Behavior of the Minimum Degree Throughout the $d$-process

Abstract: The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We show that, in the evolution of a random graph with $n$ vertices under the $d$-process with $d$ fixed, with high probability, for each $j \in {0,1,\dots,d-2}$, the minimum degree jumps from $j$ to $j+1$ when the number of steps left is on the order of $\ln(n)^{d-j-1}$. This answers a question of Ruci\'nski and Wormald. More specifically, we show that, when the last vertex of degree $j$ disappears, the number of steps left divided by $\ln(n)^{d-j-1}$ converges in distribution to the exponential random variable of mean $\frac{j!}{2(d-1)!}$; furthermore, these $d-1$ distributions are independent.