Extension of the Arrowsmith–Essam Formula to the Domany–Kinzel Model

Arrowsmith and Essam gave an expansion formula for point-to-point connectedness functions of the mixed site-bond percolation model on oriented lattices, in which each term is characterized by a graph. We extend this formula to general k-point correlation functions, which are point-to-set (with k points) connectivities in the context of percolation, of the two-neighbor discrete-time Markov process (stochastic cellular automata with two parameters) in one dimension called the Domany–Kinzel model, which includes the mixed site-bond oriented percolation model on a square lattice as a special case. Our proof of the formula is elementary and based on induction with respect to time-step, which is different from the original graph-theoretical one given by Arrowsmith and Essam. We introduce a system of m interacting random walkers called m friendly walkers (m FW) with two parameters. Following the argument of Cardy and Colaiori, it is shown that our formula is useful to derive a theorem that the correlation functions of the Domany–Kinzel model are obtained as an m0 limit of the generating functions of the m FW.     

Tsallis Entropy for Loss Models and Survival Models Involving Truncated and Censored Random Variables

The aim of this paper consists in developing an entropy-based approach to risk assessment for actuarial models involving truncated and censored random variables by using the Tsallis entropy measure. The effect of some partial insurance models, such as inflation, truncation and censoring from above and truncation and censoring from below upon the entropy of losses is investigated in this framework. Analytic expressions for the per-payment and per-loss entropies are obtained, and the relationship between these entropies are studied. The Tsallis entropy of losses of the right-truncated loss random variable corresponding to the per-loss risk model with a deductible d and a policy limit u is computed for the exponential, Weibull, χ2 or Gamma distribution. In this context, the properties of the resulting entropies, such as the residual loss entropy and the past loss entropy, are studied as a result of using a deductible and a policy limit, respectively. Relationships between these entropy measures are derived, and the combined effect of a deductible and a policy limit is also analyzed. By investigating residual and past entropies for survival models, the entropies of losses corresponding to the proportional hazard and proportional reversed hazard models are derived. The Tsallis entropy approach for actuarial models involving truncated and censored random variables is new and more realistic, since it allows a greater degree of flexibility and improves the modeling accuracy.     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. II. Extended Results for Square-Lattice Chromatic Polynomial

We study the chromatic polynomials for m×n square-lattice strips, of width 9m13 (with periodic boundary conditions) and arbitrary length n (with free boundary conditions). We have used a transfer matrix approach that allowed us also to extract the limiting curves when n. In this limit we have also obtained the isolated limiting points for these square-lattice strips and checked some conjectures related to the Beraha numbers.     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. III. Triangular-Lattice Chromatic Polynomial

We study the chromatic polynomial P _G (q) for m×n triangular-lattice strips of widths m12_P,9_F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin–Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m,n and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.     

Mean-Field Theory for Percolation Models of the Ising Type

The q=2 random cluster model is studied in the context of two mean-field models: the Bethe lattice and the complete graph. For these systems, the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high-density side, which vindicates the results of earlier studies. In particular, the exponent ~ which characterizes the divergence of the average size of finite clusters is 1/2, and ~, the exponent associated with the length scale of finite clusters, is 1/4. The full collection of exponents indicates an upper critical dimension of 6. The standard mean field exponents of the Ising system are also present in this model (=1/2, =1), which implies, in particular, the presence of two diverging length-scales. Furthermore, the finite cluster exponents are stable to the addition of disorder, which, near the upper critical dimension, may have interesting implications concerning the generality of the disordered system/correlation length bounds.     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. I. General Theory and Square-Lattice Chromatic Polynomial

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P _G (q) for m×n rectangular subsets of the square lattice, with m8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin–Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B ₂,B ₃,B ₄,B ₅ are limiting points of partition-function zeros as n whenever the strip width m is 7 (periodic transverse b.c.) or 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B ₁₀) cannot be a chromatic root of any graph.     

Entropy-Driven Phase Transitions in Multitype Lattice Gas Models

In multitype lattice gas models with hard-core interaction of Widom–Rowlinson type, there is a competition between the entropy due to the large number of types, and the positional energy and geometry resulting from the exclusion rule and the activity of particles. We investigate this phenomenon in four different models on the square lattice: the multitype Widom–Rowlinson model with diamond-shaped resp. square-shaped exclusion between unlike particles, a Widom–Rowlinson model with additional molecular exclusion, and a continuous-spin Widom–Rowlinson model. In each case we show that this competition leads to a first-order phase transition at some critical value of the activity, but the number and character of phases depend on the geometry of the model. We also analyze the typical geometry of phases, combining percolation techniques with reflection positivity and chessboard estimates.     

The Constants in the Central Limit Theorem for the One-Dimensional Edwards Model

The Edwards model in one dimension is a transformed path measure for one-dimensional Brownian motion discouraging self-intersections. We study the constants appearing in the central limit theorem (CLT) for the endpoint of the path (which represent the mean and the variance) and the exponential rate of the normalizing constant. The same constants appear in the weak-interaction limit of the one-dimensional Domb–Joyce model. The Domb–Joyce model is the discrete analogue of the Edwards model based on simple random walk, where each self-intersection of the random walk path recieves a penalty e ^–2. We prove that the variance is strictly smaller than 1, which shows that the weak interaction limits of the variances in both CLTs are singular. The proofs are based on bounds for the eigenvalues of a certain one-parameter family of Sturm–Liouville differential operators, obtained by using monotonicity of the zeros of the eigen-functions in combination with computer plots.     

Dimension in a Radiative Stellar Atmosphere

Dimensional scales are examined in an extended 3 + 1 Vaidya atmosphere surrounding a Schwarzschild source. At one scale, the Vaidya null fluid vanishes and the spacetime contains only a single spherical 2-surface. Both of these behaviors can be addressed by including higher dimensions in the spacetime metric.     

Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.     

Rigidity of the Interface in Percolation and Random-Cluster Models

We study conditioned random-cluster measures with edge-parameter p and cluster-weighting factor q satisfying q1. The conditioning corresponds to mixed boundary conditions for a spin model. Interfaces may be defined in the sense of Dobrushin, and these are proved to be rigid in the thermodynamic limit, in three dimensions and for sufficiently large values of p. This implies the existence of non-translation-invariant (conditioned) random-cluster measures in three dimensions. The results are valid in the special case q=1, thus indicating a property of three-dimensional percolation not previously noted.     

Strict inequalities for some critical exponents in two-dimensional percolation

For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL():={(x, y):x=r cos, y=r sin for some r0, or} and():=limpp _c [log(p–p _c )]^–1 log P_cr {itO is connected to by an occupied path inL()}, then() is strictly decreasing in on [0, 2]. Similarly, lim_n [–logn]^–1 logP _cr {itO is connected by an occupied path inL()() to the exterior of [–n, n]×[–n, n] is strictly decreasing in on [0, 2].     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Moyal Deformation,Seiberg–Witten Maps,and Integrable Models

A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg–Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg–Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.     

New Correlation Duality Relations for the Planar Potts Model

We introduce a new method to generate duality relations for correlation functions of the Potts model on a planar graph. The method extends previously known results, by allowing the consideration of the correlation function for arbitrarily placed vertices on the graph. We show that generally it is linear combinations of correlation functions, not the individual correlations, that are related by dualities. The method is illustrated in several non-trivial cases, and the relation to earlier results is explained. A graph-theoretical formulation of our results in terms of rooted dichromatic, or Tutte, polynomials is also given.     

Models for a Finite Universe

A method of constructing 3-dimensional hyperbolic manifolds is described. Because of their high degree of symmetry, these may be suitable models for a finite universe. Because their group of symmetries is different from any in the list of manifolds given in Hodgeson and Weeks (available at ftp://ftp.northnet.org/pub/weeks/snappea/closedcensus), it is claimed that these are new.     

Focused Information Criterion for Restricted Mean Survival Times: Non-Parametric or Parametric Estimators

Restricted Mean Survival Time (RMST), the average time without an event of interest until a specific time point, is a model-free, easy to interpret statistic. The heavy reliance on non-parametric or semi-parametric methods in the survival analysis has drawn criticism, due to the loss of efficacy compared to parametric methods. This assumes that the parametric family used is the true one, otherwise the gain in efficacy might be lost to interpretability problems due to bias. The Focused Information Criterion (FIC) considers the trade-off between bias and variance and offers an objective framework for the selection of the optimal non-parametric or parametric estimator for scalar statistics. Herein, we present the FIC framework for the selection of the RMST estimator with the best bias-variance trade-off. The aim is not to identify the true underling distribution that generated the data, but to identify families of distributions that best approximate this process. Through simulation studies and theoretical reasoning, we highlight the effect of censoring on the performance of FIC. Applicability is illustrated with a real life example. Censoring has a non-linear effect on FICs performance that can be traced back to the asymptotic relative efficiency of the estimators. FICs performance is sample size dependent; however, with censoring percentages common in practical applications FIC selects the true model at a nominal probability (0.843) even with small or moderate sample sizes.