series expansions of about p=0 and p=1 .

Dynamical symmetry breaking of extended gauge symmetries.

We construct asymptotically free gauge theories exhibiting dynamical breaking of the left-right gauge group G(LR)=SU(3)(c) x SU(2)(L) x SU(2)(R) x U(1)(B-L), and its extension to the Pati-Salam gauge group G(422)=SU(4)(PS) x SU(2)(L) x SU(2)(R). The models incorporate technicolor for electroweak breaking, and extended technicolor for the breaking of G(LR) and G422 and the generation of fermion masses. They include a seesaw mechanism for neutrino masses, without a grand unified theory (GUT) scale. These models explain why G(LR) and G422 break to SU(3)(c) x SU(2)(L) x U(1)(Y), and why this takes place at a scale (approximately 10(3) TeV) large compared to the electroweak scale, but much smaller than a GUT scale.

n-n* oscillations in models with large extra dimensions.

We analyze n-n* oscillations in generic models with large extra dimensions in which standard-model fields propagate and fermion wave functions have strong localization. We find that in these models n-n* oscillations might occur at levels not too far below the current limit.