Eternal multiplicative coalescent is encoded by its Lévy-type processes

Abstract: The multiplicative coalescent is a Markov process taking values in ordered $l^2$. It is a mean-field process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (1998) each extreme eternal version $(\mathbf{X}(t),- \infty < t < \infty)$ of the multiplicative coalescent was described in three different ways. One of these specifications matches the (marginal) law of $\mathbf{X}(t)$ to that of the ordered excursion lengths above past minima of ${L_{\mathbf{X}}(s) +ts, \,s \geq 0}$, where $L_{\mathbf{X}}$ is a certain L\'evy-type process which (modulo shift and scaling) has infinitesimal drift $-s$ at time $s$. Using a modification of the breadth-first-walk construction from Aldous (1997) and Aldous and Limic (1998), and some new insight from the thesis by Uribe (2007), this work settles an open problem (3) from Aldous (1997), in the more general context of Aldous and Limic (1998). Informally speaking, $\mathbf{X}$ is entirely encoded by $L_{\mathbf{X}}$, and contrary to Aldous' original intuition, the evolution of time for $\mathbf{X}$ does correspond to the linear increase in the constant part of the drift of $L_{\mathbf{X}}$. In the "standard multiplicative coalescent" context of Aldous (1997), this result was first announced by Armend\'ariz in 2001, and obtained in a recent preprint by Broutin and Marckert, who simultaneously account for the process of excess edge counts (or marks). The novel argument presented here is based on a sequence of relatively elementary observations. Some of its components (for example, the new dynamic random graph construction via "simultaneous" breadth-first walks) are of independent interest, and may be useful for obtaining more sophisticated asymptotic results on near critical random graphs and related processes.