On the Bowditch boundary of a free group with random cyclic peripheral groups

The topology of the Bowditch boundary of a relatively hyperbolic group pair gives information about relative splittings of the group. It is therefore interesting to ask if there is generic behavior of this boundary. The purpose of this article is to show that in the case of a non-abelian free group with cyclic peripheral structure, there is not a generic case. This goal is accomplished by showing the minimal size of cut sets in the boundary increases as the lengths of the random words generating the peripheral structure go to infinity.