Uniqueness of Unbounded Component for Level Sets of Smooth Gaussian Fields

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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, and 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.