335 research outputs found

### Directed percolation near a wall

Series expansion methods are used to study directed bond percolation clusters
on the square lattice whose lateral growth is restricted by a wall parallel to
the growth direction. The percolation threshold $p_c$ is found to be the same
as that for the bulk. However the values of the critical exponents for the
percolation probability and mean cluster size are quite different from those
for the bulk and are estimated by $\beta_1 = 0.7338 \pm 0.0001$ and $\gamma_1 =
1.8207 \pm 0.0004$ respectively. On the other hand the exponent
$\Delta_1=\beta_1 +\gamma_1$ characterising the scale of the cluster size
distribution is found to be unchanged by the presence of the wall.
The parallel connectedness length, which is the scale for the cluster length
distribution, has an exponent which we estimate to be $\nu_{1\parallel} =
1.7337 \pm 0.0004$ and is also unchanged. The exponent $\tau_1$ of the mean
cluster length is related to $\beta_1$ and $\nu_{1\parallel}$ by the scaling
relation $\nu_{1\parallel} = \beta_1 + \tau_1$ and using the above estimates
yields $\tau_1 = 1$ to within the accuracy of our results. We conjecture that
this value of $\tau_1$ is exact and further support for the conjecture is
provided by the direct series expansion estimate $\tau_1= 1.0002 \pm 0.0003$.Comment: 12pages LaTeX, ioplppt.sty, to appear in J. Phys.

### Low-density series expansions for directed percolation III. Some two-dimensional lattices

We use very efficient algorithms to calculate low-density series for bond and
site percolation on the directed triangular, honeycomb, kagom\'e, and $(4.8^2)$
lattices. Analysis of the series yields accurate estimates of the critical
point $p_c$ and various critical exponents. The exponent estimates differ only
in the $5^{th}$ digit, thus providing strong numerical evidence for the
expected universality of the critical exponents for directed percolation
problems. In addition we also study the non-physical singularities of the
series.Comment: 20 pages, 8 figure

### The exact evaluation of the corner-to-corner resistance of an M x N resistor network: Asymptotic expansion

We study the corner-to-corner resistance of an M x N resistor network with
resistors r and s in the two spatial directions, and obtain an asymptotic
expansion of its exact expression for large M and N. For M = N, r = s =1, our
result is
R_{NxN} = (4/pi) log N + 0.077318 + 0.266070/N^2 - 0.534779/N^4 + O(1/N^6).Comment: 12 pages, re-arranged section

### Phase diagram of a dilute ferromagnet model with antiferromagnetic next-nearest-neighbor interactions

We have studied the spin ordering of a dilute classical Heisenberg model with
spin concentration $x$, and with ferromagnetic nearest-neighbor interaction
$J_1$ and antiferromagnetic next-nearest-neighbor interaction $J_2$. Magnetic
phases at absolute zero temperature $T = 0$ are determined examining the
stiffness of the ground state, and those at finite temperatures $T \neq 0$ are
determined calculating the Binder parameter $g_L$ and the spin correlation
length $\xi_L$. Three ordered phases appear in the $x-T$ phase diagram: (i) the
ferromagnetic (FM) phase; (ii) the spin glass (SG) phase; and (iii) the mixed
(M) phase of the FM and the SG. Near below the ferromagnetic threshold $x_{\rm
F}$, a reentrant SG transition occurs. That is, as the temperature is decreased
from a high temperature, the FM phase, the M phase and the SG phase appear
successively. The magnetization which grows in the FM phase disappears in the
SG phase. The SG phase is suggested to be characterized by ferromagnetic
clusters. We conclude, hence, that this model could reproduce experimental
phase diagrams of dilute ferromagnets Fe$_x$Au$_{1-x}$ and Eu$_x$Sr$_{1-x}$S.Comment: 9 pages, 23 figure

### Critical frontier of the Potts and percolation models in triangular-type and kagome-type lattices I: Closed-form expressions

We consider the Potts model and the related bond, site, and mixed site-bond
percolation problems on triangular-type and kagome-type lattices, and derive
closed-form expressions for the critical frontier. For triangular-type lattices
the critical frontier is known, usually derived from a duality consideration in
conjunction with the assumption of a unique transition. Our analysis, however,
is rigorous and based on an established result without the need of a uniqueness
assumption, thus firmly establishing all derived results. For kagome-type
lattices the exact critical frontier is not known. We derive a closed-form
expression for the Potts critical frontier by making use of a homogeneity
assumption. The closed-form expression is new, and we apply it to a host of
problems including site, bond, and mixed site-bond percolation on various
lattices. It yields exact thresholds for site percolation on kagome, martini,
and other lattices, and is highly accurate numerically in other applications
when compared to numerical determination.Comment: 22 pages, 13 figure

### Nonlinear and spin-glass susceptibilities of three site-diluted systems

The nonlinear magnetic $\chi_{3}$ and spin-glass $\chi_{sg}$ susceptibilities
in zero applied field are obtained, from tempered Monte Carlo simulations, for
three different spin glasses (SGs) of Ising spins with quenched site disorder.
We find that the relation $-T^3\chi_3=\chi_{sg}-2/3$ ($T$ is the temperature),
which holds for Edwards-Anderson SGs, is approximately fulfilled in
canonical-like SGs. For nearest neighbor antiferromagnetic interactions, on a
0.4 fraction of all sites in fcc lattices, as well as for spatially disordered
Ising dipolar (DID) systems, $-T^3\chi_3$ and $\chi_{sg}$ appear to diverge in
the same manner at the critical temperature $T_{sg}$. However, $-T^3\chi_3$ is
smaller than $\chi_{sg}$ by over two orders of magnitude in the diluted fcc
system. In DID systems, $-T^3\chi_3/\chi_{sg}$ is very sensitive to the systems
aspect ratio. Whereas near $T_{sg}$, $\chi_{sg}$ varies by approximately a
factor of 2 as system shape varies from cubic to long-thin-needle shapes,
$\chi_3$ sweeps over some four decades.Comment: 7 pages, 7 figure

### Simple Asymmetric Exclusion Model and Lattice Paths: Bijections and Involutions

We study the combinatorics of the change of basis of three representations of
the stationary state algebra of the two parameter simple asymmetric exclusion
process. Each of the representations considered correspond to a different set
of weighted lattice paths which, when summed over, give the stationary state
probability distribution. We show that all three sets of paths are
combinatorially related via sequences of bijections and sign reversing
involutions.Comment: 28 page

### Directed Percolation with a Wall or Edge

We examine the effects of introducing a wall or edge into a directed
percolation process. Scaling ansatzes are presented for the density and
survival probability of a cluster in these geometries, and we make the
connection to surface critical phenomena and field theory. The results of
previous numerical work for a wall can thus be interpreted in terms of surface
exponents satisfying scaling relations generalising those for ordinary directed
percolation. New exponents for edge directed percolation are also introduced.
They are calculated in mean-field theory and measured numerically in 2+1
dimensions.Comment: 14 pages, submitted to J. Phys.

### Absorbing boundaries in the conserved Manna model

The conserved Manna model with a planar absorbing boundary is studied in
various space dimensions. We present a heuristic argument that allows one to
compute the surface critical exponent in one dimension analytically. Moreover,
we discuss the mean field limit that is expected to be valid in d>4 space
dimensions and demonstrate how the corresponding partial differential equations
can be solved.Comment: 8 pages, 4 figures; v1 was changed by replacing the co-authors name
"L\"ubeck" with "Lubeck" (metadata only

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