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...; 0st ! 1) is the Marcus-Lushnikov process associated with the additive kernel K(x; y) = x + y, with monodisperse initial conditions. This construction was apparently first explicitly given by Pitman =-=[63]-=-, although various formulas associated with it had previously been developed in the 32 combinatorial literature [62, 105] and the SM literature [40]. Here are two examples of simple formulas. The numb...

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... process (Gs; 0s1) arises in a more subtle way as a limit of the MarcusLushnikov (discrete coalescent) process with additive kernel (see [6] for background on the general Marcus-Lushnikov process and =-=[42, 43, 44, 38]-=- for recent results on the additive case). More exotic variations of (Gs), e.g. a stationary Markov process in which branches grow and are cut down upon becoming infinite, arise as other N !1 limits a...

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...such an additive coalescent with an arbitrary initial state consisting a countable number of masses. Both multiplicative and additive coalescents arise combinatorially from the study of random graphs =-=[1, 33]-=-, and this limiting regime arises naturally in that work. Section 6.2 investigates a particularly interesting feature of two of the codings of the additive coalescent as a measure--valued process desc...

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...s by type is simpler than its companion for labeled forests. But the result for labeled forests has the following simpler equivalent: Theorem 6 (Enumeration of labeled forests by out-degree sequence) =-=[51]-=-,[59, Thm. 3.4] For all sequences of non-negative integers (c 1 ; : : : ; c n ) with P i c i = n \Gamma k the number N(c 1 ; : : : ; c n ) of forests f with vertex set [n] in which vertex i has c i ch...

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... a finite state Markov chain [8],[20, §6.1]. By specializing a general Markov chain formula to the present setting, and evaluating a constant of normalization by use of Cayley’s multinomial expansion =-=[26, 25]-=-, there is the following result, an alternative proof of which is indicated after Lemma 7. Lemma 1 [13]If Y0,Y1,... are i.i.d.(p) for p with finite support S := {i : pi > 0}, then the random tree T :=...

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...f random graphs and has been studied and applied in several other contexts. The coalescent obtained by reversing the process of deleting the edges at random is the additive coalescent as discussed in =-=[23]-=-. This is what is illustrated in figure 2. 3 Fragmentation Processes Definition 5. Call a sequence of P [n]-valued random variables (Π1,Π2,... ,Πn) a fragmentation process or a refining process if wit...