1D Potts, Yang-Lee Edges and Chaos

It is known that the (exact) renormalization transformations for the one-dimensional Ising model in field can be cast in the form of a logistic map f(x) = 4 x (1 - x) with x a function of the Ising couplings. Remarkably, the line bounding the region of chaotic behaviour in x is precisely that defining the Yang-Lee edge singularity in the Ising model. In this paper we show that the one dimensional q-state Potts model for q greater than or equal to 1 also displays such behaviour. A suitable combination of Potts couplings can again be used to define an x satisfying f(x) = 4 x (1 -x). The Yang-Lee zeroes no longer lie on the unit circle in the complex z = exp (h) plane, but their locus is still reproduced by the boundary of the chaotic region in the logistic map.Comment: 6 pages, no figure

Scaling and Density of Lee-Yang Zeroes in the Four Dimensional Ising Model

The scaling behaviour of the edge of the Lee--Yang zeroes in the four dimensional Ising model is analyzed. This model is believed to belong to the same universality class as the $\phi^4_4$ model which plays a central role in relativistic quantum field theory. While in the thermodynamic limit the scaling of the Yang--Lee edge is not modified by multiplicative logarithmic corrections, such corrections are manifest in the corresponding finite--size formulae. The asymptotic form for the density of zeroes which recovers the scaling behaviour of the susceptibility and the specific heat in the thermodynamic limit is found to exhibit logarithmic corrections too. The density of zeroes for a finite--size system is examined both analytically and numerically.Comment: 17 pages (4 figures), LaTeX + POSTSCRIPT-file, preprint UNIGRAZ-UTP 20-11-9

Identity of the universal repulsive-core singularity with Yang-Lee edge criticality

Lattice and continuum fluid models with repulsive-core interactions typically display a dominant, critical-type singularity on the real, negative activity axis. Lai and Fisher recently suggested, mainly on numerical grounds, that this repulsive-core singularity is universal and in the same class as the Yang-Lee edge singularities, which arise above criticality at complex activities with positive real part. A general analytic demonstration of this identification is presented here using a field-theory approach with separate representation of the repulsive and attractive parts of the pair interactions.Comment: 6 pages, 3 figure

Fisher zeros of the Q-state Potts model in the complex temperature plane for nonzero external magnetic field

The microcanonical transfer matrix is used to study the distribution of the Fisher zeros of the $Q>2$ Potts models in the complex temperature plane with nonzero external magnetic field $H_q$. Unlike the Ising model for $H_q\ne0$ which has only a non-physical critical point (the Fisher edge singularity), the $Q>2$ Potts models have physical critical points for $H_q<0$ as well as the Fisher edge singularities for $H_q>0$. For $H_q<0$ the cross-over of the Fisher zeros of the $Q$-state Potts model into those of the ($Q-1$)-state Potts model is discussed, and the critical line of the three-state Potts ferromagnet is determined. For $H_q>0$ we investigate the edge singularity for finite lattices and compare our results with high-field, low-temperature series expansion of Enting. For $3\le Q\le6$ we find that the specific heat, magnetization, susceptibility, and the density of zeros diverge at the Fisher edge singularity with exponents $\alpha_e$, $\beta_e$, and $\gamma_e$ which satisfy the scaling law $\alpha_e+2\beta_e+\gamma_e=2$.Comment: 24 pages, 7 figures, RevTeX, submitted to Physical Review

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

Yang-Lee Zeros of the Q-state Potts Model on Recursive Lattices

The Yang-Lee zeros of the Q-state Potts model on recursive lattices are studied for non-integer values of Q. Considering 1D lattice as a Bethe lattice with coordination number equal to two, the location of Yang-Lee zeros of 1D ferromagnetic and antiferromagnetic Potts models is completely analyzed in terms of neutral periodical points. Three different regimes for Yang-Lee zeros are found for Q>1 and 0<Q<1. An exact analytical formula for the equation of phase transition points is derived for the 1D case. It is shown that Yang-Lee zeros of the Q-state Potts model on a Bethe lattice are located on arcs of circles with the radius depending on Q and temperature for Q>1. Complex magnetic field metastability regions are studied for the Q>1 and 0<Q<1 cases. The Yang-Lee edge singularity exponents are calculated for both 1D and Bethe lattice Potts models. The dynamics of metastability regions for different values of Q is studied numerically.Comment: 15 pages, 6 figures, with correction

Spanning forests and the q-state Potts model in the limit q \to 0

We study the q-state Potts model with nearest-neighbor coupling v=e^{\beta J}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \le L \le 10, as well as the limiting curves of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp. w_0 = -0.1753 \pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w = w_0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w_0, while the correlation length diverges as w \downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the conformal charge is c = -1, the leading thermal scaling dimension is x_{T,1} = 0, and the critical exponents are \nu = 1/d = 1/2 and \alpha = 1.Comment: 131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal versio

Deficient mismatch repair system in patients with sporadic advanced colorectal cancer

A deficient mismatch repair system (dMMR) is present in 10–20% of patients with sporadic colorectal cancer (CRC) and is associated with a favourable prognosis in early stage disease. Data on patients with advanced disease are scarce. Our aim was to investigate the incidence and outcome of sporadic dMMR in advanced CRC. Data were collected from a phase III study in 820 advanced CRC patients. Expression of mismatch repair proteins was examined by immunohistochemistry. In addition microsatellite instability analysis was performed and the methylation status of the MLH1 promoter was assessed. We then correlated MMR status to clinical outcome. Deficient mismatch repair was found in only 18 (3.5%) out of 515 evaluable patients, of which 13 were caused by hypermethylation of the MLH1 promoter. The median overall survival in proficient MMR (pMMR), dMMR caused by hypermethylation of the MLH1 promoter and total dMMR was 17.9 months (95% confidence interval 16.2–18.8), 7.4 months (95% CI 3.7–16.9) and 10.2 months (95% CI 5.9–19.8), respectively. The disease control rate in pMMR and dMMR patients was 83% (95% CI 79–86%) and 56% (30–80%), respectively. We conclude that dMMR is rare in patients with sporadic advanced CRC. This supports the hypothesis that dMMR tumours have a reduced metastatic potential, as is observed in dMMR patients with early stage disease. The low incidence of dMMR does not allow drawing meaningful conclusions about the outcome of treatment in these patients

Measurement of exclusive pion pair production in proton–proton collisions at √s=7 TeV with the ATLAS detector

The exclusive production of pion pairs in the process pp→ ppπ+π- has been measured at s=7TeV with the ATLAS detector at the LHC, using 80μb-1 of low-luminosity data. The pion pairs were detected in the ATLAS central detector while outgoing protons were measured in the forward ATLAS ALFA detector system. This represents the first use of proton tagging to measure an exclusive hadronic final state at the LHC. A cross-section measurement is performed in two kinematic regions defined by the proton momenta, the pion rapidities and transverse momenta, and the pion–pion invariant mass. Cross-section values of 4.8±1.0(stat)-0.2+0.3(syst)μb and 9±6(stat)-2+2(syst)μb are obtained in the two regions; they are compared with theoretical models and provide a demonstration of the feasibility of measurements of this type

Search for resonant WZ production in the fully leptonic final state in proton–proton collisions at √s=13 TeV with the ATLAS detector

A search for a WZ resonance, in the fully leptonic final state (electrons or muons), is performed using 139&nbsp;fb - 1 of data collected at a centre-of-mass energy of 13&nbsp;TeV by the ATLAS detector at the Large Hadron Collider. The results are interpreted in terms of a singly charged Higgs boson of the Georgi–Machacek model, produced by WZ fusion, and of a Heavy Vector Triplet, with the resonance produced by WZ fusion or the Drell–Yan process. No significant excess over the Standard Model prediction is observed and limits are set on the production cross-section times branching ratio as a function of the resonance mass for these processes