The eternal multiplicative coalescent encoding via excursions of L\'evy-type processes, with Supplement

TL;DR

This paper provides a comprehensive encoding of the eternal multiplicative coalescent process using excursions of Lévy-type processes, resolving an open problem and clarifying the relationship between process time and Lévy process drift.

Contribution

It establishes that each eternal version of the multiplicative coalescent is fully encoded by its Lévy-type process, confirming the process time corresponds to the linear drift increase.

Findings

Eternal multiplicative coalescent is encoded by Lévy-type excursions.

Process time matches linear drift increase in Lévy process.

Resolves an open problem from Aldous (1997).

Abstract

The multiplicative coalescent is a mean-field Markov 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 of the multiplicative coalescent was described in three different ways, one of which matched its (marginal) law to that of the ordered excursion lengths above past minima of a certain L\'evy-type process. 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, each eternal version is entirely encoded by its L\'evy-type process, and contrary to Aldous' original intuition, the time for the multiplicative coalescent does correspond to the linear increase in the constant part of the drift of the L\'e}y-type process. In the "standard multiplicative coalescent" context of Aldous (1997), this result was first announced by Armend\'ariz in 2001, while its first published proof is due to Broutin and Marckert (2016), who simultaneously account for the process of excess (or surplus) edge counts.

Equations

$$\begin{array}[]{c}\mbox{ each pair of blocks of mass $x$ and $y$ merges at rate $xy$ }\\ \mbox{into a single block of mass $x+y$.}\end{array}$$

$$W^t(s)=W(s)-\frac{1}{2}{s}^2+ts,s\ge 0,$$

$$B^t(s)=W^t(s)-\underset{0\le s^{\prime }\le s}{\min }{W}^t(s^{\prime }),s\ge 0,$$

$$V^{\mathbf{c}}(s)=\sum _j\left(c_j{1}_{({\xi }_j\le s)}-c_j^{2}s\right),s\ge 0.$$

$${\cal I}:=\left((0,\infty)\times(-\infty,\infty)\times l^{3}_{\mbox{{\footnotesize$\searrow$}}}\right)\cup\left(\{0\}\times(-\infty,\infty)\times{l^{3}_{\mbox{{\footnotesize$\searrow$}}}\backslash\,\,l^{2}_{\mbox{{\footnotesize$\searrow$}}}}\right).$$

$${\tilde{W}}^{\kappa ,\tau }(s)={\kappa }^{1/2}W(s)+\tau s-\frac{1}{2}\kappa s^2,s\ge 0$$

$$W^{\kappa ,\tau ,\mathbf{c}}(s)={\tilde{W}}^{\kappa ,\tau }(s)+V^{\mathbf{c}}(s),s\ge 0$$

$$B^{\kappa ,\tau ,\mathbf{c}}(s)=W^{\kappa ,\tau ,\mathbf{c}}(s)-\underset{0\le s^{\prime }\le s}{\min }{W}^{\kappa ,\tau ,\mathbf{c}}(s^{\prime }),s\ge 0.$$

$$\mathbf{X}(t)=(y,0,0,0,\dots ),-\infty <t<\infty$$

$$S(t)=S_2(t)=\sum _i{X}_i^2(t),\text{mbox}and{S}_3(t)=\sum _i{X}_i^3(t).$$

$$|t{|}^3{S}_3(t)$$

$$t+\frac{1}{S(t)}$$

$$|t|X_j(t)$$

$$Z^q(s):=\sum _{i=1}^n{1}_{({\xi }_i\le q\cdot s)}-s,s\ge 0,$$

$$W^{\kappa ,t-\tau ,\mathbf{c}}(s):=W^{\kappa ,-\tau ,\mathbf{c}}(s)+ts,s\ge 0.$$

$$Z^{\mathbf{x},q}(s):=\sum _{i=1}^{\mathrm{len}(\mathbf{x})}{x}_i{1}_{({\xi }_i^q\le s)}-s=\sum _{i=1}^{\mathrm{len}(\mathbf{x})}{x}_{(i)}{1}_{({\xi }_{(i)}^q\le s)}-s,s\ge 0,q>0.$$

$${\pi }_1:=i\text{mbox}ifandonlyif{\xi }_i={\xi }_{(1)},\text{mbox}and{J}_1:=J_0\setminus \{{\pi }_1\},$$

$$\pi_{l}:=i\mbox{ if and only if }\xi_{i}=\xi_{(l)},\ l\in\{1,\ldots,\rm len({\bf x})\},\mbox{ and $J_{l}:=J_{l-1}\setminus\{\pi_{l}\}$}.$$

$$P(\xi_{i}^{q}-\xi^{q}_{(1)}>u|\mbox{${\cal F}$}^{q}_{\xi_{(1)}})1_{(i\in J_{1})}=e^{-x_{i}qu}1_{(i\in J_{1})}=P(\xi^{q}_{i}>u)1_{(i\in J_{1})}.$$

$$\text{mbox}edgej\leftrightarrow {\pi }_1\text{mbox}appearsbeforetimeq\text{mbox}inthemul\text{-}ti\text{-}pli\text{-}ca\text{-}tiveco\text{-}a\text{-}le\text{-}scent.$$

$$(a,b]\oplus [c,d]:=(a+c,b+d].$$

$$I_{l}^{q}\equiv I_{l}:=\left\{\begin{array}[]{ll}I_{l-1}\oplus[0,x_{\pi_{l}}],&\mbox{provided }\frac{\xi_{(l)}}{q}\in I_{l-1}\\ {\rm undefined},&\mbox{otherwise}\end{array}\right.,$$

$$N_{l}^{q}\equiv N_{l}:=\mbox{ the number of $\xi^{q}$s that rung during $I_{l}$},$$

$$\displaystyle P(\xi^{q}_{k}\in I_{l}\setminus I_{l-1}|\mbox{${\cal F}$}_{\xi_{(1)}^{q}+x_{\pi_{1}}+\cdots+x_{\pi_{l-1}}})1_{(k\in J_{N_{l-1}})}=$$

$$\displaystyle P(k\in J_{N_{l-1}}\setminus J_{N_{l}}|\mbox{${\cal F}$}_{\xi_{(1)}^{q}+x_{\pi_{1}}+\cdots+x_{\pi_{l-1}}})1_{(k\in J_{N_{l-1}})}=e^{-qx_{k}x_{\pi_{l}}}1_{(k\in J_{N_{l-1}})}\ \ \mbox{a.s.}$$

$$Z^{\mathbf{x},q}(s)>Z^{\mathbf{x},q}(a_1)=Z^{\mathbf{x},q}(b_1),\forall s\in (a_1,b_1).$$

$$\mathbf{X}:=(\mathbf{X}(q),q>0)\text{mbox}and\mathbf{X}(0)=\mathbf{x}$$

$$L_1^{\prime }:$$

$$L_2^{\prime }:$$

$$L_3^{\prime }:$$

$$L_n^{\prime }:$$

$$s_k:=\underset{j<k}{\min }{s}_{k,j},{\ell }_k:=\{1\le j<k:s_k=s_{k,j}\}.$$

$$L_1:$$

Full text

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, each eternal version is entirely encoded by its Lévy-type process, and contrary to Aldous’ original intuition, the time for the multiplicative coalescent does correspond to the linear increase in the constant part of the drift of the Lévy-type process. In the “standard multiplicative coalescent” context of Aldous (1997), this result was first announced by Armendáriz in 2001, while its first published proof is due to Broutin and Marckert (2016), who simultaneously account for the process of excess (or surplus) edge counts.

entrance law,

excursion,

Lévy process,

multiplicative coalescent,

near-critical,

random graph,

stochastic coalescent,

arXiv:0000.0000 \startlocaldefs

\endlocaldefs

**The writing of this article was sponsored in part by the Alexander von Humbold Foundation’s Friedrich Wilhelm Bessel Research Award. *

1 Introduction

Erdős-Rényi [32] (binomial) random graph $G(n,p)$ is one of the most studied objects of probabilistic combinatorics. In this model there are $n\geq 2$ vertices labeled by $\{1,2,\ldots,n\}$, and each of the ${n\choose 2}$ edges is present with probability $p\in[0,1]$ (and absent otherwise), independently of each other. So $G(n,p)$ is a random sub-graph of a complete graph, that looks (qualitatively, in the sense of distribution) the same when viewed from any of its vertices.

The most natural coupling of ($G(n,p)$, $p\in[0,1]$) is in terms of a family of ${n\choose 2}$ independent uniform random variables, indexed by the undirected edges $\{i,j\}$; an edge $e=\{i,j\}$ is declared “open (present) in $G(n,p)$” on the event $\{U_{\{i,j\}}\leq p\}$, and “closed in (absent from) $G(n,p)$” on the complement. In this way, if $p_{1}\leq p_{2}$, then $G(n,p_{1})$ is a random subgraph of $G(n,p_{2})$. A time-change $q:=-\log(1-p)$ transforms this model into a Markov chain running in continuous time. Its transitions are particularly simple: each undirected edge $\{i,j\}$ arrives as an exponential (rate 1) random variable, and stays in the graph forever after. Two different connected components of this (growing) random graph process will merge at the minimal connection time of a pair of vertices (or particles) $(k,l)$, where $k$ is from one, and $l$ from the other component. The mass (or size) of any connected component (or block) equals the number of its particles. Using elementary properties of independent exponentials, it is simple to see that the vector of block sizes of $(G(n,1-e^{-q}),\,q\geq 0)$ is also a continuous-time Markov chain, evolving according to the multiplicative coalescent dynamics:

[TABLE]

1.1 The multiplicative coalescent in 1997

Suppose a slightly more general setting: let $(x_{1},x_{2},\ldots,x_{m})$ be the vector of initial block masses, where $x_{i}$ is now a positive integer for each $i$. We can represent each initial block as a collection of $x_{i}$ different particles of mass $1$, which have been pre-connected in some specified arbitrary way. In particular, the total number of vertices is now $n=\sum_{i=1}^{m}x_{i}$. As in $(G(n,1-e^{-q}),\,q\geq 0)$, for each edge $\{k,l\}$ let $\xi_{k,l}$ be an exponential (rate $1$) arrival time of the edge connecting $k$ and $l$ (independent over $\{k,l\}$). Then the process of connected component masses again evolves according to (1.1).

A general multiplicative coalescent takes values in the space of collections of blocks, where each block has mass in $(0,\infty)$. Informally, it is a stochastic process with transitions specified by (1.1). For a given initial state with a finite number of blocks (where block masses are not necessarily integer-valued), it is easy to formalize (1.1), e.g. via a similar “graph-construction”, in order to define a continuous-time finite-state Markov process. Furthermore, Aldous [6] extended the state space to include $l_{2}$ configurations. More precisely, if (l^{2}_{\mbox{{\footnotesize\searrow}}},d) is the metric space of infinite sequences ${\bf x}=(x_{1},x_{2},\ldots)$ with $x_{1}\geq x_{2}\geq\ldots\geq 0$ and $\sum_{i}x_{i}^{2}<\infty$, where $d({\bf x},{\bf y})=\sqrt{\sum_{i}(x_{i}-y_{i})^{2}}$, then the multiplicative coalescent is a Feller process on l^{2}_{\mbox{{\footnotesize\searrow}}} (see [6] Proposition 5, or Section 2.1 in [45] for an alternative argument), evolving according to description (1.1). The focus in [6] was on the existence and properties of the multiplicative coalescent, as well as on the construction of a particular eternal version $({\bf X}^{*}(t),-\infty<t<\infty)$, called the (Aldous’) standard multiplicative coalescent. The standard version arises as a limit of the classical random graph process near the phase transition (each particle has initial mass $n^{-2/3}$ and the random graph is viewed at times $n^{1/3}+O(1)$). In particular, the marginal distribution ${\bf X}^{*}(t)$ of ${\bf X}^{*}$ was described in [6] as follows: if $(W(s),0\leq s<\infty)$ is standard Brownian motion and

[TABLE]

and $B^{t}$ is its “reflection above past minima”

[TABLE]

then (see [6] Corollary 2) the ordered sequence of excursion (away from [math]) lengths of $B^{t}$ has the same distribution as ${\bf X}^{*}(t)$. Note in particular that the total mass $\sum_{i}X^{*}_{i}(t)$ is infinite. The entrance law ${\bf X}^{*}$ “starts from dust” and ends by forming “a giant”: $\lim_{t\to-\infty}\|{\bf X}^{*}(t)\|_{2}=0$ and $\lim_{t\to\infty}\|{\bf X}^{*}(t)\|_{2}=\infty$.

The author’s thesis [45] was based on a related question: are there any other eternal versions of the multiplicative coalescent, and, provided that the answer is positive, what are they? Paper [7] completely described the entrance boundary of the multiplicative coalescent (or equivalently, the set of all of its extreme eternal laws). The extreme eternal laws or versions are conveniently characterized by the property that their corresponding tail $\sigma$-fields at time $-\infty$ are trivial. Any (other) eternal versions must be a mixture of extreme ones, see e.g. [31] Section 10. Note that the word “version” is not used here in the classical (Markov process theory) sense.

1.2 Characterizations of eternal versions in 1998

The notation to be introduced next is inherited from [7] . We write l^{3}_{\mbox{{\footnotesize\searrow}}} for the space of infinite sequences ${\bf c}=(c_{1},c_{2},\ldots)$ with $c_{1}\geq c_{2}\geq\ldots\geq 0$ and $\sum_{i}c_{i}^{3}<\infty$. For {\bf c}\in l^{3}_{\mbox{{\footnotesize\searrow}}}, let $(\xi_{j},j\geq 1)$ be independent with exponential (rate $c_{j}$) distributions and consider

[TABLE]

We may regard $V^{\bf c}$ as a Lévy-type process, where for each $x$ only the first jump of size $x$ is kept (cf. Section 2.5 of [7] , and Bertoin [14] for background on Lévy processes). In reality, the finite number $n_{x}$ of jumps of size $x$ is kept, where $n_{x}$ is the number of indices $j$ for which $x={\bf c}_{j}$. It is easy to see that $\sum_{i}c_{i}^{3}<\infty$ is precisely the condition for (1.4) to yield a well-defined process (see also Section 2.1 of [7] or [48] Section 2).

Define the parameter space

[TABLE]

Now modify (1.2,1.3) by defining, for each $(\kappa,\tau,{\bf c})\in{\cal I}$,

[TABLE]

[TABLE]

[TABLE]

So $B^{\kappa,\tau,{\bf c}}(s)$ is again the reflected process with some set of (necessarily all finite, see Theorem 1.1 below) excursions away from [math].

Denote by $\hat{\mu}(y)$ the distribution of the constant process

[TABLE]

where $y\geq 0$ is arbitrary but fixed.

Let {\bf X}(t)=(X_{1}(t),X_{2}(t),\ldots)\in l^{2}_{\mbox{{\footnotesize\searrow}}} be the state of a particular eternal version of the multiplicative coalescent. Then $X_{j}(t)$ is the mass of its $j$’th largest block at time $t$. Write

[TABLE]

The main results of [7] are stated next.

Theorem 1.1** ([7] , Theorems 2–4).**

*(a) For each $(\kappa,\tau,{\bf c})\in{\cal I}$ there exists an eternal multiplicative coalescent ${\bf X}$ such that for each $-\infty<t<\infty$, ${\bf X}(t)$ is distributed as the ordered sequence of excursion lengths of $B^{\kappa,t-\tau,{\bf c}}$.

(b) Denote by $\mu(\kappa,\tau,{\bf c})$ the distribution of ${\bf X}$ from (a). The set of extreme eternal multiplicative coalescent distributions is precisely

$\{\mu(\kappa,\tau,{\bf c}):\ (\kappa,\tau,{\bf c})\in{\cal I}\}$ $\cup\ \{\hat{\mu}(y):0\leq y<\infty\}$.

(c) Let $(\kappa,\tau,{\bf c})\in{\cal I}$. An (extreme) eternal multiplicative coalescent ${\bf X}$ has distribution $\mu(\kappa,\tau,{\bf c})$ if and only if*

[TABLE]

In terms of the above defined parametrization, the Aldous [6] standard (eternal) multiplicative coalescent has distribution $\mu(1,0,{\bf 0})$. The parameters $\tau$ and $\kappa$ correspond to time-centering and time/mass scaling respectively: if ${\bf X}$ has distribution $\mu(1,0,{\bf c})$, then $\widetilde{{\bf X}}(t)=\kappa^{-1/3}{\bf X}(\kappa^{-2/3}(t-\tau))$ has distribution $\mu(\kappa,\tau,\kappa^{1/3}{\bf c})$. Due to (1.11), the components of ${\bf c}$ may be interpreted as the relative sizes of distinguished large blocks in the $t\to-\infty$ limit.

1.3 The main results

The rest of this work will mostly ignore the constant eternal multiplicative coalescents. For a given $(\kappa,\tau,{\bf c})\in{\cal I}$ we can clearly write $W^{\kappa,t-\tau,{\bf c}}(s)=W^{\kappa,-\tau,{\bf c}}(s)+ts,\,s\geq 0.$ The Lévy-type process $W^{\kappa,-\tau,{\bf c}}$ is particularly important for this work. As we are about to see, $W^{\kappa,-\tau,{\bf c}}$ corresponds to the Lévy-type process from the abstract as soon as ${\bf X}$ has law $\mu(\kappa,\tau,{\bf c})$.

As noted in [7] and in [6] beforehand, at the time there was no appealing intuitive explanation of why excursions of a stochastic process would be relevant in describing the marginal laws in Theorem 1.1(a). One purpose of this work is to offer a convincing explanation (see Proposition 5 below and also in Section 4, then Lemma 9 in Section 5, and Lemma 11 in Section 6). Furthermore, open problem (3) of [6] asks about the existence of a two parameter (non-negative) process $(B^{t}(s),\,s\geq 0,\,t\in\mathbb{R})$ such that the excursion (away from [math]) lengths of $(B^{\,\cdot}(s),\,s\geq 0)$ evolve as $X^{*}(\cdot)$. The statement of this problem continues by offering an intuitive explanation for why $R^{1,0,{\bf 0}}:=\{{\rm reflected}(W^{1,0,{\bf 0}}(s)+ts),\ s\geq 0,\ t\geq 0\}$ should not be the answer to this problem. Aldous’ argument is more than superficially convincing, but the striking reality is that, on the contrary, the simplest guess $R^{1,0,{\bf 0}}$ is the correct answer. Armendáriz [11] obtained but never published this result, and Broutin and Marckert [27] recently derived it, via a different approach from either [11] or the one presented below, while considering in addition the excess-edge data in agreement with [6] (thus improving on the Armendáriz claim).

Popular belief judges the breadth-first-walk construction, on which [6] ; [7] reside, as “inadequate” and the main reason for the just described “confusion” in the statement of [6] , open problem (3). One of the main points of this work is to show the contrary. Indeed, a modification of the original (Aldous’) breadth-first-walk from [6] ; [7] , combined with a rigorous formulation (see Proposition 3) of Uribe’s [62] graphical interpretation of the Armendáriz’ representation [11] , yields the following claim of independent interest, here stated for readers’ benefit in the simplest (purely) homogeneous setting.

Claim (Proposition 5, special case) Suppose that $x_{1}=x_{2}=\ldots=x_{n}=1$, for some $n\in\mathbb{N}$, and define for $q>0$

[TABLE]

*where $\xi_{i}$, $i=1.\ldots,n$ is a family of i.i.d. exponential (rate $1$) random variables. For each $q>0$, let “blocks at time $q$” be the finite collection of excursions (above past minima) of $Z^{q}$, and for each block let its mass be the corresponding excursion length. Set $X(0)$ to be the configuration of $n$ blocks of mass $1$, and for $q>0$ let $X(q)$ be the configuration of masses of blocks at time $q$ ($X(q)$ is a vector with components listed in non-increasing order, and infinitely many [math]s may be appended to make it an element of l^{2}_{\mbox{{\footnotesize\searrow}}}). Then $(X(q),\,q\geq 0)$ evolves according to (1.1). *

To the best of our knowledge, even the “static” statement that matches the law of $X(q)$ to the law of the component sizes of the continuous-time homogeneous Erdős-Rényi random graph for each fixed time $q$ separately, was not previously recorded (even though the analysis of [62] , on pages 111-112 is implicitly equivalent). To have a glimpse at the power of this approach, the reader is invited to fix $c>1$, and consider the asymptotic behavior (as $n\to\infty$) of a related process $Z^{(1/n,\ldots,1/n),cn}(s):=\sum_{i=1}^{n}\frac{1}{n}1_{(n\xi_{i}\,\leq\ cns)}-s$ , $s\geq 0$, (see Section 2 or Proposition 5 for notation) in order to determine (in a few lines only) the asymptotic size of the giant component in the supercritical regime. In addition, the simultaneous breadth-first walks framework allows for a particularly elegant treatment of surplus edges, carried out in [47] .

The analysis similar to that of [7] (to be done in Sections 5 and 6) now yields:

Theorem 1.2**.**

Fix a Lévy-type process $W^{\kappa,-\tau,{\bf c}}$, and for any $t\in(-\infty,\infty)$ define

[TABLE]

Let $B^{\kappa,t-\tau,{\bf c}}$ be defined as in (1.7). For each $t$, let ${\bf X}(t)={\bf X}^{\kappa,\tau,{\bf c}}(t)$ be the infinite vector of ordered excursion lengths of $B^{\kappa,t-\tau,{\bf c}}$ away from [math]. Then $({\bf X}(t),t\in(-\infty,\infty))$ is a càdlàg realization of $\mu(\kappa,\tau,{\bf c})$.

1.4 Further comments on the literature and related work

For almost two decades the only stochastic merging process widely studied by probabilists was the (Kingman) coalescent [43] ; [44] . Starting with Aldous [5] ; [6] , and Pitman [55] , Sagitov [58] , and Donnelly and Kurtz [29] , the main-stream probability research on coalescents was much diversified.

The Kingman coalescent and, more generally, the mass-less (exchangeable) coalescents of [55] ; [58] ; [29] mostly appear in connection to the mathematical population genetics, as universal (robust) scaling limits of genealogical trees (see for example [53] ; [59] ; [19] ; [60] , or a survey [13] ).

The standard multiplicative coalescent is the universal scaling limit of numerous stochastic (typically combinatorial or graph-theoretic) homogeneous (or symmetric) merging-like models [6] ; [1] ; [10] ; [20] ; [21] ; [22] ; [57] ; [61] . The “non-standard” eternal extreme laws from [7] are also scaling limits of inhomogeneous random graphs and related processes, under appropriate assumptions [7] ; [23] ; [24] ; [26] .

The two nice graphical constructions for coalescents with masses were discovered early on: by Aldous in [6] for the multiplicative case, and almost simultaneously by Aldous and Pitman [8] for the additive case (here any pair of blocks of mass $x$ and $y$ merges at rate $x+y$), via cutting the continuum random tree [4] (see also Remark 2.1(c) in the next section). The analogue of [7] in the additive coalescent case is again due to Aldous and Pitman [9] . No nice graphical construction for another (merging rate) coalescent with masses seems to have been found since. For studies of stochastic coalescents with general kernel see Evans and Pitman [34] and Fournier [36] ; [37] . Interest for probabilistic study of related Smoluchowski’s equations (with general merging kernels) was also sparked by [5] , see for example Norris [54] , Jeon [41] , then Fournier and Laurençot [38] ; [39] and Bertoin [18] for more recent, and Merle and Normand [51] ; [52] for even more recent developments. All of the above mentioned models are mean-field. See for example [49] ; [12] ; [35] for studies of (mass-less) coalescent models in the presence of spatial structure.

As already mentioned, Broutin and Marckert [27] obtain Theorem 1.2 in the standard multiplicative coalescent case, via Prim’s algorithm construction invented for the purpose of their study, and notably different from the approach presented here. Before them Bhamidi et al. [20] ; [21] proved f.d.d. convergence for models similar to Erdős-Rényi random graph. For the standard additive coalescent, analogous results were obtained rather early by Bertoin [15] ; [16] and Chassaing and Louchard [28] , and are rederived in [27] , again via an appropriate Prim’s algorithm representation.

In parallel to and independently from the research presented here, both Martin and Ráth [50] and Uribe Bravo [63] have been studying closely related models and questions. Their approaches seem to be quite different from the one taken here, with some notable similarities. James Martin and Balázs Ráth [50] introduce a coalescence-fragmentation model called the multiplicative coalescent with linear deletion (MCLD). Here in addition to (and independently of) the multiplicative coalescence, each component is permanently removed from the system at a rate proportional to its mass (this proportionality parameter is denoted by $\lambda$). In the absence of deletion (i.e. when $\lambda=0$), their “tilt (and shift) operator” representation of the MCLD leads to an alternative proof of Theorem 1.2, sketched in detail in [50] , Section 6.1 (see [50] , Corollary 6.6). Further comments on links and similarities to [50] will be made along the way, most frequently in Section 3. Gerónimo Uribe [63] relies on a generalization of the construction from [27] , explains its links to Armendáriz’ representation, and works towards another derivation of Theorem 1.2.

The arguments presented in the sequel are partially relying on direct applications of a non-trivial result from [7] , Section 2.6 (depending on [7] , Section 2.5) in Section 6 (more precisely, Corollary 10). In comparison, (a) [27] also rely on the convergence results of [6] in the standard multiplicative coalescent setting, as well as additional estimates proved in [1] , and (b) the analysis done in [50] , Sections 4 and 5 seems to be a formal analogue of that in [7] , Sections 2.5-2.6.

The present approach to Theorem 1.2 is of independent interest even in the standard multiplicative coalescent setting (where Section 5 would simplify further, since ${\bf c}={\bf 0}$, and already Lemma 8 from [6] would be sufficient for making conclusions in Section 6). In addition, it may prove useful for continued analysis of the multiplicative coalescents, as well as various other processes in the multiplicative coalescent “domain of attraction”.

The reader is referred to Bertoin [17] and Pitman [56] for further pointers to stochastic coalescence literature, and to Bollobas [25] and Durrett [30] for the random graph theory and literature.

The rest of the paper is organized as follows: Section 2 introduces the simultaneous breadth-first walks and explains their link to the (marginal) law of the multiplicative coalescent, and the original breadth-first walks of [6] ; [7] . Section 3 recalls Uribe’s diagrams and includes Proposition 3, that connects the diagrams to the multiplicative coalescent. In Section 4 the simultaneous BFWs and Uribe’s diagrams are linked, and as a result an important conclusion is made in Proposition 5 (the generalized version of the claim preceding Theorem 1.2). All the processes considered in Sections 2–4 have finite initial states. Section 5 serves to pass to the limit where the initial configuration is in l^{2}_{\mbox{{\footnotesize\searrow}}}. The similarities to and differences from [7] are discussed along the way, and in the accompanying paper [48] . Theorem 1.2 is proved in Section 6. Supplementary material [48] is described in the paragraph LABEL:suppA preceding the bibliography.

2 Simultaneous breadth-first walks

This section revisits the Aldous [6] breadth-first walk construction of the multiplicative coalescent started from a finite vector ${\bf x}$ (see for example [48] Section 3), with two important differences (or modifications), to be described along the way.

Recall that “breadth-first” refers here to the order in which the vertices of a given connected graph (or one of its spanning trees) are explored. Such exploration process starts at the root, visits all of its children (these vertices become the 1st generation), then all the children of all the vertices from the 1st generation (these vertices become the 2nd generation), then all the children of the 2nd generation, and keeps going until all the vertices (of all the generations) are visited, or until forever (if the tree is infinite).

Refer to {\bf x}=(x_{1},x_{2},x_{3}\ldots)\in l^{2}_{\mbox{{\footnotesize\searrow}}} as finite, if for some $i\in\mathbb{N}$ we have $x_{i}=0$. Let the length of ${\bf x}$ be the number $\rm len({\bf x})$ of non-zero coordinates of ${\bf x}$. Fix a finite initial configuration {\bf x}\in l^{2}_{\mbox{{\footnotesize\searrow}}}. For each $i\leq\rm len({\bf x})$ let $\xi_{i}$ have exponential (rate $x_{i}$) distribution, independently over $i$.

As before, ${\bf X}$ is used to denote a stochastic process evolving according to the multiplicative coalescent dynamics. In this section the initial configuration ${\bf X}(0)={\bf x}$ is discrete (finite), so ${\bf X}$ is the law of the continuous-time random graph connected component masses. Given $\xi$, simultaneously for all $q>0$, we next construct the modified (with respect to [6] ; [7] ) breadth-first walk, coupled with ${\bf X}(q)$ started from ${\bf X}(0)={\bf x}$ at time [math] (see Propositions 1 and 5). This simultaneity in $q$ is a new feature with respect to [6] ; [7] . To the best of our knowledge, the powerful (full law) coupling of Proposition 5 (see also the simplified Claim in the introduction) was previously unknown. The family of processes defined in (2.1) below will be henceforth called the simultaneous breadth-first walks.

Fix $q>0$, and consider the sequence $(\xi_{i}/q)_{i\leq\rm len({\bf x})}$. Let us introduce the abbreviation $\xi_{i}^{q}:=\xi_{i}/q$. The order statistics of $(\xi^{q}_{i})_{i\leq\rm len({\bf x})}$ are $(\xi^{q}_{(i)})_{i\leq\rm len({\bf x})}$. Define

[TABLE]

In words, $Z^{{\bf x},q}$ has a unit negative drift and successive positive jumps, which occur precisely at times $(\xi^{q}_{(i)})_{i\leq\rm len({\bf x})}$, and where the magnitude of the $i$th successive jump is denoted by $x_{(i)}$. The next figure shows graphs of $Z^{{\bf x},q}$ and of $Z^{{\bf x},4q/3}$ for the same realization as that depicted on Figure 2. The three ticks on the $x$-axis of each graph correspond to $\xi_{1}^{q}$, $\xi_{5}^{q}$ and $\xi_{6}^{q}$ (resp. to $\xi_{1}^{4q/3}$, $\xi_{5}^{4q/3}$ and $\xi_{6}^{4q/3}$). The meaning of the intervals indicated in gray or blue below each graph will become clear shortly (see also Figure 2).

Here is the first important observation. For each $q$, the multiplicative coalescent started from ${\bf x}$ and evaluated at time $q$ can be constructed in parallel to $Z^{{\bf x},q}$ via a breadth-first walk coupling, similar to the one from [6] ; [7] . The interval $F_{1}^{q}:=[0,\xi^{q}_{(1)}]$ is the first “load-free” period. Set $J_{0}:=\{1,2,\ldots,\rm len({\bf x})\}$. At the time of the first jump of $Z^{{\bf x},q}$ we note

[TABLE]

so that $\pi_{1}$ is the index of the first size-biased pick from $({\bf x}_{i})_{i=1}^{\rm len({\bf x})}$ using $\xi$s (or equally, $\xi^{q}$s).

s$$1$$2$$3$$4$$5$$6$$7$$s$$1$$2$$3$$4$$5$$6$$7

Figure 1

Furthermore, let us define for $l\leq\rm len({\bf x})$

[TABLE]

In this way, $(x_{\pi_{1}},x_{\pi_{2}},\ldots,x_{\pi_{\rm len({\bf x})}})$ is the size-biased random ordering of the initial non-trivial block masses, and in particular $x_{\pi_{i}}$ equals $x_{(i)}$ from (2.1). As already noted, the random permutation $\pi$ does not depend on $q$. Let \mbox{{\cal F}}^{q}_{s}:=\sigma\{\{\{\xi^{q}_{i}>u\}:\,i\in J_{0}\},\,u\leq s\}. Then \mbox{{\cal F}}^{q}=\{\mbox{{\cal F}}^{q}_{s},\,s\geq 0\} is the filtration generated by the arrivals of $\xi^{q}$s. Due to elementary properties of independent exponentials, it is clear that the above defined process $Z^{{\bf x},q}$ is a continuous-time Markov chain with respect to \mbox{{\cal F}}^{q}. Indeed, given \mbox{{\cal F}}^{q}_{s}, the (residual) clocks $\xi^{q}_{i}\vee s-s$ are again mutually independent, and moreover on the event $\{\xi^{q}_{i}>s\}$ we clearly have P(\xi^{q}_{i}-s>u|\mbox{{\cal F}}^{q}_{s})=e^{-x_{i}qu}=P(\xi_{i}^{q}>u). Furthermore, $\xi^{q}_{(1)}$ is a finite stopping time with respect to \mbox{{\cal F}}^{q} and

[TABLE]

Let $I_{0}=\emptyset$ and $I_{1}:=(\xi^{q}_{(1)},\xi^{q}_{(1)}+x_{\pi_{1}}]$. Note that the length of the interval $I_{1}$ is the same (positive) quantity $x_{\pi_{1}}$ for all $q>0$. During the time interval $I_{1}$ the dynamics “listens for the children of $\pi_{1}$”. More precisely, if for some $j$ we have $\xi^{q}_{j}\in I_{1}$, or equivalently, if $\xi^{q}_{j}-\xi^{q}_{(1)}\leq x_{\pi_{1}}$, we can interpret this as

[TABLE]

Indeed, as argued above, P(\xi^{q}_{j}-\xi^{q}_{(1)}>x_{\pi_{1}}|\mbox{{\cal F}}^{q}_{\xi^{q}_{(1)}})=e^{-qx_{j}x_{\pi_{1}}}, and this is precisely the multiplicative coalescent probability of the $j$th and the $\pi_{1}$st block not merging before time $q$. For any two reals $a<b$ and an interval $[c,d]$ where $0\leq c<d$, define the concatenation

[TABLE]

Recall that $I_{1}=(\xi^{q}_{(1)},\xi^{q}_{(1)}+x_{\pi_{1}}]$, and define $N_{1}$ to be the number of $\xi^{q}$s that rung during $I_{1}$ (this is the size of the 1st generation in the exploration process). For any $l\geq 2$ define recursively: if $I_{l-1}$ is defined

[TABLE]

and if $I_{l}$ is defined in (2.4), let

[TABLE]

and otherwise let $N_{l}$ be (temporarily) undefined. Since $\xi^{q}$s decrease in $q$, the intervals $I^{q}_{\cdot}$ defined in this (coupling) construction do vary over $q$ (their endpoints decrease in $q$), but all of their lengths are constant in $q$. In fact, if defined, $I_{l}$ equals $(\xi_{(1)}^{q},\xi_{(1)}^{q}+\sum_{m=1}^{l}x_{\pi_{m}}]$. We henceforth abuse the notation and mostly omit the superscript $q$ when referring to $I$s or $N$s. During each $I_{l}\setminus I_{l-1}$ the coupling dynamics “listens for the children of $\pi_{l}$”, among all the $\xi^{q}$s which have not been heard before (i.e. they did not ring during $I_{l-1}$). If $I_{l}$ is defined in (2.4), the set of children of $\pi_{l}$ in the above breadth-first order is precisely $J_{N_{l-1}}\setminus J_{N_{l}}$ ($J$s were defined in (2.2)), which will be empty if and only if $N_{l}=N_{l-1}$. The same memoryless property of exponential random variables as used above (e.g. in (2.3)) ensures that

[TABLE]

Due to independence of $\xi$s, the residual clocks have again the (conditional) multi-dimensional product law. So for each $l$, the set of children of $\pi_{l}$ equals in law to the set of blocks which are connected by an edge to the $\pi_{l}$th block in the multiplicative coalescent at time $q$, given that they did not get connected by an edge (before time $q$) to any of the previously recorded blocks $\pi_{1},\ldots,\pi_{l-1}$. The above procedure may (and typically will) stop at some $l_{1}\leq\rm len({\bf x})$, due to $\xi^{q}_{(l_{1})}$ not arriving in $I_{l_{1}-1}$. This will happen if and only if the whole connected component of the $\pi_{1}$st initial block (in the multiplicative coalescent, observed at time $q$) was explored during $I_{l_{1}-1}$, and the $\pi_{l_{1}-1}$st initial block was its last visited “descendant”, while the rest of the graph was not yet “seen” during $F_{1}^{q}\cup I_{l_{1}-1}$. Indeed, if $a_{1}=\xi^{q}_{(1)}$ and $b_{1}=\xi^{q}_{(1)}+x_{\pi_{1}}+\ldots+x_{\pi_{l_{1}-1}}$, it is straight-forward to see that

[TABLE]

In words, the interval $Cl(I_{l_{1}-1})=[a_{1},b_{1}]$ is an excursion of $Z^{q,{\bf x}}$ above past minima of length $b_{1}-a_{1}=x_{\pi_{1}}+\ldots+x_{\pi_{l_{1}-1}}$, which is the total mass of the first (explored) spanning tree in the breadth-first walk. Due to (2.3,2.6) and the related observations made above, this (random) tree matches the spanning tree of the connected component of $\pi_{1}$ in the coupled multiplicative coalescent, observed at time $q$. This (first) spanning tree is rooted at $\pi_{1}$ (cf. Figures 2 and 3) for all $q>0$. It will be clear from construction, that the roots of subsequently explored spanning trees can (and inevitably do) change at some $q>0$. The next interval of time $F_{2}^{q}:=(\xi^{q}_{(1)}+x_{\pi_{1}}+\cdots+x_{\pi_{l_{1}-1}},\xi^{q}_{(l_{1})}]$ is again “load-free” for the breadth-first walk. Repeating the above exploration procedure starting from $\xi^{q}_{(l_{1})}$ amounts to defining $I_{l_{1}}^{q}\equiv I_{l_{1}}:=(\xi^{q}_{(l_{1})},\xi^{q}_{(l_{1})}+x_{\pi_{l_{1}}}]$ and listening for the children of $\pi_{l_{1}}$st block during $I_{l_{1}}$, and then running the recursion (2.4,2.5) for $l\geq l_{1}+1$ until it stops, which occurs when all the vertices (blocks) of the second connected component are explored. This exploratory coupling construction continues until all the initial blocks of positive mass are accounted for, or equivalently until $\xi^{q}_{(\rm len({\bf x}))}$. Clearly no $\xi$ can ring during $I_{\rm len({\bf x})}\setminus I_{\rm len({\bf x})-1}$ (which is open on the left), and $Z^{{\bf x},q}$ continues its evolution as a deterministic process (line of slope $-1$) starting from the left endpoint of $I_{\rm len({\bf x})}$.

s$$1$$2$$3$$4$$5$$6$$7$$\xi_{(1)}^{q}$$\xi_{(5)}^{q}$$\xi_{(6)}^{q}legend to $F_{1}^{q},\,I_{1}^{q},\,I_{2}^{q}\setminus I_{1}^{q},\,I_{3}^{q}\setminus I_{2}^{q},\,I_{4}^{q}\setminus I_{3}^{q},\,F_{2}^{q},\,I_{5}^{q},\,F_{3}^{q},\,I_{6}^{q}$, and $I_{7}^{q}\setminus I_{6}^{q}$

Figure 2

Figures 1 and 2 illustrate the just described coupling. In the current notation ${\bf x}=(1.1,0.8,0.5,0.4,0.4,0.3,0.2,0,$ $0,\dots)$ so that $\rm len({\bf x})=7$. The three “load-free” intervals $F_{i}^{q}$, $i=1,2,3$ are indicated in gray. The interval $I_{i}^{q}$ or $I_{i}^{q}\setminus I_{i-1}^{q}$ (the latter corresponds to non-leading blocks) is indicated in blue with marker $i$ on top. The excursions of $Z^{{\bf x},q}$ above past minima are the (closed) disjoint unions of blue intervals. Each excursion of $Z^{{\bf x},q}$ above past minima corresponds uniquely to a connected component in the coupled multiplicative coalescent evaluated at time $q$. It is clear from (2.4,2.5) that the order of blocks visited within any given connected component is breadth-first. Note as well that the connected components are explored in the size-biased order. Indeed, the fact that the initial block of the next component to be explored is picked in a size-biased way, with respect to block masses, induces size-biasing of connected components (again with respect to mass) in the multiplicative coalescent at time $q$. The above reasoning can be also summarized as follows:

Proposition 1*.*

Let ${\bf x}$ be finite, and $q>0$. Let the breadth-first walk $Z^{{\bf x},q}$ encode {\bf X}(q)\in l^{2}_{\mbox{{\footnotesize\searrow}}} as follows: for each excursion (above past minima) of $Z^{{\bf x},q}$ record its length, and let ${\bf X}(q)$ be the vector of thus obtained decreasingly ordered excursion lengths, appended with infinitely many [math]s. Then ${\bf X}(q)$ has the marginal law of the multiplicative coalescent  started from ${\bf X}(0)={\bf x}$ and observed at time $q$.

Since $(Z^{{\bf x},q})_{q>0}$ exist on one and the same probability space, the process

[TABLE]

is well-defined, and we refer to it temporarily as the multiplicative coalescent marginals coupled to $Z^{{\bf x},\cdot}$.

\xi_{(2)}^{q}-\xi_{(1)}^{q}$$\xi_{(3)}^{q}-\xi_{(1)}^{q}$$\xi_{(4)}^{q}-\xi_{(1)}^{q}$$\xi_{(7)}^{q}-\xi_{(6)}^{q}+\sum_{i=1}^{5}x_{\pi_{i}}

Figure 3

In the original breadth-first construction (coupling) in [6] ; [7] , the leading block of each component did not correspond to a jump of the walk. It was chosen instead via an auxiliary source of randomness. In comparison, each non-leading block was uniquely matched to a jump of the breadth-first walk, and for a non-leading block of mass $m$, this jump had size $m$ and was exponentially distributed with rate $m$. As in the simultaneous construction, all the exponential jumps were mutually independent. The reader is also referred to [48] Section 3 for further details. Figure 3 (without the vertical dashed lines and their labels) is a duplicate of [7] , Figure 1. Here we assume that it shows the graph of the original breadth-first walk (at multiplicative coalescent-time $q$), corresponding to the same realization as the one used in Figure 2. In particular ${\bf x}=(1.1,0.8,0.5,0.4,0.4,0.3,0.2,0,$ $0,\dots)$ and $x_{\pi_{i}}\equiv v(i)$. Note that the segment marked by $i$ in Figure 2 has exactly the same length as the segment marked by $v(i)$ in Figure 3. One can read off from Figure 3 that $\pi$ belongs to $\{(2,4,3,7,6,5,1)$, $(2,5,3,7,6,4,1)\}$. However, $\xi_{(1)}=\xi_{2}$, $\xi_{(5)}=\xi_{6}$ and $\xi_{(6)}=\xi_{5}$ (or $\xi_{(6)}=\xi_{4}$, depending on $\pi$) are not observed. In comparison, in the simultaneous breadth-first walk construction, the $\xi_{(1)}^{q}$ (here it equals $\xi_{2}$), $\xi_{(5)}$ (here it equals $\xi_{6}$) and $\xi_{(6)}$ all influence (see (2.1) or Figure 2) the walk. The additional vertical dashed lines (not existing in [7] , Figure 1) and their labels illustrate the link between the two breadth-first walk constructions (see also Figure 2 and the explanations provided below it). In particular, the first jump of the original breadth-first walk happens at time (distributed as) $\xi_{(2)}^{q}-\xi_{(1)}^{q}$, the second one happens at time (distributed as) $\xi_{(3)}^{q}-\xi_{(1)}^{q}$, and this continues until the first component is exhausted. The next jump happens at the time the next non-leading block is encountered. Moreover, due to elementary properties of residual exponentials, the following is true.

Lemma 2*.*

Fix a finite initial configuration ${\bf x}$ and (multiplicative coalescent) time $q>0$, recall $Z^{{\bf x},q}$ from (2.1), and the load-free intervals $F_{i}^{q}$, $i\geq 1$. If $F_{i}^{q}$, $i\geq 1$ are all cut from the abscissa, and the jumps which happen at the end points of $F_{\cdot}^{q}$ are all ignored (deleted), then (the graph of) $Z^{{\bf x},q}$ transformed in this way has the law of (the graph of) the breadth-first walk from [6] ; [7] , corresponding to $q$.

Most of the argument is included in the above made observations and explanations. Figures 2 and 3 illustrate the claim. Time $s$ for $Z^{{\bf x},q}$ (on Figure 2) corresponds to time $s-\xi_{(L+1)}^{q}+\sum_{i=1}^{K}x_{\pi_{i}}$ in the original breadth-first walk (on Figure 3), where $L$ is the number of connected components completely explored via $Z^{{\bf x},q}$ before time $s$, and $(x_{\pi_{i}})_{i=1}^{K}$ is the total mass of these $L$ connected components (on $L=0$ this mass is naturally [math]). The consecutive load-free intervals and their final jumps, which are cut out by the transformation, serve as the auxiliary source of randomness used for choosing leading blocks in the exploration of [6] ; [7] . The details are left to the reader.

Remark 2.1*.*

(a) At the moment it may seem that the main (potential) gain of the just described modified breadth-first construction is in “compactifying” the input data (compare with [7] Section 2.3 or [27] , Section 6.1 for alternatives). It will become apparent in the sequel (see Proposition 5 and Section 5) that this construction is quite natural, in that stronger convergence results can be obtained from it with less effort.

(b) As $q\searrow 0$, the $\xi^{q}$ diverge to $\infty$, but more importantly they diverge from each other, so ${\bf X}(q)\to{\bf x}={\bf X}(0)$ almost surely. It is not difficult to see that for any $q\geq 0$, ${\bf X}$ is also almost surely right-continuous at $q$ (see Section 4).

(c) A little thought is needed to realize that as $q$ increases, the excursion families of $Z^{{\bf x},\cdot}$ are “nested”: with probability $1$, if $q_{1}<q_{2}$ and two blocks $k,l$ are merged in ${\bf X}(q_{1})$, they are also merged in ${\bf X}(q_{2})$ (an example of this is depicted in Figure 1). This fact is encouraging, but cannot ensure on its own that the multiplicative coalescent marginals coupled to $Z^{{\bf x},\cdot}$ is in fact a multiplicative coalescent process. Moreover, while the nesting is encouraging, the following observation will likely increase the level of reader’s skepticism about ${\bf X}$ having the multiplicative coalescent law: if $e_{1}^{q_{1}}$, $e_{2}^{q_{1}}$ and $e_{3}^{q_{1}}$ are three different excursions of $Z^{{\bf x},q_{1}}$ explored in the increasing order of their indices, and if the initial blocks $k,l,m$ are contained in the connected components matched to $e_{1}^{q_{1}},e_{2}^{q_{1}},e_{3}^{q_{1}}$, respectively, then it is impossible that $k$ and $m$ are merged in $X(q_{2})$ without $l$ being merged with $k$ (and therefore with $m$) in $X(q_{2})$. If there is a simultaneous (for all $q$) scaling limit of $(Z^{{\bf x},q},{\bf X}(q))$ (under well chosen hypotheses), the just mentioned property persists in the limit. This observation is perhaps the strongest intuitive argument pointing against the claim of Proposition 5 and Theorem 1.2. On the other hand, analogous representations of the standard additive coalescent are well-known (cf. [28] ; [15] ; [16] ). One may be less surprised there, due to the “cutting the CRT” dual (from [8] ), and the well-known connection between the exploration process of continuum trees and forests on the one hand, and Brownian excursions on the other (cf. [4] ; [56] ; [17] ). As Nicolas Broutin (personal communication) points out, any (binary) fragmentation can be formally represented as a “stick-breaking” process, in which the two broken pieces of any split block remain nearest neighbors (in some arbitrary but fixed way). The reversed “coalescent” will then have the above counterintuitive property by definition. However, one is particularly fortunate if both of these processes (time-reversals of each other) are Markov, and if in addition the “sticks” are the excursions of a (generalized) random walk or a related process.

(d) Let us denote by ${\mathcal{C}}$ the operation on paths (i.e. the cutting and pasting transformation) from Lemma 2. In [7] the non-trivial multiplicative coalescent extreme entrance laws were obtained by taking limits of Aldous’ breadth-first walks (see Section 5 below), and the limits of their excursions (nearly) above past minima. It was shown that these excursion lengths, considered as an l^{2}_{\mbox{{\footnotesize\searrow}}}-valued random object, converge in law to the excursion lengths above past minima of the limiting “walk” (a member of the family defined in (1.6)). Lemma 2 makes this latter (somewhat technical) step redundant in the present setting. More precisely, if one can show that under the same hypotheses as those in [7] , $(Z^{t+n^{1/3},{\bf x}^{(n)}})_{n}$ (see Section 5 for precise definitions) converges to the same $W^{\kappa,t-\tau,{\bf c}}$ as their $({\mathcal{C}}(Z^{t+n^{1/3},{\bf x}^{(n)}}))_{n}$, the conclusion about the ordered excursion lengths is immediate. Indeed, the sequence of excursion above past minima of $Z^{t+n^{1/3},{\bf x}^{(n)}}$ almost surely matches the sequence of excursions (nearly) above past minima of ${\mathcal{C}}(Z^{t+n^{1/3},{\bf x}^{(n)}})$, for which Propositions 7 and 9 of [7] (including the results in [7] , Section 2.6) apply verbatim.

3 Uribe’s diagram

We start by recalling the insight given in Chapter 4 of Uribe [62] , in the notation analogous to that of Section 2. In particular, ${\bf x}=(x_{1},x_{2},\ldots,x_{n})$ is a finite-dimensional vector with $n\geq 2$, $\xi_{i}$ is an exponential (rate $x_{i}$) random variable, and $\xi$s are mutually independent. Denote by $\pi$ the size-biased random reordering of ${\bf x}$, which is determined by $\xi$s, so that $\xi_{\pi_{i}}\equiv\xi_{(i)}$, $\forall i$ (almost surely). Define $n$ different half-lines: for $s\geq 0$

[TABLE]

Consider two integers $k,j$ such that $1\leq j<k\leq n$. Since $L_{k}^{\prime}$ starts (a.s.) at a strictly larger value than $L_{j}^{\prime}$, and it has (absolute) slope strictly greater than $L_{j}^{\prime}$, it is clear that $L_{k}^{\prime}$ and $L_{j}^{\prime}$ intersect at some $s_{k,j}>0$. For each $k=2,\ldots,n$ define

[TABLE]

There are (almost surely) no ties among $s_{k}$ or $s_{k,j}$ for different indices $k,j$ (see also Remark 3.1 below). Uribe’s diagram consists of line segments (see Figures 4 and 5)

[TABLE]

The image of a realization of $L$ in Figure 4 is inspired by [62] , Chapter 4, Figure 1. It is interesting to note that its reflection in the $x$ axis is a redrawing of [50] Figure 1.3. A more detailed figure, which will correspond in terms of the values of ${\bf x}$, $q$ and $\xi$s to the images in Figures 2 and 3 is provided in Section 4. According to [62] , Chapter 4, Section 2, Armendáriz’ representation of the multiplicative coalescent [11] is this graphical summary (introduced in [62] for enhanced understanding) joint with an informal description of the following kind:

• the state space is $\mathbb{R}_{+}^{n}$, its elements are interpreted as lists of block masses, not necessarily ordered,

• at time [math] match the block of mass $x_{\pi_{i}}$ to $L_{i}^{\prime}$,

• whenever two of the lines intersect, merge the corresponding blocks, form the new vector of block masses accordingly, continue drawing the lowest indexed line (now matched to the new block), as well as any line that has not participated in the intersection,

• keep going as long as there is more than one line remaining.

Another construction, advocated as “essentially Armendáriz’ representation” (but better for certain applications) is described at the beginning of [62] , Chapter 4, Section 3. This latter construction strongly resembles the “tilt” representation of Martin and Ráth (see [50] , Definition 2.7 and Theorem 2.8). The rest of this section is only one possible rigorous formulation of the aforementioned picture. Here Proposition 3 is stated and proved (in [48] ) in a particularly convenient partition-valued framework. Corollary 4 features a process that should be the analogue of the Armendáriz representation according to [62] . It serves here as a step in obtaining Theorem 1.2, through the equivalence obtained in Section 4, suggesting its potential relevance elsewhere. An alternative approach is the particle representation of Martin and Ráth [50] , Section 3.2, developed in the more general setting of multiplicative coalescent with linear deletion. Interestingly, the analysis of [50] is again based on an analogue of (a reflection of) Uribe’s diagram. Uribe’s diagram could be interpreted as a “genealogical tree”. More precisely, let us match each point on the diagram $\cup_{i=1}^{n}\cup_{s\in[0,s_{i}]}L_{i}(s)$ to a subset of $\{1,2,\ldots,n\}$ in the following way. For each $i$, set $T_{i}(0):=\{\pi_{i}\}$. Each $T_{i}$ is piece-wise constant, and jumps according to the following algorithm:

[TABLE]

and $T_{i}(s):=\emptyset$, $\forall s\geq s_{i}$. In words, for each $i$, the contents of $T_{i}$ are moved (without replacement) to $T_{\ell_{i}}$ at time $s_{i}$, and there is no other copying, cutting or pasting done. Define $T(s):=\{T_{1}(s),\ldots,T_{n}(s)\}$, $s\geq 0$, where the empty sets are ignored. Note that in this way each $s\geq 0$ is mapped to a partition $T(s)$ of $\{1,2,\ldots,n\}$. The reader is referred to an analogous look-down construction of [29] , which has since been extensively used in the setting of massless (usually called exchangeable) coalescents. Clearly the partitions along each path of $T$ are nested: if $s_{1}<s_{2}$ and $l$ and $k$ are in the same equivalence class of $T(s_{1})$, they are (almost surely) in the same equivalence class of $T(s_{2})$. So $T$ can also be regarded as a random coalescent on the space of partitions of $\{1,2,\ldots,n\}$. Its initial state is the trivial partition $\theta_{0}:=\{\{\pi_{1}\},\{\pi_{2}\},\ldots,\{\pi_{n}\}\}=\{\{1\},\{2\},\ldots,\{n\}\}$. Denote by \mbox{{\cal G}}_{t}:=\sigma\{T(s),\ s\leq t\}, $t\geq 0$, so that \mbox{{\cal G}}:=(\mbox{{\cal G}}_{t})_{t\geq 0} is the filtration generated by $T$. The process $T$ will be referred to in the sequel as Uribe’s coalescent process. It is evident that Uribe’s diagram $L$ is a deterministic function of ${\bf x}$ and $\xi$s, and when needed we shall underline this fact by writing $L(\xi_{1},\xi_{2},\ldots,\xi_{n};{\bf x})$.

Remark 3.1*.*

The $\xi$s are independent and continuous, and therefore (with probability $1$) no two pairs of lines in $L^{\prime}$ can meet simultaneously. Therefore $T$ (viewed as path-valued) takes value in the space of step functions, such that successive values on a typical path are nested (sub)partitions of $\theta_{0}$, each having exactly one fewer equivalence class than the prior one. In particular, \mbox{{\cal G}}_{t} is generated by events of the following type: for $k\geq 1$

[TABLE]

where $\theta_{j+1}$ is either equal to $\theta_{j}$ or to a “coarsening” of $\theta_{j}$ obtained by merging two different equivalent classes in $\theta_{j}$, $0\leq j\leq k-1$.

Let us now account for the masses: for any $i$ and $s\geq 0$ define $M_{i}(s):=\sum_{l\in T_{i}(s)}x_{l}$, with the convention that a sum over an empty set equals [math]. In this way, to each non-trivial equivalence class of $T(s)$ a positive mass is uniquely assigned, and the sum of the masses $\sum_{i}M_{i}(s)$ is the identity $\sum_{i=1}^{n}x_{i}$, almost surely. Suppose for a moment that $n=2$. For Uribe’s diagram, there are two possibilities: either $\pi$ is the identity, or $\pi$ is the transposition. In either case, the two initial equivalence classes $\{1\}$ and $\{2\}$ merge at random time $s_{2}$ which we denote by $S$. Note that the event $\{S>s\}$ is (almost surely) identical to the union of the following two disjoint events $\{\xi_{2}>\xi_{1}+sx_{1}\}$ and $\{\xi_{1}>\xi_{2}+sx_{2}\}$. Thus

[TABLE]

and the reader can easily verify that the RHS equals $e^{-x_{1}x_{2}s}$. So if $n=2$, the coalescent time is distributed equally in the random graph (component masses) and in Uribe’s coalescent. Even with this hint in mind, the next result will likely seem at least counterintuitive if not striking to an non-expert reader.

Proposition 3*.*

Uribe’s coalescent process $T$ has the law of the partition-valued process generated by the connected components of the continuous-time random graph. More precisely, it is a continuous-time Markov chain, such that any two equivalence classes in $T$ merge, independently of all the other merger events, at the rate equal to the product of their masses (where the mass of an equivalence class is the sum of $x$s over its elements).

The argument given in the supplement [48] Section 4 is based on a sequence of elementary observations, and its outline is comparable to that of [11] Lemma 15. Some care is however needed in correctly setting up the conditioning (otherwise the statement of the proposition would seem obvious from the start). In particular, there seems to be no way of a priori knowing that $T$ with respect to ${\cal G}$ has the Markov property. The argument exhibits the transition rates in the process of checking for Markovianity. Recall that, for $i=1,2,\ldots,n$, $M_{i}(s)$ is the mass of the equivalence class $T_{i}(s)$, provided that $T_{i}(s)\neq\emptyset$, and it is defined to be [math] otherwise. Now let ${\bf Y}(s)$ be a l^{2}_{\mbox{{\footnotesize\searrow}}}-valued random variable, formed by listing the components of $(M_{1}(s),M_{2}(s),\ldots,M_{n}(s))$ in decreasing order, and appending infinitely many zeros (to obtain a vector in l^{2}_{\mbox{{\footnotesize\searrow}}}). It is clear that ${\bf Y}=({\bf Y}(s),\,s\geq 0)$ is adapted to ${\cal G}$. Moreover Proposition 3 can be restated as

Corollary 4*.*

The process ${\bf Y}$ is a multiplicative coalescent started from the decreasing ordering of $(x_{1},x_{2},\ldots,x_{n},0,\ldots)$.

In the case where all the $n$ initial masses are equal, a subset of the just derived identities was already known to Gumbel [40] . It is well-known (see for example the discussion in [62] , Chapter 4, or [25] , Chapter 7, or [30] , Chapter 2, Section 8) that the connectivity time of the (classical) Erdős-Rényi random graph, is of the order $(\log{n}+G+o(1))/n$, where $G$ has Gumbel’s law $P(G\leq g)=e^{-e^{-g}}$, $g\in\mathbb{R}$.

4 Breadth-first walks meet Uribe’s diagram

In this section we will compare the simultaneous breadth-first random walks of Section 2 with Uribe’s diagram of Section 3. More precisely, a coupling of these random objects will be realized on one and the same probability space, so that the multiplicative coalescent marginals ${\bf X}$ coupled to $Z^{{\bf x},\cdot}$ (see Section 2) can be matched to ${\bf Y}$ derived from Uribe’s diagram (see Section 3). As an immediate corollary we obtain

Proposition 5*.*

Let ${\bf x}$ be finite. Then the multiplicative coalescent marginals ${\bf X}$ coupled to $Z^{{\bf x},\cdot}$ has the law of a multiplicative coalescent started from ${\bf x}$.

Let ${\bf x}$ be finite, set $n:=\rm len({\bf x})$, and recall the construction from Section 2. Use the same $(\xi_{1},\ldots,\xi_{n})$ to form the corresponding Uribe diagram $L(\xi_{1},\xi_{2},\ldots,\xi_{n};{\bf x})$. The notation is slightly abused here since in Section 2 (resp. 3) vectors have infinite (resp. finite) length, but the correspondence between the two is clear (appending infinitely many zeros to the finite vector will give the infinite one). Assume for a moment that ${\bf x}=(1.1,0.8,0.5,0.4,0.4,0.3,0.2)$, and let us pretend that the realization from Figures 2 and 3 corresponds to the time parameter $q$ equal to $2$. Let us assume in addition that $\pi=\tau:=(2,4,3,7,6,5,1)$. This means that $(\xi_{(1)},\xi_{(2)},\ldots,\xi_{(7)})=(\xi_{2},\xi_{4},\xi_{3},\xi_{7},\xi_{6},\xi_{5},\xi_{1})=(0.2,0.7,1.4,3.4,4.6,5.6,6)$.

s$$T_{1}(2)=\{2,4,3,7\}$$T_{5}(2)=\{6\}$$T_{6}(2)=\{5,1\}$$\{2\}$$\{4\}$$\{3\}$$\{7\}$$\{6\}$$\{5\}$$\{1\}

Figure 5

The corresponding Uribe’s diagram $L(\xi,{\bf x})$ is shown in Figure 5. For any time $s$, the partition $T(s)$ can be read from the graph as it could be read from a genealogical tree. Each of the “active” lines represents a different equivalence class. The blue vertical dashed line marks time $q=2$. In $T(2)$ there are three equivalence classes, matched to $L_{1},L_{5}$ and $L_{6}$, as shown in the figure. This partition is, of course, the same as the one given in terms of trees depicted on Figure 3 (recall that the same realization is being illustrated in Figures 2, 3 and 5). The coupling stated at the beginning of the section is realized in the most natural way. Recall that both $(Z^{{\bf x},q},\,q>0)$ and $L(\xi;{\bf x})$ (and therefore $T$ and ${\bf Y}$) are functions of $\xi$s and ${\bf x}$. It is important here to let the finite family of independent exponential random variables $\xi_{\cdot}$, used in the construction of $Z^{{\bf x},\cdot}$ and $L(\xi;{\bf x})$, be the same, almost surely. As already noted (see Remark 2.1 (c)), the partition structure induced by the evolution of $Z^{{\bf x},\cdot}$ gets coarser as $q$ increases. In addition, only pairs of neighboring blocks or families of blocks, with respect to the random order established by $\pi$, can coalesce either in ${\bf X}$ or in $T$ (that is, in ${\bf Y}$). Note that, for each multiplicative coalescent time $q$, the relation of being connected by a path of edges $\leftrightarrow$ that occurred before time $q$ is an equivalence relation on the initial set of blocks. Hence it suffices to show that, almost surely, for each $q>0$ and $i\in\{1,2,\ldots,n-1\}$ it is true that $\pi_{i}\leftrightarrow\pi_{j}$ with respect to $Z^{{\bf x},q}$ (see Section 2) if and only if $\pi_{i}\sim\pi_{j}$ with respect to $T$ (see Section 3). At time $q=0$ the just made claim is clearly correct, since there are no edges $\leftrightarrow$ in ${\bf X}$, and the partition of $T$ is trivial. Suppose that random time $Q_{1}>0$ is such that $T(Q_{1}-)=\theta_{0}$ and $T(Q_{1})$ contains $\{\pi_{i},\pi_{i+1}\}$. This means that the lines $L_{i}^{\prime}$ and $L_{i+1}^{\prime}$ intersect at time $Q_{1}$, and no other pair of lines intersects before time $Q_{1}$. Or equivalently, $\xi_{(i+1)}-\xi_{(i)}=Q_{1}x_{\pi_{i}}$ and $\xi_{(j+1)}-\xi_{(j)}>Q_{1}x_{\pi_{j}}$ for $j\neq i$. Or equivalently, $\xi_{(i+1)}^{Q_{1}}-\xi_{(i)}^{Q_{1}}=x_{\pi_{i}}$, and $\xi_{(j+1)}^{Q_{1}}-\xi_{(j)}^{Q_{1}}>x_{\pi_{j}}$ for $j\neq i$. A quick check of the construction in Section 2 suffices to see that, on the above event, the edge $\pi_{i}\leftrightarrow\pi_{i+1}$ arrives at time $Q_{1}$ in ${\bf X}$, and no edge arrives to ${\bf X}$ before time $Q_{1}$. At time $Q_{1}$, the line $L_{i+1}^{\prime}$ stops being active in Uribe’s diagram. The new neighbors of $\{\pi_{i},\pi_{i+1}\}$ are $\pi_{i+2}$ and $\pi_{i-1}$. All the active lines above $L_{i}^{\prime}$ account for the new mass $x_{\pi_{i}}+x_{\pi_{i+1}}$ of $\{\pi_{i},\pi_{i+1}\}$, since this quantity is built into their slope (together with masses corresponding to any other active lines underneath them). Similarly, $Z^{{\bf x},q}$ for $q>Q_{1}$ does not need to observe $\xi_{(i+1)}^{q}$ any longer, it suffices to attribute the cumulative “listening length” $x_{\pi_{i}}+x_{\pi_{i+1}}$ to the breadth-first walk time $\xi_{(i)}^{q}$ at which the leading particle of the component $\{\pi_{i},\pi_{i+1}\}$ is seen by the walk. Due to these two observations, one can continue the comparison of the coalescence of the remaining blocks driven by $(T(q),\,q\geq Q_{1})$ to that driven by $(Z^{{\bf x},q},\,q\geq Q_{1})$, and conclude by induction that in both processes the sequence of pairs of blocks that coalesce, and their respective times of coalescence, are identical, almost surely.

As already noted, Proposition 5 is a direct consequence. Note that in the sense of the just produced coupling, the simultaneous breadth-first walks of Section 2 are equivalent to Uribe’s diagram. In comparison, the original breadth-first walk of [6] coupling is “static” (it works for one $q$ at a time), and it seems difficult to turn it into a “dynamic” version due to a certain (small but present) loss of information (see Lemma 2).

5 Scaling limits for simultaneous breadth-first walks

This section imitates the approach of Section 2.4 in [7] . It is interesting to note that these scaling limits are simpler to derive here than they were for the original multiplicative coalescent encoding walks in [7] . Given {\bf x}\in l^{2}_{\mbox{{\footnotesize\searrow}}} let

[TABLE]

For each $n\geq 1$, let ${\bf x}^{(n)}$ be a finite vector (in the sense of Section 2). Let $((Z^{{\bf x}^{(n)},q}(s),\,s\geq 0),\,q\geq 0)$ and $({\bf X}^{(n)}(q),q\geq 0)$ be the simultaneous breadth-first walks, and the multiplicative coalescent coupled to $Z^{{\bf x}^{(n)},\cdot}$ (see Section 2 and Proposition 5), respectively. Suppose that for some $\kappa\in[0,\infty)$ and {\bf c}\in l^{3}_{\mbox{{\footnotesize\searrow}}}, the following hypotheses are true:

[TABLE]

as $n\rightarrow\infty$. It is easy to convince oneself (or see Lemma 8 of [7] or [48] Section 6) that for any $(\kappa,0,{\bf c})\in{\cal I}$ there exists a finite vector valued sequence $({\bf x}^{(n)})_{n\geq 1}$ satisfying (5.1)–(5.3). As in [7] , we furthermore pick an integer valued sequence $(m(n))_{n\geq 1}$, which increases to infinity sufficiently slowly so that

[TABLE]

[TABLE]

Fix $t\in\mathbb{R}$ and let $q_{n}:={\textstyle\frac{1}{\sigma_{2}({\bf x}^{(n)})}}+t$. Recall (2.1), and define

[TABLE]

[TABLE]

It is implicit in the notation that $\xi_{i}^{q_{n}}:=\xi_{i}^{(n)}/q_{n}$, where $\xi_{i}^{(n)}$ has exponential (rate $x_{i}^{(n)}$) distribution, and where $(\xi_{i}^{(n)})_{i}$ are independent over $i$, for each $n$. Define ${\bar{Z}}_{n},{\bar{R}}_{n},{\bar{Y}}_{n}$ to be respectively $Z_{n},R_{n},Y_{n}$ multiplied by $\frac{1}{\sigma_{2}({\bf x}^{(n)})}$, so that ${\bar{Z}}_{n}\equiv{\bar{Y}}_{n}+{\bar{R}}_{n}$. It should not be surprising that both the shift in the multiplicative coalescent time and the spatial scaling applied to the walks are the same as in [7] . It is clear that, for each $n$, $R_{n}$ and $Y_{n}$ are independent (the former depends only on the first $m(n)$ terms of the sequence $(\xi^{(n)}_{i})_{i}$, and the latter only on the other terms). Recall the definitions (1.5–1.7). The following result is a direct analogue of [7] , Proposition 9.

Proposition 6*.*

If $(\kappa,0,{\bf c})\in{\cal I}$, and provided (5.1–5.3) are satisfied as $n\to\infty$, then

[TABLE]

where ${\widetilde{W}}^{\kappa,t}$ and $V^{\bf c}$ are independent, and therefore ${\bar{Z}}_{n}\ \stackrel{{\scriptstyle d}}{{\rightarrow}}\ W^{\kappa,t,{\bf c}}.$

The rest of this section is devoted to the proof of the above proposition, and some of its consequences. As already mentioned, the argument is a simplification of that given in Section 2.4 of [7] , for the main reason that the current $Y_{n}$ has a simpler explicit form. From now on assume that $t\in\mathbb{R}$ is the one fixed in Proposition 6 via the definition of $q_{n}$. Note that the independence of $Y_{n}$ and $R_{n}$ clearly implies that of ${\bar{Y}}_{n}$ and ${\bar{R}}_{n}$. So provided that each of the sequences converges in law, the joint convergence in law to the product limit law is a trivial consequence. Furthermore, the convergence of ${\bar{R}}_{n}$ can be verified in a standard way (for each $k$, the $k$th largest jump of ${\bar{R}}_{n}$ converges to the $k$th largest jump of $V^{\bf c}$, and the second-moment MG estimates are used to bound the tails), as was already done in [7] (see the supplement [48] Section 7).

Lemma 7*.*

We have

[TABLE]

It remains to study the convergence of $({\bar{Y}}_{n})_{n}$. This sequence of processes differs from the equally named sequence in [7] . Write $\sigma_{r}^{n}$ for $\sigma_{r}({\bf x}^{(n)})$, $r=1,2,3$ in the sequel. An important observation is that

[TABLE]

The infinitesimal drift and variance calculations are now straightforward. Let \mbox{{\cal F}}_{s}^{n}:=\sigma\{{\bar{Y}}_{n}(u):u\leq s\}, so that \mbox{{\cal F}}^{n}:=(\mbox{{\cal F}}_{s}^{n})_{s\geq 0} is the filtration generated by ${\bar{Y}}_{n}$. The proof of the following result is also provided in [48] Section 7.

Lemma 8*.*

For each fixed $s$,

[TABLE]

Since the largest jump of ${\bar{Y}}_{n}$ is of size $x_{m(n)+1}^{(n)}/\sigma_{2}^{n}=o_{n}(1)$, the classical martingale central limit theorem (cf. [33] ) implies that

[TABLE]

and, as already argued, this concludes the proof of the proposition.

Remark 5.1*.*

By the Cauchy-Schwarz inequality, $(\sigma_{2}^{n})^{2}\leq\sigma_{1}^{n}\sigma_{3}^{n}$, and so (5.1,5.3) imply that $\sigma_{1}^{n}\to\infty$ as $n\to\infty$. While this fact was needed in the proof of the analogous [7] , Proposition 9, here it could slip by unnoticed. If $\kappa>0$, it is easy to see that the limit $W^{\kappa,t,{\bf c}}$ of ${\bar{Z}}_{n}$ has (countably) infinitely many excursions above past minima. If $\kappa=0$ and {\bf c}\in l^{3}_{\mbox{{\footnotesize\searrow}}}\setminus l^{2}_{\mbox{{\footnotesize\searrow}}}, the same was proved in [7] , Proposition 14.

Using the Skorokhod representation theorem, we may assume that the convergence stated in Proposition 6 holds in the almost sure sense. To state the next result (essential for the conclusions to be made in Section 6), redefine $q_{n}(t):=t+\frac{1}{\sigma_{2}^{n}}$, for $t\in\mathbb{R}$. Then let $Z_{n}^{t}:=Z^{{\bf x}^{(n)},q_{n}(t)}$ and $\bar{Z}_{n}^{t}:=Z_{n}^{t}/\sigma_{2}^{n}$. The (almost sure version of) Proposition 6 says that there exists a Brownian motion $W$ and an independent jump process $V^{\bf c}$, such that

[TABLE]

where the convergence of paths is considered in the Skorokhod $J_{1}$ topology. Let

[TABLE]

Lemma 9*.*

On the event $A_{t}$, for any $z\in\mathbb{R}$

[TABLE]

in the Skorokhod $J_{1}$ topology.

Proof. Recall the explicit form (2.1) of $Z^{\,\cdot,\cdot}$ Observe the following identity:

[TABLE]

Since clearly $\lim_{n\to\infty}\frac{q_{n}(t)}{q_{n}(z)}=1$, and moreover since

[TABLE]

the convergence stated in the lemma follows omega-by-omega on $A_{t}$.

6 Conclusions

Propositions 7 and 9 from [7] are stated in [48] Section 5. Here is an immediate consequence of them and Lemma 2, as announced in Remark 2.1 (d).

Corollary 10*.*

For each fixed $t$, under the hypotheses of Proposition 6, the sequence $({\bf X}^{(n)}(q_{n}(t)))_{n}$ converges in law (with respect to l^{2}_{\mbox{{\footnotesize\searrow}}}-metric) to the sequence of ordered excursions of $B^{\kappa,t,{\bf c}}$ (away from [math]).

But in fact more is true in view of Lemma 9. From now on we take the families $(W^{\kappa,t,{\bf c}},\,t\in\mathbb{R})$ and $(B^{\kappa,t,{\bf c}},\,t\in\mathbb{R})$ to be jointly defined on a common probability space, as in Section 1.3 (and in Theorem 1.2) via a given pair $(W,V^{\bf c})$, where $W$ is Brownian motion and $V^{\bf c}$ is an independent jump process from (1.4). Let us denote by $A$ the event $A_{t}$ of full probability from Lemma 9. For each $t\in\mathbb{R}$, define $\Xi^{(n)}(t)$ to be the point process on $[0,\infty)\times(0,\infty)$ such that $(x,y)$ is in $\Xi^{(n)}(t)$ if and only if there is an excursion above past minima of $\bar{Z}_{n}^{t}$ (see (2.7) and Figure 2), starting from $x$ and ending at $x+y$. Similarly, let $\Xi^{(\infty)}(t)$ be the point process on $[0,\infty)\times(0,\infty)$ such that $(x,y)$ is in $\Xi^{(\infty)}(t)$ if and only if there is an excursion away from [math] of $B^{\kappa,t,{\bf c}}$, starting from $x$ and ending at $x+y$. One can then apply deterministic result stated as [6] , Lemma 7 to conclude the following: on the event $A$ of full probability, for each $t\in\mathbb{R}$, one has

[TABLE]

in the sense of vague convergence of counting measures on $[0,\infty)\times(0,\infty)$ (see e.g. [42] ). As in [6] ; [7] , write $\pi$ for the “project onto the $y$-axis” defined on $\mathbb{R}^{2}$, and “${\rm ord}$” for the “decreasing ordering” map defined on infinite-length vectors, respectively. For a fixed (think large) $K<\infty$, define in addition $\pi_{K}$ to be the “project the strip $[0,K]\times(0,\infty)$ onto the $y$-axis” analogue of $\pi$. Then one can recognize ${\rm ord}(\pi(\Xi^{(n)}(t)))$ as ${\bf X}^{(n)}(q_{n}(t))$, and ${\bf X}^{(\infty)}(t):={\rm ord}(\pi(\Xi^{(\infty)}(t)))$ as the infinite vector of excursion lengths of $B^{\kappa,t,{\bf c}}$. Similarly $\pi_{K}(\Xi^{(\infty)}(t))$ (resp. $\pi_{K}(\Xi^{(n)}(t))$) is the collection of all the excursions of $B^{\kappa,t,{\bf c}}$ (resp. $\bar{Z}_{n}^{t}$), which start before time $K$. We already know that the law of ${\bf X}^{(\infty)}(t)$ is that of the marginal of $\mu(\kappa,0,{\bf c})$ at time $t$ (or equivalently, the marginal of $\mu(\kappa,t,{\bf c})$ at time [math]). The vague convergence (6.1) now easily implies that there exists a (random) order of $\pi_{K}(\Xi^{(n)}(t))$, here temporarily denoted by ${\rm ord}_{\Xi^{(n)},\Xi^{(\infty)}}$, since it is induced by the similarity of the $\Xi$s, such that

[TABLE]

In words, if considering only the starts before time $K$, it is possible to order the excursions of $\bar{Z}_{n}^{t}$ so that the corresponding infinite vector (obtained by appending an infinite sequence of [math]s to the elements of $\pi_{K}(\Xi^{(n)}(t))$) matches the infinite vector ${\rm ord}(\pi_{K}(\Xi^{(\infty)}(t)))$ in $l_{1}$-norm up to an $o_{n}(1)$ error term. Moreover, convergence in $l_{1}$-norm implies convergence in $l_{2}$-norm. Take $z>t$, another real number, and let $\varepsilon>0$ be fixed but arbitrarily small. From now on $u$ will denote either $t$ or $z$. In order to upgrade (6.2) to the convergence of ${\bf X}^{(n)}(q_{n}(u))$ to ${\bf X}^{(\infty)}(u)$ with respect to distance $d(\cdot,\cdot)$, one can make the following observations. For {\bf x}\in l^{2}_{\mbox{{\footnotesize\searrow}}}, let $f_{m}({\bf x}):=(x_{1},\ldots,x_{m},0,0,\ldots)$ be the “projection” onto the first $m$ components. Take some arbitrarily large integer $k$, and choose $m_{k}\in\mathbb{N}$ such that

[TABLE]

Since ${\bf X}^{(n)}(q_{n}(t))\ \stackrel{{\scriptstyle d}}{{\rightarrow}}\ {\bf X}^{(\infty)}(t)$ with respect to $d$, for this $k$ and possibly larger but still finite $m_{k}$ we can have in addition

[TABLE]

In words, with an overwhelming probability, all the (random) infinite vectors under consideration are well-approximated (in the $l_{2}$-norm) by their first $m_{k}$ components. Since $\pi(\Xi^{(\infty)}(\cdot))$ is $l_{2}$-valued, and since its elements are listed in size-biased order, one can easily deduce that for the above $m_{k}$, there exists some large time $K_{k}:=K(m_{k})<\infty$, such that

[TABLE]

In words, $K$ is sufficiently large so that with high probability the largest $m_{k}$ elements of $\pi(\Xi^{(\infty)}(u))$, all correspond to excursions that started before time $K$. Again due to ${\bf X}^{(n)}(q_{n}(u))\ \stackrel{{\scriptstyle d}}{{\rightarrow}}\ {\bf X}^{(\infty)}(u)$, the analogous

[TABLE]

is implied for some (possibly larger but) finite $K=K(m_{k})$. Apply the triangle inequality to bound $d({\bf X}^{(n)}(q_{n}(u)),{\bf X}^{(\infty)}(u))$ by the sum of the following terms: $d({\bf X}^{(n)}(q_{n}(u)),f_{m_{k}}({\bf X}^{(n)}(q_{n}(u))))$, $d(f_{m_{k}}({\bf X}^{(n)}(q_{n}(u))),$ $f_{m_{k}}({\bf X}^{(\infty)}(u)))$, and $d(f_{m_{k}}({\bf X}^{(\infty)}(u)),$ ${\bf X}^{(\infty)}(u))$. The initial and the final term are controlled by (6.3–6.4), while the middle term is controlled by (6.5–6.6) and (6.2), where one makes use of the elementary inequality: for ${\bf x},{\bf y}\in l^{2}$,

[TABLE]

regardless of the choice of bijection $b:\mathbb{N}\to\mathbb{N}$.

Remark 6.1*.*

It is clear (for example from (5.8–5.9), think about redefining $q_{n}(t)$ as ${\textstyle\frac{1}{\sigma_{2}({\bf x}^{(n)})}}+t-\tau$), that the parameter $\tau$ corresponds to the time-shift of the eternal multiplicative coalescent, and so the above conclusions automatically extend to the setting where $\tau\neq 0$.

An immediate conclusion is

Lemma 11*.*

If $(({\bar{Z}}^{x^{(n)},q_{n}(t)}(s),\,s\geq-1/\sigma_{2}^{n}),\,t\in\mathbb{R})\longrightarrow((W^{\kappa,t-\tau,{\bf c}}(s),\,s\geq 0)\,,t\in\mathbb{R}),\mbox{ as }n\to\infty,$ in the sense of Lemma 9, and if {\bf X}^{(\infty)}(t)\in l^{2}_{\mbox{{\footnotesize\searrow}}} is the vector of ordered excursion lengths of $B^{\kappa,t-\tau,{\bf c}}$, then

(i) for any $t\in\mathbb{R}$

[TABLE]

(ii) for any finite sequence of times $t_{1}<t_{2},\ldots<t_{m}$, one can find a subsequence $(n_{j})_{j}$ such that almost surely

[TABLE]

Recalling that $({\bf X}^{(n)}(q_{n}(s)),\,s\geq-1/\sigma_{2}^{n})$ has the law of the multiplicative coalescent (see Proposition 5), and applying the Feller property together with Lemma 11(i), where one should identify ${\bf X}^{(\infty)}$ with ${\bf X}$, will complete the proof of the claim about the distribution of ${\bf X}$ in Theorem 1.2. It is easy to see (using arguments analogous to those given above) that the realization ${\bf X}^{(\infty)}\equiv{\bf X}$ of each eternal version from Theorem 1.2 is a càdlàg (rcll) process on an event of full probability.

Remark 6.2*.*

Recall the COL operation of [7] , Section 5. In particular, each $c_{i}>0$ is interpreted as the rate of Poisson coloring (per unit mass) by marks of the $i$th “color”, applied to the standard Aldous’ multiplicative coalescent ${\bf X}^{*}$. Once all the color marks are deposited, any two blocks of ${\bf X}^{*}$ that share at least one mark of the same color are instantaneously and simultaneously merged together. The jump in $W^{\kappa,\cdot-\tau,{\bf c}}$ at time $\xi_{i}$ of size $c_{i}$ corresponds precisely to the effect of coloring by the $i$th color. Moreover, one could argue that if $\widetilde{W}^{\kappa,\cdot-\tau,{\bf 0}}$ and $\widetilde{W}^{\kappa,\cdot-\tau,{\bf c}}$ are given in (1.6) using the same Brownian motion $W$, then the excursions of the corresponding $(B^{\kappa,t-\tau+\|{\bf c}\|_{2},{\bf c}},\,t\in\mathbb{R})$ (suppose for simplicity that {\bf c}\in l^{2}_{\mbox{{\footnotesize\searrow}}}) away from [math] are almost surely the result of the above ${\rm COL}$ operation executed on the excursions of $(B^{\kappa,t-\tau,{\bf 0}},\,t\in\mathbb{R})$. The fact that, as time increases, each color “spreads” in this coupling (almost surely) only over the “neighboring” blocks may again seem counterintuitive. The point is that ${\rm COL}$ commutes with the multiplicative coalescent dynamics, and that therefore it can be pushed to $-\infty$. The infinitesimally small dust particles of ${\bf X}^{*}(-\infty)$ are mutually interchangeable. The $i$th color at time $-\infty$ is represented as an additional dust particle, of mass much superior to standard dust, but still negligible (a formal statement of this is (1.11) or (5.2)). One can naturally couple the representation of the multiplicative coalescent using simultaneous breadth-first walks (or Uribe’s diagram) started only from standard dust as $t\to-\infty$, with the same representation of the multiplicative coalescent started from the union of two types of dust as $t\to-\infty$. Proposition 6 and Lemma 9 do this formally (their predecessor is [7] , Proposition 41). In this coupling every color gradually spreads only to neighboring blocks of those already marked by it.

Acknowledgements

The research reported here, and more concisely in [46] , was initiated due to a series of author’s conversations with Nicolas Fournier, Mathieu Merle and Justin Salez, Jean Bertoin, and Gerónimo Uribe Bravo. Subsequent exchanges with Gerónimo Uribe, with Nicolas Broutin and Jean-François Marckert, and with James Martin and Balázs Ráth, accelerated and improved the writing (a longer list of acknowledgments is provided in [46] ). The current expanded version, intended for broader audience, is written in response to the feedback from the anonymous reviewer and editors of Bernoulli journal. {supplement}[id=suppA] \snameSupplement A \stitleSupplement to “The eternal multiplicative coalescent encoding via excursions of Lévy-type processes” \slink[doi]COMPLETED BY THE TYPESETTER \sdatatype.pdf \sdescription The accompanying text consists of eight sections (including a short introductory note). The title of each section summarizes its contents. A fraction of the material presented ([48] Sections 2, 3, 5 and 6) is intended to help readers gain time (reduce the need for consulting external literature) while reading this article. The rest ([48] Sections 4 and 7) contains novel arguments or open problems ([48] Sections 8).