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O’Connell, Some large deviation results for sparse random graphs. Probab. Theory Relat. Fields 110(3), 277–285 (1998). pdf 13. J. Pitman, Coalescent random forests. J. Combin. Theory Ser. A 85, 165–193 (1999). pdf 14. A. Rényi, Some remarks on the theory of trees. Magyar Tud. Akad. Mat. Kutató Int. Közl. 4, 73–85 (1959) 15. A. Rényi, G. Szekeres, On the height of trees. J. Austral. Math. Soc. 7, 497–507 (1967). N;k be the distance-k graph of the N-fold star power of G. N;k converges to a centered Bernoulli distribution, 1=2ı 1 C 1=2ı1.

Kn ; W/, and refer to it as the random MST of Kn . n/ . Kn /. Letting Gk have edges e1 ; : : : ; ek thus yields an important instantiation of our coupling of the Erd˝osRényi coalescent and the multiplicative coalescent; we return to this in Sect. 5. 3 Other Features of the Multiplicative Coalescent The remainder of the section is not essential to the main development. The following exercise was inspired by a discussion with Remco van der Hofstad. i; j/ D min e2 We , the minimum taken over paths from i to j in Kn .

5. 3 Other Features of the Multiplicative Coalescent The remainder of the section is not essential to the main development. The following exercise was inspired by a discussion with Remco van der Hofstad. i; j/ D min e2 We , the minimum taken over paths from i to j in Kn . Show that the minimum is attained by a unique path i; j . Kn /g for which, for each for each 1 Ä i < j Ä n, i; j is a path of T1 . Discrete Coalescents 19 Finally, we turn to Version 3 of the process, in which we view arbitrary iid nonnegative weights X D fXi; j ; 1 Ä i < j Ä ng as rates for edge addition.

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A case study in non-centering for data augmentation: Stochastic epidemics by Neal P.


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