A random graph growth model

A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process, the pool contains (Formula presented.) virtual vertices and no edges. Each time a vertex is sampled and occupied, the edges linking the vertex to previously occupied vertices are added to the pool of virtual elements. We focus on the edge-counting at times when the graph has (Formula presented.) occupied vertices. Two different Poisson limits are identified for (Formula presented.) and (Formula presented.). For the bulk of the process, when (Formula presented.), the scaled number of edges is shown to fluctuate about a deterministic curve, with fluctuations being of the order of (Formula presented.) and approximable by a Gaussian bridge.