338 research outputs found
Exact bond percolation thresholds in two dimensions
Recent work in percolation has led to exact solutions for the site and bond
critical thresholds of many new lattices. Here we show how these results can be
extended to other classes of graphs, significantly increasing the number and
variety of solved problems. Any graph that can be decomposed into a certain
arrangement of triangles, which we call self-dual, gives a class of lattices
whose percolation thresholds can be found exactly by a recently introduced
triangle-triangle transformation. We use this method to generalize Wierman's
solution of the bow-tie lattice to yield several new solutions. We also give
another example of a self-dual arrangement of triangles that leads to a further
class of solvable problems. There are certainly many more such classes.Comment: Accepted for publication in J. Phys
Rigorous confidence intervals for critical probabilities
We use the method of Balister, Bollobas and Walters to give rigorous 99.9999%
confidence intervals for the critical probabilities for site and bond
percolation on the 11 Archimedean lattices. In our computer calculations, the
emphasis is on simplicity and ease of verification, rather than obtaining the
best possible results. Nevertheless, we obtain intervals of width at most
0.0005 in all cases
Predictions of bond percolation thresholds for the kagom\'e and Archimedean lattices
Here we show how the recent exact determination of the bond percolation
threshold for the martini lattice can be used to provide approximations to the
unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one
of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2)
bond problems, and the other gives estimates of for the homogeneous
kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively
agree with numerical results to five and six significant figures.Comment: 4 pages, 5 figure
Exact Site Percolation Thresholds Using the Site-to-Bond and Star-Triangle Transformations
I construct a two-dimensional lattice on which the inhomogeneous site
percolation threshold is exactly calculable and use this result to find two
more lattices on which the site thresholds can be determined. The primary
lattice studied here, the ``martini lattice'', is a hexagonal lattice with
every second site transformed into a triangle. The site threshold of this
lattice is found to be , while the others have and
. This last solution suggests a possible approach to establishing
the bound for the hexagonal site threshold, . To derive these
results, I solve a correlated bond problem on the hexagonal lattice by use of
the star-triangle transformation and then, by a particular choice of
correlations, solve the site problem on the martini lattice.Comment: 12 pages, 10 figures. Submitted to Physical Review
Random Cluster Models on the Triangular Lattice
We study percolation and the random cluster model on the triangular lattice
with 3-body interactions. Starting with percolation, we generalize the
star--triangle transformation: We introduce a new parameter (the 3-body term)
and identify configurations on the triangles solely by their connectivity. In
this new setup, necessary and sufficient conditions are found for positive
correlations and this is used to establish regions of percolation and
non-percolation. Next we apply this set of ideas to the random cluster
model: We derive duality relations for the suitable random cluster measures,
prove necessary and sufficient conditions for them to have positive
correlations, and finally prove some rigorous theorems concerning phase
transitions.Comment: 24 pages, 1 figur
The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices
We give a conditional derivation of the inhomogeneous critical percolation manifold of the bow-tie lattice with five different probabilities, a problem that does not appear at first to fall into any known solvable class. Although our argument is mathematically rigorous only on a region of the manifold, we conjecture that the formula is correct over its entire domain, and we provide a non-rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface. We discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow-tie lattices, and that, if it could be put on a firm foundation, the negative probability portion of our method would extend this class to many further systems, including F Y Wu’s checkerboard formula for the square lattice. We conclude by showing that this latter problem can in fact be proved using a recent result of Grimmett and Manolescu for isoradial graphs, lending strong evidence in favor of our other conjectured results. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/98528/1/1751-8121_45_49_494005.pd
A pulsed, mono-energetic and angular-selective UV photo-electron source for the commissioning of the KATRIN experiment
The KATRIN experiment aims to determine the neutrino mass scale with a
sensitivity of 200 meV/c^2 (90% C.L.) by a precision measurement of the shape
of the tritium -spectrum in the endpoint region. The energy analysis of
the decay electrons is achieved by a MAC-E filter spectrometer. To determine
the transmission properties of the KATRIN main spectrometer, a mono-energetic
and angular-selective electron source has been developed. In preparation for
the second commissioning phase of the main spectrometer, a measurement phase
was carried out at the KATRIN monitor spectrometer where the device was
operated in a MAC-E filter setup for testing. The results of these measurements
are compared with simulations using the particle-tracking software
"Kassiopeia", which was developed in the KATRIN collaboration over recent
years.Comment: 19 pages, 16 figures, submitted to European Physical Journal
Critical surfaces for general inhomogeneous bond percolation problems
We present a method of general applicability for finding exact or accurate
approximations to bond percolation thresholds for a wide class of lattices. To
every lattice we sytematically associate a polynomial, the root of which in
is the conjectured critical point. The method makes the correct
prediction for every exactly solved problem, and comparison with numerical
results shows that it is very close, but not exact, for many others. We focus
primarily on the Archimedean lattices, in which all vertices are equivalent,
but this restriction is not crucial. Some results we find are kagome:
, , ,
, , :
. The results are generally within of numerical
estimates. For the inhomogeneous checkerboard and bowtie lattices, errors in
the formulas (if they are not exact) are less than .Comment: Submitted to J. Stat. Mec
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