Abstract
In these expository notes, we describe some features of the multiplicative coalescent and its connection with random graphs and minimum spanning trees. We use Pitman’s proof (Pitman, J Combin Theory Ser A 85:165–193, 1999) of Cayley’s formula, which proceeds via a calculation of the partition function of the additive coalescent, as motivation and as a launchpad. We define a random variable which may reasonably be called the empirical partition function of the multiplicative coalescent, and show that its typical value is exponentially smaller than its expected value. Our arguments lead us to an analysis of the susceptibility of the Erdős-Rényi random graph process, and thence to a novel proof of Frieze’s ζ(3)-limit theorem for the weight of a random minimum spanning tree.
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- 1.
We find this proof of the ζ(3) limit for the MST weight pleasing, as it avoids lemmas which involve estimating the number of unicyclic and complex components in G(n, p); morally, the cycle structure of components of G(n, p) should be unimportant, since cycles are never created in Kruskal’s algorithm!
- 2.
It is more common to order by reverse order of addition, so that labels increase along root-to-leaf paths; this change of perspective may help with Exercise 4.
- 3.
Until further notice, we omit ceilings and floors for readability.
- 4.
We omit the dependence on n in the notation for E 1; similar infractions occur later in the proof.
- 5.
To maximize ∑ j x j 2 subject to the conditions that ∑ j x j = 1 and that max j x j ≤ δ, take x j = δ for 1 ≤ j ≤ δ −1.
- 6.
Stirling’s approximation says that \(m!/(\sqrt{2\pi m}(m/e)^{m}) \rightarrow 1\) as m → ∞; in fact the (much less precise) fact that log(m! ) = mlogm − m + o(m) is enough for the current situation.
- 7.
A glance back at Figs. 2, 3 and 4 gives a hint as to the relative heights of the three trees.
Fig. 2 
One of the 3, 0002, 998 labeled trees with 3, 000 vertices, selected uniformly at random
Fig. 3 
One of the 2, 999! rooted trees on 3, 000 vertices with a decreasing edge labelling (labels suppressed)
Fig. 4 
The tree resulting from the multiplicative coalescent on 3, 000 points
- 8.
- 9.
In fact, if edge lengths in T ac (n) are multiplied by n −1∕2 then the resulting object converges in distribution to a random compact metric space called the Brownian continuum random tree (or CRT), and h(T ac )∕n 1∕2 converges in distribution to the height of the CRT. For more on this important result, we refer the reader to [4, 10].
- 10.
With more care, one can show that with high probability H contains tree components containing around n 2∕3 vertices and with height around n 1∕3, which yields that with high probability T mc (n) has height of order at least n 1∕3.
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Acknowledgements
My thanks go to a very careful and astute referee. I would also like to thank several people who saw me present parts of this material in the form of mini-courses, for thought-provoking questions and feedback. During the preparation of this work I was supported by funding from both NSERC and the FRQNT.
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Addario-Berry, L. (2015). Partition Functions of Discrete Coalescents: From Cayley’s Formula to Frieze’s ζ(3) Limit Theorem. In: Mena, R., Pardo, J., Rivero, V., Uribe Bravo, G. (eds) XI Symposium on Probability and Stochastic Processes. Progress in Probability, vol 69. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-13984-5_1
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