Thermodynamics and statistical analysis of Gaussian random fields

Summary Gaussian fields are considered as Gibbsian fields. Thermodynamic functions are calculated for them and the variational principle is proved. As an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for Gaussian Markov fields.     

Randomized consistent statistical inference for random processes and fields

Statistical Inference for Stochastic Processes - We propose a randomized approach to the consistent statistical analysis of random processes and fields on $${mathbb {R}}^m$$ and $${mathbb {Z}}^m,...     

Poisson fields and distribution functions in the statistical mechanics of charged particles

A brief survey of the Poisson analysis on the spaces of tempered distributions is given and the generalized Wick theorem for Poisson fields is formulated. For systems of charged particles, new representations in terms of integrals with respect to the Poisson measure are obtained for distribution functions and diagonal elements of a reduced density matrix; these representations are convenient for investigation of model systems of statistical mechanics by the cluster expansion method. In the quantum case, the Boltzmann, Fermi-Dirac, and Bose-Einstein statistics are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 710–719, May, 1995.The author expresses his deep gratitude to A. L. Rebenko for posing the problem and for his attention to the work and to G. F. Us for helpful discussions.This research was partially supported by the International Science Foundation, grant No. UB3000.     

Markov random fields and gibbs random fields

Spitzer has shown that every Markov random field (MRF) is a Gibbs random field (GRF) and vice versa when (i) both are translation invariant, (ii) the MRF is of first order, and (iii) the GRF is defined by a binary, nearest neighbor potential. In both cases, the field (iv) is defined onZ ^v, and (v) at anyxεZ^v, takes on one of two states. The current paper shows that a MRF is a GRF and vice versa even when (i)−(v) are relaxed, i.e., even if one relaxes translation invariance, replaces first order bykth order, allows for many states and replaces finite domains of Z^v by arbitrary finite sets. This is achieved at the expense of using a many body rather than a pair potential, which turns out to be natural even in the classical (nearest neighbor) case when Z^v is replaced by a triangular lattice. The contents of this paper were presented in August, 1971, at a seminar of the Battelle Rencontre in Statistical Mechanics and also at a pair of seminars in December, 1971, at the Weizmann Institute of Science. Partially supported by NSF GP 7469 and a Weizmann Institute senior fellowship while on sabbatical leave from Indiana University.     

Limit theorems for sums of dependent random variables occurring in statistical mechanics

Summary By the use of conditioning, we extend previously obtained results on the asymptotic behavior of partial sums for certain triangular arrays of dependent random variables, known as Curie-Weiss models. These models arise naturally in statistical mechanics. The relation of these results to multiple phases, metastable states, and other physical phenomena is explained.Alfred P. Sloan Research Fellow. Research supported in part by National Science Foundation Grant MPS 76-06644A01Alfred P. Sloan Research Fellow. Research supported in part by National Science Foundation Grant MCS 77-20683 and by U.S.-Israel Binational Science FoundationResearch supported in part by National Science Foundation Grant PHY77-02172     

Limit theorems for sums of dependent random variables occurring in statistical mechanics

Summary We study the asymptotic behavior of partial sums S _nfor certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number , and a positive integer k so that (S _n–nm)/n^1–1/2k converges weakly to a random variable with density proportional to exp(–¦s¦ ^2k/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.Alfred P. Sloan Research Fellow. Research supported in part by a Broadened Faculty Research Grant at the University of Massachusetts and by National Science Foundation Grant MPS 76-06644Research supported in part by National Science Foundation Grants MPS 74-04870 A01 and MCS 77-20683     

Gaussian random fields

Nonanticipative representations of Gaussian random fields equivalent to the two-parameter Wiener process are defined, and necessary and sufficient conditions for their existence derived. When such representations exist they provide examples of canonical representations of multiplicity one. In contrast to the one-parameter case, examples are given where nonanticipative representations do not exist. Nonanticipative representations along increasing paths are also studied.University of Minnesota and University of North Carolina. Presented at theWorkshop on Stochastic Processes in Infinite Dimensional Spaces and Random Fields, held at UCLA in April 1979.     

Operator-stable and operator-self-similar random fields

Two classes of multivariate random fields with operator-stable marginals are constructed. The random fields $\mathbb{X}=\{X\left(t\right):t\in {\mathbb{R}}^d\}$ with values in ${\mathbb{R}}^m$ are invariant in law under operator-scaling in both the time-domain and the state-space. The construction is based on operator-stable random measures utilizing certain homogeneous functions.     

Invariant random boolean fields

In the set of finite binary sequences a Markov process is defined with discrete time in which each symbol of the binary sequence at time t+1 depends on the two neighboring symbols at time t. A proof is given of the existence and uniqueness of an invariant distribution, and its derivation is also given in a number of cases.Translated from Matematicheskie Zametki, Vol. 6. No. 5, pp. 555–566, November, 1969.     

Multivariate operator-self-similar random fields

Multivariate random fields whose distributions are invariant under operator-scalings in both the time domain and the state space are studied. Such random fields are called operator-self-similar random fields and their scaling operators are characterized. Two classes of operator-self-similar stable random fields X={X(t),t∈R^d} with values in R^m are constructed by utilizing homogeneous functions and stochastic integral representations.     

Inequalities for random fields

This paper gives extensions of the Doob and Burkholder inequalities for certain classes of random fields. The Brennan-Doob inequality for V-quasimartingales is extended to the case p > 1 and is shown to hold for the class of decomposable processes satisfying the Doob inequality of Wong and Zakai [10]. A Doob inequality for the class of i-martingales having finite quadratic variation in the non-martingale coordinate is shown. For the class of quasi martingales having independent increments two Burkholder-type inequalities are derived     

Two-parameter diffusions random fields

This paper presents an alternative method for calculating the diffusion, drift, and mixed coefficients of an example of biparameter Gaussian diffusion defined as a solution of a linear hyperbolic stochastic partial differential equation (Nualart & Sanz , 1979). To derive the expression of these coefficients, we part from an integral stochastic repre , sentation given by these authors for this class of biparameter diffusion processes arising from biparameter Gaussian random fields verifying a particular Markov property     

Multi-operator scaling random fields

In this paper, we define and study a new class of random fields called harmonizable multi-operator scaling stable random fields. These fields satisfy a local asymptotic operator scaling property which generalizes both the local asymptotic self-similarity property and the operator scaling property. Actually, they locally look like operator scaling random fields, whose order is allowed to vary along the sample paths. We also give an upper bound of their modulus of continuity. Their pointwise Hölder exponents may also vary with the position x and their anisotropic behavior is driven by a matrix which may also depend on x.     

Several constants arising in statistical mechanics

This is a brief survey of certain constants associated with random lattice models, including self-avoiding walks, polyominoes, the Lenz-Ising model, monomers and dimers, ice models, hard squares and hexagons, and percolation models.