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.

Inflation in multi-field modified DBM potentials

“…In our models, as the number of fields increases so does the power in isocurvature modes, but the percentage of cases when the Planck bound is exceeded is below 2% for the largest N f = 100. This small percentage of cases seems consistent with Bjorkmo and Marsh [40] though we are unable to compare the exact percentage. There is no inconsistency with Masoumiet al [37] either since they predict a single field behavior whenever Λ h N −1/4 f 1 and the values of Λ h and N f that we considered in this work do not satisfy this bound.…”

“…These approaches can be divided into two groups. In [35,36,39,40] the potential around the inflection or saddle point is smooth and given by a polynomial potential with random coefficients following (1.2). Though a polynomial can't represent a RGF (it is not translationally invariant), the short-field excursion during inflation justifies its use.…”

Slepian models for Gaussian random landscapes

“…We will leave this analysis for a later publication. 15 It is worth mentioning that the present work has potentially very interesting applications to characterise the landscape of 4d effective field theories in String Theory flux compactifications at tree-level. Actually, as was discussed in [1], the superpotential defining the effective supergravity description of flux compactifications can be modelled as a (complex) Gaussian random field with a specific covariance function determined by the geometry of the compact dimensions.…”

“…The parameter U 0 sets the energy scale of the potential while Λ represents the correlation length in field space. It is important to realize that the techniques used in this paper are generic and can be applied to other interesting situations like, for example, non-Gaussian covariance functions so in this sense these constructions are quite more generic than the ones presented in [15]. We have decided…”

“…(4.9), in combination with the Slepian techniques, might also provide a very good prediction for the decay rates of high false vacua in higher dimensional landscapes. For this purpose, the alternative methods proposed in [15] to generate constrained multidimensional Gaussian random landscapes might also proof very useful.…”