Relation between Wick powers and excursion clusters of the 2D GFF

Abstract

We study the decomposition of the Wick powers of the continuum GFF in dimension $2$ via the first passage sets (FPS) and the excursion clusters (sign components) of the GFF. These sets are non-thin for the GFF, that is to say the field has non-trivial restriction to such a set, which is a measure, negative or positive depending on the sign. In this work we show that all the odd Wick powers of the GFF can be restricted to the FPS and the excursion clusters, and the restrictions are generalized functions supported on these fractal sets. By contrast, the restriction of an even Wick power to an FPS or excursion cluster is diverging, and to get something converging an additional compensation is required, which is provided by a smooth function living outside of the set and blowing up in a non-integrable way when approaching the set. We further provide expressions of restricted odd Wick powers and restricted-compensated even Wick powers as limits of functions living outside the FPS/excursion cluster. Then, we study the $\varepsilon$-neighborhoods, in the sense of conformal radius, of first passage sets and excursion clusters. We show that such $\varepsilon$-neighborhoods admit asymptotic expansions in $L^2$ into half-integer powers $\vert\log \varepsilon\vert^{-(n+1/2)}$, $n\in\mathbb{N}$, of $1/\vert\log \varepsilon\vert$. The coefficients of the expansion involve the restrictions of the odd Wick powers. By contrast, the even Wick powers do not appear in the expansion. Our expansion is reminiscent of Le Gall's expansion for the Wiener sausage in dimension 2, with however some important differences. The most important one is that the powers of $1/\vert\log \varepsilon\vert$ are different. In the case of the Wiener sausage the powers are integer, $\vert\log \varepsilon\vert^{-n}$, $n\in\mathbb{N}\setminus \{0\}$.