Restricted odd Wick powers of the 2D GFF appear as coefficients of the half-integer logarithmic asymptotics of conformal-radius neighborhoods of first passage sets and sign clusters; even powers appear only after compensation.
First passage sets of the 2 D G aussian free field
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Relation between Wick powers and excursion clusters of the 2D GFF
Restricted odd Wick powers of the 2D GFF appear as coefficients of the half-integer logarithmic asymptotics of conformal-radius neighborhoods of first passage sets and sign clusters; even powers appear only after compensation.