12,647 research outputs found
The traveling salesman problem, conformal invariance, and dense polymers
We propose that the statistics of the optimal tour in the planar random
Euclidean traveling salesman problem is conformally invariant on large scales.
This is exhibited in power-law behavior of the probabilities for the tour to
zigzag repeatedly between two regions, and in subleading corrections to the
length of the tour. The universality class should be the same as for dense
polymers and minimal spanning trees. The conjectures for the length of the tour
on a cylinder are tested numerically.Comment: 4 pages. v2: small revisions, improved argument about dimensions d>2.
v3: Final version, with a correction to the form of the tour length in a
domain, and a new referenc
Unbiased sampling of globular lattice proteins in three dimensions
We present a Monte Carlo method that allows efficient and unbiased sampling
of Hamiltonian walks on a cubic lattice. Such walks are self-avoiding and visit
each lattice site exactly once. They are often used as simple models of
globular proteins, upon adding suitable local interactions. Our algorithm can
easily be equipped with such interactions, but we study here mainly the
flexible homopolymer case where each conformation is generated with uniform
probability. We argue that the algorithm is ergodic and has dynamical exponent
z=0. We then use it to study polymers of size up to 64^3 = 262144 monomers.
Results are presented for the effective interaction between end points, and the
interaction with the boundaries of the system
On the universality of compact polymers
Fully packed loop models on the square and the honeycomb lattice constitute
new classes of critical behaviour, distinct from those of the low-temperature
O(n) model. A simple symmetry argument suggests that such compact phases are
only possible when the underlying lattice is bipartite. Motivated by the hope
of identifying further compact universality classes we therefore study the
fully packed loop model on the square-octagon lattice. Surprisingly, this model
is only critical for loop weights n < 1.88, and its scaling limit coincides
with the dense phase of the O(n) model. For n=2 it is exactly equivalent to the
selfdual 9-state Potts model. These analytical predictions are confirmed by
numerical transfer matrix results. Our conclusions extend to a large class of
bipartite decorated lattices.Comment: 13 pages including 4 figure
Selfduality for coupled Potts models on the triangular lattice
We present selfdual manifolds for coupled Potts models on the triangular
lattice. We exploit two different techniques: duality followed by decimation,
and mapping to a related loop model. The latter technique is found to be
superior, and it allows to include three-spin couplings. Starting from three
coupled models, such couplings are necessary for generating selfdual solutions.
A numerical study of the case of two coupled models leads to the identification
of novel critical points
Exact valence bond entanglement entropy and probability distribution in the XXX spin chain and the Potts model
By relating the ground state of Temperley-Lieb hamiltonians to partition
functions of 2D statistical mechanics systems on a half plane, and using a
boundary Coulomb gas formalism, we obtain in closed form the valence bond
entanglement entropy as well as the valence bond probability distribution in
these ground states. We find in particular that for the XXX spin chain, the
number N_c of valence bonds connecting a subsystem of size L to the outside
goes, in the thermodynamic limit, as = (4/pi^2) ln L, disproving a recent
conjecture that this should be related with the von Neumann entropy, and thus
equal to 1/(3 ln 2) ln L. Our results generalize to the Q-state Potts model.Comment: 4 pages, 2 figure
The packing of two species of polygons on the square lattice
We decorate the square lattice with two species of polygons under the
constraint that every lattice edge is covered by only one polygon and every
vertex is visited by both types of polygons. We end up with a 24 vertex model
which is known in the literature as the fully packed double loop model. In the
particular case in which the fugacities of the polygons are the same, the model
admits an exact solution. The solution is obtained using coordinate Bethe
ansatz and provides a closed expression for the free energy. In particular we
find the free energy of the four colorings model and the double Hamiltonian
walk and recover the known entropy of the Ice model. When both fugacities are
set equal to two the model undergoes an infinite order phase transition.Comment: 21 pages, 4 figure
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
Combining tree decomposition and transfer matrix techniques provides a very
general algorithm for computing exact partition functions of statistical models
defined on arbitrary graphs. The algorithm is particularly efficient in the
case of planar graphs. We illustrate it by computing the Potts model partition
functions and chromatic polynomials (the number of proper vertex colourings
using Q colours) for large samples of random planar graphs with up to N=100
vertices. In the latter case, our algorithm yields a sub-exponential average
running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the
exponential running time ~ exp(0.245 N) provided by the hitherto best known
algorithm. We study the statistics of chromatic roots of random planar graphs
in some detail, comparing the findings with results for finite pieces of a
regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded.
Version 3 shows that the worst-case running time is sub-exponential in the
number of vertice
Role of conformational entropy in force-induced bio-polymer unfolding
A statistical mechanical description of flexible and semi-flexible polymer
chains in a poor solvent is developed in the constant force and constant
distance ensembles. We predict the existence of many intermediate states at low
temperatures stabilized by the force. A unified response to pulling and
compressing forces has been obtained in the constant distance ensemble. We show
the signature of a cross-over length which increases linearly with the chain
length. Below this cross-over length, the critical force of unfolding decreases
with temperature, while above, it increases with temperature. For stiff chains,
we report for the first time "saw-tooth" like behavior in the force-extension
curves which has been seen earlier in the case of protein unfolding.Comment: 4 pages, 5 figures, ReVTeX4 style. Accepted in Phys. Rev. Let
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