881 research outputs found
Self-Attracting Walk on Lattices
We have studied a model of self-attracting walk proposed by Sapozhnikov using
Monte Carlo method. The mean square displacement
and the mean number of visited sites are calculated for
one-, two- and three-dimensional lattice. In one dimension, the walk shows
diffusive behaviour with . However, in two and three dimension, we
observed a non-universal behaviour, i.e., the exponent varies
continuously with the strength of the attracting interaction.Comment: 6 pages, latex, 6 postscript figures, Submitted J.Phys.
Minimum spanning trees on random networks
We show that the geometry of minimum spanning trees (MST) on random graphs is
universal. Due to this geometric universality, we are able to characterise the
energy of MST using a scaling distribution () found using uniform
disorder. We show that the MST energy for other disorder distributions is
simply related to . We discuss the relationship to invasion
percolation (IP), to the directed polymer in a random media (DPRM) and the
implications for the broader issue of universality in disordered systems.Comment: 4 pages, 3 figure
Structural Properties of Self-Attracting Walks
Self-attracting walks (SATW) with attractive interaction u > 0 display a
swelling-collapse transition at a critical u_{\mathrm{c}} for dimensions d >=
2, analogous to the \Theta transition of polymers. We are interested in the
structure of the clusters generated by SATW below u_{\mathrm{c}} (swollen
walk), above u_{\mathrm{c}} (collapsed walk), and at u_{\mathrm{c}}, which can
be characterized by the fractal dimensions of the clusters d_{\mathrm{f}} and
their interface d_{\mathrm{I}}. Using scaling arguments and Monte Carlo
simulations, we find that for u<u_{\mathrm{c}}, the structures are in the
universality class of clusters generated by simple random walks. For
u>u_{\mathrm{c}}, the clusters are compact, i.e. d_{\mathrm{f}}=d and
d_{\mathrm{I}}=d-1. At u_{\mathrm{c}}, the SATW is in a new universality class.
The clusters are compact in both d=2 and d=3, but their interface is fractal:
d_{\mathrm{I}}=1.50\pm0.01 and 2.73\pm0.03 in d=2 and d=3, respectively. In
d=1, where the walk is collapsed for all u and no swelling-collapse transition
exists, we derive analytical expressions for the average number of visited
sites and the mean time to visit S sites.Comment: 15 pages, 8 postscript figures, submitted to Phys. Rev.
Infinite-cluster geometry in central-force networks
We show that the infinite percolating cluster (with density P_inf) of
central-force networks is composed of: a fractal stress-bearing backbone (Pb)
and; rigid but unstressed ``dangling ends'' which occupy a finite
volume-fraction of the lattice (Pd). Near the rigidity threshold pc, there is
then a first-order transition in P_inf = Pd + Pb, while Pb is second-order with
exponent Beta'. A new mean field theory shows Beta'(mf)=1/2, while simulations
of triangular lattices give Beta'_tr = 0.255 +/- 0.03.Comment: 6 pages, 4 figures, uses epsfig. Accepted for publication in Physical
Review Letter
Random manifolds in non-linear resistor networks: Applications to varistors and superconductors
We show that current localization in polycrystalline varistors occurs on
paths which are, usually, in the universality class of the directed polymer in
a random medium. We also show that in ceramic superconductors, voltage
localizes on a surface which maps to an Ising domain wall. The emergence of
these manifolds is explained and their structure is illustrated using direct
solution of non-linear resistor networks
Complete wetting in the three-dimensional transverse Ising model
We consider a three-dimensional Ising model in a transverse magnetic field,
and a bulk field . An interface is introduced by an appropriate choice
of boundary conditions. At the point spin configurations
corresponding to different positions of the interface are degenerate. By
studying the phase diagram near this multiphase point using quantum-mechanical
perturbation theory we show that that quantum fluctuations, controlled by ,
split the multiphase degeneracy giving rise to an infinite sequence of layering
transitions.Comment: 16 pages (revtex) including 8 figs; to appear in J. Stat. Phy
Swelling-collapse transition of self-attracting walks
We study the structural properties of self-attracting walks in d dimensions
using scaling arguments and Monte Carlo simulations. We find evidence for a
transition analogous to the \Theta transition of polymers. Above a critical
attractive interaction u_c, the walk collapses and the exponents \nu and k,
characterising the scaling with time t of the mean square end-to-end distance
~ t^{2 \nu} and the average number of visited sites ~ t^k, are
universal and given by \nu=1/(d+1) and k=d/(d+1). Below u_c, the walk swells
and the exponents are as with no interaction, i.e. \nu=1/2 for all d, k=1/2 for
d=1 and k=1 for d >= 2. At u_c, the exponents are found to be in a different
universality class.Comment: 6 pages, 5 postscript figure
Combinatorial models of rigidity and renormalization
We first introduce the percolation problems associated with the graph
theoretical concepts of -sparsity, and make contact with the physical
concepts of ordinary and rigidity percolation. We then devise a renormalization
transformation for -percolation problems, and investigate its domain of
validity. In particular, we show that it allows an exact solution of
-percolation problems on hierarchical graphs, for . We
introduce and solve by renormalization such a model, which has the interesting
feature of showing both ordinary percolation and rigidity percolation phase
transitions, depending on the values of the parameters.Comment: 22 pages, 6 figure
Stressed backbone and elasticity of random central-force systems
We use a new algorithm to find the stress-carrying backbone of ``generic''
site-diluted triangular lattices of up to 10^6 sites. Generic lattices can be
made by randomly displacing the sites of a regular lattice. The percolation
threshold is Pc=0.6975 +/- 0.0003, the correlation length exponent \nu =1.16
+/- 0.03 and the fractal dimension of the backbone Db=1.78 +/- 0.02. The number
of ``critical bonds'' (if you remove them rigidity is lost) on the backbone
scales as L^{x}, with x=0.85 +/- 0.05. The Young's modulus is also calculated.Comment: 5 pages, 5 figures, uses epsfi
Fast Algorithms For Josephson Junction Arrays : Bus--bars and Defects
We critically review the fast algorithms for the numerical study of
two--dimensional Josephson junction arrays and develop the analogy of such
systems with electrostatics. We extend these procedures to arrays with
bus--bars and defects in the form of missing bonds. The role of boundaries and
of the guage choice in determing the Green's function of the system is
clarified. The extension of the Green's function approach to other situations
is also discussed.Comment: Uuencoded 1 Revtex file (11 Pages), 3 Figures : Postscript Uuencode
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