181 research outputs found
Stretching of a chain polymer adsorbed at a surface
In this paper we present simulations of a surface-adsorbed polymer subject to
an elongation force. The polymer is modelled by a self-avoiding walk on a
regular lattice. It is confined to a half-space by an adsorbing surface with
attractions for every vertex of the walk visiting the surface, and the last
vertex is pulled perpendicular to the surface by a force. Using the recently
proposed flatPERM algorithm, we calculate the phase diagram for a vast range of
temperatures and forces. The strength of this algorithm is that it computes the
complete density of states from one single simulation. We simulate systems of
sizes up to 256 steps.Comment: 13 pages, 7 figure
On trivial words in finitely presented groups
We propose a numerical method for studying the cogrowth of finitely presented
groups. To validate our numerical results we compare them against the
corresponding data from groups whose cogrowth series are known exactly.
Further, we add to the set of such groups by finding the cogrowth series for
Baumslag-Solitar groups and prove
that their cogrowth rates are algebraic numbers.Comment: This article has been rewritten as two separate papers, with improved
exposition. The new papers are arXiv:1309.4184 and arXiv:1312.572
Scaling of the atmosphere of self-avoiding walks
The number of free sites next to the end of a self-avoiding walk is known as
the atmosphere. The average atmosphere can be related to the number of
configurations. Here we study the distribution of atmospheres as a function of
length and how the number of walks of fixed atmosphere scale. Certain bounds on
these numbers can be proved. We use Monte Carlo estimates to verify our
conjectures. Of particular interest are walks that have zero atmosphere, which
are known as trapped. We demonstrate that these walks scale in the same way as
the full set of self-avoiding walks, barring an overall constant factor
Permutations generated by a depth 2 stack and an infinite stack in series are algebraic
Β© 2015, Australian National University. All rights reserved. We prove that the class of permutations generated by passing an ordered sequence 12... n through a stack of depth 2 and an in nite stack in series is in bi-jection with an unambiguous context-free language, where a permutation of length n is encoded by a string of length 3n. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free grammar to compute the generating function:(formula presented) where cn is the number of permutations of length n that can be generated, and (formula presented) is a simple variant of the Catalan generating function. This in turn implies that (formula presented
On the universality of knot probability ratios
Let denote the number of self-avoiding polygons of length on a
regular three-dimensional lattice, and let be the number which have
knot type . The probability that a random polygon of length has knot
type is and is known to decay exponentially with length.
Little is known rigorously about the asymptotics of , but there is
substantial numerical evidence that grows as , as , where is the
number of prime components of the knot type . It is believed that the
entropic exponent, , is universal, while the exponential growth rate,
, is independent of the knot type but varies with the lattice.
The amplitude, , depends on both the lattice and the knot type.
The above asymptotic form implies that the relative probability of a random
polygon of length having prime knot type over prime knot type is
. In the thermodynamic limit this probability ratio becomes an amplitude
ratio; it should be universal and depend only on the knot types and . In
this letter we examine the universality of these probability ratios for
polygons in the simple cubic, face-centered cubic, and body-centered cubic
lattices. Our results support the hypothesis that these are universal
quantities. For example, we estimate that a long random polygon is
approximately 28 times more likely to be a trefoil than be a figure-eight,
independent of the underlying lattice, giving an estimate of the intrinsic
entropy associated with knot types in closed curves.Comment: 8 pages, 6 figures, 1 tabl
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