Nonstandard conditions for the global Markov property for lattice fields

The global Markov property (GMP) has been shown in various cases under various conditions by quite different methods. Here we investigate the problem of the GMP (for lattice spin systems) from the nonstandard point of view. By embedding the given system into a hyperfinite system we are able to approximate the conditional expectations that are involved in the formulation of the GMP by internal conditional expectations. This leads to a nonstandard equivalent to the GMP as well as to sufficient nonstandard conditions that are easy to formulate. Finally, we then determine the interrelations between these conditions and some of the standard criteria, thus making their relative position somewhat clearer.     

The global Markov property for lattice systems

We prove the global Markov property for lattice systems of classical statistical mechanics, with bounded spins and finite range interactions. The method uses the one developed by two of us to prove the global Markov property of Euclidean generalized random fields. The result shows that the systems considered have a transition matrix, which together with a distribution on a hyperplane, describes completely the system.     

The Spearman footrule and a Markov chain property

An equivalent representation of the Spearman footrule is considered and a characterization in terms of a Markov chain is established. A martingale approach is thereby incorporated in the study of the asymptotic normality of the statistics.     

Large deviations for lattice systems I. Parametrized independent fields

Summary We prove large deviation theorems for empirical measures of independent random fields whose distributions depend measurably on an auxiliary parameter. This dependence respects the action of the shift group, and a large deviation principle holds whenever a certain ergodicity condition is satisfied. We also investigate the entropy functions for these processes, especially in relation to the usual relative entropy.     

Diophantine equations over global function fields I: The Thue equation

We solve completely Thue equations in function fields over arbitrary finite fields. In the function field case such equations were formerly only solved over algebraically closed fields (of characteristic zero and positive characteristic). Our method can be applied to similar types of Diophantine equations, as well.     

The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces

The asymptotic distribution of orbits for discrete subgroups of motions in Euclidean and non-Euclidean spaces are found; our principal tool is the wave equation. The results are new for the crystallographic groups in Euclidean space and for those groups in non-Euclidean spaces which have fundamental domains of infinite volume. In the latter case we show that the only point spectrum of the Laplace-Beltrami operator lies in the interval ($\text{?(}\text{(m ? 1)}\text{2}\text{)}{}^2\text{,0}$]; furthermore we show that when the subgroup is nonelementary and the fundamental domain has a cusp, then there is at least one eigenvalue in this interval.     

The lumpability property for a family of Markov chains on poset block structures

We construct different classes of lumpings for a family of Markov chain products which reflect the structure of a given finite poset. We essentially use combinatorial methods. We prove that, for such a product, every lumping can be obtained from the action of a suitable subgroup of the generalized wreath product of symmetric groups, acting on the underlying poset block structure, if and only if the poset defining the Markov process is totally ordered, and one takes the uniform Markov operator in each factor state space. Finally we show that, when the state space is a homogeneous space associated with a Gelfand pair, the spectral analysis of the corresponding lumped Markov chain is completely determined by the decomposition of the group action into irreducible submodules.     

Markov random fields,Markov cocycles and the 3-colored chessboard

The well-known Hammersley–Clifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley–Clifford Theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce “Markov cocycles”, reparametrizations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley–Clifford Theorem for a family of Markov random fields which are outside the theorem’s purview where the underlying graph is Z_d. This family includes all Markov random fields whose support is the d-dimensional “3-colored chessboard”. On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction.     

The ideal property, the projection property, continuous fields and crossed products

Let A be the C^∗-algebra associated to an arbitrary continuous field of C^∗-algebras. We give a necessary and sufficient condition for A to have the ideal property and, if moreover A is separable, we give a necessary and sufficient condition for A to have the projection property. Some applications of these results are given. We also prove that “many” crossed products of commutative C^∗-algebras by discrete, amenable groups have the projection property, generalizing some of our previous results.     

On a strong property of the weak subalgebra lattice

In the present paper, we apply results from [Pió1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak subalgebra lattice uniquely determines its strong subalgebra lattice (recall that in the case of total algebras the strong subalgebra lattice is the well-known lattice of all (total) subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak subalgebra lattices of A and B are isomorphic (with A as above), then the strong subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A. Received September 5, 1997; accepted in final form October 7, 1998.     

The semigroup characterization of Osterwalder-Schrader path spaces and the construction of Euclidean fields

Axioms are given for relativistic quantum fields so the corresponding Schwinger functions are the expectation values of Euclidean fields. The main ingredient is the characterization of Osterwalder-Schrader path spaces by the associated semigroup structure.     

Stationary self-similar random fields on the integer lattice

We establish several methods for constructing stationary self-similar random fields (ssf's) on the integer lattice by “random wavelet expansion”, which stands for representation of random fields by sums of randomly scaled and translated functions, or more generally, by composites of random functionals and deterministic wavelet expansion. To construct ssf's on the integer lattice, random wavelet expansion is applied to the indicator functions of unit cubes at integer sites. We demonstrate how to construct Gaussian, symmetric stable, and Poisson ssf's by random wavelet expansion with mother wavelets having compact support or non-compact support. We also generalize ssf's to stationary random fields which are invariant under independent scaling along different coordinate axes. Finally, we investigate the construction of ssf's by combining wavelet expansion and multiple stochastic integrals.     

Shifted lattice rules based on a general weighted discrepancy for integrals over Euclidean space

We approximate weighted integrals over Euclidean space by using shifted rank-1 lattice rules with good bounds on the “generalised weighted star discrepancy”. This version of the discrepancy corresponds to the classic L_∞ weighted star discrepancy via a mapping to the unit cube. The weights here are general weights rather than the product weights considered in earlier works on integrals over R^d. Known methods based on an averaging argument are used to show the existence of these lattice rules, while the component-by-component technique is used to construct the generating vector of these shifted lattice rules. We prove that the bound on the weighted star discrepancy considered here is of order O(n^−1+δ) for any δ>0 and with the constant involved independent of the dimension. This convergence rate is better than the O(n^−1/2) achieved so far for both Monte Carlo and quasi-Monte Carlo methods.