Uniqueness of unbounded component for level sets of smooth Gaussian fields

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 $\{f\ge \ell \}$) 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.

arXiv comment8 pages, minor corrections in the proof, introduction rewritten and assumptions relaxed, now including some strongly correlated and non-FKG fields