  The q-state Potts model can be formulated in geometric terms, with
Fortuin-Kasteleyn (FK) clusters as fundamental objects. If the phase transition
of the model is second order, it can be equivalently described as a percolation
transition of FK clusters. In this work, we study the percolation structure
when the model undergoes a first order phase transition. In particular, we
investigate numerically the percolation behaviour along the line of first order
phase transitions of the 3d 3-state Potts model in an external field and find
that the percolation strength exhibits a discontinuity along the entire line.
The endpoint is also a percolation point for the FK clusters, but the
corresponding critical exponents are neither in the Ising nor in the random
percolation universality class.
