Coupling of stationary fields with application to arithmetic waves

TL;DR

This paper develops methods to couple Gaussian fields with similar covariance structures, enabling the approximation of arithmetic random waves by the random plane wave with quantifiable convergence rates.

Contribution

It introduces a coupling technique for Gaussian fields with close covariances and applies it to demonstrate local uniform convergence of arithmetic random waves to the random plane wave.

Findings

Coupling Gaussian fields with close covariance kernels is feasible.

Arithmetic random waves can be coupled to converge to the random plane wave.

The rate of convergence for the coupling is quantitatively estimated.

Abstract

In this paper we obtain a range of quantitative results of the following type: given two centered Gaussian fields with close covariance kernels we construct a coupling such that the fields are uniformly close on some compact with probability very close to one. As an application, we show that it is possible to couple arithmetic random waves so that they converge locally uniformly to the random plane wave and estimate the rate of convergence.