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

Uniqueness of unbounded component for level sets of smooth Gaussian fields