$\mathbb{U}^{\downarrow}=\left\\{(x,s)\in\ell^{2}_{\downarrow}\times\mathbb{Z}_{\geqslant
0}^{\infty}:\sum_{i\geqslant 1}x_{i}s_{i}<\infty\mbox{ and }s_{i}=0\mbox{
whenever }x_{i}=0\right\\}.$

The process $\lambda\mapsto(\mathscr{M}(\lambda),\mathscr{Sp}(\lambda))$ is
the augmented multiplicative coalescent introduced in [3, 18] and appears as
the scaling limit999Actually, we deal with a slightly different version of the
$G(n,p)$ model since we have a fixed number $m$ of edges and we allow self-
loops and multiple edges, but this model is considered in [18, 59] and the
result applies. of the renormalized component sizes and surplus in
$G_{n}(\lambda)$, see [18, 24].

This convergence holds for the Skorokhod topology on
$\mathbb{C}\mathrm{adlag}(\mathbb{R},\mathbb{U}^{\downarrow})$ where
$\mathbb{U}^{\downarrow}$ is endowed with the metric

$\mathrm{d}_{\mathbb{U}^{\downarrow}}\left((x,s),(x^{\prime},s^{\prime})\right)=\left(\sum_{i\geqslant
1}|x_{i}-x_{i}^{\prime}|^{2}\right)^{1/2}+\sum_{i\geqslant
1}|x_{i}s_{i}-x_{i}^{\prime}s_{i}^{\prime}|.$

Since $(x,s)\in\mathbb{U}^{\downarrow}\mapsto\sum x_{i}\mathbf{1}_{s_{i}>0}$
is continuous for this topology, the previous convergence implies the
convergence of the total renormalized size
$n^{-2/3}\cdot\|\mathrm{O}_{n}(0)\|_{\bullet}$ of all components at time
$\lambda=0$ carrying a surplus at time $0$, towards its continuous counterpart

$\displaystyle n^{-2/3}\cdot\|\mathrm{O}_{n}(0)\|_{\bullet}$
$\displaystyle\xrightarrow[n\to\infty]{(d)}$ $\displaystyle\sum_{i\geqslant
1}\mathscr{M}_{i}(0)\mathbf{1}_{\mathscr{Sp}_{i}(0)>0}.$ (37)

Furthermore, we have seen above that every atom $(\tau,y)$ of $\Xi$
corresponds, in the augmented coalescent to a surplus of one in a particle and
we can define the time of appearance of this surplus as the smallest
$\lambda\in\mathbb{R}$ so that $B_{\tau}^{(\lambda)}\leqslant y$, notice that
$\lambda>-\infty$ almost surely for each atom. It follows from these
observations, and the proof of Theorem 4 in [24, Section 7.2] that we have the
convergence in law for each $\lambda_{0}<0$ fixed

$\displaystyle
n^{-2/3}\cdot\|\mathrm{O}_{n}(\lambda_{0})\|_{\bullet}\xrightarrow[n\to\infty]{(d)}|\mathscr{O}(\lambda_{0})|,$
(38)

where $|\mathscr{O}(\lambda_{0})|$ is the total mass of all particles of
$\mathscr{M}(0)$ which have a surplus appeared before time $\lambda_{0}$.
Also, $|\mathscr{O}(\lambda_{0})|\to 0$ as $\lambda_{0}\to-\infty$ almost
surely by dominated convergence. Together with the last display, this proves
the first point of the proposition and gives the existence of $\lambda_{0}$
and $n_{0}$.

For the second and third item, notice that once $\lambda_{0}$ has been fixed,
the convergence to the augmented multiplicative coalescent [19] implies that
the masses of the specks (i.e. the renormalized sizes of the components of
$\mathrm{G}_{n}^{[\lambda_{0}]}(\lambda_{0})=[\mathrm{G}_{n}(\lambda_{0})]_{\mathrm{tree}}$)
converge in distribution in the $\ell^{2}$ sense to

$\displaystyle\left(\mathscr{M}_{i}(\lambda_{0})\mathbf{1}_{\mathscr{Sp}_{i}(\lambda_{0})=0}:i\geqslant
1\right)^{\downarrow}.$ (39)

After an inoffensive Poissonization of the time (i.e. by letting the edges
arrive according to a Poisson process instead of discrete time steps) the
second point is a consequence of the Feller property of the multiplicative
coalescent [3, Proposition 5] together with the last display: in words the
renormalized component sizes of $\mathrm{G}^{[\lambda_{0}]}_{n}(0)$ are well
approximated by restricting to specks of mass $\geqslant\eta$ in the
$\ell^{2}$ sense (uniformly in $n$) hence in the $\ell^{\infty}$ sense so that
$\sup_{n\geqslant 1}n^{-2/3}\Delta_{n}(\lambda_{0},\eta)\to 0$ in probability
as $\eta\to 0$. The third convergence is similar and follows from the Feller
property of the augmented coalescent [19, Theorem 3.1]. See also [73,
Corollary 5.6]. ∎

####  6.4 Proof of Theorem 2

We now gather the deterministic controls established in Sections 6.1 and 6.2
together with the probabilistic estimates of Proposition 13 to prove Theorem
2. As announced, we start with a weaker convergence for the supremum norm.

###### Convergence for the supremum norm.

We consider
$\mathcal{E}_{0}=\ell_{\downarrow,0}^{\infty}\times\ell_{\downarrow,0}^{\infty}$
where $\ell_{\downarrow,0}^{\infty}$ is the space of non-increasing sequences
tending to $0$ endowed with $\mathrm{d}_{\sup}$ (see (29)) which is a Polish
space. Clearly $\mathcal{E}\subset\mathcal{E}_{0}$ and the convergence for the
$\mathrm{d}_{\mathcal{E}}$ distance is stronger than for $\mathrm{d}_{\sup}$.

Fix $\varepsilon>0$ and find $\lambda_{0},\eta,\delta>0$ and $n_{0}\geqslant
1$ as in Proposition 13. On the event described in this proposition of
probability at least $1-\varepsilon$ for $n$ large enough we have for every
$\lambda\in[\lambda_{0},0]$

$\displaystyle\mathrm{d}_{\sup}\left(\mathbb{F}_{n}(\lambda);\mathbb{F}_{n}^{[\lambda_{0},\delta]}(\lambda)\right)$
(40) $\displaystyle\underset{\mathrm{trig.\leavevmode\nobreak\
ineg}}{\leqslant}$
$\displaystyle\mathrm{d}_{\sup}\left(\mathbb{F}_{n}(\lambda);\mathbb{F}_{n}^{[\lambda_{0}]}(\lambda)\right)+\mathrm{d}_{\sup}\left(\mathbb{F}_{n}^{[\lambda_{0}]}(\lambda);\mathbb{F}_{n}^{[\lambda_{0},\delta]}(\lambda)\right)$
$\displaystyle\underset{\eqref{eq:firstcutoff},\eqref{eq:secondcutoff}}{\leqslant}$
$\displaystyle
n^{-2/3}\left(\|\mathrm{O}_{n}(\lambda_{0})\|_{\bullet}+\Delta_{n}(\lambda_{0},\eta)\right)$
$\displaystyle\underset{\mathrm{Prop.\ }\ref{prop:controls}}{\leqslant}$
$\displaystyle 2\varepsilon.$

On the other hand, by (39), the starting configuration of
$\mathbb{F}_{n}^{[\lambda_{0},\delta]}$ converges in law towards some vector
having a finite number of non-zero components. Since there are only finitely
many particles to take care of, applying the dynamics of the frozen coalescent
it should be clear that for fixed $\lambda_{0}<0$ and $\delta>0$

$\left(\mathbb{F}_{n}^{[\lambda_{0},\delta]}(\lambda):\lambda\in[\lambda_{0},0]\right)$

converges in distribution for the Skorokhod topology on
$\mathbb{C}\mathrm{adlag}([\lambda_{0},0],\mathcal{E}_{0})$. If
$\mathrm{d}_{LP}$ denotes the Lévy–Prokhorov distance associated to the
convergence in law for the Skorokhod topology on
$\mathbb{C}\mathrm{adlag}([\lambda_{0},0],\mathcal{E}_{0})$ then restricting
(40) to $\lambda\in[-1,0]$ we deduce

$\mathrm{d}_{LP}\left(\left(\mathbb{F}_{n}(\lambda)\right)_{\lambda\in[-1,0]};\left(\mathbb{F}_{n}^{[\lambda_{0},\delta]}(\lambda)\right)_{\lambda\in[-1,0]}\right)\leqslant
2\varepsilon,$

for all $n\geqslant n_{0}$ (this actually holds for the supremum norm which is
stronger than the Skorokhod distance). Since
$(\mathbb{F}_{n}^{[\lambda_{0},\delta]}(\lambda):\lambda\in[-1,0])$ is
converging in law as $n\to\infty$, we can combine this with the last display
to deduce that $\left(\mathbb{F}_{n}(\lambda)\right)_{\lambda\in[-1,0]}$ is
Cauchy for $\mathrm{d}_{LP}$ and so converges as desired. Its limit is
obtained by first letting $n\to\infty$, then $\delta\to 0$ and finally
$\lambda_{0}\to-\infty$ in the process
$n^{-2/3}\cdot\mathrm{F}_{n}^{[\lambda_{0},\delta]}$. ∎

###### Convergence in $\ell^{1}_{\downarrow}\times\ell^{2}_{\downarrow}$.

To upgrade the previous convergence for the $\mathrm{d}_{\sup}$ distance to a
convergence for the distance $\mathrm{d}_{\mathcal{E}}$, we need to prove
tightness i.e. to control uniformly over $\lambda\in[-1,0]$ the cumulative
effect of the small component sizes in our frozen coalescent processes. More
precisely, for any $\xi\geqslant 0$ and
$\mathbf{z}=(\mathbf{x},\mathbf{y})\in\mathcal{E}$ we denote by

$R_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ},\xi}(\mathbf{z})=\sum_{i\geqslant
1}x_{i}\mathbf{1}_{x_{i}\leqslant\xi}\quad\mbox{ and }\quad
R_{\circ,\xi}(\mathbf{z})=\sum_{i\geqslant
1}y_{i}^{2}\mathbf{1}_{y_{i}\leqslant\xi},$

respectively the sum of the masses of the blue particles and sum of the
squares of the masses of the white particles of mass smaller than $\xi$. We
also put
$R_{\xi}(\mathbf{z})=R_{\xi,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}(\mathbf{z})+R_{\xi,\circ}(\mathbf{z})$.
We then have :

######  Lemma 9 (Towards tightness of $\mathbb{F}_{n}(\cdot)$).

For any $\varepsilon>0$ there exists $\xi>0$ and $n_{0}\geqslant 1$ such that
for all $n\geqslant n_{0}$ we have with probability at least $1-\varepsilon$

$\sup_{\lambda\in[-1,0]}\left(R_{\xi}(\mathbb{F}_{n}(\lambda))\right)\leqslant\varepsilon.$

###### Proof.

Let us begin with the $\ell^{2}$-part. By the inclusion of the frozen
exploration process in the standard Erdős–Rényi process, all the components of
$\mathrm{F}_{n}(\lambda)$ for $\lambda\in(-\infty,0]$ -frozen or not- are
contained in $\mathrm{G}_{n}(0)$. Next, if $0\leqslant
f_{1},\dots,f_{k}\leqslant\xi$ and $f_{1}+\dots+f_{k}\leqslant y$ then we have

$\displaystyle
f_{1}^{2}+\cdots+f_{k}^{2}\leqslant(f_{1}+\cdots+f_{k})\cdot(\xi\wedge
y)\leqslant\left(\xi\cdot y\right)\wedge y^{2}.$ (41)

We apply this inequality when $f_{1},\dots,f_{k}$ are the renormalized sizes
of the components in $\mathrm{F}_{n}(\lambda)$ which are included in the same
component of $\mathrm{G}_{n}(0)$ with renormalized size $y$ and this for each
component of $\mathrm{G}_{n}(0)$ which is made of small components of
$\mathrm{F}_{n}(\lambda)$: if we denote by
$\boldsymbol{Y^{(n)}}=(Y^{(n)}_{i}\leavevmode\nobreak\ :\leavevmode\nobreak\
i\leavevmode\nobreak\ \geqslant\leavevmode\nobreak\ 1)$ the decreasing sizes
of the components of $\mathrm{G}_{n}(0)$ renormalized by $n^{-2/3}$ then we
have

$\sup_{\lambda<0}R_{\circ,\xi}(\mathbb{F}_{n}(\lambda))\leqslant\sum_{i\geqslant
1}\left(\xi\cdot Y^{(n)}_{i}\right)\wedge\left(Y_{i}^{(n)}\right)^{2}.$

By the result of Aldous [3], the sequence $(Y_{i}^{(n)}:i\geqslant 1)$
converges in distribution for the $\ell^{2}_{\downarrow}$ distance towards the
multiplicative coalescent $\mathscr{M}(0)$ at time $0$. Since
$\psi_{\xi}:\ell^{2}_{\downarrow}\to\mathbb{R}_{+}$ defined by
$\psi_{\xi}((y_{i}:i\geqslant 1))=\sum_{i\geqslant
1}y_{i}\cdot(y_{i}\wedge\xi)$ is continuous for the
$\ell^{2}_{\downarrow}$-distance we deduce from the previous convergence that
$\psi_{\xi}(\boldsymbol{Y^{(n)}})$ converges in law towards
$\psi_{\xi}(\mathscr{M}(0))$. Furthermore, by dominated convergence we have
$\psi_{\xi}(\mathscr{M}(0))\to 0$ a.s. as $\xi\to 0$. We deduce that

$\forall\varepsilon>0,\quad\sup_{n\geqslant 1}\mathbb{P}\left(\sum_{i\geqslant
1}\left(\xi\cdot
Y^{(n)}_{i}\right)\wedge\left(Y_{i}^{(n)}\right)^{2}\geqslant\varepsilon\right)\xrightarrow[\xi\to
0]{}0,$

and this takes care of the $R_{\circ,\cdot}$ part of the lemma.

The $\ell^{1}$-part is a bit trickier. Recall from Section 6.1 that for any
$\lambda\in(-\infty,0]$, the frozen components of $\mathrm{F}_{n}(\lambda)$
are included in $\mathrm{O}_{n}(0)$, the union of the components of
$\mathrm{G}_{n}(0)$ that have a surplus. Notice that if $k\geqslant 1$ frozen
components of $\mathrm{F}_{n}(\lambda)$ belong to the same component of
$\mathrm{G}_{n}(0)$, then this component must have surplus at least $k$
(recall that each frozen component contains exactly one cycle). Hence if
$\boldsymbol{X^{(n)}}=(X_{i}^{(n)}:i\geqslant 1)$ are the decreasing sizes of
the components of $\mathrm{O}_{n}(0)$ renormalized by $n^{-2/3}$ and if
$K_{n}$ is the maximum surplus of a component in $\mathrm{G}_{n}(0)$ then we
have for all $\lambda\in(-\infty,0]$

$R_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ},\xi}(\mathbb{F}_{n}(\lambda))\leqslant
K_{n}\cdot\sum_{i\geqslant 1}\left(X_{i}^{(n)}\wedge\xi\right).$

By [63, Theorem 1], the sequence $(K_{n}:n\geqslant 1)$ is tight. From the
discussion just before (37) we get that $\boldsymbol{X^{(n)}}$ converges in
law for the $\ell^{1}_{\downarrow}$-topology towards the masses
$\left(\mathscr{M}_{i}(0)\mathbf{1}_{\mathscr{Sp}_{i}(0)>0}\right)^{\downarrow}$
of the particles in the augmented multiplicative coalescent at time $0$ that
carry a surplus. By the same argument as above we deduce that for every
$\varepsilon>0$, there exists $\xi>0$ such that
$\mathbb{P}\left(K_{n}\cdot\sum_{i\geqslant 1}\left(\xi\cdot
X^{(n)}_{i}\right)\geqslant\varepsilon\right)\leqslant\varepsilon$ for all
$n\geqslant 1$ and this finishes the proof of the lemma. ∎

We can now finish the proof of Theorem 2: Recall that
$\mathbb{F}_{n}(\lambda)$ is the renormalized sizes of the frozen and standard
components in $\mathrm{F}_{n}(\lambda)$. In a nutshell, the tightness of
$\left(\mathbb{F}_{n}\left(\lambda\right):\lambda\in[-1,0]\right)$ for the
Skorokhod topology with values in $\mathcal{E}_{0}$ together with the last
lemma establishes the tightness of
$\left(\mathbb{F}_{n}\left(\lambda\right):\lambda\in[-1,0]\right)$ for the
Skorokhod topology with values in $\mathcal{E}$. Since the convergence in
$\mathcal{E}_{0}$ determines the law, we are done. Let us provide some
details. Recall that we already proved that

$\displaystyle\left(\mathbb{F}_{n}\left(\lambda\right):\lambda\in[-1,0]\right)\xrightarrow[n\to\infty]{(d)}\left(\mathscr{F}\mathscr{M}(\lambda):\lambda\in[-1,0]\right),$
(42)

for the Shokorhod topology on the space
$\mathbb{C}\mathrm{adlag}(\mathbb{R},\mathcal{E}_{0})$ of càdlàg functions
with values in $\mathcal{E}_{0}$ endowed with $\mathrm{d}_{\sup}$. By
Skrokorhod representation theorem, we can then assume that for each
$n\geqslant 1$, the processes $\mathbb{F}_{n}$ and $\mathscr{F}\mathscr{M}$
are coupled in such a way that the Skorokhod distance between
$\mathbb{F}_{n}(\cdot)$ and $\mathscr{F}\mathscr{M}$ converges almost surely
to $0$ for $\mathrm{d}_{\mathrm{sup}}$ as $n\to\infty$. This means that we can
find increasing time shifts $\psi_{n}:[-1,0]\to[-1,0]$ with
$\|\psi_{n}-\mathrm{Id}\|\to 0$ a.s. and such that

$\displaystyle\sup_{\lambda\in[-1,0]}\mathrm{d}_{\mathrm{sup}}(\mathbb{F}_{n}(\lambda),\mathscr{F}\mathscr{M}(\psi_{n}(\lambda))\xrightarrow[n\to\infty]{a.s.}0.$
(43)

Recalling the notation introduced before Lemma 9 and using this lemma, up-to
rebuilding a new coupling, one can furthermore suppose that we have for every
$\xi>0$

$\limsup_{\xi\downarrow 0}\sup_{n\geqslant
1}\sup_{\lambda\in[-1,0]}R_{\xi}(\mathbb{F}_{n}(\lambda))\xrightarrow[n\to\infty]{a.s.}0.$

In particular, by Fatou’s lemma, this implies a similar estimate for
$\mathscr{F}\mathscr{M}$ namely,

$\limsup_{\xi\downarrow
0}\sup_{\lambda\in[-1,0]}R_{\xi}(\mathscr{F}\mathscr{M}(\lambda))\xrightarrow[n\to\infty]{a.s.}0.$

One can now use our coupling to evaluate the $\mathcal{E}$-distance between
$\mathbb{F}_{n}$ and $\mathscr{F}\mathscr{M}$, namely

$\mathrm{d}_{\mathcal{E}}(\mathbb{F}_{n}(\lambda),\mathscr{F}\mathscr{M}(\psi_{n}(\lambda)))\leqslant
2\sup_{\lambda\in[-1,0]}(R_{\xi}(\mathscr{F}\mathscr{M}(\lambda))+R_{\xi}(\mathbb{F}_{n}(\lambda)))+\mathrm{d}_{\mathcal{E}}(\mathbb{F}_{n}^{\\{\xi\\}}(\lambda),\mathscr{F}\mathscr{M}^{\\{\xi\\}}(\psi_{n}(\lambda))).$

Using the second and third to last displays, the first term on the right-hand
side can be made small uniformly in $n$ and $\lambda\in[-1,0]$ by choosing
$\xi$ small enough. The second term also tends to $0$ thanks to (43): since
$\mathscr{F}\mathscr{M}$ is càdlàg with values in $\mathcal{E}_{0}$, the
maximal size of a particle in $\mathscr{F}\mathscr{M}$ (and in
$\mathbb{F}_{n}(\cdot)$) and the maximal number of particles of mass $>\xi$ is
bounded over $[-1,0]$. We have indeed proved that in this coupling we have
$\mathbb{F}_{n}\to\mathscr{F}\mathscr{M}$ almost surely for the
$\mathrm{d}_{\mathcal{E}}$ metric. This implies the desired result.

###  7 Markovian properties of the freezer and the flux

Since the mapping
$\mathbf{z}\in\mathcal{E}\to\|\mathbf{z}\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
is continuous for the topology induced by $\mathrm{d}_{\mathcal{E}}$, our
Theorem 2 implies that

$\displaystyle(n^{-2/3}\cdot\|\mathrm{F}_{n}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}:\lambda\in\mathbb{R})\xrightarrow[n\to\infty]{(d)}(\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}:\lambda\in\mathbb{R}),$
(44)

for the Skorokhod topology where we recall that
$\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
is the total mass of the frozen particles in the frozen multiplicative
coalescent. In this section, we use the Markov properties of the process
$\|F(n,\cdot)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
–or more precisely of
$(\|F(n,\cdot)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},\|F(n,\cdot)\|_{\bullet-\bullet})$–
given in Proposition 8 to prove (Proposition 14) the joint convergence of the
number of discarded edges $D(n,\cdot)$ in the scale $n^{1/3}$ which, thanks to
our coupling construction, will give us the flux of outgoing cars in the
parking process on Cayley trees.

Using this Markovian point of view, we also give in Proposition 15 a new and
perhaps more concrete construction of the process
$\lambda\mapsto\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
as a pure-jump Feller process with a time-inhomogeneous infinite jump measure

$\displaystyle\mathbf{n}_{\lambda}(x,\mathrm{d}y)=\frac{1}{2}\cdot\frac{\mathrm{d}y}{\sqrt{2\pi
y^{3}}}g_{x,\lambda}(y),$ (45)

where $g_{x,\lambda}(y)$ was defined in (14). We complete this alternative
Markovian description of the frozen multiplicative coalescent by computing the
law of $[\mathscr{F}\mathscr{M}(\lambda)]_{\circ}$ given
$\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
(see Proposition 16): By passing Proposition 8 to the scaling limit,
conditionally on
$\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$,
the $\ell^{2}$-part of $\mathscr{F}\mathscr{M}(\lambda)$ has the law of the
scaling limit of component sizes in a critical random forest. This law has
been described in [64] using excursion lengths of a time inhomogeneous
diffusion with a reflection term. We give here an alternative (and quicker)
description of this law using conditioned $3/2$-stable Lévy processes. The
last two results are not used for the parking process but we include them to
motivate further the study of the frozen Erdős–Rényi process, see Part III for
perspectives.

####  7.1 Scaling limits for the freezer and flux

The main result of this section is the joint convergence of the renormalized
number of discarded edges $D(n,m)$ together with (44), see Figure 21.

######  Proposition 14 (Joint convergence of discarded edges).

Jointly with the convergence of Theorem 2 we have the following convergence in
distribution for the uniform topology on $\mathcal{C}(\mathbb{R},\mathbb{R})$

$\left(n^{-1/3}\cdot\mathrm{D}_{n}(\lambda)\right)_{\lambda\in\mathbb{R}}\xrightarrow[n\to\infty]{(d)}\left(\frac{1}{2}\int_{-\infty}^{\lambda}\mathrm{d}s\,\|\mathscr{F}\mathscr{M}(s)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}\right)_{\lambda\in\mathbb{R}.}$

Figure 21: A simulation of the renormalized processes
$n^{-2/3}\cdot\|\mathrm{F}_{n}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
(in orange) and $n^{-1/3}\cdot\mathrm{D}_{n}(\lambda)$ (in blue) for
$\lambda\in[-5,5]$ and $n=3000$. Notice that the first process is pure jump in
the scaling limit and that the second one is the integral of the first.

The main ingredient to prove the joint convergence of the number of discarded
edges together with Theorem 2 is the following consequence of Corollary 1

$\mathbb{P}\left(\Delta
D(n,m)=1\Big{|}\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}\right)=\frac{\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}}{n},$
(46)

so that the above result formally follows from integrating and passing to the
limit. To make this precise, we shall start with a lemma controlling the flux
at the bottom of the critical window:

######  Lemma 10.

There exists a constant $C>0$ such that for any $n\geqslant 1$ and any
$\lambda<-1$

$\mathbb{E}[\mathrm{D}_{n}(\lambda)]\leqslant\frac{C}{|\lambda|}n^{1/3}.$

###### Proof.

In this proof, for $n\geqslant 1$ and $m\geqslant 0$ we denote by
$\mathrm{C}\ell(n,m)$ the cluster of the vertex $1$ in $G(n,m)$ and by
$\mathrm{Spl}(n,m)=\|\mathrm{C}\ell(n,m)\|_{\bullet-\bullet}-\|\mathrm{C}\ell(n,m)\|_{\bullet}+1$
its surplus. Recall from (2) that the number of blue vertices in $F(n,m)$ is
less than the number of vertices which belong to a component which has a cycle
in $G(n,m)$. By (46) and taking expectations we deduce that

$\displaystyle\mathbb{E}\left[\Delta
D(n,m)\right]=\frac{1}{n}\mathbb{E}\left[\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}\right]\leqslant\mathbb{P}\left(\mathrm{Spl}(n,m)\geqslant
1\right)\leqslant\mathbb{E}\left[\mathrm{Spl}(n,m)\right].$ (47)

Now we provide an upper bound for $\mathbb{E}\left[\mathrm{Spl}(n,m)\right]$
following the proof of [41, Theorem 1.2]. Note that $\mathrm{Spl}(n,m)$ is
bounded from above by the number of vertex-disjoint cycles (including self-
loops) in the cluster of $1$. Given a graph $\mathfrak{g}$ and $k$ distinct
vertices $(v_{1},\dots,v_{k})$, we say that the graph $\mathfrak{g}$ _contains
the cycle_ $(v_{1},\dots v_{k})$ of length $k\geqslant 1$ if $\mathfrak{g}$
contains the (unoriented) edges $(v_{i},v_{i+1\,\mathrm{mod}\,k})$ for
$1\leqslant i\leqslant k$. For $k=1$, the expected number of self-loops (or
cycle of length $1$) in $\mathrm{C}\ell(n,m)$ is bounded above by
$m\mathbb{E}\left[\|\mathrm{C}\ell(n,m)\|_{\bullet}\right]/n^{2}.$ Since every
cycle of length $k\geqslant 3$ (resp. 2) corresponds to exactly $2k$ (resp. 2)
$k$-uplet $(v_{1},\dots,v_{k})$, we have for $2m<n$,

$\displaystyle\sum_{k=2}^{n}\sum_{L_{k}\mbox{\tiny{ cycle of length
}}k}\mathbb{P}\left(G(n,m)\mbox{ contains }L_{k}\mbox{ and }1\mbox{ is
connected to }L_{k}\right)$
$\displaystyle\leqslant\sum_{k=2}^{n}\sum_{L_{k}=(v_{1},\dots,v_{k})}\frac{1}{k}\sum_{I\subset\\{1,\dots,m\\},|I|=k}\mathbb{P}((\vec{E}_{i})_{i\in
I}\mbox{ form }L_{k}\mbox{ and }(\vec{E}_{i})_{i\leqslant m,i\notin I}\mbox{
connect }1\mbox{ with }\\{v_{1},\dots v_{k}\\})$
$\displaystyle\leqslant\sum_{k=2}^{n}\sum_{(v_{1},\dots,v_{k})}\frac{1}{k}\sum_{I\subset\\{1,\dots,m\\},|I|=k}k!\left(\frac{2}{n}\right)^{k}\cdot\frac{2k}{n}\mathbb{E}\left[\|\mathrm{C}\ell(n,m)\|_{\bullet}\right]$
$\displaystyle=\sum_{k=2}^{n}\frac{n!}{k(n-k)!}{m\choose
k}k!\left(\frac{2}{n^{2}}\right)^{k}\cdot\frac{2k}{n}\mathbb{E}\left[\|\mathrm{C}\ell(n,m)\|_{\bullet}\right]$
$\displaystyle\leqslant\frac{2}{n}\sum_{k=2}^{n}\left(\frac{2m}{n}\right)^{k}\mathbb{E}\left[\|\mathrm{C}\ell(n,m)\|_{\bullet}\right]\leqslant\frac{C}{n\left(1-\frac{2m}{n}\right)}\mathbb{E}\left[\|\mathrm{C}\ell(n,m)\|_{\bullet}\right],$

for some constant $C>0$ that may vary in the following lines. An easy
adaptation of [48, Theorem 1.1] using [48, Remark 1.6] to our model $G(n,m)$
shows that

$\mathbb{E}\left[\|\mathrm{C}\ell(n,m)\|_{\bullet}\right]\leqslant\frac{C}{\left(1-\frac{2m}{n}\right)},$
(48)

so that combining these inequalities we deduce that for $2m<n$

$\displaystyle\mathbb{E}[\mathrm{Spl}(n,m)]\leqslant\frac{C}{n(1-\frac{2m}{n})^{2}}.$
(49)

Coming back to $D$ and writing
${m}_{n}(\lambda)=0\wedge\lfloor\frac{n}{2}+\frac{\lambda}{2}n^{2/3}\rfloor$,
we obtain using (47) that for $\lambda<-1$,

$\displaystyle\mathbb{E}\left[\mathrm{D}_{n}(\lambda)\right]=\mathbb{E}\left[D(n,\mathrm{m}_{n}(\lambda))\right]=\sum_{m=0}^{\mathrm{m}_{n}(\lambda)-1}\mathbb{E}\left[\Delta
D(n,m)\right]\leqslant
C\sum_{m=0}^{\mathrm{m}_{n}(\lambda)-1}\frac{1}{n(1-\frac{2m}{n})^{2}}\leqslant
C\frac{n^{1/3}}{|\lambda|},$

which concludes the proof. ∎

###### Proof of Proposition 14.

Recall that the convergence of Theorem 2 implies the convergence of of
$(n^{-2/3}\|\mathrm{F}_{n}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}:\lambda\in\mathbb{R})$.
We now prove the joint convergence of
$(n^{-1/3}\mathrm{D}_{n}(\lambda):\lambda\in\mathbb{R})$ using the probability
transitions given in (46). Indeed, writing
$0\wedge\mathrm{m}_{n}(\lambda)=\lfloor\frac{n}{2}+\frac{\lambda}{2}n^{2/3}\rfloor$
as above, we have

$\mathrm{D}_{n}(\lambda)-\mathrm{D}_{n}(\lambda_{0})=\sum_{m=\mathrm{m}_{n}(\lambda_{0})}^{\mathrm{m}_{n}(\lambda)-1}\Delta
D(n,m),$

and for all $m\geqslant 0$, conditionally on
$\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$,
the variable $\Delta D(n,m)$ is a Bernoulli variable with parameter
$\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}/n$.
By the convergence of the process
$(n^{-2/3}\|\mathrm{F}_{n}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}:\lambda\in\mathbb{R})$
to
$(\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}:\lambda\in\mathbb{R})$
in the Skorokhod sense, we have for every
$-\infty<\lambda_{0}<\lambda<+\infty$,

$\sum_{m=\mathrm{m}_{n}(\lambda_{0})}^{\mathrm{m}_{n}(\lambda)-1}n^{-4/3}\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}=\frac{1}{2}n^{-2/3}\int_{n^{-2/3}(\mathrm{m}_{n}(\lambda_{0})-n/2)}^{n^{-2/3}(\mathrm{m}_{n}(\lambda)-n/2)}\|\mathrm{F}_{n}(s)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}\mathrm{d}s\xrightarrow[n\to\infty]{(d)}\frac{1}{2}\int_{\lambda_{0}}^{\lambda}\mathrm{d}s\,\|\mathscr{F}\mathscr{M}(s)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}.$

In addition, since $(\Delta
D(n,m)-\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}/n:m\geqslant
0)$ are the increments of a martingale,

$\displaystyle\mathbb{E}$
$\displaystyle\left[\left(n^{-1/3}\sum_{m=\mathrm{m}_{n}(\lambda_{0})}^{\mathrm{m}_{n}(\lambda)-1}\Delta
D(n,m)-\sum_{m=\mathrm{m}_{n}(\lambda_{0})}^{\mathrm{m}_{n}(\lambda)-1}n^{-4/3}\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}\right)^{2}\right]$
$\displaystyle=n^{-2/3}\sum_{m=\mathrm{m}_{n}(\lambda_{0})}^{\mathrm{m}_{n}(\lambda)-1}\mathbb{E}\left[\Big{(}\Delta
D(n,m)-\frac{\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}}{n}\Big{)}^{2}\right]$
$\displaystyle=n^{-2/3}\sum_{m=\mathrm{m}_{n}(\lambda_{0})}^{\mathrm{m}_{n}(\lambda)-1}\mathbb{E}\left[\frac{\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}}{n}\left(1-\frac{\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}}{n}\right)^{2}+\left(1-\frac{\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}}{n}\right)\cdot\left(\frac{\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}}{n}\right)^{2}\right]$
$\displaystyle\leqslant
n^{-2/3}\sum_{m=\mathrm{m}_{n}(\lambda_{0})}^{\mathrm{m}_{n}(\lambda)-1}\mathbb{E}\left[\frac{\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}}{n}\right]$

which converges to $0$ as $n\to\infty$ by the estimates of (the proof of) the
previous lemma. It follows that

$\mathrm{D}_{n}(\lambda)-\mathrm{D}_{n}(\lambda_{0})\xrightarrow[n\to\infty]{(d)}\frac{1}{2}\int_{\lambda_{0}}^{\lambda}\mathrm{d}s\,\|\mathscr{F}\mathscr{M}(s)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}.$

Now we use Lemma 10, which shows that
$\mathbb{E}\left[n^{-1/3}\mathrm{D}_{n}(\lambda_{0})\right]$ can be made
arbitrarily small if we choose $\lambda_{0}$ small enough, and this uniformly
in $n$. Hence, by Fatou’s lemma, we obtain for all
$\lambda_{0}\in(-\infty,\lambda]$,

$\frac{1}{2}\int_{\lambda_{0}}^{\lambda}\mathrm{d}s\,\|\mathscr{F}\mathscr{M}(s)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}<\infty\qquad\mbox{a.s.},$

and letting $\lambda_{0}\to-\infty$, we get

$\displaystyle\mathrm{D}_{n}(\lambda)\xrightarrow[n\to\infty]{(d)}\frac{1}{2}\int_{-\infty}^{\lambda}\mathrm{d}s\,\|\mathscr{F}\mathscr{M}(s)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}=\mathscr{D}(\lambda).$
(50)

The above reasoning can be extended to prove that, jointly with the
convergence of the first coordinate in Proposition 14, for each
$-\infty<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{k}<\infty$ we have
$n^{-1/3}\cdot\mathrm{D}_{n}(\lambda_{i})\to\mathscr{D}(\lambda_{i})$. Since
the processes $\mathrm{D}_{n}(\cdot)$ are increasing and since
$\lambda\mapsto\mathscr{D}(\lambda)$ is continuous and increasing as well,
this is sufficient to imply the joint convergence of
$n^{-1/3}\cdot\mathrm{D}_{n}(\cdot)$ to $\mathscr{D}(\cdot)$ for the uniform
norm over every compact of $\mathbb{R}$. ∎

####  7.2 Proof of Theorem 1

###### Near components.

Recall the notation $\mathbb{F}_{n}(\lambda)\in\mathcal{E}$ for the
renormalized components sizes (first frozen, followed by the standard ones) in
$\mathrm{F}_{n}(\lambda)$. Accordingly, we write
$\mathbb{T}_{\mathrm{near},n}(\lambda)$ for the vector

$\left(n^{-2/3}\mathrm{C}_{*,n}(\lambda);\left(n^{-2/3}\cdot\mathrm{C}_{i,n}(\lambda):i\geqslant
1\right)\right),$

where $\mathrm{C}_{*,n}(\lambda)$ and $\mathrm{C}_{i,n}(\lambda)$ are the
sizes of the (blue) root component followed by the other components in
decreasing order of size in $\mathrm{T}_{\mathrm{near},n}(\lambda)$. Recall
that by Proposition 6, the white components of $F(n,m)$ are exactly the
components of $T_{\mathrm{near}}(n,m)$ which do not contain the root and the
blue vertices of $F(n,m)$ are exactly the vertices of the parked component of
the root. Furthermore, the flux of outgoing cars $D(n,m)$ corresponds to the
number of discarded edges in $F(n,m)$. Since the mapping
$\mathbf{z}\in\mathcal{E}\mapsto(\|\mathbf{z}\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},[\mathbf{z}]_{\circ})\in\mathbb{R}_{+}\times\ell^{2}_{\downarrow}$
is continuous, we can combine Theorem 2 and Proposition 14 to deduce that

$\left(n^{-1/3}\cdot\mathrm{D}_{n}(\lambda);\mathbb{T}_{\mathrm{near},n}(\lambda)\right)_{\lambda\in\mathbb{R}}\xrightarrow[n\to\infty]{(d)}\left(\mathscr{D}(\lambda),\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},[\mathscr{F}\mathscr{M}(\lambda)]_{\circ}\right)_{\lambda\in\mathbb{R}},$

for the Skorokhod topology on
$\mathbb{C}\mathrm{adlag}(\mathbb{R}\times\mathbb{R}\times\ell^{2}_{\downarrow})$.
∎

###### Full components.

Let us sketch how to obtain the equivalent of Theorem 1 for the full
components of $\mathrm{T}_{\mathrm{full}}(n,\cdot)$ rather than the near
components. To extend the above convergence to the case of full components,
notice first that $\mathrm{D}_{n}(\lambda)$ stays the same for near and full
components, and that as soon as $\mathrm{D}_{n}(\lambda)>0$ (which is the case
with high probability in the whole critical window) we have with an obvious
notation

$\|\mathbb{T}_{\mathrm{near},n}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}=\|\mathbb{T}_{\mathrm{full},n}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},$

since the blue components of the root with flux are the same in
$\mathrm{T}_{\mathrm{near}}$ and in $\mathrm{T}_{\mathrm{full}}$. We just have
to show that for any fixed compact time interval $I$ we have

$\sup_{\lambda\in
I}\mathrm{d}_{\ell^{2}_{\downarrow}}\left([\mathbb{T}_{\mathrm{near},n}(\lambda)]_{\circ},[\mathbb{T}_{\mathrm{full},n}(\lambda)]_{\circ}\right)\xrightarrow[n\to\infty]{(\mathbb{P})}0,$

with an obvious notation. For any fixed $\lambda$, the fully parked trees of
$\mathrm{T}_{\mathrm{full},n}(\lambda)$ are obtained by splitting the nearly
parked trees of $\mathrm{T}_{\mathrm{near},n}(\lambda)$ at their root
vertices. Since by Proposition 12, conditionally on their sizes, those are
uniform nearly parked trees (this can also be seen by combining our coupling
construction with Proposition 8 and Proposition 7), we deduce from Proposition
11 that each large nearly parked tree of
$\mathrm{T}_{\mathrm{near},n}(\lambda)$ contains a unique fully parked tree of
roughly the same size. Since $[\mathbb{T}_{\mathrm{near},n}(\lambda)]_{\circ}$
converges in $\ell^{\infty}_{\downarrow,0}$ we easily deduce that for each
$\lambda\in\mathbb{R}$ we have

$\mathrm{d}_{\ell^{\infty}_{\downarrow,0}}\left([\mathbb{T}_{\mathrm{near},n}(\lambda)]_{\circ},[\mathbb{T}_{\mathrm{full},n}(\lambda)]_{\circ}\right)\xrightarrow[n\to\infty]{(\mathbb{P})}0.$

Actually, the last display also holds for any stopping time $\Lambda$ which
belongs to some fixed time interval. Combining this we the monotony property
of the processes $\mathrm{T}_{\mathrm{near}}$ and $T_{\mathrm{full}}$,
standard but tedious arguments (which we shamefully leave to the reader) show
that we in fact have

$\sup_{\lambda\in
I}\mathrm{d}_{\ell^{\infty}_{\downarrow,0}}\left([\mathbb{T}_{\mathrm{near},n}(\lambda)]_{\circ},[\mathbb{T}_{\mathrm{full},n}(\lambda)]_{\circ}\right)\xrightarrow[n\to\infty]{(\mathbb{P})}0.$

To boostrap the above convergence by replacing the
$\ell^{\infty}_{\downarrow,0}$ metric with the $\ell^{2}_{\downarrow}$ metric,
we use Lemma 9 on $[\mathbb{T}_{\mathrm{near},n}(\lambda)]_{\circ}$ and remark
that the proof straightforwardly extend to
$[\mathbb{T}_{\mathrm{full},n}(\lambda)]_{\circ}$. ∎

###### Strong components.

Obviously a version of Theorem 1 holds if we consider the strong components in
the parking process to the cost of multiplying the scaling limit of the
components sizes by $1/2$, since by Proposition 11 each large nearly parked
tree contains a giant strongly parked tree of roughly half its size. To be
precise, one would need to establish the same behavior for the component of
the root (which may have some outgoing flux). We refrain from doing so to keep
the paper’s length acceptable.

####  7.3 The freezer as a Lévy-type process

In this section we give an alternative description of the process
$\|\mathscr{F}\mathscr{M}(\cdot)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
by “passing Corollary 8 to the scaling limit”. Since we shall not use this in
the rest of the paper, the proofs are only sketched and this section can be
skipped at first reading. Recall the function $g_{x,\lambda}(y)$ from (14).

######  Proposition 15 (A pure-jump description of
$\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$).

The process
$\lambda\mapsto\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
is a Markov Feller processs with inhomogeneous jump measure

$\mathbf{n}_{\lambda}(x,\mathrm{d}y)=\frac{1}{2}\frac{\mathrm{d}y}{\sqrt{2\pi
y^{3}}}g_{x,\lambda}(y)$

started “from $0$ at time $-\infty$”.

Heuristically, this means that the process
$\lambda\mapsto\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
has no drift, no Brownian part and jumps according to a modification of the
(infinite) measure $y^{-3/2}{\mathrm{d}y}\mathbf{1}_{y>0}$ depending on time
$\lambda$ and location $x$. This is an example of a so-called Lévy-type
process (quite simple in our case since we only have positive jumps) we refer
to the monograph [20] for survey. We shall rather see it as the solution of a
pure-jump stochastic differential equation driven by some Poisson measure.

###### Sketch of proof..

Let us first see why we can define a Feller Markov process $\mathscr{P}$ with
the above jump kernel over a time interval
$[\lambda_{0},\lambda_{1}]\subset\mathbb{R}$ starting from the initial value
$x_{0}\geqslant 0$ at time $\lambda_{0}$. To do this, we consider a Poisson
point process $\Pi$ over
$\mathbb{R}_{+}\times\mathbb{R}_{+}\times[\lambda_{0},\lambda_{1}]$ with
(infinite) intensity

$\frac{1}{2}\frac{\mathrm{d}y}{\sqrt{2\pi
y^{3}}}\cdot\mathrm{dz}\mathbf{1}_{z\geqslant 0}\cdot\mathrm{d}\lambda.$

We then consider the solution $\mathscr{P}$ to a pure-jump stochastic
differential equation driven by $\Pi$, obtained by starting from $x_{0}$ at
time $\lambda_{0}$ and from every atom $(y,z,\lambda)$ of $\Pi$, the process
$\mathscr{P}$ has a jump of height $y$ at time $\lambda^{-}$ i.e.
$\mathscr{P}_{\lambda}=\mathscr{P}_{\lambda-}+y$ if

$z\leqslant g_{\mathscr{P}_{\lambda^{-}},\lambda}(y),$

so that the jump kernel is indeed given by
$\mathbf{n}_{\lambda}(x,\mathrm{d}y)$. We now verify the usual Lipschitz
conditions so that strong solution and pathwise uniqueness holds. For this, we
shall first gather a few remarks on the function $p_{1}$:

(P1) The function $x\mapsto p_{1}(x)$ is bimodal: increasing from $-\infty$ to
$x_{\max}\approx-0.886$ and then decreasing from then on.

(P2) For all $\lambda\in[\lambda_{0},\lambda_{1}]$ and $x,y\geqslant 0$, the
ratio $p_{1}(\lambda-x-y)/p_{1}(\lambda-x)$ is bounded by a constant
$C=1\wedge(p_{1}(x_{\max})/p_{1}(\lambda_{1}))>0$ depending only on
$\lambda_{1}$.

(P3) The function $g_{x,\lambda}(y)$ is a smooth function of any of its
variable $x\geqslant 0,y\geqslant 0$ and $\lambda\in\mathbb{R}$.

Those properties are easily proven using a Math software such as Mathematica
or Maple. In particular, using $(P2)$ we see that

$\forall\lambda\in[\lambda_{0},\lambda_{1}],\forall x\geqslant
0,\quad\int_{0}^{1}y\cdot\mathbf{n}_{\lambda}(x,\mathrm{d}y)\leqslant
c_{1}(1+x),$

for some $c_{1}>0$ depending on $\lambda_{1}$ only so that the “linear growth
condition is satisfied” and the process does not explode in finite time. By
$(P3)$, it follows that for any $A>0$ we have

$\forall\lambda\in[\lambda_{0},\lambda_{1}],\forall
x,x^{\prime}\in[0,A],\quad\int_{0}^{1}y\cdot\frac{\mathrm{d}y}{\sqrt{2\pi
y^{3}}}|g_{x,\lambda}(y)-g_{x^{\prime},\lambda}(y)|\leqslant
c_{2}|x-x^{\prime}|,$

for some $c_{2}>0$ depending on $\lambda_{1}$ and $A$. We are thus in the
classical Lipschitz and linear growth condition so that we have strong
solution and pathwise uniqueness for $\mathscr{P}$, see [45, Theorem 9.1, page
245] or [46, Chap III.2.c, page 155]. It is also easy to check that the
resulting process is a Feller Markov process $\mathscr{P}$. Furthermore, the
process $\mathscr{P}$ is the scaling limit of the chain
$\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
in the sense that if we start the Markov chain
$(\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},\|F(n,m)\|_{\bullet-\bullet})$
from $m=\frac{n}{2}+\frac{\lambda_{0}}{2}n^{2/3}$ with
$\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}=x_{0}n^{2/3}$
and $m-\|F(n,m)\|_{\bullet-\bullet}=o(n^{2/3})$ then

$\displaystyle(n^{-2/3}\mathrm{F}_{n}(\lambda):\lambda\in[\lambda_{0},\lambda_{1}])\xrightarrow[n\to\infty]{}(\mathscr{P}(\lambda):\lambda\in[\lambda_{0},\lambda_{1}])\mbox{
with }\mathscr{P}(\lambda_{0})=x_{0}$ (51)

in the sense of Skorokhod. Indeed, the asymptotics (14) shows that the jump
kernels of
$(n^{-2/3}\|\mathrm{F}_{n}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},n^{-2/3}\mathrm{D}_{n}(\lambda))$
converge towards $(\mathbf{n}_{\lambda}(x,\mathrm{d}y),\mathbf{0})$ and for
any $\varepsilon>0$, all $m=\frac{n}{2}+\frac{\lambda}{2}n^{2/3}$ for
$\lambda\in[\lambda_{0},\lambda_{1}]$ and $n$ large enough

$\mathbb{E}\left[\begin{array}[]{rcl}\min(\Delta\|F(n,m+1)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},\varepsilon
n^{2/3})\\\
\Delta(m-\|F(n,m)\|_{\bullet-\bullet})\end{array}\left|\begin{array}[]{l}\|F(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}=xn^{2/3}\\\
m-\|F(n,m)\|_{\bullet-\bullet}\leqslant
n^{2/3}\end{array}\right]\right.\leqslant\left(\begin{array}[]{c}C\sqrt{\varepsilon}\\\
2(1+x)n^{-1/3}\end{array}\right),$

for some constant $C>0$ independent of $n$. The convergence (51) is then a
consequence of general convergence results on Feller processes, see [50,
Chapter 19] or [46, Chapter IX, 4]. We leave the verifications to the reader.

Finally, let us see why we can define the Feller process $\mathscr{P}$ by
starting from $0$ at time $-\infty$. To prove this, we need to show
convergence of $\mathscr{P}$ at a fixed time, say $\lambda=0$, when
$\mathscr{P}$ is started from $0$ at a very negative time $\lambda_{0}\ll 0$.
This can be deduced by rather tedious calculations using
$\mathbf{n}_{\lambda}$ and asymptotics of $p_{1}$ but let us sketch another
route using our cutoff construction of Section 6.1. Specifically, recall the
construction of the process $\mathrm{F}_{n}^{[\lambda_{0}]}(\lambda)$ obtained
by throwing all components with cycles in $\mathrm{G}_{n}(\lambda_{0})$ and
starting the construction of the frozen process from there. We shall use a
variant of this construction by considering a random stopping time
$\Lambda_{0}$ (with an implicit dependence in $n$) associated to
$M_{0}=\frac{n}{2}+\frac{\Lambda_{0}}{2}n^{2/3}$ defined as follows

$M_{0}=\inf\left\\{m\geqslant
0:2\left(m-\frac{n}{2}\right)-\|G(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}\geqslant\lambda_{0}n^{2/3}\right\\}.$

In words, $M_{0}$ is the first instant $m$ where the felt time-parameter in
the forest part $[G(n,m)]_{\mathrm{tree}}$ is above $\lambda_{0}$. We first
claim that for $\lambda_{0}$ negative enough, $M_{0}\leqslant n/2$ and is
actually close to $\frac{n}{2}+\frac{\lambda_{0}}{2}n^{2/3}$ with high
probability: indeed it follows from (49) that
$n^{-2/3}\cdot\|\mathrm{G}_{n}(\lambda_{0})\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
is of order $\lambda_{0}^{-2}$ and so the function $m\mapsto
2\left(m-\frac{n}{2}\right)-\|G(n,m)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}\geqslant\lambda_{0}n^{2/3}$
crosses $\lambda_{0}n^{2/3}$ around time $\lambda_{0}\pm\lambda_{0}^{-2}$. On
this event, by Proposition 8, the process
$\mathrm{F}_{n}^{[\Lambda_{0}]}(\cdot+\Lambda_{0})$ has the same transitions
as $F$ started from $0$ at time
$m^{\prime}=\frac{n^{\prime}}{2}+\frac{\lambda_{0}}{2}n^{\prime
2/3}+o(n^{2/3})$ over
$n^{\prime}=n-\|G(n,M_{0})\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
vertices with a slight time change coming from the fact that certain edges are
discarded (this does not persist in the limit). So by (51) it converges after
scaling towards the process $\mathscr{P}(\cdot+\lambda_{0})$ started from $0$
at time $\lambda_{0}$. The convergence of
$n^{-2/3}\cdot(\mathrm{F}_{n}^{[\Lambda_{0}]}(\lambda):\lambda\geqslant 0)$
proved in Section 6.1 together with the fact that
$n^{-2/3}\|\mathrm{G}_{n}(\Lambda_{0})\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}\to
0$ as $\lambda_{0}\to\infty$ and the above convergence imply that the law of
the process $\mathscr{P}$ started from $0$ at time $\lambda_{0}$ does converge
as $\lambda_{0}\to\infty$ and this enables us to start $\mathscr{P}$ from $0$
at time $-\infty$. Combining those observations we deduce that the process
$\mathscr{P}$ started from $0$ at time $-\infty$ has the same law as
$\|\mathscr{F}\mathscr{M}(\cdot)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$.
We leave the many details to the fearless reader.

####  7.4 Scaling limit of random forest

In this section we revisit the result of Martin and Yeo [64] to complete the
Markovian description of the scaling limit of the frozen multiplicative
coalescent. As for the preceding section, the results are not used in the rest
of the paper and so this part may be skipped at first reading.

As in the proof of Corollary 8, for $n\geqslant 1$ and $m\geqslant 0$ we
denote by $W(n,m)\in\mathfrak{F}(n,m)$ a uniform random forest over the $n$
labeled vertices $\\{1,2,\dots,n\\}$ with $m$ edges in total. We chose the
letter $W$ for the German “Wald” because there are already too many f’s in the
paper :) In particular, the forest $W(n,m)$ has $n-m$ components. Although
there is _no_ obvious coupling of $W(n,m)$ for varying $m\geqslant 0$ (see
[64, Section 1.4.2]), we shall use our usual notation (1) and write
$\mathrm{W}_{n}(\lambda)$ for a random forest with
$m=\lfloor\frac{n}{2}+\frac{\lambda}{2}n^{2/3}\rfloor$ edges and by
$\mathbb{W}_{n}(\lambda)\in\ell^{\infty}_{\downarrow,0}$ the renormalized
sequence of its component sizes in non-increasing order.

Recall from Section 3.1 that $(\mathscr{S}_{t})_{t\geqslant 0}$ denotes the
stable Lévy process with index $3/2$ and only positive jumps, which starts
from $0$ and normalized so that
$\mathbb{E}[\exp(-\ell\mathscr{S}_{t})]=\exp(\tfrac{2^{3/2}}{3}t\ell^{3/2})$
for any $\ell,t\geqslant 0$, see Figure 22 for a simulation. The density of
$\mathscr{S}_{t}$ is $p_{t}(\cdot)$ for $t>0$. For any $\lambda\in\mathbb{R}$
we can use this function to define the process
$(\mathscr{S}^{\lambda}_{t}:0\leqslant t\leqslant 1)$ called the
$(0,0)\to(1,\lambda)$ bridge, obtained by conditioning
$(\mathscr{S}_{t}:0\leqslant t\leqslant 1)$ to be equal to $\lambda$ at time
$1$. Of course this is a degenerate conditioning, but it can be obtained by
performing an inhomogeneous $h$-transform with respect to the function

$\frac{p_{1-t}(\lambda-\mathscr{S}_{t})}{p_{1}(\lambda)},$

see [58, Theorem 4].

######  Proposition 16 (Another route towards critical random forests).

Fix $\lambda\in\mathbb{R}$. For all $\varepsilon>0$, we have the following
convergence in distribution for the $\ell^{3/2+\varepsilon}_{\downarrow}$\-
topology

$\displaystyle\mathbb{W}_{n}(\lambda)\xrightarrow[n\to\infty]{(d)}\big{(}\Delta\mathscr{S}^{\lambda}_{t}:0\leqslant
t\leqslant 1\big{)}^{\downarrow}$ (52)

where $(x_{i}:i\geqslant 1)^{\downarrow}$ is the non-increasing rearrangement
of the sequence $(x_{i}:i\geqslant 1)$.

Figure 22: A simulation of a $\tfrac{3}{2}$-stable spectrally positive Lévy
process over the time interval $[0,1]$. The jumps are displayed in orange.

###### Proof.

From Proposition 7, the sizes of the components in the random forest $W(n,m)$
has the same law as the increments $+2$ of a random walk
$(\tilde{S}_{i}:0\leqslant i\leqslant n-m)$ started from $0$, conditioned to
hit $n-2(n-m)=\lambda n^{2/3}+o(n^{2/3})$ at time $n-m$ and with independent
increments of law $\mu(k+2)$ for $k\in\\{-1,0,1,2,\dots\\}$ introduced in (9).
Recall that the variable $\tilde{S}_{1}$ is centered and in the domain of
attraction of the $3/2$-stable spectrally positive random variable. The
following convergence for the Skorokhod topology on
$\mathbb{C}\mathrm{adlag}([0,1],\mathbb{R})$ follows from the conditional
invariance principle of Liggett [58]

$\left(n^{-2/3}\tilde{S}_{[(n-m)t]}:0\leqslant t\leqslant
1\right)\xrightarrow[n\to\infty]{(d)}\left(\mathscr{S}^{\lambda}_{t}:0\leqslant
t\leqslant 1\right).$

In particular by [46, Corollary 2.8], the random point measure
$\sum_{0\leqslant t\leqslant
1}\delta_{n^{-2/3}\cdot\Delta\tilde{S}_{[(n-m)t]}}$ converges weakly towards
$\sum_{0\leqslant t\leqslant 1}\delta_{\Delta\mathscr{S}^{\lambda}_{t}}$ from
which we deduce the convergence (52) for the $\ell_{\downarrow,0}^{\infty}$
topology. To bootstrap this into a convergence for the
$\ell^{3/2+\varepsilon}_{\downarrow}$ topology, it suffices to establish
tightness in the later (since convergence in $\ell^{\infty}_{\downarrow,0}$
characterizes the limit in $\ell^{3/2+\varepsilon}_{\downarrow}$). For this we
claim that it is sufficient to prove that

$\displaystyle\sup_{n\geqslant
1}\mathbb{E}\left[\sum_{i=0}^{n-m-1}\left(n^{-2/3}\cdot\Delta{S}_{i}\right)^{3/2+\varepsilon}\Big{|}{S}_{n-m}=n\right]<\infty,$
(53)

where $S$ has increments of law $\mu(k)$ given by (9). Indeed, for $\xi>0$ and
$\ \varepsilon>0$

$\sum_{i\geqslant
1}\left(n^{-2/3}\cdot\Delta{S}_{i}\right)^{3/2+2\varepsilon}\mathbf{1}_{n^{-2/3}\cdot\Delta{S}_{i}>\xi}\leqslant\xi^{\varepsilon}\sum_{i=0}^{n-m-1}\left(n^{-2/3}\cdot\Delta{S}_{i}\right)^{3/2+\varepsilon},$

so that using (53) the expectation of the right-hand side of the last display
can be made arbitrarily small by choosing $\xi$ small. Combining this with the
tightness in $\ell^{\infty}_{\downarrow,0}$, it is easy to deduce tightness in
$\ell_{\downarrow}^{3/2+2\varepsilon}$ of
$\left(\mathbb{W}_{n}(\lambda):n\geqslant 1\right)$. To prove our claim,
notice that by cyclic exchangeability we have

$\displaystyle\mathbb{E}\left[\sum_{i=0}^{n-m-1}\left(n^{-2/3}\cdot\Delta{S}_{i}\right)^{3/2+\varepsilon}\Big{|}{S}_{n-m}=n\right]$
$\displaystyle=$ $\displaystyle(n-m)\sum_{k\geqslant
1}\mu(k)\left|kn^{-2/3}\right|^{3/2+\varepsilon}\frac{\mathbb{P}(S_{n-m-1}=n-k)}{\mathbb{P}(S_{n-m}=n)}.$

Let us focus first on the $k$’s such that $k\ll n$. In this case, writing
$k=yn^{2/3}$ and using (9) and (12), we deduce that there are constants
$C,C^{\prime}>0$ (which may depend on our fixed $\lambda$) such that the last
display is bounded above by

$Cn^{-2/3}\sum_{k=1}^{\infty}(kn^{{-2/3}})^{-1+\varepsilon}\cdot\frac{p_{1}(\lambda-
kn^{{-2/3}})}{p_{1}(\lambda)}\leqslant
C^{\prime}\int_{0}^{\infty}y^{-1+\varepsilon}\ \mathrm{d}y<\infty.$

On the other hand, if $k$ if of order $n$, rough large deviations estimates
show that $\frac{\mathbb{P}(S_{n-m-1}=n-k)}{\mathbb{P}(S_{n-m}=n)}$ is
exponentially small (in $k$ and so in $n$) so that the contribution to the sum
is negligible. This finishes the proof of (53) and of the proposition. ∎

As an application of this methodology, let us revisit a few of the results of
[62] discussed in [64, Section 1.4.3]. Consider a slightly supercritical
random forest $W(n,m)$ with $m=\frac{n}{2}+\frac{s}{2}$ with $n^{2/3}\ll s\ll
n$ whose component sizes are coded by $(\Delta{\tilde{S}}_{i}+2:0\leqslant
n-m-1)$ conditioned on $\tilde{S}_{n-m}=s$. According to standard “big-jump”
principles, since $\mu$ is a subexponential distribution such a random walk
has a unique “big-jump” of height of order $s$ and once this jump has been
removed, the remaining random walk is close in total variation distance to an
unconditioned $\mu$-random walk, see [7, Theorem 1]. This gives another way to
prove that the largest cluster in $W(n,m)$ is of size $(1+o(1))s$ and the
remaining components converge after normalization by $n^{2/3}$ to the jumps of
the _unconditioned_ Lévy process $(\mathscr{S}_{t}:0\leqslant t\leqslant 1)$.

##  Part III Comments and perspectives

We end this paper by presenting several research directions and connections of
our work. This part is informal and we do not claim any mathematical
statement. We first draw a parallel between the enumeration of (strongly,
fully or nearly) parked trees and random planar maps which gives another
support for Conjecture 1. We then present a few fallouts of the study of
(generalized) frozen process $F(n,\cdot)$ on the Erdős–Rényi random graph
$G(n,\cdot)$. We end with extensions of our work concerning the parking
process on random trees.

###  8 Links with planar maps and growth-fragmentation trees

We shall consider strongly parked trees with _outgoing flux_. More precisely,
for $n,p\geqslant 0$ we denote by $\mathrm{SP}(n,n+p)$ the number of labeled
rooted Cayley trees of size $n$ with $n+p$ labeled cars so that after parking,
all edges have a positive flux and exactly $p$ cars exit the tree. We encode
these numbers into the generating function

$\mathbf{S}(x,y)=\sum_{n\geqslant 1,p\geqslant
0}\frac{\mathrm{SP}(n,n+p)}{n!(n+p)!}x^{n}y^{p},$

which replaces the univariate generating function $\mathbf{S}(x)$ which we
considered in Section 4. In particular since King & Yan [52] computed
$\mathrm{SP}(n,n)=(2n-2)!$ we have (Proposition 10) that

$\displaystyle\mathbf{S}(x,0)=1-\ln(2)-\sqrt{1-4x}+\ln\left(1+\sqrt{1-4x}\right),\quad\mbox{
for }0\leqslant x\leqslant x_{c}=\frac{1}{4}.$ (54)

####  8.1 Tutte’s equation

To get a functional equation on $\mathbf{S}$ one considers the decomposition
of strongly parked trees at the root vertex (see Figure 23 left) which shows
that $\mathrm{SP}(n,n+p)$ is equal to

$\displaystyle\sum_{a\geqslant 0}\sum_{k\geqslant
0}\sum_{\begin{subarray}{c}n_{1},\dots,n_{k}\geqslant 1\\\
p_{1},\dots,p_{k}\geqslant 1\end{subarray}}\frac{1}{k!}{n\choose
1,n_{1},n_{2},\dots,n_{k}}{n+p\choose
a,n_{1}+p_{1},\dots,n_{k}+p_{k}}\prod_{i=1}^{k}\mathrm{SP}(n_{i},n_{i}+p_{i})\mathbf{1}_{\begin{subarray}{l}n_{1}+\dots+n_{k}=n-1\\\
a+p_{1}+\dots+p_{k}=p+1.\end{subarray}}$

Indeed, the integer $a$ counts the number of cars arriving at the root, the
integer $k$ is the number of children of the root vertex and $n_{i},p_{i}$ are
the characteristics (number of vertices and outgoing flux) of the subtrees
above it. This equation translates into the following equation on $\mathbf{S}$

$\displaystyle\mathbf{S}(x,y)=\frac{x}{y}\left(\mathrm{e}^{y}\mathrm{e}^{\mathbf{S}(x,y)-\mathbf{S}(x,0)}-1\right).$
(55)

At first sight, one may think that the series $\mathbf{S}(x,0)$ is a necessary
input (which we do have) to solve the equation, but a close inspection shows
that the equation actually determines the coefficients of $\mathbf{S}$ by
induction on $n+p$.

This type of equation is very common in the map enumeration literature where
they are called “Tutte” equations, see [21] for a comprehensive survey. More
precisely, recall that a map is a planar graph properly embedded in the plane
given with one distinguished oriented edge. Following Tutte, when enumerating
(various classes of) planar maps by their size $n$, it is convenient to
introduce an external parameter $p$, the perimeter of the external face (lying
on the right of the root edge). When performing the root erasure, certain
situations yield a splitting of a map of size $n$ and perimeter $p$ into two
components of size $n_{1}$ and $n_{2}$ having perimeter $p_{1}$ and $p_{2}$ so
that we have (on a high level) $n_{1}+n_{2}\approx n$ and $p_{1}+p_{2}\approx
p$ which is similar to penultimate equation above, see Figure 23 right.
Similar equations arose in [34, 40].

Figure 23:  Left: Illustration of the recursive decomposition at the root of
strongly parked trees to get a functional equation on $\mathbf{S}$. Right:
heuristic representation of Tutte’s equation in the theory of planar map
enumeration.

In our case, Equation (55) can be solved using the Lambert function. Fix
$(x,y)$ and observe that $\mathbf{S}=\mathbf{S}(x,y)$ is solution to an
equation of the form $a\mathrm{e}^{\mathbf{S}}+b\mathbf{S}+c=0$. If we put

$\Delta=-\exp\left(y-\mathbf{S}(x,0)-\frac{x}{y}\right)\frac{x}{y}\leqslant 0$

when $x\geqslant 0$ and $y>0$, then the above equation has solutions if
$\Delta\geqslant-\mathrm{e}^{-1}$ which are

$-W_{i}(\Delta)-\frac{x}{y}$

where $W_{i}$ is the $i$th branch of the Lambert function. There is actually a
singularity and we need to change branch (see Figure 24), more precisely, when
$x<x_{c}=1/4$ we have

$\displaystyle\mathbf{S}(x,y)=\left\\{\begin{array}[]{ll}-W_{-1}(\Delta)-\frac{x}{y}&\mbox{
if }y\leqslant\frac{1}{2}\left(1-\sqrt{1-4x}\right)\\\
-W_{0}(\Delta)-\frac{x}{y}&\mbox{ if }\left(1-\sqrt{1-4x}\right)\leqslant
y\leqslant y_{c}(x),\end{array}\right.$ (58)

where $y_{c}(x)$ is the radius of convergence of the series in $y$ when $x$ is
fixed which is the maximal solution of $\Delta=-\mathrm{e}^{-1}$. At
$x=x_{c}=1/4$, then $\Delta+\mathrm{e}^{-1}$ vanishes at
$y_{c}=y_{c}(1/4)=1/2$ and yields a singularity of type $(y-y_{c})^{3/2}$.

Figure 24: Plot of the function $y\mapsto\mathbf{S}(x,y)$ for $x=0.245$. To
get an analytic function, $\mathbf{S}$ changes from the blue to the orange
branch at $y\approx 0.41$.

####  8.2 Lackner & Panholzer’s decomposition and the KP hierarchy?

Actually, there is another completely different way to get a functional
equation on $\mathbf{S}$. Adapting an idea of [56] (see in particular Equation
(4) there) one can decompose a strongly parked tree according to the travel of
the car labeled $n+p$ (the last car) in a sequence of strongly parked trees
each given with a distinguished point, see Figure 25. This last car
decomposition yields to the following equation on $\mathbf{S}(x,y)$:

$\displaystyle
y\partial_{y}\mathbf{S}(x,y)+x\partial_{x}\mathbf{S}(x,y)-\mathbf{S}^{\bullet}(x,0)=\frac{xy\partial_{x}\mathbf{S}(x,y)}{1-\mathbf{S}^{\bullet}(x,0)},$
(59)

where $\mathbf{S}^{\bullet}(x,y)=x\partial_{x}\mathbf{S}(x,y)$ is the
generating series of strongly parked trees with an additional distinguished
vertex. It should be possible to solve the above equation using the method of
characteristics to recover (58) but we have not been able to carry the
calculations.

Figure 25:  Illustration of the decomposition of a strongly parked tree
according to the ride of the last car. If we remove the edges through which
the last car had to go, then we end up with a sequence of strongly parked
trees with distinguished points, where the last of those may have a positive
flux at the root.

Also, applying the last car decomposition to the case $y=0$ (no flux) one
finds the equation

$\textbf{S}^{\bullet}(x,0)=\frac{x}{1-\textbf{S}^{\bullet}(x,0)}$

involving $\mathbf{S}(x,0)$ only and enables us to recover (54) very easily.
In the theory of planar maps, there are similar inductive decompositions of
planar maps of size $n$ involving planar maps of size $n_{1}$ and $n_{2}$ with
$n_{1}+n_{2}\approx n$ _without_ boundary. Those decompositions are obtained
via the KP hierarchy, see [44] or [61, Corollary 1] for details. We plan on
adapting the “last car decomposition” of Lackner & Panholzer to the
enumeration of random planar maps.

We expect that the information on the bivariate generating function
$\mathbf{S}(x,y)$ will enable us to perform the asymptotic enumeration of
strongly parked trees with flux and prove similar results as in the planar map
setting, i.e.

$\displaystyle\frac{\mathrm{SP}(n,n+p)}{n!(n+p)!}$ $\displaystyle\sim$
$\displaystyle c_{1}\cdot 4^{n}\cdot 2^{p}\cdot n^{-5/2}\cdot
p^{1/2}\exp\left(-c_{2}\frac{p^{2}}{n}\right)$ (60)

for some constants $c_{1},c_{2}>0$ as long as $p^{2}/n$ stays in a fixed
compact interval of $(0,\infty)$. Those asymptotics are necessary to progress
towards Conjecture 1 but also would be a crucial input to compute, for fixed
$\lambda\in\mathbb{R}$, the exact distribution of
$(\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},\mathscr{D}(\lambda))$.
The behavior (60) has been observed in a great generality for a related model
[27] which we now describe.

###### Chen’s and Panholzer’s generalizations of fully parked trees.

Panholzer [67] studies the model of fully parked trees (with no flux) when the
underlying Cayley tree is replaced by a combinatorial model such as $d$-ary
trees, ordered trees… and he obtains remarkable explicit formulas. In
particular, he finds connections with models of planar maps (OEIS A000139, or
OEIS A000260 via OEIS A294084), see Remark 2 in [67]. It is natural to extend
the above discussion to those models. In [27], Chen considers the enumeration
of _plane_ trees (rather than Cayley trees) decorated with i.i.d. (not
necessarily Poisson) car arrivals conditioned to be fully parked and with a
possible flux at the root. He proves a phase transition for the enumeration of
such structures appearing at the same location as the phase transition for the
parking process [37]. In the case of bounded car arrivals, he proves in [27,
Theorem 3] an asymptotic enumeration of plane fully parked trees with flux of
the form (60). This supports the belief that the parking processes are in the
same universality classes, see below.

####  8.3 Growth-fragmentation trees and conjectural scaling limits.

Let us now give some background for Conjecture 1. By the discussion in the
last paragraph, the generating series of strongly parked trees is finite at
$x=x_{c}=\frac{1}{4}$. Hence, for each $p\geqslant 0$, we can define as in
Section 5.1 a random strongly parked tree $S_{p}$ with flux $p$ at the root
under Boltzmann critical distribution whose law is simply

$\mathbb{P}(S_{p}=\mathfrak{s}_{p})=\frac{1}{[y^{p}]\mathbf{S}(x_{c},y)}\frac{1}{\|\mathfrak{s}_{p}\|_{\bullet}!(\|\mathfrak{s}_{p}\|_{\bullet}+p)!}\left(\frac{1}{4}\right)^{\|\mathfrak{s}_{p}\|_{\bullet}},$

for each strongly parked tree $\mathfrak{s}_{p}$ with flux $p$ at the root.
Let us forget the labels of the vertices and the cars and see such a tree as a
rooted unordered tree where each vertex is labeled by the flux of car
emanating from it (so that the root has label $p$). It is easy to see that
such trees are _Markov branching trees_ , that is, have the same law as the
family tree of a system of particles evolving independently of each other.
Each particle carries a non-negative integer label (the emanating flux of car
from that vertex) and at each step, a particle of label $p$ “splits” into $k$
particles of labels $p_{1},p_{2},\dots,p_{k}$ (ordered uniformly at random)
with probability

$\frac{1}{[y^{p}]\mathbf{S}(x_{c},y)}\frac{1}{(p-(p_{1}+\cdots+p_{k})+1)!}\frac{1}{k!}\prod_{i=1}^{k}[x^{p_{i}}]\mathbf{S}(x_{c},y).$

When the scaling limit of the labels along a branch101010to be specific, one
define a branch by following the locally largest label at each splitting is
given by a positive self-similar Markov process, the scaling limit of those
trees are described by the growth-fragmentation trees111111to be precise,
Bertoin defines a growth-fragmentation process from to which we can associate
a continuum random tree by [71, Corollary 4.2] of Bertoin [13]. In our case of
random strongly parked trees, the labels evolve in the scaling limit as
(versions of) the $3/2$-stable Lévy process, exactly as for the Markov
branching trees appearing the peeling exploration of random planar maps, see
[16] and [14, Section 6]. To be more specific, the growth-fragmentation
mechanism involved in Conjecture 1 is the one “canonically associated” to the
spectrally positive $3/2$-stable Lévy process i.e. with the cumulant function

$\kappa(q)=\frac{\Gamma(q-\tfrac{3}{2})}{\Gamma(q-3)}$

for $q>3/2$ and self-similarity index $\alpha=-3/2$, see [14, Section 5]. Our
Conjecture 1 concerns _conditioned version_ of those Markov branching trees,
see the forthcoming work [15] for details.

###  9 Back to Erdős–Rényi

Let us now formulate a few possible consequences of our work on the classical
Erdős–Rényi random graph. For this we need to generalize a little the frozen
process by introducing a parameter $p\in[0,1]$.

####  9.1 Generalized frozen process

Given the sequence of _unoriented_ edges $(E_{i}:i\geqslant 1)$ and an
independent sequence of uniform random variables $(U_{i}:i\geqslant 1)$ we
construct a generalization of the frozen Erdős–Rényi process as follows. Fix a
parameter $p\in[0,1]$ and define a growing graph process $F_{p}(n,m)$ with two
colors, white and blue, in a way very similar to $F(n,m)$: Initially
$F_{p}(n,0)$ is made of the $n$ labeled white vertices $\\{1,2,\dots,n\\}$ and
for $m\geqslant 1$

  * •

if both endpoints of $E_{m}$ are white vertices then the edge $E_{m}$ is added
to $F_{p}(n,m-1)$ to form $F_{p}(n,m)$. If this addition creates a cycle in
the graph then the vertices of its component are declared frozen and colored
in blue.

  * •

if both endpoints of $E_{m}$ are blue (frozen vertices), then $E_{m}$ is
discarded.

  * •

if $E_{m}$ connects a white and a blue vertex, then $E_{m}$ is discarded _if
$U_{m}>p$_ and kept otherwise, in which case the new connected component is
declared frozen and colored in blue.

Figure 26:  Transitions in the frozen Erdős-Rényi with parameter $p$.

Obviously, in the case $p=\frac{1}{2}$, the process $F_{1/2}$ has the same law
as the frozen Erdős–Rényi that we used in this paper121212in the paper we used
the orientation of the edges $\vec{E}_{i}$ and did not require the additional
randomness of the $U_{i}$.. For $p=0$, the process corresponds to completely
stopping the connected components once they create a cycle. In the case $p=1$,
the process is obtained from $G(n,m)$ by discarding the edges which would
create a surplus of $2$. In particular, we have

$\displaystyle[F_{1}(n,m)]_{\circ}=[F_{1}(n,m)]_{\mathrm{tree}}=[G(n,m)]_{\mathrm{tree}},\quad\mbox{
for all }m\geqslant 0.$ (61)

Notice however, that the obvious coupling of $F_{p}$ for all $p\in[0,1]$ is
not monotonic in $p$.

It should be easy to extend our analysis to the frozen Erdős–Rényi processes
with parameter $p$ and in particular Theorem 2, Propositions 8 and 15 and
Corollary 1 should hold with the proper changes. E.g., the scaling limit of
the rescaled total size of the frozen components in
$\mathrm{F}_{n,p}(\lambda)$ should be a pure-jump Feller process
$\mathscr{X}_{p}(\lambda)$ with jump kernel given by

$\displaystyle\frac{1}{2}\frac{1}{\sqrt{2\pi}}\frac{\mathrm{d}y}{y^{3/2}}(y+2p\cdot
x)\frac{p_{1}(\lambda-x-y)}{p_{1}(\lambda-x)}.$ (62)

Specifying those results for $p=1$ and using (61) we deduce that the process
of the total mass of the particles with surplus in the multiplicative
coalescent $\mathscr{M}(\lambda)$ has law $\mathscr{X}_{1}(\lambda)$ and that
conditionally on it, the remaining components are distributed as the jump of
the conditioned Lévy process $\mathscr{S}^{\lambda-\mathscr{X}_{p}(\lambda)}$.
We were not aware of such a description prior to this work.

####  9.2 Asymptotics when $\lambda\to\infty$

If the above description of the scaling limit $\mathscr{X}_{p}(\lambda)$ of
$n^{-2/3}\cdot\|\mathrm{F}_{n,p}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$
is granted, then one can perform the analysis in the near supercritical regime
$\lambda\to\infty$. It is easy to see that $\mathscr{X}_{p}(\lambda)$ tends to
$\infty$ and it is not hard to see that it is asymptotically larger than
$\lambda$. Using (11) and the definition of the jump kernel (62) we can
compute formally

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\lambda}\mathbb{E}[\mathscr{X}_{p}(\lambda)]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}y}{\sqrt{2\pi}y^{3/2}}\left(2p\mathscr{X}_{p}(\lambda)+y\right)\cdot
y\cdot\frac{p_{1}(\lambda-\mathscr{X}_{p}(\lambda)-y)}{p_{1}(\lambda-\mathscr{X}_{p}(\lambda))}\right]$
$\displaystyle\underset{\begin{subarray}{c}\lambda\to\infty\\\
\mathscr{X}_{p}(\lambda)-\lambda\to\infty\end{subarray}}{\sim}$
$\displaystyle\mathbb{E}\left[\frac{p\mathscr{X}_{p}(\lambda)}{\sqrt{2\pi}}\int_{0}^{\infty}\frac{\mathrm{d}y}{y^{3/2}}\cdot
y\cdot\mathrm{e}^{-y(\lambda-\mathscr{X}_{p}(\lambda))^{2}/2}\right]=\mathbb{E}\left[\frac{p\mathscr{X}_{p}(\lambda)}{\mathscr{X}_{p}(\lambda)-\lambda}\right].$

From this we conjecture the asymptotic rate of growth of the process
$\lambda\mapsto\mathscr{X}_{p}(\lambda)$ for $p\in[0,1]$

$\displaystyle\frac{\mathscr{X}_{p}(\lambda)}{\lambda}\xrightarrow[\lambda\to\infty]{(\mathbb{P})}1+p.$
(63)

In particular, when $p=0$ i.e. when we stop the cluster growth when they have
a positive surplus, we believe that the total mass of the frozen part is of
order $\lambda$ and furthermore that $\mathscr{X}_{0}(\lambda)-\lambda$
converge in distribution as $\lambda\to\infty$ towards a stationary law (an
example of self-organized criticality).

In the case $p=1$ we should have $\mathscr{X}_{1}(\lambda)\approx 2\lambda$.
This is coherent with the result of Luczak [63] saying that the largest
cluster in $\mathrm{G}_{n}(\lambda)$ is of order $2\lambda n^{2/3}$ when
$\lambda\to\infty$ (this cluster is likely to be formed by the majority of the
connected components of the frozen part).

####  9.3 Process construction

In the case of the multiplicative coalescent, there has been a substantial
amount of work describing the process in terms of collections of excursion
lengths of evolving random functions [6, 24, 60, 65, 74]. We do not know
whether such a construction is doable for our frozen processes.

###  10 Extension of parking process

In this work we used a coupling between the Erdős–Rényi random graph and
uniform parking on Cayley trees to study the later. Our results obviously call
for generalizations for other models of random trees and other arrival
distributions of cars. The ideas of this paper can indeed be extended to cover
the model of [32] and this is the subject of a forthcoming work of the first
author [31]. In particular, although the precise location of the phase
transition depends on the combinatorial details, we believe that the scaling
limits unraveled in this paper are common to a large class of models as long
as the degree distribution and the car arrivals have a sufficiently light
tail. In the presence of group arrival of cars with heavy tail, new scaling
limits should occur related to the different universality classes observed for
component sizes in configuration models [19, 23, 30, 49].

Once Conjecture 1 has been addressed, one hope to describe a dynamical scaling
limit for the geometry of the parking process involving the geometry of the
components as well as the flux of cars. This should involve spiraling frozen
fractals all around [39]. We we will try to address those questions in future
works.

### Table of notation

#### General notation

$\mathrm{X}_{n}(\lambda)$ |  $\mathrm{X}_{n}(\lambda)=X\left(n,\left\lfloor\frac{n}{2}+\frac{\lambda}{2}n^{2/3}\right\rfloor\vee 0\right)$ shorthand notation for a process $X(n,m)$
---|---
$\Delta X(n,m)$ | $\Delta X(n,m)=X(n,m+1)-X(n,m)$
| shorthand notation for the increments of a process $X(n,m)$
$T_{n}$ | uniform rooted Cayley tree over $\\{1,2,\dots,n\\}$
$T_{\star}(n,m)$ | for $\star\in\\{\mathrm{near},\mathrm{full},\mathrm{strong}\\}$ different types of components
| in $T_{n}$ after parking $m$ cars, see Figure 7.
$X_{1},Y_{1},X_{2},Y_{2}\dots$ | independent uniform numbers over $\\{1,2,\dots,n\\}$
| the $X_{i}$’s are seen as car arrivals and are independent of $T_{n}$
| while the $Y_{i}$’s are coupled non-trivially with $T_{n}$
$\vec{E}_{i}=(X_{i},Y_{i})$ |  $i$th oriented edge
$G(n,m)$ | Erdős–Rényi random graph built by adding the first $m$ unoriented edges
$F(n,m)$ | frozen Erdős–Rényi random graph built from the first $m$ edges
$D(n,m)$ | number of discarded edges in the construction of $F(n,m)$
| or equivalently of cars that did not manage to park on $T_{n}$
$W(n,m)$ | uniform unrooted labeled forest with $n$ vertices and $m$ edges
$\mathbb{W}_{n}(\cdot)$ | sequence of renormalized sizes of components in $\mathrm{W}_{n}(\cdot)$.
$\mathfrak{g},[\mathfrak{g}]_{\circ},[\mathfrak{g}]_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},[\mathfrak{g}]_{\mathrm{tree}}$ | a multigraph, its subgraph made of white/blue vertices, and its forest part
$\|\mathfrak{g}\|_{\bullet},\|\mathfrak{g}\|_{\circ},\|\mathfrak{g}\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},\|\mathfrak{g}\|_{\bullet-\bullet}$ | number of vertices, white vertices, blue vertices and edges of $\mathfrak{g}$
$\mathfrak{F}(n,m)$ | unrooted forests over $\\{1,2,\dots,n\\}$ with $m$ edges
$\\#\mathfrak{F}(n,m)$ | number of unrooted forests over $\\{1,2,\dots,n\\}$ with $m$ edges
$\mu(k)=2\mathrm{e}^{-k}\frac{k^{k-2}}{k!}$ | step distribution in the random walk $S$ coding the forests

#### Continuous processes notation

The random variables in the “continuous world” are usually denoted with a
mathscr font.

$\mathscr{S}$ |  $3/2$-stable spectrally positive Lévy process with Lévy measure $\frac{\mathrm{d}y}{\sqrt{2\pi}y^{5/2}}$
---|---
$\mathscr{S}^{u}$ | version of $\mathscr{S}$ conditioned on $\mathscr{S}_{1}=u$
$p_{1},p_{s}$ | density of $\mathscr{S}$ at time $1$ (Airy distribution) resp. $s\geqslant 0$
$\mathbf{n}_{\lambda}(x,\mathrm{d}y),g_{x,\lambda}(y)$ | jump kernel, see Section 7.3 and (14)
$(\mathscr{M}(\lambda):\lambda\in\mathbb{R})$ | (standard) multiplicative coalescent
$(\mathscr{F}\mathscr{M}(\lambda):\lambda\in\mathbb{R})$ | frozen multiplicative coalescent
$[\mathscr{F}\mathscr{M}(\lambda)]_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$ |  $\ell^{1}$\- part of $\mathscr{F}\mathscr{M}(\lambda)$ and its total mass
$[\mathscr{F}\mathscr{M}(\lambda)]_{\circ}$ |  $\ell^{2}$\- part of $\mathscr{F}\mathscr{M}(\lambda)$

#### Generating functions and counting functions

Generating functions are denoted by a mathbf symbol.

$\mathrm{PF}(n,m)$ | for $0\leqslant m\leqslant n$ number of parking functions,
---|---
| i.e. of Cayley trees of size $n$ and $m$ cars so that all cars park
$\mathrm{PF}_{\mathrm{root}}(n,m)$ | for $0\leqslant m\leqslant n$ number of parking functions with empty root,
| i.e. of Cayley trees of size $n$ and $m$ cars so that all cars park and the
root stays void
$\mathrm{FP}(n,n+p)$ | for $n,p\geqslant 0$ number of fully parked trees with flux $p$
| i.e. of Cayley tree of size $n$ and $n+p$ cars so that exactly $p$ cars do
not park
$\mathrm{SP}(n,n+p)$ | for $n,p\geqslant 0$ number of strongly parked trees with flux $p$
| i.e. of Cayley trees of size $n$ and $n+p$ cars so that exactly $p$ cars do
not park
| and so that all edges have positive flux
$\mathbf{T}(x)$ | Exponential GF $\sum_{n\geqslant 1}\frac{x^{n}n^{n-2}}{n!}$ for unrooted Cayley trees
$\mathbf{N}(x)$ | Exponential GF $\sum_{n\geqslant 1}\frac{\mathrm{PF}(n,n-1)}{n!(n-1)!}x^{n}$ for nearly parked trees
| which is equal to $\frac{1}{2}\mathbf{T}(2x)$ by Proposition 9
$\mathbf{F}(x)$ | Exponential GF $\sum_{n\geqslant 1}\frac{\mathrm{FP}(n,n)}{(n!)^{2}}x^{n}$ for fully parked trees
$\mathbf{S}(x)$ | Exponential GF $\sum_{n\geqslant 1}\frac{\mathrm{SP}(n,n)}{(n!)^{2}}x^{n}$ for strongly parked trees
| which is equal to $1-\ln(2)-\sqrt{1-4x}+\ln\left(1+\sqrt{1-4x}\right)$ by
Proposition 10
$\mathbf{S}(x,y)$ | Exponential GF $\sum_{n\geqslant 1}\frac{\mathrm{SP}(n,n+p)}{n!(n+p)!}x^{n}y^{p}$ for strongly parked trees with flux

#### Notation for Section 6

$\mathbb{C}\mathrm{adlag}(I,\mathrm{Pol})$ | Space of càdlàg function from an interval $I\subset\mathbb{R}$ to some Polish space $\mathrm{Pol}$
---|---
$(\mathscr{M}(\lambda):\lambda\in\mathbb{R})$ | (standard) multiplicative coalescent
$\mathcal{E}=\ell^{1}_{\downarrow}\times\ell^{2}_{\downarrow},\quad\mathrm{d}_{\mathcal{E}}$ | state space of the frozen multiplicative coalescent and its metric
$\mathcal{E}_{0}=\ell^{\infty}_{\downarrow,0}\times\ell^{\infty}_{\downarrow,0},\quad\mathrm{d}_{\mathrm{sup}}$ | proxy state space of pair of decreasing sequences tending to $0$ and its metric
$(\mathscr{F}\mathscr{M}(\lambda):\lambda\in\mathbb{R})$ | frozen multiplicative coalescent
$[\mathscr{F}\mathscr{M}(\lambda)]_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}},\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$ |  $\ell^{1}$\- part of $\mathscr{F}\mathscr{M}(\lambda)$ and its total mass
$\mathscr{X}(\lambda)=\|\mathscr{F}\mathscr{M}(\lambda)\|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}\mathllap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ}}$ | shorthand notation for the total mass of the frozen particles
$\mathrm{O}_{n}(\lambda_{0})$ | number of vertices of $\mathrm{G}_{n}(0)$ whose components
| carry surplus appeared before time $\lambda_{0}$
$\mathrm{F}_{n}^{[\lambda_{0}]},\mathrm{G}_{n}^{[\lambda_{0}]}$ | frozen (resp standard Erdős–Rényi process) process started from time $\lambda_{0}$
| by removing the components with surplus at time $\lambda_{0}$
$\mathrm{F}_{n}^{[\lambda_{0},\delta]},\mathrm{G}_{n}^{[\lambda_{0},\delta]}$ | same process as above restricted to the specks
| (i.e. components of $[\mathrm{G}_{n}(\lambda_{0})]_{\mathrm{tree}})$ of size
at least $\delta n^{2/3}$
$\mathbb{F}_{n},\mathbb{F}_{n}^{[\lambda_{0}]},\mathbb{F}_{n}^{[\lambda_{0},\delta]}$ | Sequences of renormalized sizes of components (frozen followed by standard)
$\eta$-skeleton | graph spanned by the specks of $\mathrm{G}_{n}(0)$ of mass at least $\eta$
| or belonging to a cycle of $\mathrm{G}_{n}^{[\lambda_{0}]}(0)$
$\gamma=\gamma_{n}(\lambda_{0},\eta)$ | minimal weight of a speck of the $\eta$-skeleton

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