A random field, such as the GFF or the Ising magnetisation field (IMF), has an excursion de-
composition if it can be written as an infinite sum of (random) positive measures each multiplied
by their own sign. These decompositions often result in geometric and probabilistic interpretations
for predictions of conformal field theory. In recent work, we constructed a natural decomposition of
the IMF which is distinct from the earlier ‘FK-Ising’ representation, thereby illustrating that such
decompositions are not unique.
We ask if it is possible for two different random fields to have a common excursion decomposition,
i.e. starting from an excursion decomposition of the first field, is there a random procedure to alter
just the signs, leaving the measures the same, which results in the second field? We show in upcoming
works that this is possible in two cases. Firstly, for two instances of the GFF with different boundary
conditions relying on a new adaptation of Lupu and Le Jan’s isomorphism theorems. Secondly, we
prove that (perhaps more suprisingly) two IMFs with ‘Dobrushin’ and ‘plus’ boundary conditions can
be coupled together by just changing the signs of the boundary excursions.
Based on joint works with Tomás Alcalde López and Marcin Lis.