  We derive the scaling dimension associated with crossing bonds in the
random-cluster representation of the two-dimensional Potts model, by means of a
mapping on the Coulomb gas. The scaling field associated with crossing bonds
appears to be irrelevant, on the critical as well as on the tricritical branch.
The latter result stands in a remarkable contrast with the existing result for
the tricritical O(n) model that crossing bonds are relevant. In order to obtain
independent confirmation of the Coulomb gas result for the crossing-bond
exponent, we perform a finite-size-scaling analysis based on numerical
transfer-matrix calculations.
