Vertex counting: Statistical distribution of vertices in large sets of tree diagrams

We study the problem of determining the distribution of vertices of a particular given type in the set of all Feynman tree graphs in quantum field theories. We show that in almost all cases a Gaussian distribution arises asymptotically, and we compute the mean and variance of this distribution for several theories. We show that the distribution's ”fine structure”, arising from topological sum rules, can be obtained. Received: 12 September 2002 / Published online: 24 January 2003 RID="a" ID="a" e-mail: petros@sci.kun.nl RID="b" ID="b" e-mail: kleiss@sci.kun.nl