  In [Muhl2019], Peter M\"uhlbacher showed that in the random loop model
without loop weights, a loop phase transition (assuming it exists) cannot occur
at the same parameter as the percolation phase transition of the occupied
edges. In this work, we give a quantitative version of this result, specifying
a minimal gap between the percolation phase transition and a possible loop
phase transition. A substantial part of our argument also works for weighted
loop models.
