Uniqueness of unbounded component for level sets of smooth Gaussian fields

Franco Severo

Published 2022-08-08Version 1

We consider the level-sets of a stationary, continuous, centered Gaussian field $f$ on $\mathbb{R}^d$. Under certain regularity, positivity and integrability conditions on the correlations of $f$ (in particular, this includes the Bargmann-Fock field), we prove that for every level $\ell\in \mathbb{R}$ there exist at most one unbounded connected component in $\{f=\ell\}$ (as well as in $\{f\geq\ell\}$) almost surely, 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 by using a soft shift argument based on the Cameron-Martin theorem.