19 research outputs found
Four-dimensional polymer collapse II: Interacting self-avoiding trails
We have simulated four-dimensional interacting self-avoiding trails (ISAT) on
the hyper-cubic lattice with standard interactions at a wide range of
temperatures up to length 4096 and at some temperatures up to length 16384. The
results confirm the earlier prediction (using data from a non-standard model at
a single temperature) of a collapse phase transition occurring at finite
temperature. Moreover they are in accord with the phenomenological theory
originally proposed by Lifshitz, Grosberg and Khokhlov in three dimensions and
recently given new impetus by its use in the description of simulational
results for four-dimensional interacting self-avoiding walks (ISAW). In fact,
we argue that the available data is consistent with the conclusion that the
collapse transitions of ISAT and ISAW lie in the same universality class, in
contradiction with long-standing predictions. We deduce that there exists a
pseudo-first order transition for ISAT in four dimensions at finite lengths
while the thermodynamic limit is described by the standard polymer mean-field
theory (giving a second-order transition), in contradiction to the prediction
that the upper critical dimension for ISAT is .Comment: 23 pages, 8 figure
Collapse transition of self-avoiding trails on the square lattice
The collapse transition of an isolated polymer has been modelled by many
different approaches, including lattice models based on self-avoiding walks and
self-avoiding trails. In two dimensions, previous simulations of kinetic growth
trails, which map to a particular temperature of interacting self-avoiding
trails, showed markedly different behaviour for what was argued to be the
collapse transition than that which has been verified for models based of
self-avoiding walks. On the other hand, it has been argued that kinetic growth
trails represent a special simulation that does not give the correct picture of
the standard equilibrium model. In this work we simulate the standard
equilibrium interacting self-avoiding trail model on the square lattice up to
lengths over steps and show that the results of the kinetic growth
simulations are, in fact, entirely in accord with standard simulations of the
temperature dependent model. In this way we verify that the collapse transition
of interacting self-avoiding walks and trails are indeed in different
universality classes in two dimensions
Critical Percolation in High Dimensions
We present Monte Carlo estimates for site and bond percolation thresholds in
simple hypercubic lattices with 4 to 13 dimensions. For d<6 they are
preliminary, for d >= 6 they are between 20 to 10^4 times more precise than the
best previous estimates. This was achieved by three ingredients: (i) simple and
fast hashing which allowed us to simulate clusters of millions of sites on
computers with less than 500 MB memory; (ii) a histogram method which allowed
us to obtain information for several p values from a single simulation; and
(iii) a new variance reduction technique which is especially efficient at high
dimensions where it reduces error bars by a factor up to approximately 30 and
more. Based on these data we propose a new scaling law for finite cluster size
corrections.Comment: 5 pages including figures, RevTe
Stretched Polymers in a Poor Solvent
Stretched polymers with attractive interaction are studied in two and three
dimensions. They are described by biased self-avoiding random walks with
nearest neighbour attraction. The bias corresponds to opposite forces applied
to the first and last monomers. We show that both in and a phase
transition occurs as this force is increased beyond a critical value, where the
polymer changes from a collapsed globule to a stretched configuration. This
transition is second order in and first order in . For we
predict the transition point quantitatively from properties of the unstretched
polymer. This is not possible in , but even there we can estimate the
transition point precisely, and we can study the scaling at temperatures
slightly below the collapse temperature of the unstretched polymer. We find
very large finite size corrections which would make very difficult the estimate
of the transition point from straightforward simulations.Comment: 10 pages, 16 figure
Order Parameters of the Dilute A Models
The free energy and local height probabilities of the dilute A models with
broken \Integer_2 symmetry are calculated analytically using inversion and
corner transfer matrix methods. These models possess four critical branches.
The first two branches provide new realisations of the unitary minimal series
and the other two branches give a direct product of this series with an Ising
model. We identify the integrable perturbations which move the dilute A models
away from the critical limit. Generalised order parameters are defined and
their critical exponents extracted. The associated conformal weights are found
to occur on the diagonal of the relevant Kac table. In an appropriate regime
the dilute A model lies in the universality class of the Ising model in a
magnetic field. In this case we obtain the magnetic exponent
directly, without the use of scaling relations.Comment: 53 pages, LaTex, ITFA 93-1
Mechanical unfolding of directed polymers in a poor solvent: Critical exponents
We study the thermodynamics of an exactly solvable model of a self-interacting, partially directed self-avoiding walk in two dimensions when a force is applied on one end of the chain. The critical force for the unfolding is determined exactly, as a function of the temperature, below the Theta transition. The transition is of second order and is characterized by new critical exponents that are determined by a careful numerical analysis. The usual polymer critical index nu on the critical line, and another one which we call zeta, takes a nontrivial value that is numerically close to 2/3