Local, algebraic simplifications of Gaussian random fields

Abstract

Many applications of Gaussian random fields and Gaussian random processes are limited by the computational complexity of evaluating the probability density function, which involves inverting the relevant covariance matrix. In this work, we show how that problem can be completely circumvented for the local Taylor coefficients of a Gaussian random field with a Gaussian (or `square exponential') covariance function. Our results hold for any dimension of the field and to any order in the Taylor expansion. We present two applications. First, we show that this method can be used to explicitly generate non-trivial potential energy landscapes with many fields. This application is particularly useful when one is concerned with the field locally around special points (e.g.~maxima or minima), as we exemplify by the problem of cosmic `manyfield' inflation in the early universe. Second, we show that this method has applications in machine learning, and greatly simplifies the regression problem of determining the hyperparameters of the covariance function given a training data set consisting of local Taylor coefficients at single point. An accompanying Mathematica notebook is available at https://doi.org/10.17863/CAM.22859 .

15 pages, 2 figures

References in corpus (6)

• Exploring a string-like landscape
• Manyfield Inflation in Random Potentials
• Seven Lessons from Manyfield Inflation in Random Potentials
• Inflation in multi-field random Gaussian landscapes
• Random Functions via Dyson Brownian Motion: Progress and Problems
• Inflation in Random Landscapes with two energy scales

Cited by in corpus (3)

• Hyperinflation generalised: from its attractor mechanism to its tension with the `swampland conjectures'
• Towards a complete mass spectrum of type-IIB flux vacua at large complex structure
• Slepian models for Gaussian Random Landscapes