By Morris J.S.

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Example text

N/ exchangeable, distinct edge weights W. In this coupling, for m 2 Œ n2 , we took Gm to be the subgraph of Kn consisting of the m edges of smallest W-weight. In the current section, it is useful to be more specific. We suppose the entries of W are iid UniformŒ0; 1 random variables. n; p/ for the graph with vertices Œn and edges fej W Wej Ä pg. n/ probability p. n; p/; 0 Ä p Ä 1/. The next exercise is standard, but important. n; p/ is the same as that of Gm . c/ be the largest real solution of e section is to prove the following result.

X/j Ä C for all x ¤ 1. 1 ˛ . 1 Â log n ˛ . 4. 1/ ; t u which completes the proof. 1. Recall that we are trying to approximate i ! n 1/ log 2. X 2/ 1 Â 1 „D mD0 n C 2m n2 Ã " 1 E 1 ! 4 and Exercise 24 to approximate this sum by an integral. Before proceeding to the ’s and ı’s, we evaluate the integral. 1 ˛ 2 . 1 ˛ 2 . //d D 2. 4, though decidedly less pleasing. Since ˛. / D 0 for Ä 1, we may change the domain of integration to Œ1; 1/. Then use the identity ˛0 . / D ˛. 1 ˛. // ˛. 1 ˛. 1 ˛ 2 . 1 ˛ 2 .

Let F1 be a forest of n isolated vertices 1; : : : ; n. For 1 Ä i < n: ? Kn / minimize fWf j;kg W j; k in distinct trees of Fi g. Form FiC1 from Fi by adding f j; kg. Kn /g again yield a process with the law of the multiplicative coalescent. At step i, the edge-weight driven multiplicative coalescent simply adds the smallest weight edge whose endpoints lie in distinct components of Fi . In other words, it adds the smallest weight edge whose addition will not create a cycle in the growing graph.