We present a class of variational wave functions for strong-coupling Heisenberg Hubbard models. These are written in the form of three factors—a pair of determinants and a Jastrow function—and are made out of orbitals, a la Hartree-Fock theory, which solve a fictitious one-body problem. The wave functions respect various constraints known from general principles and appear to be potentially useful in understanding the possible behavior of the models in quantitative terms.

The inverse problem of determining the spatial dependence of random fields (inference of the spatial model) from experimental samples is a central issue in geostatistics. We propose a computationally efficient approach based on Spartan Gibbs random fields. Their probability density function depends on a small set of parameters that can be determined by matching sample constraints with corresponding model constraints based on the stochastic moments. We investigate a specific Spartan probability density with spatial dependence derived from generalized gradient and Laplacian operators, and we derive permissibility conditions for the model parameters. The optimal values of the model parameters are determined by minimizing a normalized metric measuring the "distance" between stochastic moments and the respective sample constraints. The computational complexity of the minimization depends on the number of parameters but not on the sample size. The computational complexity of the constraint calculation increases linearly with the sample size. In contrast, the classical variogram calculation is considerably less efficient and its complexity increases as the square of the sample size. We propose a numerical approach for inferring the Spartan model parameters and illustrate it using simulated (synthetic) samples for regular (lattice) and irregular sample distributions. Based on our analysis, Spartan Gibbs random fields provide computationally efficient spatial models, which are especially useful if the sample size is large or reliable estimation of the variogram is not possible. Estimation of the field values at unsampled positions, conditional simulations, anisotropic spatial dependence, and non-Gaussian probability densities are briefly discussed.

Spartan spatial random fields (SSRFs) are generalized Gibbs random fields, equipped with a coarse-graining kernel that acts as a low-pass filter for the fluctuations. SSRFs are defined by means of physically motivated spatial interactions and a small set of free parameters (interaction couplings). This paper focuses on the fluctuation-gradient-curvature (FGC) SSRF model, henceforth referred to as FGC-SSRF. This model is defined on the Euclidean space R by means of interactions proportional to the squares of the field realizations, as well as their gradient and curvature. The permissibility criteria of FGC-SSRFs are extended by considering the impact of a finite-bandwidth kernel. It is proved that the FGC-SSRFs are almost surely differentiable in the case of finite bandwidth. Asymptotic explicit expressions for the Spartan covariance function are derived for d = 1 and d= 3; both known and new covariance functions are obtained depending on the value of the FGC-SSRF shape parameter. Nonlinear dependence of the covariance integral scale on the FGC-SSRF characteristic length is established, and it is shown that the relation becomes linear asymptotically. The results presented in this paper are useful in random field parameter inference, and in spatial interpolation of irregularly spaced samples.