  Consider a unimodular random graph, or just a finitely generated Cayley
graph. When does its cycle space have an invariant random generating set of
cycles such that every edge is contained in finitely many of the cycles?
Generating the free Loop $O(1)$ model as a factor of iid is closely connected
to having such a generating set for FK-Ising cluster. We show that geodesic
cycles do not always form such a generating set, by showing for a parameter
regime of the FK-Ising model on the lamplighter group the origin is contained
in infinitely many geodesic cycles. This answers a question by Angel, Ray and
Spinka. Then we take a look at how the property of having an invariant locally
finite generating set for the cycle space is preserved by Bernoulli
percolation, and apply it to the problem of factor of iid presentations of the
free Loop $O(1)$ model.
