First Order Probabilities for Galton–Watson Trees
In the regime of Galton–Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such finite subtrees will be almost surely present when the tree is infinite. Introducing the notion of universal trees, we show that all first order sentences of quantifier depth k depend only on local neighbourhoods of the root of sufficiently large radius depending on k. We compute the probabilities of these neighbourhoods conditioned on the tree being infinite. We give an almost sure theory for infinite trees.
- 3.J. Spencer, L. Thoma, On the limit values of probabilities for the first order properties of graphs, in Contemporary Trends in Discrete Mathematics (Štiřín Castle, 1997). Volume 49 of DIMACS Series in Discrete Mathematics & Theoretical Computer Science (American Mathematical Society, Providence, 1999), pp. 317–336Google Scholar
- 4.R. van der Hofstad, Random Graphs and Complex Networks, vol. 1. Lecture Notes (2015). http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf