490 research outputs found
Critical manifold of the kagome-lattice Potts model
Any two-dimensional infinite regular lattice G can be produced by tiling the
plane with a finite subgraph B of G; we call B a basis of G. We introduce a
two-parameter graph polynomial P_B(q,v) that depends on B and its embedding in
G. The algebraic curve P_B(q,v) = 0 is shown to provide an approximation to the
critical manifold of the q-state Potts model, with coupling v = exp(K)-1,
defined on G. This curve predicts the phase diagram both in the ferromagnetic
(v>0) and antiferromagnetic (v<0) regions. For larger bases B the
approximations become increasingly accurate, and we conjecture that P_B(q,v) =
0 provides the exact critical manifold in the limit of infinite B. Furthermore,
for some lattices G, or for the Ising model (q=2) on any G, P_B(q,v) factorises
for any choice of B: the zero set of the recurrent factor then provides the
exact critical manifold. In this sense, the computation of P_B(q,v) can be used
to detect exact solvability of the Potts model on G.
We illustrate the method for the square lattice, where the Potts model has
been exactly solved, and the kagome lattice, where it has not. For the square
lattice we correctly reproduce the known phase diagram, including the
antiferromagnetic transition and the singularities in the Berker-Kadanoff
phase. For the kagome lattice, taking the smallest basis with six edges we
recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases
provide successive improvements on this formula, giving a natural extension of
Wu's approach. The polynomial predictions are in excellent agreement with
numerical computations. For v>0 the accuracy of the predicted critical coupling
v_c is of the order 10^{-4} or 10^{-5} for the 6-edge basis, and improves to
10^{-6} or 10^{-7} for the largest basis studied (with 36 edges).Comment: 31 pages, 12 figure
Logarithmic observables in critical percolation
Although it has long been known that the proper quantum field theory
description of critical percolation involves a logarithmic conformal field
theory (LCFT), no direct consequence of this has been observed so far.
Representing critical bond percolation as the Q = 1 limit of the Q-state Potts
model, and analyzing the underlying S_Q symmetry of the Potts spins, we
identify a class of simple observables whose two-point functions scale
logarithmically for Q = 1. The logarithm originates from the mixing of the
energy operator with a logarithmic partner that we identify as the field that
creates two propagating clusters. In d=2 dimensions this agrees with general
LCFT results, and in particular the universal prefactor of the logarithm can be
computed exactly. We confirm its numerical value by extensive Monte-Carlo
simulations.Comment: 11 pages, 2 figures. V2: as publishe
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
Combining tree decomposition and transfer matrix techniques provides a very
general algorithm for computing exact partition functions of statistical models
defined on arbitrary graphs. The algorithm is particularly efficient in the
case of planar graphs. We illustrate it by computing the Potts model partition
functions and chromatic polynomials (the number of proper vertex colourings
using Q colours) for large samples of random planar graphs with up to N=100
vertices. In the latter case, our algorithm yields a sub-exponential average
running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the
exponential running time ~ exp(0.245 N) provided by the hitherto best known
algorithm. We study the statistics of chromatic roots of random planar graphs
in some detail, comparing the findings with results for finite pieces of a
regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded.
Version 3 shows that the worst-case running time is sub-exponential in the
number of vertice
Prognostic value of lymph node ratio and extramural vascular invasion on survival for patients undergoing curative colon cancer resection
There was no study funding. We are grateful to Tony Rafferty (Tailored Information for the People of Scotland, TIPs) for providing survival data.Peer reviewedPublisher PD
Critical Behavior of Random Bond Potts Models
The effect of quenched impurities on systems which undergo first-order phase
transitions is studied within the framework of the q-state Potts model. For
large q a mapping to the random field Ising model is introduced which provides
a simple physical explanation of the absence of any latent heat in 2D, and
suggests that in higher dimensions such systems should exhibit a tricritical
point with a correlation length exponent related to the exponents of the random
field model by \nu = \nu_RF / (2 - \alpha_RF - \beta_RF). In 2D we analyze the
model using finite-size scaling and conformal invariance, and find a continuous
transition with a magnetic exponent \beta / \nu which varies continuously with
q, and a weakly varying correlation length exponent \nu \approx 1. We find
strong evidence for the multiscaling of the correlation functions as expected
for such random systems.Comment: 13 pages, RevTeX. 4 figures included. Submitted to Phys.Rev.Let
A Transfer Matrix for the Backbone Exponent of Two-Dimensional Percolation
Rephrasing the backbone of two-dimensional percolation as a monochromatic
path crossing problem, we investigate the latter by a transfer matrix approach.
Conformal invariance links the backbone dimension D_b to the highest eigenvalue
of the transfer matrix T, and we obtain the result D_b=1.6431 \pm 0.0006. For a
strip of width L, T is roughly of size 2^{3^L}, but we manage to reduce it to
\sim L!. We find that the value of D_b is stable with respect to inclusion of
additional ``blobs'' tangent to the backbone in a finite number of points.Comment: 19 page
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions
We study the chromatic polynomial P_G(q) for m \times n square- and
triangular-lattice strips of widths 2\leq m \leq 8 with cyclic boundary
conditions. This polynomial gives the zero-temperature limit of the partition
function for the antiferromagnetic q-state Potts model defined on the lattice
G. We show how to construct the transfer matrix in the Fortuin--Kasteleyn
representation for such lattices and obtain the accumulation sets of chromatic
zeros in the complex q-plane in the limit n\to\infty. We find that the
different phases that appear in this model can be characterized by a
topological parameter. We also compute the bulk and surface free energies and
the central charge.Comment: 55 pages (LaTeX2e). Includes tex file, three sty files, and 22
Postscript figures. Also included are Mathematica files transfer4_sq.m and
transfer4_tri.m. Journal versio
Excess Mortality Rate During Adulthood Among Danish Adoptees
BACKGROUND AND OBJECTIVE: Adoption studies have been used to disentangle the influence of genes from shared familial environment on various traits and disease risks. However, both the factors leading to adoption and living as an adoptee may bias the studies with regard to the relative influence of genes and environment compared to the general population. The aim was to investigate whether the cohort of domestic adoptees used for these studies in Denmark is similar to the general population with respect to all-cause mortality and cause-specific mortality rates. METHODS: 13,111 adoptees born in Denmark in 1917, or later, and adopted in 1924 to 1947 were compared to all Danes from the same birth cohorts using standardized mortality ratios (SMR). The 12,729 adoptees alive in 1970 were similarly compared to all Danes using SMR as well as cause-specific SMR. RESULTS: The excess in all-cause mortality before age 65 years in adoptees was estimated to be 1.30 (95% CI 1.26-1.35). Significant excess mortality before age 65 years was also observed for infections, vascular deaths, cancer, alcohol-related deaths and suicide. Analyses including deaths after age 65 generally showed slightly less excess in mortality, but the excess was significant for all-cause mortality, cancer, alcohol-related deaths and suicides. CONCLUSION: Adoptees have an increased all-cause mortality compared to the general population. All major specific causes of death contributed, and the highest excess is seen for alcohol-related deaths
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