Analytic Properties and Covariance Functions of a New Class of Generalized Gibbs Random Fields

TL;DR

This paper introduces a new class of Spartan Spatial Random Fields (SSRFs), analyzes their mathematical properties, derives explicit covariance functions, and discusses implications for spatial data modeling and interpolation.

Contribution

It extends the theory of FGC-SSRFs by establishing differentiability criteria, deriving explicit covariance functions for specific dimensions, and analyzing the relationship between covariance scale and characteristic length.

Findings

FGC-SSRFs are almost surely differentiable with finite bandwidth.

Explicit covariance functions are derived for dimensions 1 and 3.

Nonlinear and asymptotically linear relations between covariance scale and characteristic length.

Abstract

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 FGC-SSRF model, which is defined on the Euclidean space $\mathbb{R}^{d}$ 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, as well as in spatial interpolation of irregularly-spaced samples.