Uniqueness of unbounded component for level sets of smooth Gaussian fields

TL;DR

This paper proves that for a broad class of smooth Gaussian fields, there is almost surely at most one unbounded component in their level and excursion sets, confirming a conjecture and employing a novel soft shift method.

Contribution

It establishes the uniqueness of unbounded components in level sets for a wide class of Gaussian fields, including the Bargmann-Fock and Cauchy fields, using a new approach.

Findings

Uniqueness of unbounded component proven for various Gaussian fields

Method applicable despite lack of finite energy property

Confirms conjecture by Duminil-Copin et al.

Abstract

For a large family of stationary continuous Gaussian fields $f$ on $\mathbb{R}^d$, including the Bargmann-Fock and Cauchy fields, we prove that there exists at most one unbounded connected component in the level set $\{f=\ell\}$ (as well as in the excursion set $\{f\geq\ell\}$) almost surely for every level $\ell\in \mathbb{R}$, thus proving a conjecture proposed by Duminil-Copin, Rivera, Rodriguez & Vanneuville. As the fields considered are typically very rigid (e.g.~analytic almost surely), there is no sort of finite energy property available and the classical approaches to prove uniqueness become difficult to implement. We bypass this difficulty using a soft shift argument based on the Cameron-Martin theorem.