The evolution of the min–min random graph process

We study the following min–min random graph process G=(G0,G1,…)G=(G0,G1,…): the initial state G0G0 is an empty graph on nn vertices (nn even). Further, GM+1GM+1 is obtained from GMGM by choosing a pair {v,w}{v,w} of distinct vertices of minimum degree   uniformly at random among all such pairs in GMGM and adding the edge {v,w}{v,w}. The process may produce multiple edges. We show that GMGM is asymptotically almost surely disconnected if M≤nM≤n, and that for M=(1+t)nM=(1+t)n, 0