1,365 research outputs found
The Expected Perimeter in Eden and Related Growth Processes
Following Richardson and using results of Kesten on First-passage
percolation, we obtain an upper bound on the expected perimeter in an Eden
Growth Process. Using results of the author from a problem in Statistical
Mechanics, we show that the average perimeter of the lattice animals resulting
from a very natural family of "growth histories" does not obey a similar bound.Comment: 11 page
Exact Scaling Functions for Self-Avoiding Loops and Branched Polymers
It is shown that a recently conjectured form for the critical scaling
function for planar self-avoiding polygons weighted by their perimeter and area
also follows from an exact renormalization group flow into the branched polymer
problem, combined with the dimensional reduction arguments of Parisi and
Sourlas. The result is generalized to higher-order multicritical points,
yielding exact values for all their critical exponents and exact forms for the
associated scaling functions.Comment: 5 pages; v2: factors of 2 corrected; v.3: relation with existing
theta-point results clarified, some references added/update
Thin Animals
Lattice animals provide a discretized model for the theta transition
displayed by branched polymers in solvent. Exact graph enumeration studies have
given some indications that the phase diagram of such lattice animals may
contain two collapsed phases as well as an extended phase. This has not been
confirmed by studies using other means. We use the exact correspondence between
the q --> 1 limit of an extended Potts model and lattice animals to investigate
the phase diagram of lattice animals on phi-cubed random graphs of arbitrary
topology (``thin'' random graphs). We find that only a two phase structure
exists -- there is no sign of a second collapsed phase.
The random graph model is solved in the thermodynamic limit by saddle point
methods. We observe that the ratio of these saddle point equations give
precisely the fixed points of the recursion relations that appear in the
solution of the model on the Bethe lattice by Henkel and Seno. This explains
the equality of non-universal quantities such as the critical lines for the
Bethe lattice and random graph ensembles.Comment: Latex, 10 pages plus 6 ps/eps figure
Scaling prediction for self-avoiding polygons revisited
We analyse new exact enumeration data for self-avoiding polygons, counted by
perimeter and area on the square, triangular and hexagonal lattices. In
extending earlier analyses, we focus on the perimeter moments in the vicinity
of the bicritical point. We also consider the shape of the critical curve near
the bicritical point, which describes the crossover to the branched polymer
phase. Our recently conjectured expression for the scaling function of rooted
self-avoiding polygons is further supported. For (unrooted) self-avoiding
polygons, the analysis reveals the presence of an additional additive term with
a new universal amplitude. We conjecture the exact value of this amplitude.Comment: 17 pages, 3 figure
Expansion for -Core Percolation
The physics of -core percolation pertains to those systems whose
constituents require a minimum number of connections to each other in order
to participate in any clustering phenomenon. Examples of such a phenomenon
range from orientational ordering in solid ortho-para mixtures to
the onset of rigidity in bar-joint networks to dynamical arrest in
glass-forming liquids. Unlike ordinary () and biconnected ()
percolation, the mean field -core percolation transition is both
continuous and discontinuous, i.e. there is a jump in the order parameter
accompanied with a diverging length scale. To determine whether or not this
hybrid transition survives in finite dimensions, we present a expansion
for -core percolation on the -dimensional hypercubic lattice. We show
that to order the singularity in the order parameter and in the
susceptibility occur at the same value of the occupation probability. This
result suggests that the unusual hybrid nature of the mean field -core
transition survives in high dimensions.Comment: 47 pages, 26 figures, revtex
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