2,606 research outputs found
Exact enumeration of Hamiltonian circuits, walks, and chains in two and three dimensions
We present an algorithm for enumerating exactly the number of Hamiltonian
chains on regular lattices in low dimensions. By definition, these are sets of
k disjoint paths whose union visits each lattice vertex exactly once. The
well-known Hamiltonian circuits and walks appear as the special cases k=0 and
k=1 respectively. In two dimensions, we enumerate chains on L x L square
lattices up to L=12, walks up to L=17, and circuits up to L=20. Some results
for three dimensions are also given. Using our data we extract several
quantities of physical interest
2-Dimensional Polymers Confined in a Strip
Single two dimensional polymers confined to a strip are studied by Monte
Carlo simulations. They are described by N-step self-avoiding random walks on a
square lattice between two parallel hard walls with distance 1 << D << N^\nu
(\nu = 3/4 is the Flory exponent). For the simulations we employ the
pruned-enriched-Rosenbluth method (PERM) with Markovian anticipation. We
measure the densities of monomers and of end points as functions of the
distance from the walls, the longitudinal extent of the chain, and the forces
exerted on the walls. Their scaling with D and the universal ratio between
force and monomer density at the wall are compared to theoretical predictions.Comment: 5 pages RevTex, 7 figures include
Secondary Structures in Long Compact Polymers
Compact polymers are self-avoiding random walks which visit every site on a
lattice. This polymer model is used widely for studying statistical problems
inspired by protein folding. One difficulty with using compact polymers to
perform numerical calculations is generating a sufficiently large number of
randomly sampled configurations. We present a Monte-Carlo algorithm which
uniformly samples compact polymer configurations in an efficient manner
allowing investigations of chains much longer than previously studied. Chain
configurations generated by the algorithm are used to compute statistics of
secondary structures in compact polymers. We determine the fraction of monomers
participating in secondary structures, and show that it is self averaging in
the long chain limit and strictly less than one. Comparison with results for
lattice models of open polymer chains shows that compact chains are
significantly more likely to form secondary structure.Comment: 14 pages, 14 figure
Crossover phenomena involving the dense O() phase
We explore the properties of the low-temperature phase of the O() loop
model in two dimensions by means of transfer-matrix calculations and
finite-size scaling. We determine the stability of this phase with respect to
several kinds of perturbations, including cubic anisotropy, attraction between
loop segments, double bonds and crossing bonds. In line with Coulomb gas
predictions, cubic anisotropy and crossing bonds are found to be relevant and
introduce crossover to different types of behavior. Whereas perturbations in
the form of loop-loop attractions and double bonds are irrelevant, sufficiently
strong perturbations of these types induce a phase transition of the Ising
type, at least in the cases investigated. This Ising transition leaves the
underlying universal low-temperature O() behavior unaffected.Comment: 12 pages, 8 figure
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