Split-Merge Processes on Graphs and Matrices

We consider a model of split-merge processes on intervals, more precisely a continuous version of a split-merge process on intervals and a discrete version, the latter having a natural number n as parameter. These processes are well known and quite famous in the literature. It is known that the discrete process converges to the continuous one in the sense of stationary distributions. In this thesis we establish a kind of convergence which involves the finite dimensional distributions. Furthermore, we define a new model of split-merge processes which act on a more complex state space and can be seen as a generalization of the processes on intervals. More precisely, the processes act on weighted graphs. Since each weighted graph can be described by a matrix, we can also think of the process as if it would act on matrices. The idea of such a process is that at each time step we either merge two nodes into one or split one node into two. Then, the weights are updated in an adequate way. In this model, we equip a split-merge process with a parameter, which describes how to update the weights in case of a split. It turns out that for some parameters the resulting process is isomorphic to the continuous or discrete process on intervals, respectively. So we easily can derive properties for these processes. Moreover, we show that for a certain class of parameters the resulting processes might not be isomorphic to one of the interval model, but still converge to the process that is isomorphic to the continuous process on intervals.