| Abstract: | In this seminar, I will present a phase transition for random generated graphs with respect to the graph size. The analysis will be made using Grassmann field theory defined on the graph’s vertexes. More precisely I will discuss the H0|4 sigma model with respect to its graphical representation over the complete graph. The H0|2n refers to an hyperbolic target space generated on n-pairs of fermionic variables (H0|2 is a fermionic version of 2D Minkowski). In the graphical representation, the H0|4 is a certain measure of un-rooted sub-graphs with arboricity constraint. I will begin with toy models over small graphs, thus providing some intuition to the H0|2 and H0|4 models. The main focus will then be on the asymptotic behavior of the model as the graph size tends to infinity. I will contrast the H0|4 model to its predecessor, the H0|2 model, a uniform measure to all un-rooted sub-forests of a graph. |