# Stable quadrangulations and stable spheres

Eleanor Archer111Modal’X, UMR CNRS 9023, UPL, Univ. Paris-Nanterre, F92000
Nanterre, France.
aaa<EMAIL_ADDRESS><EMAIL_ADDRESS>Ariane
Carrance222CMAP, UMR CNRS 7641, École polytechnique, F9112 Palaiseau, France.
<EMAIL_ADDRESS>and Laurent Ménard11footnotemark: 1

(August 28, 2024)

###### Abstract

We consider scaling limits of random quadrangulations obtained by applying the
Cori–Vauquelin–Schaeffer bijection to Bienaymé–Galton–Watson trees with
stably-decaying offspring tails with an exponent $\alpha\in(1,2)$. We show
that these quadrangulations admit subsequential scaling limits which all have
Hausdorff dimension $\frac{2\alpha}{\alpha-1}$ almost surely. We conjecture
that the limits are unique and spherical, and we introduce a candidate for the
limit that we call the $\alpha$-stable sphere. In addition, we conduct a
detailed study of volume fluctuations around typical points in the limiting
maps, and show that the fluctuations share similar characteristics with those
of stable trees.

(a)

(b)

Figure 1: A stable tree with index $\alpha=1.2$ and 12699 edges and its
associated quadrangulation.

(c)

(d)

Figure 2: A stable tree with index $\alpha=1.6$ and 22185 edges and its
associated quadrangulation.

###### Contents

  1. 1 Introduction
  2. 2 Background on stable trees and snakes
    1. 2.1 Stable trees
    2. 2.2 Stable snake with Brownian spatial displacements
  3. 3 Trees, maps and scaling limits
    1. 3.1 The pointed Gromov-Hausdorff-Prokhorov topology
    2. 3.2 Discrete trees
    3. 3.3 Random planar maps
    4. 3.4 The Cori–Vauquelin–Schaeffer bijection
    5. 3.5 Subsequential scaling limits of random quadrangulations
  4. 4 Large volumes
    1. 4.1 Upper tail bounds for large volumes
    2. 4.2 Lower tail bounds for large volumes
  5. 5 Small volumes
    1. 5.1 Upper tail bounds for small volumes
    2. 5.2 Lower tail bounds for small volumes
  6. 6 Hausdorff dimension
    1. 6.1 Hausdorff dimension of $(M,D)$
    2. 6.2 Hausdorff dimension of $\left(\mathbf{m}_{\alpha},D_{\alpha}^{*}\right)$
  7. Misc

##  1 Introduction

The purpose of this paper is to introduce a stable model of random planar
quadrangulations and to start to investigate their scaling limits. The
starting point is the celebrated convergence of uniform planar
quadrangulations to the Brownian sphere [LG13, Mie13], which utilises the
well-known Cori–Vauquelin–Schaeffer (CVS) bijection between labelled trees and
planar quadrangulations. In particular, a quadrangulation can be represented
as the image of a labelled tree under the CVS bijection. Each edge of the tree
becomes a face of the quadrangulation and the labels determine distances in
the latter object. In this paper, we consider sequences of random
quadrangulations obtained by applying the CVS bijection to labelled
Bienaymé–Galton–Watson trees for which the offspring distribution has stably-
decaying tails.

The Brownian sphere was first constructed in [MM06] as a candidate for the
scaling limit of uniform quadrangulations with $n$ faces, along with various
associated convergence results. This followed on from the work of Chassaing
and Schaeffer who already showed that the rescaled radius of a uniform
quadrangulation converges to the width of the support of the Brownian snake
[CS04]. The breakthrough in terms of its metric structure came in the landmark
works of Le Gall [LG07, LG10, LG13] and Miermont [Mie13] who showed that the
Brownian sphere is the unique scaling limit, in the Gromov-Hausdorff sense, of
uniform $2p$-angulations, and triangulations, with $n$ faces. Moreover, it is
almost surely homeomorphic to the sphere [LG07, LGP08, Mie08], and its
Hausdorff dimension is equal to 4 almost surely [LG07]. Since these pioneering
results, the universality class of random models of planar maps whose scaling
limit is the Brownian sphere has been extended to general maps [BJM14],
bipartite maps [Abr16], simple triangulations and quadrangulations [ABA17],
Boltzmann maps and maps with prescribed degree sequences in the Brownian
regime [Mar22], all odd $p$-angulations [ABA21], bicolored triangulations
[Car21], cubic planar graphs [Stu22, AFL23] and block-weighted
quadrangulations [FS23].

The crucial ingredient in the proofs of both Le Gall and Miermont were
bijections between labelled trees and random planar maps obtained earlier by
Cori-Vauqielin [CV81], Schaeffer [Sch98] and generalised by Bouttier, Di
Francesco and Guitter [BDFG04]. In particular, a uniform quadrangulation with
$n$ faces can be sampled by taking a uniform rooted random plane tree with $n$
edges, sampling a so-called label function that evolves as a simple random
walk along each branch of the tree, and applying the CVS bijection to this
rooted labelled tree. It should therefore be of no surprise that the Brownian
sphere can be constructed from scaling limits of such labelled trees. The
scaling limit of such a label process is known as the Brownian snake, which is
informally Brownian motion indexed by the Brownian continuum random tree (CRT)
[LG99]. The Brownian sphere can be constructed as a quotient space of the CRT,
and the precise form of the quotient is determined by the associated Brownian
snake. Moreover, the CRT plays the role of the cut-locus for the Brownian
sphere.

The fact that the label process converges to the Brownian snake is therefore
completely key to the scaling limit results of Le Gall and Miermont. Thanks to
recent results of Marzouk [Mar20], we also now understand how to take scaling
limits of labelled trees in a much more general setting. In particular, he
considers labelled critical Bienaymé–Galton–Watson trees with offspring
distribution in the domain of attraction of an $\alpha$-stable law for some
$\alpha\in(1,2)$. The metric space scaling limits of such trees (without
labels) are well-known to be the stable trees introduced by Le Gall and Le Jan
[LGLJ98] and developed by Duquesne and Le Gall [DLG02] – see Figure 2 for
simulations. Like the CRT, these are continuous random trees but unlike the
CRT they have the property that they contain branch points of infinite degree,
known as hubs. In [Mar20] Marzouk shows that these discrete trees can be
additionally endowed with a label process that converges to “Brownian motion
indexed by the stable tree”. This will be referred to as the stable snake
throughout this article.

This therefore begs the question of whether it is possible to emulate the
construction of the Brownian sphere using a stable snake and to obtain
convergence of the associated discrete model of random quadrangulations to
this limiting object. The present paper is a natural step towards that aim. We
consider convergence in the pointed Gromov-Hausdorff-Prokhorov topology (this
is a straightforward extension of the Gromov-Hausdorff topology in which we
also consider convergence of measures and roots). Just as in the Brownian
case, the convergence of snakes of Marzouk ensures that we have tightness in
this topology. The space of rooted compact finite metric measure spaces is
Polish [ALW16], hence this implies that these maps have subsequential limits,
but we do not a priori know that limits are unique. However we conjecture
that, similarly to the Brownian case, the limits are indeed unique and we
moreover have a candidate for the limit, which is constructed from the stable
snake analogously to the construction of the Brownian sphere from the Brownian
snake (we plan to address this in future papers). We call this limit the
stable sphere. In the present paper, we show that any subsequential limit of
the discrete quadrangulations has Hausdorff dimension equal to
$\frac{2\alpha}{\alpha-1}$ almost surely, and obtain precise results on the
corresponding volume fluctuations.

Let us now briefly describe our model. For $\alpha\in(1,2)$, we write
$\mathcal{T}_{\alpha}$ for the stable tree with index $\alpha$ as introduced
by Le Gall and Le Jan [LGLJ98] and developed by Duquesne and Le Gall [DLG02];
see Section 2 for details. The case $\alpha=2$ corresponds to the CRT,
introduced by Aldous [Ald91, Ald91a, Ald93]. We will denote by $\rho$ the root
of $\mathcal{T}_{\alpha}$ and by $d_{\alpha}$ the distance on the tree
$\mathcal{T}_{\alpha}$. It will be useful to have in mind that
$\mathcal{T}_{\alpha}$ has both a genealogical and a lexicographical order,
which is inherited from the left to right ordering of its branches in the
plane.

The stable snake of Duquesne and Le Gall [DLG02] allows us to assign Brownian
labels to the vertices of $\mathcal{T}_{\alpha}$ as follows : given
$\mathcal{T}_{\alpha}$, we consider a centred Gaussian process
$(Z_{v})_{v\in\mathcal{T}_{\alpha}}$ such that $Z_{\rho}=0$ and such that
$\mathrm{Var}(Z_{v}-Z_{v^{\prime}})=d_{\alpha}(v,v^{\prime})$ for every
$v,v^{\prime}\in\mathcal{T}_{\alpha}$. We will review this process in detail
in Section 2.2.1. Similarly to the Brownian sphere, we can define a pseudo-
distance $D^{*}_{\alpha}$ on $\mathcal{T}_{\alpha}$ from these labels. First,
we define the following mapping

$\displaystyle
D_{\alpha}^{\circ}\,:\,\mathcal{T}_{\alpha}\times\mathcal{T}_{\alpha}$
$\displaystyle\to\mathbb{R}_{+}$ $\displaystyle(v,v^{\prime})$
$\displaystyle\mapsto
D_{\alpha}^{\circ}(v,v^{\prime})=Z_{v}+Z_{v^{\prime}}-2\inf_{u\in
I_{v,v^{\prime}}}Z_{u},$ (1)

where $I_{v,v^{\prime}}$ denotes the lexicographical interval between $v$ and
$v^{\prime}$ that minimises $D_{\alpha}^{\circ}(v,v^{\prime})$. Indeed, for
any pair of vertices $v,v^{\prime}\in\mathcal{T}_{\alpha}$, there are always
two lexicographical intervals between $v$ and $v^{\prime}$ corresponding to
the two possible ways to go from $v$ to $v^{\prime}$ around the tree.

The function $D_{\alpha}^{\circ}$ does not satisfy the triangle inequality,
which leads us to introduce

$D_{\alpha}^{*}(v,v^{\prime}):=\inf\left\\{\sum_{i=1}^{k}D_{\alpha}^{\circ}(v_{i-1},v_{i})\,\Big{\lvert}\,k\geq
1,\,v_{i}\in\mathcal{T}_{\alpha}\,\forall
i\in\\{0,\dots,k\\},v_{0}=v,v_{k}=v^{\prime}\right\\},$ (2)

which is by construction the largest pseudo-distance that satisfies
$D_{\alpha}^{*}\leq D_{\alpha}^{\circ}$. We also let $\approx$ denote the
equivalence relation on $\mathcal{T}_{\alpha}$ defined by $v\approx
v^{\prime}$ if and only if $D_{\alpha}^{*}(v,v^{\prime})=0$ and denote by
$\mathbf{m}_{\alpha}$ the quotient space $\mathcal{T}_{\alpha}/\approx$. We
can also endow it with a measure $\nu_{\alpha}$ and a root $\rho_{\alpha}$,
defined as the pushforwards of the canonical measure and root of
$\mathcal{T}_{\alpha}$ under the quotient operation. The stable sphere is the
random compact rooted metric measure space
$\left(\mathbf{m}_{\alpha},D_{\alpha}^{*},\nu_{\alpha},\rho_{\alpha}\right)$.
Note that the special case $\alpha=2$ is the Brownian sphere, and that we will
focus on $\alpha<2$. Like in the Brownian setting, we expect the stable tree
$\mathcal{T}_{\alpha}$ to play the role of the cut-locus of the stable sphere.

Our discrete model of random quadrangulations can be defined as follows. Fix
$\alpha\in(1,2)$ and let $T_{n}^{(\alpha)}$ be a critical Bienaymé-Galton-
Watson tree with offspring distribution $\mu_{\alpha}$ in the domain of
attraction of an $\alpha$-stable law conditioned to have $n$ edges. For ease
of notation we will in fact assume that there exists a constant
$c\in(0,\infty)$ such that $\mu_{\alpha}([x,\infty))\sim cx^{-\alpha}$ as
$x\to\infty$ (we anticipate however that all of our results can be adapted to
incorporate slowly-varying corrections). We add labels to the vertices of
$T_{n}^{(\alpha)}$ by labelling the root $0$ and choosing i.i.d. centred
increments in $\\{-1,0,+1\\}$ along edges. Denote by $M_{n}^{(\alpha)}$ the
random rooted quadrangulation with $n$ faces associated to $T_{n}^{(\alpha)}$
by the Cori–Vauquelin–Schaeffer correspondence, $d_{n}$ the graph distance on
$M_{n}^{(\alpha)}$, $\nu_{n}$ the uniform probability measure on its vertices,
and $\rho_{n}$ its root vertex. We conjecture the following convergence:

######  Conjecture 1.1.

For $\alpha\in(1,2)$, there exists a constant $\kappa>0$ that depends on
$\mu_{\alpha}$, such that the following convergence in law holds in the
pointed Gromov-Hausdorff-Prokhorov topology:

$\left(M_{n}^{(\alpha)},n^{-\frac{\alpha-1}{2\alpha}}\,d_{n},\nu_{n},\rho_{n}\right)\underset{n\to\infty}{\rightarrow}\,\left(\mathbf{m}_{\alpha},\kappa
D_{\alpha}^{*},\nu_{\alpha},\rho_{\alpha}\right).$

The starting point of this paper is the following result.

######  Theorem 1.2.

The sequence
$\left(M_{n}^{(\alpha)},n^{-\frac{\alpha-1}{2\alpha}}\,d_{n},\nu_{n},\rho_{n}\right)_{n\geq
0}$ is tight in the pointed Gromov-Hausdorff-Prokohorov topology. Moreover,
any subsequential limit $(M,D,\nu,\rho)$ can be defined as a quotient space of
$\left(\mathbf{m}_{\alpha},D^{*}_{\alpha},\nu_{\alpha},\rho_{\alpha}\right)$.

The main contribution of the paper is in fact a study of volumes of balls at
typical points in a limit $(M,D,\nu,\rho)$. We start with the Hausdorff
dimension.

######  Theorem 1.3.

Any subsequential limit $(M,D,\nu,\rho)$ has Hausdorff dimension
$\frac{2\alpha}{\alpha-1}$ almost surely, as does
$\left(\mathbf{m}_{\alpha},D^{*}_{\alpha},\nu_{\alpha},\rho_{\alpha}\right)$.

Our volume results are actually a lot more precise than this. As is the case
for stable trees (and for similar reasons), the volumes of balls at typical
points experience logarithmic fluctuations above their typical value.

######  Theorem 1.4.

Take any $\varepsilon>0$. Let $(M,D,\nu,\rho)$ be a subsequential limit
appearing in Theorem 1.2. Then, almost surely, it holds for $\nu$-almost every
$x$ in $M$ that

$\limsup_{r\downarrow 0}\frac{\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log
r^{-1})^{\frac{2\alpha+\varepsilon}{\alpha-1}}}=0,\qquad\limsup_{r\downarrow
0}\frac{\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log
r^{-1})^{\frac{1-\varepsilon}{\alpha-1}}}=\infty.$

One can ask whether there is a precise regularly varying gauge function $g(r)$
such that the limit supremum appearing in Theorem 1.4 is equal to a finite
positive constant almost surely. In the case of stable trees, this question
has been answered negatively by Duquesne [Duq10, Proposition 1.9] (see also
[Duq10, Theorem 1.10] for a similar statement for the exact Hausdorff
measure). We have no reason to expect otherwise for $(M,D,\nu,\rho)$.

Since $D\leq D_{\alpha}^{*}$, the first result of Theorem 1.4 clearly also
holds for
$\left(\mathbf{m}_{\alpha},D^{*}_{\alpha},\nu_{\alpha},\rho_{\alpha}\right)$,
and in fact an inspection of the proof shows that the same is true for the
second result.

We now turn to a discussion of fluctuations below typical values. The
fluctuations are log-logarithmic, as is again the case for stable trees.

######  Theorem 1.5.

Let $(M,D,\nu,\rho)$ be a subsequential limit appearing in Theorem 1.2. Then,
almost surely, for $\nu$-almost every $x$ in $M$

$\liminf_{r\downarrow
0}\frac{\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log\log
r^{-1})^{-\frac{1}{\alpha-1}}}<\infty,\qquad\liminf_{r\downarrow
0}\frac{\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log\log
r^{-1})^{-\frac{32\alpha}{\alpha-1}}}=\infty.$

Moreover, almost surely,

$\liminf_{r\downarrow 0}\frac{\inf_{x\in
M}\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log
r^{-1})^{-\frac{32\alpha}{\alpha-1}}}>0.$

It was similarly shown in [Duq12] that stable trees admit
$r^{\frac{\alpha}{\alpha-1}}(\log\log r)^{-\frac{1}{\alpha-1}}$ as a regular
packing gauge function, meaning that the limit infimum when renormalising
volumes by this function is almost surely a finite, non-zero constant. In
addition, Duquesne [Duq12, Theorem 1.1] showed that the packing measure
associated with the gauge function $r^{\frac{\alpha}{\alpha-1}}(\log\log
r^{-1})^{-\frac{1}{\alpha-1}}$ coincides with the natural volume measure on
stable trees; we leave it open whether a similar result is true for
$(M,D,\nu,\rho)$. The exponent of $\frac{32\alpha}{\alpha-1}$ appearing in the
second and third results is certainly not optimal.

We remark that do not expect global supremal volumes to be as tightly
concentrated as global infima; in fact it is fairly easy to show that there
will exist balls $B(x,r)$ in $M$ of volume at least $r^{2}$. This is due to
the fact that there are also polynomial volume fluctuations at hubs in the
underlying stable tree (see Remark 4.14).

The main tool to obtain lower bounds on volumes is the fact that $D\leq
D_{\alpha}^{*}$ and hence it is sufficient to lower bound the volumes of balls
measured with respect to the metric $D_{\alpha}^{*}$. This can be achieved by
controlling the fluctuations of the snake in various intervals. In order to
obtain upper bounds on volumes of balls, we will instead need a lower bound on
$D$. This is done by relating $D$ to the metric on the associated cactus; see
Lemma 3.14 for the precise result.

The heavier upper tails on the volumes allows us to make an important
distinction between $(M,D)$ and the Brownian sphere. Let $v^{*}$ denote the
point in $\mathcal{T}_{\alpha}$ where the minimum of $Z$ is attained (like in
the Brownian case, it is unique almost surely, see Proposition 2.2). By
comparing the bounds in Theorem 1.4 with bounds on $\nu(B(v^{*},r))$ obtained
in [ACM24], we deduce that $v^{*}$ is not a typical point of $(M,D,\nu,\rho)$.

######  Corollary 1.6.

The vertex $v^{*}$ is not a typical point in $(M,D)$, meaning that $v^{*}$
cannot be coupled with an independent point chosen according to $\nu$.

We end the introduction with some comments on related models. At first sight
one might guess that the subsequential scaling limits of
$\left(M_{n}^{(\alpha)},n^{-\frac{\alpha-1}{2\alpha}}\,d_{n},\nu_{n},\rho_{n}\right)_{n\geq
0}$ may be related to the dual maps, or the images by Tutte’s bijection, of
the stable maps of [LGM10]. Indeed, the latter is a random map model in which
the size of typical _faces_ have stably-decaying tails, while the typical
_vertex_ degrees are heavy-tailed in our model (see Remark 3.8 for a more
detailed discussion). However, our result on the Hausdorff dimension clearly
rules this out; while our limits have Hausdorff dimension
$\frac{2\alpha}{\alpha-1}$, the stable duals have dimension
$\frac{\alpha}{\alpha-3/2}$ when $\frac{3}{2}<\alpha<2$, and in fact have
exponential volume growth when $\alpha<\frac{3}{2}$ [BC17] (in the critical
case of $\alpha=\frac{3}{2}$ the volume growth is stretched exponential
[Kam23]). Nonetheless, like the limits of Theorem 1.2, scaling limits of these
dual graphs are believed to be spherical when $\frac{3}{2}<\alpha<2$. The
stable maps themselves, on the other hand, are not believed to be spherical,
and instead have more complicated fractal-like topology.

The tightness of Theorem 1.2 is very straightforward to establish; the main
contributions of the paper are the volume results. Throughout most of the
paper we will assume implicitly that we have already restricted to an
appropriate subsequence where the maps converge, without explicitly stating it
every time.

Organisation. The paper is organised as follows. In Section 2 we give some
background on stable trees and stable snakes, and in Section 3 we give the
definitions of all of the map models. In Section 4 and 5 we respectively
consider the upper and lower fluctuations of volumes around uniform points.
Finally we conclude in Section 6 with a quick deduction of the Hausdorff
dimension.

Acknowledgements. We would like to thank Marie Albenque, Meltem Ünel, Armand
Riera and Alejandro Rosales-Ortiz for helpful conversations, and Nicolas
Curien for a comment which led to Corollary 1.6. EA and LM were supported by
the ANR grant ProGraM (ANR-19-CE40-0025).

##  2 Background on stable trees and snakes

###  2.1 Stable trees

####  2.1.1 Stable Lévy processes and the Itô excursion measure

We consider a spectrally positive stable process Lévy $X$ with index
$\alpha\in(1,2)$. Its law started at $0$ is denoted by $P$. This process is
characterised by its Laplace transform and the fact that it has no negative
jumps. For $\lambda>0$ and $t\geq 0$ we will assume that $X$ is normalised so
that

$\mathbb{E}\\!\left[\exp\\{-\lambda
X_{t}\\}\right]=\exp\\{-t\lambda^{\alpha}\\}.$ (3)

(See [Ber96, Sections VII and VIII] for more background.)

We will write $\psi(\lambda)=\lambda^{\alpha}$ to denote this Laplace
exponent. The function $\psi$ is known as the branching mechanism. By the
Lévy–Khinchin formula, $\psi$ can equivalently be written in the form

$\psi(\lambda)=\int_{0}^{\infty}(e^{-\lambda r}-1+\lambda
r)\pi(dr)=C_{\alpha}\lambda+\int_{0}^{\infty}(e^{-\lambda r}-1+\lambda
r\mathbbm{1}\\{y\leq 1\\})\pi(dr),$ (4)

where $\pi$ is the jump measure of $X$ and $-C_{\alpha}$ is the drift
coefficient. The fact that $\psi(\lambda)=\lambda^{\alpha}$ entails that

$C_{\alpha}=\frac{\alpha-1}{\Gamma(2-\alpha)}\quad\text{ and
}\quad\pi(dr)=\alpha C_{\alpha}r^{-\alpha-1}dr.$ (5)

In standard terminology, this means that the process $X$ corresponds to the
Lévy triple $(0,-C_{\alpha},\pi)$.

To define the stable tree, we will in fact use a normalised Lévy excursion
rather than a Lévy process. Let $X$ be a spectrally positive $\alpha$-stable
Lévy process, satisfying (3), and let $I_{t}=\inf_{s\in[0,t]}X_{s}$ denote its
running infimum process. Define $g_{1}$ and $d_{1}$ by

$\displaystyle g_{1}$ $\displaystyle=\sup\\{s\leq 1:X_{s}=I_{s}\\}$
$\displaystyle d_{1}$ $\displaystyle=\inf\\{s>1:X_{s}=I_{s}\\}.$

Note that $X_{g_{1}}=X_{d_{1}}$ almost surely since $X$ almost surely has no
jump at time $g_{1}$ and no negative jumps. We define the normalised excursion
$X^{\text{exc}}$ of $X$ above its infimum at time $1$ by

$X_{s}^{\text{exc}}=(d_{1}-g_{1})^{\frac{-1}{\alpha}}(X_{g_{1}+s(d_{1}-g_{1})}-X_{g_{1}})$

for every $s\in[0,1]$. Then $X^{\text{exc}}$ is almost surely an
$\alpha$-stable càdlàg function on $[0,1]$ with $X^{\text{exc}}(s)>0$ for all
$s\in(0,1)$, and $X_{0}^{\text{exc}}=X_{1}^{\text{exc}}=0$.

We also take this opportunity to introduce the Itô excursion measure. For full
details, see [Ber96, Chapter IV], but the measure is defined by applying
excursion theory to the process $X-I$, which is strongly Markov and for which
the point $0$ is regular for itself. We normalise local time so that $-I$
denotes the local time of $X-I$ at its infimum, and let
$(g_{j},d_{j})_{j\in\mathcal{I}}$ denote the excursion intervals of $X-I$ away
from zero. For each $i\in\mathcal{I}$, the process $(e^{i})_{0\leq s\leq
d_{i}-g_{i}}$ defined by $e^{i}(s)=X_{g_{i}+s}-X_{g_{i}}$ is an element of the
excursion space

$E=\bigcup_{l>0}D^{\text{exc}}([0,l],\mathbb{R}^{\geq 0}),$

where for every $l>0$, $D^{\text{exc}}([0,l],\mathbb{R}^{\geq 0})$ is the
space of all _càdlàg_ nonnegative functions $f$ such that $f(0)=f(l)=0$. We
let $\sigma(e)=\sup\\{s>0:e(s)>0\\}$ denote the lifetime (duration) of the
excursion $e$. It was shown in [Itô72] that the measure on
$\mathbb{R}_{+}\times E$

$\sum_{i\in\mathcal{I}}\delta(-I_{g_{i}},e^{i})$

is a Poisson point measure of intensity $dtN(de)$, where $N$ is a
$\sigma$-finite measure on the set $E$ known as the Itô excursion measure.

The measure $N(\cdot)$ inherits a scaling property from that of $X$. For any
$\lambda>0$ we define a mapping $\Phi_{\lambda}:E\rightarrow E$ by
$\Phi_{\lambda}(e)(t)=\lambda^{\frac{1}{\alpha}}e(\frac{t}{\lambda})$, so that
$N\circ\Phi_{\lambda}^{-1}=\lambda^{\frac{1}{\alpha}}N$ (e.g. see [Wat10]). It
follows from the results of [Ber96, Section IV.4] that we can uniquely define
a set of measures $(N^{(s)},s>0)$ on $E$ such that:

  1. (i)

For every $s>0$, $N^{(s)}(\sigma=s)=1$.

  2. (ii)

For every $\lambda>0$ and every $s>0$, $\Phi_{\lambda}(N^{(s)})=N^{(\lambda
s)}$.

  3. (iii)

For every measurable $A\subset E$,

$N(A)=\int_{0}^{\infty}\frac{N^{(s)}(A)}{\alpha\Gamma(1-\frac{1}{\alpha})s^{\frac{1}{\alpha}+1}}ds.$
(6)

$N^{(s)}$ is thus used to denote the law $N(\cdot|\sigma=s)$. The probability
distribution $N^{(1)}$ coincides with the law of $X^{\text{exc}}$ constructed
above. Note that (see [GH10, Proposition 5.6]), for all $s\geq 0$,

$N(\sigma>s)=cs^{-\frac{1}{\alpha}}.$ (7)

####  2.1.2 Stable height processes and exploration processes

We will see later in Section 3 that discrete plane trees can be coded by
different functions. The same is true in the continuum; in the case of
$\alpha$-stable trees, we first code using a spectrally positive
$\alpha$-stable Lévy process or excursion $X$ (this plays the same role as the
Łukasiewicz path in the discrete setting - we will define this in Section
3.2). The height function $H$ is then defined by setting, for $t\geq 0$,

$H_{t}=\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\int_{0}^{t}\mathbbm{1}\\{X_{s}<I_{s}^{t}+\varepsilon\\}ds,$
(8)

where $I_{s,t}=\inf_{s\leq r\leq t}X_{r}$. For each $t\geq 0$, this limit
exists in probability by [DLG02, Lemma 1.2.1]. Moreover, it follows from
[DLG02, Theorem 1.4.3] that $H$ almost surely has a continuous modification,
and we will assume henceforth that $H$ is indeed continuous.

The height function is not in general Markovian; to overcome this difficulty
we introduce the exploration process, a Markovian random measure that encodes
the height process. Let $M_{f}(\mathbb{R}_{+})$ denote the set of finite
measures on $\mathbb{R}_{+}$. The exploration process $(\rho_{t})_{t\geq 0}$
is a process such that for each $t\geq 0$, $\rho_{t}$ is the measure in
$M_{f}(\mathbb{R}_{+})$ satisfying

$\displaystyle\langle\rho_{t},f\rangle=\int_{0}^{t}f(H_{s})d_{s}(I_{s,t})$ (9)

for all bounded measurable functions $f:\mathbb{R}_{+}\to\mathbb{R}$. This
implies that

$\displaystyle\langle\rho_{t},1\rangle=I_{t,t}-I_{0,t}=X_{t}-I_{t}.$ (10)

It follows from the definition that for each $t\geq 0$, $\text{supp
}\rho_{t}=[0,H_{t}]$. In particular this means that $H$ can be defined as a
functional of $\rho$; for a given realisation $\rho^{\prime}$ we denote the
associated height process by $H(\rho^{\prime})$.

We also introduce the dual exploration process. Note that it follows from (9)
that $\rho_{t}$ can also be defined by

$\displaystyle\rho_{t}(dr)=\sum_{0<s\leq t:X_{s-}\leq
I_{s,t}}(I_{s,t}-X_{s-})\delta_{H_{s}}(dr)$

for all $t>0,r\in[0,H_{t}]$. The dual exploration process is similarly defined
by

$\displaystyle{\hat{\rho}}_{t}(dr)=\sum_{0<s\leq t:X_{s-}\leq
I_{s,t}}(X_{s}-I_{s,t})\delta_{H_{s}}(dr).$ (11)

The height function appearing in (8) in fact codes a whole forest of trees. We
will sometimes instead code a single tree by replacing $X$ with
$X^{\text{exc}}$ in (8). This corresponds to the law of $H$ under the measure
$N^{(1)}$. By scaling, this means that we can also make sense of it under
$N^{(s)}$ for any $s>0$, and therefore under $N$ using (6). The pair
$(\rho_{t},{\hat{\rho}}_{t})_{t\geq 0}$ can be similarly defined under
$N^{(1)}$ or $N$. Under $N$, it holds that (recall that $\sigma$ denotes the
lifetime of the excursion)

$(\rho_{t},{\hat{\rho}}_{t})_{t\geq
0}\overset{(d)}{=}({\hat{\rho}}_{(\sigma-t)-},\rho_{(\sigma-t)-})_{t\geq 0}.$
(12)

####  2.1.3 Stable trees

To define the stable tree under $N^{(1)}$, we first define $H$ using (8) but
with $X$ replaced by $X^{\text{exc}}$, then for $s<t$ define

$m_{s,t}=\inf_{s\wedge t\leq r\leq s\vee t}H_{r}.$ (13)

We then define a pseudodistance on $[0,1]$ by

$d_{\alpha}(s,t)=H_{s}+H_{t}-2m_{s,t}$ (14)

whenever $s\leq t$. We then define an equivalence relation on $[0,1]$ by
saying $s\sim t$ if and only if $d_{\alpha}(s,t)=0$, and set
$\mathcal{T}_{\alpha}$ to be the quotient space $([0,1]/\sim,d_{\alpha})$. We
also define a measure $\lambda$ on $\mathcal{T}_{\alpha}$ as the image of
Lebesgue measure on $[0,1]$ under the quotient operation.

It follows from the construction that $\mathcal{T}_{\alpha}$ is an
$\mathbb{R}$-tree in the sense of [DMT96], essentially meaning that it is a
continuum object sharing standard properties of discrete trees (see [DLG05,
Definition 2.1] for the precise definition).

Properties of $\mathcal{T}_{\alpha}$ are determined by the process
$X^{\text{exc}}$. Letting $p:[0,1]\rightarrow\mathcal{T}_{\alpha}$ be the
canonical projection, there is a distinguished vertex $\rho=p(0)$ which we
call the root of $\mathcal{T}_{\alpha}$. Also, for
$u,v\in\mathcal{T}_{\alpha}$, we denote by $[[u,v]]$ the unique geodesic
between $u$ and $v$, and we say $u\preceq v$ if and only if $u\in[[\rho,v]]$.
Given $u,v\in\mathcal{T}_{\alpha}$, we say $z$ is the most recent common
ancestor of $u$ and $v$, written $z=u\wedge v$, if $z$ is the unique element
in $\mathcal{T}_{\alpha}$ with $[[\rho,u]]\cap[[\rho,v]]=[[\rho,z]]$. We say
that a subtree is grafted to the path $[[u,v]]$ if it is coded by an excursion
interval of $X^{\text{exc}}$ away from its infimum, and its root (i.e. the
projection of the start point of this excursion) is in
$[[u,v]]\setminus\\{u,v\\}$.

The relation $\preceq$ on $\mathcal{T}_{\alpha}$ can be recovered from
$X^{\text{exc}}$ by defining

$s\preceq t\text{ if and only if }s\leq t\text{ and
}X^{\text{exc}}_{s^{-}}\leq I_{s}^{t}.$

It is possible to show that the two definitions of $\preceq$ given above
define partial orders on $\mathcal{T}_{\alpha}$ and $[0,1]$ respectively, and
are compatible with $p$ in the sense that, if $u,v\in\mathcal{T}_{\alpha}$,
then $u\preceq v$ if and only if there exist $s,t\in[0,1]$ with $p(s)=u$,
$p(t)=v$ and such that $s\preceq t$. Additionally, if $s\wedge t$ is the most
recent common ancestor of $s$ and $t$ (with respect to $\preceq$), then it can
be verified that $p(s\wedge t)=p(s)\wedge p(t)$.

The multiplicity of a vertex $u\in\mathcal{T}_{\alpha}$ is defined as the
number of connected components of $\mathcal{T}_{\alpha}\setminus\\{u\\}$. A
vertex $u$ is called a leaf if it has multiplicity one, and a branch point if
it has multiplicity at least three. It can be shown that
$\mathcal{T}_{\alpha}$ only has branch points with infinite multiplicity,
called hubs, and that $u\in\mathcal{T}_{\alpha}$ is a hub if and only if there
exists $s\in[0,1]$ such that $p(s)=u$ and
$\Delta_{s}:=X^{\text{exc}}_{s}-X^{\text{exc}}_{s^{-}}>0$. See for instance
Theorem 4.6 and Theorem 4.7 of [DLG05]. The quantity $\Delta_{s}$ gives a
measure of the number of children of $u$.

For any fixed $t\geq 0$, the quantity appearing in (10) can be interpreted as
the sum of the sizes of the hubs appearing on the right hand side of the
branch from the root to $p(t)$. Similarly $\langle{\hat{\rho}}_{t},1\rangle$
can be interpreted as the sum of the sizes of the hubs appearing on the left
hand side of the branch from the root to $p(t)$.

We can similarly define $\mathcal{T}_{\alpha}$ under $N$, rather than
$N^{(1)}$, by replacing $X^{\text{exc}}$ with an excursion defined under $N$
in the definition of $H$, or by using (6). We can similarly define a forest of
trees by retaining $X$ in (8) (each excursion of $X-I$ above its infimum then
codes a single tree). It is also a well-known fact that the genealogy of
$\mathcal{T}_{\alpha}$ encodes a continuous-state branching process with
branching mechanism $\psi$, but we will not use this connection and refer to
[DLG02] for further background.

###  2.2 Stable snake with Brownian spatial displacements

####  2.2.1 Definition

We now give a brief introduction to Lévy snakes in the particular case of
snakes on stable trees with Brownian spatial displacements. We will often
simply refer to this as a “stable snake” in what follows. A full introduction
in the more general setting of Lévy trees is given in [DLG02, Section 4]; we
refer there for further background.

Informally, a stable snake is simply a real-valued stochastic process indexed
by a stable tree. Just as we had to introduce the exploration process as a
Markovian version of the height process, it will be more convenient to keep
track of the history of this stochastic process and we therefore instead
consider a path-valued process. In particular, we let $\mathcal{W}$ denote the
set of continuous real-valued functions defined on a compact interval of the
form $[0,\zeta]$ for some $\zeta>0$. Given $w\in\mathcal{W}$, we denote its
domain by $[0,\zeta_{w}]$ and set $\widehat{w}=w(\zeta_{w})$. We call
$\zeta_{w}$ the lifetime of $w$ and call $\widehat{w}$ the tip of $w$.

We then define a topology on $\mathcal{W}$ using the distance function

$d_{\mathcal{W}}(w,w^{\prime})=|\zeta_{w}-\zeta_{w^{\prime}}|+\sup_{r\geq
0}|w(r\wedge\zeta_{w})-w^{\prime}(r\wedge\zeta_{w^{\prime}})|.$

Before defining the stable snake, we first define a snake with Brownian
spatial displacements driven by a deterministic continuous function
$h:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with initial condition
$w_{0}:[0,h(0)]\to\mathbb{R}_{+}$. This is a random variable taking values in
$\mathcal{W}$, or in other words in the space of path-valued processes (one
should have in mind that $h$ plays the role of a height function of a
deterministic forest, and the snake is obtained by adding random spatial
displacements).

For every $x\in\mathbb{R}$, we denote by $\Pi_{x}$ the distribution of a
standard linear Brownian motion $\xi=(\xi_{t})_{t\geq 0}$ started at $x$. It
is shown in [DLG02, Section 4.1.1] that for any such choice of $(h,w_{0})$
there exists a unique probability measure on $\mathcal{W}$ such that the
following three properties hold.

  1. 1.

$\zeta_{W_{s}}=h(s)$ for all $s\in[0,\sigma]$.

  2. 2.

For all $s,s^{\prime}\geq 0$, we have that $W_{s}(t)=W_{s^{\prime}}(t)$ for
all $t\leq m_{s^{\prime},s}$.

  3. 3.

Conditionally on $W_{s^{\prime}}$, the path $W_{s}:[0,\zeta_{s}]\to\mathbb{R}$
is such that
$(W_{s}(m_{s^{\prime},s}+t)-W_{s}(m_{s^{\prime},s}))_{t\in[\zeta_{s}-m_{s^{\prime},s}]}$
has the same law as the Brownian motion
$(\xi_{t})_{t\in[0,\zeta_{s}-m_{s^{\prime},s}]}$ under
$\Pi_{W_{s^{\prime}}(m_{s^{\prime},s})}$.

In other words, conditionally on $W_{s^{\prime}}$, the path $W_{s}$ satisfies
the second point for $t\leq m_{s,s^{\prime}}$ (which informally corresponds to
the snake along the path from the root to the most recent common ancestor of
$p(s)$ and $p(s^{\prime})$), and for $t>m_{s,s^{\prime}}$ evolves as an
independent linear Brownian motion (which corresponds to the rest of the
branch to $p(s)$). Given $h$ and $w_{0}$ as above we let $Q_{w_{0}}^{h}$
denote the law of this process.

The stable snake with Brownian spatial displacements is the pair $(\rho,W)$
obtained by replacing $h$ with the random height function $H$ defined from a
Lévy process $X$ as in (8).

We will also use the notation

$\widehat{W}_{s}=W_{s}(\zeta_{s})$ (15)

to denote the tip of the snake at time $s$.

We will sometimes refer to this construction as simply the “the snake” or “the
stable snake” in this article.

####  2.2.2 Excursion measures for snakes

We will need to consider the law of a snake where the exploration process
$\rho$ is first sampled under $N$ or $N^{(1)}$, and with initial condition
$W_{0}(0)=x\in\mathbb{R}$. Informally, under $N^{(1)}$ this is the law of a
stable snake on a single stable tree of total mass $1$, started from the point
$x$. Under $N$ this is the law of a stable snake on a stable tree sampled
according to the Itô measure, started from the point $x$. We denote these
excursion measures for snakes by $\mathbb{N}_{x}$ and $\mathbb{N}^{(1)}_{x}$,
so that

$\displaystyle\mathbb{N}_{x}(d\rho,dW)=N_{0}(d\rho)Q_{x}^{H(\rho)}(dW).$ (16)

$\mathbb{N}^{(1)}_{x}$ is defined similarly by replacing $N$ with $N^{(1)}$
above. It is explained in [RRO22, Section 2.4] that the point $(0,x)$ is
regular and recurrent for the process $(\rho,W)$ (which is actually Markov),
and moreover that the process $(-I_{t})_{t\geq 0}$ is a local time at $0$ for
this process, and hence this definition makes sense as the excursion measure
of $(\rho,W)$ away from $(0,x)$ associated with the local time $-I$.

By excursion theory, it moreover follows that, if
$(\alpha_{i},\beta_{i})_{i\in\mathcal{I}}$ denote the excursion intervals of
$(\rho,W)$ away from $(0,x)$, with corresponding subtrajectories
$(\rho^{(i)},W^{(i)})$, then under $\mathbb{P}_{x}$, the measure

$\sum_{i\in\mathcal{I}}\delta(-I_{\alpha_{i}},\rho^{(i)},W^{(i)})$

is a Poisson point measure with intensity
$\mathbbm{1}_{[0,\infty)}(\ell)d\ell\mathbb{N}_{x}(d\rho,dW)$ (where $d\ell$
denotes Lebesgue measure).

Note that, under $\mathbb{N}_{x}$, the function
$(\widehat{W}_{s})_{s\in[0,\sigma]}$ can also be viewed as a function of the
tree $\mathcal{T}_{\alpha}$ coded by the height function $H(\rho)$ driving the
snake that appears in (16). We will follow the convention of [LG07] and denote
this function by $(Z_{a})_{a\in\mathcal{T}_{\alpha}}$, so that
$Z_{a}=\widehat{W}_{t}$ for any $a\in\mathcal{T}_{\alpha}$ such that $a=p(t)$.
We also will sometimes abuse notation and interchangeably use $Z_{a}$, $Z_{t}$
and $\widehat{W}_{t}$.

We denote by $\mathcal{R}$ the range of the stable snake:

$\mathcal{R}:=\left[\inf_{a\in\mathcal{T}_{\alpha}}Z_{a},\sup_{a\in\mathcal{T}_{\alpha}}Z_{a}\right].$
(17)

The measure $\mathbb{N}_{x}$ also inherits a scaling property from that of
$N(\cdot)$ and the Brownian scaling property. For any $\lambda>0$ we define a
mapping $\Phi_{\lambda}:(\rho,W)\to(\rho^{\lambda},W^{\lambda})$ by

$\displaystyle\rho^{\lambda}_{t}(dr)=\lambda^{\frac{1}{\alpha}}\rho_{t/\lambda}(d(r/\lambda^{1-\frac{1}{\alpha}})),\qquad
W_{t}^{\lambda}(s)=\lambda^{\frac{1}{2}\left(1-\frac{1}{\alpha}\right)}W_{t/\lambda}(s/\lambda^{1-\frac{1}{\alpha}}),$
(18)

so that the law of $(\rho^{\lambda},W^{\lambda})$ under $\mathbb{N}_{x}$ is
equal to
$\lambda^{1/\alpha}\mathbb{N}_{\lambda^{\frac{1}{2}\left(1-\frac{1}{\alpha}\right)}x}$.

Note that we can use (6) and (18) to calculate that

$\displaystyle N\left(1-e^{-\lambda\sigma}\right)=\lambda^{1/\alpha}.$ (19)

We also briefly recall a few useful results. The first is the following from
[ACM24, Equation (38)] (recall the definition of the range from (17)):

$\mathbb{N}_{x}\left[\mathbbm{1}\\{0\in\mathcal{R}\\}\right]=\left(\frac{\alpha+1}{(\alpha-1)^{2}}\,\frac{1}{x^{2}}\right)^{\frac{1}{\alpha-1}},$
(20)

The second concerns the Hölder continuity of the stable snake:

######  Lemma 2.1.

(Lemma 6.1 in [ACM24]).

  1. (a)

$N^{(1)}$ almost surely, for any $\gamma<1-\frac{1}{\alpha}$, the height
function is Hölder continuous with parameter $\gamma$.

  2. (b)

Take any $b\in(0,1/2)$. Then $\mathbb{N}^{(1)}$ a.e. there exists
$\varepsilon_{0}>0$ such that for every $s,t\geq 0$ with
$d(s,t)\leq\varepsilon_{0}$,

$|\widehat{W}_{t}-\widehat{W}_{s}|\leq d(s,t)^{b}.$

The third concerns the location of the minimum of the snake over the interval
$[0,\sigma]$, which is denoted by $s^{*}$.

######  Proposition 2.2 (Proposition 2.8 in [ACM24]).

$\mathbb{N}$-almost everywhere, the time $s^{*}\in[0,\sigma]$ at which
$\widehat{W}$ attains its minimum is unique.

We also take this opportunity to define the occupation measure of the snake
re-rooted at its minimum as the random measure $\overline{\mathcal{I}}$
satisfying

$\langle\overline{\mathcal{I}},f\rangle=\int_{0}^{\sigma}f(\widehat{W}_{s}-\widehat{W}_{s^{*}})ds.$
(21)

####  2.2.3 Uniform re-rooting invariance

We will use a property of stable snakes know as uniform re-rooting invariance.
To define this under $\mathbb{N}_{0}$ for an excursion with lifetime
$\sigma>0$, we follow the presentation of [LGW06, Section 2.3] for the
Brownian case. We first need to define the notion of a stable tree and of a
snake rooted at a time point $s\in[0,\sigma]$. To this end, given $\sigma>0$
we first set

$s\oplus r=\begin{cases}s+r&\text{ if }s+r\leq\sigma,\\\ s+r-\sigma&\text{ if
}s+r>\sigma.\end{cases}$

We then define the re-rooted lifetime and snake-tip processes by

$\displaystyle\zeta^{[s]}_{r}$ $\displaystyle=\zeta_{s}+\zeta_{s\oplus
r}-2\inf_{u\in[s\wedge(s\oplus r),s\vee(s\oplus r)]}\zeta_{u}\qquad\text{ and
}\qquad\widehat{W}^{[s]}_{r}=\widehat{W}_{s\oplus r}-\widehat{W}_{s}$ (22)

for all $r\in[0,\sigma]$. Note that the entire sequence of snake trajectories
can be recovered directly from (22) via

$\displaystyle{W}^{[s]}_{r}(t)=\widehat{W}^{[s]}_{\sup\\{u\leq
r:\zeta^{[s]}_{u}=t\\}},$

or equivalently (22) does indeed suffice to define the entire re-rooted
process.

Informally, $(\zeta^{[s]}_{r})_{r\geq 0}$ codes a stable tree rooted at the
vertex previously labelled $s$, and $\widehat{W}^{[s]}_{r}$ codes the spatial
displacements along the branches of the new tree from the new root to the
vertex which now has label $r$.

The re-rooting invariance statement is as follows.

######  Proposition 2.3 (Proposition 2.1 in [ACM24]).

For every non-negative function $F$ on $\mathbb{R}_{+}\times
C(\mathbb{R}_{+},\mathcal{W})$ it holds that

$\mathbb{N}_{0}\left(\int_{0}^{\sigma}F(s,{W}^{[s]})ds\right)=\mathbb{N}_{0}\left(\int_{0}^{\sigma}F(s,W)ds\right).$

####  2.2.4 Spinal decompositions

We now present an important spinal decomposition formula, which will allow us
to express certain integral functionals of the stable snake under
$\mathbb{N}_{0}$ using a decomposition over subtrees and subtrajectories along
a branch to a uniform point in the underlying tree.

To this end, we let $(U^{(1)},U^{(2)})$ be a two dimensional subordinator
defined on some probability space $(\Omega_{0},\mathcal{F}_{0},P^{0})$,
started at $(0,0)$ and with Laplace transform (recall that
$\psi(\lambda)=\lambda^{\alpha}$)

$\displaystyle
E^{0}[\exp\\{-\lambda_{1}U^{(1)}_{t}-\lambda_{2}U^{(2)}_{t}\\}]=\begin{cases}\exp\left\\{-\frac{t(\psi(\lambda_{1})-\psi(\lambda_{2}))}{\lambda_{1}-\lambda_{2}}\right\\}&\text{
if }\lambda_{1}\neq\lambda_{2},\\\
\exp\\{-t\psi^{\prime}(\lambda_{1})\\}&\text{ if
}\lambda_{1}=\lambda_{2}.\end{cases}$ (23)

The marginal of each of $U^{(1)}$ and $U^{(2)}$ is that of a subordinator with
Laplace exponent
$\tilde{\psi}(\lambda):=\frac{\psi(\lambda)}{\lambda}=\lambda^{\alpha-1}$, and
the sum $U^{(1)}+U^{(2)}$ is a subordinator with Laplace exponent
$\psi^{\prime}(\lambda)=\alpha\,\lambda^{\alpha-1}$. Similarly to (4), this
entails that

$\displaystyle\begin{split}\tilde{\psi}(\lambda)&=\int_{0}^{\infty}(1-e^{-\lambda
r})\tilde{\pi}(dr)=\frac{C_{\alpha}\lambda}{2-\alpha}+\int_{0}^{\infty}(1-e^{-\lambda
r}-\lambda r\mathbbm{1}\\{r\leq 1\\})\tilde{\pi}(dr),\\\
\psi^{\prime}(\lambda)&=\int_{0}^{\infty}(1-e^{-\lambda
r})\pi^{\prime}(dr)=\frac{\alpha
C_{\alpha}\lambda}{2-\alpha}+\int_{0}^{\infty}(1-e^{-\lambda r}-\lambda
r\mathbbm{1}\\{r\leq 1\\})\pi^{\prime}(dr).\end{split}$ (24)

where $\tilde{\pi}(dr)=C_{\alpha}r^{-\alpha}dr$ and $\pi^{\prime}(dr)=\alpha
C_{\alpha}r^{-\alpha}dr$ (these can be calculated directly from (4) and (5)
but see [Ber96, Section III.1] for further background on stable
subordinators). We note only that $\tilde{\psi}$ can be easily recovered from
$\psi^{\prime}$ by using Fubini’s theorem and noting that $U^{(1)}$ is
obtained from $U^{(1)}+U^{(2)}$ through the fact that a jump at time $t$ of
$U^{(1)}$ corresponds to a uniform portion of the corresponding jump at time
$t$ of $U^{(1)}+U^{(2)}$

We will continue to use the notation $\psi,\psi^{\prime}$ and $\tilde{\psi}$
throughout the paper to refer to these three functions above.

For each $a>0$, let $(J_{a},\hat{J}_{a})$ denote a pair of random measures
given by

$\displaystyle(J_{a},\hat{J}_{a})(dt)=(\mathbbm{1}_{[0,a]}(t)dU_{t}^{(1)},\mathbbm{1}_{[0,a]}(t)dU_{t}^{(2)})$
(25)

For $h>0$, under $E^{0}\otimes\Pi_{x}$, conditionally on the measure
$(J_{h}+\hat{J}_{h})$ and the Brownian trajectory $(\xi_{s})_{0\leq s\leq h}$,
we consider an independent Poisson point process on $[0,h]\times\mathcal{W}$,
denoted by

$\mathcal{P}:=\sum_{i\in I}\delta_{a_{i},w_{i}},$

and with intensity

$(J_{h}+\hat{J}_{h})(da)\otimes\mathbb{N}_{\xi_{a}}(dw).$ (26)

The key result is as follows. In the proposition we let $\mathcal{R}$ denote
the range of the snake $W$, and we let $\mathbb{E}_{\mathcal{P}}$ denote
expectation with respect to the point process $\mathcal{P}$ defined above.
Also recall that $\Pi_{x}$ denotes the law of a standard linear Brownian
motion started at $x$.

######  Proposition 2.4 (Proposition 2.4 in [ACM24]).

For every $x\in\mathbb{R}$ and every non-negative measurable functional $\phi$
taking values in $M_{f}(\mathbb{R}_{+})^{2}\times\mathcal{W}$, we have

$\displaystyle\mathbb{N}_{x}\left[\int_{0}^{\sigma}\phi(\rho_{s},{\hat{\rho}}_{s},W_{s},\mathcal{R},\sigma)ds\right]$
$\displaystyle\qquad=\int_{0}^{\infty}dh\,E^{0}\otimes\Pi_{x}\left[\mathbb{E}_{\mathcal{P}}\left[\phi\left(J_{h},\hat{J}_{h},(\xi_{s})_{0\leq
s\leq h},\overline{\bigcup_{i\in
I}\left(\zeta_{a_{i}}+\mathcal{R}(w_{i})\right)},\sum_{i\in
I}\sigma(w_{i})\right)\middle|(J,\hat{J},\xi)\right]\right].$

Informally, the proposition says that the sizes hubs along the branch from
$\rho$ to a uniform point are given by the jumps of the subordinator
$(J,\hat{J})$, and that the subtrees grafted to the hubs form a Poisson
process coded by the Itô excursion measure.

In order to carry out calculations for Poisson processes such as those
appearing in Proposition 2.4, and with respect to the measures $J$ and
$\hat{J}$, we remind the reader of two standard formulas.

Firstly note that, using the Markov property and stationarity of increments,
the formula (23) easily generalises to the following for continuous
nonnegative functions $f$.

$\displaystyle\begin{split}E^{0}\left[\exp\left\\{-\int_{0}^{t}f(s)d(U^{(1)}+U^{(2)})_{s}\right\\}\right]&=\exp\left\\{-\int_{0}^{t}\psi^{\prime}(f(s))ds\right\\}\\\
E^{0}\left[\exp\left\\{-\int_{0}^{t}f(s)dU^{(1)}_{s}\right\\}\right]&=\exp\left\\{-\int_{0}^{t}\tilde{\psi}(f(s))ds\right\\}.\end{split}$
(27)

Secondly, we will use the following formula for Poisson point processes
several times.

######  Proposition 2.5.

Let $P=\sum_{i\in\mathcal{I}}\delta(\omega_{i})$ denote a Poisson point
process on a space $\Omega$ with intensity measure $\hat{\mu}$ and let $f$ be
a function $\Omega\to\mathbb{R}$. Then

$\mathbb{E}\\!\left[\prod_{i\in\mathcal{I}}f(\omega_{i})\right]=\exp\left\\{\int_{\Omega}(f(\omega)-1)\hat{\mu}(d\omega)\right\\}.$

####  2.2.5 Local time at level $a$

It will also be useful to introduce the notion of the local time at a certain
level of $\mathcal{T}_{\alpha}$. For $a>0$, Duquesne and Le Gall [DLG05]
showed that one can construct a local time measure, supported on vertices at
height $a$ in $\mathcal{T}_{\alpha}$, and moreover such that the canonical
measure $\upsilon$ on $\mathcal{T}_{\alpha}$ satisfies

$\upsilon=\int_{0}^{\infty}\ell^{a}da,$ (28)

$N$-almost everywhere. A vertex chosen according to $\ell^{a}$ can therefore
be interpreted as a vertex chosen “uniformly at level $a$” in
$\mathcal{T}_{\alpha}$.

We will use the following first moment formula, which is a special case of
[DLG02, Proposition 4.3.2] or [RRO22, Equation (3.4)].

######  Proposition 2.6.

For every domain $D$, every $x\in D$, and every non-negative measurable
functional $\phi$ taking values in
$M_{f}(\mathbb{R}_{+})^{2}\times\mathcal{W}$, we have

$\displaystyle\mathbb{N}_{x}\left[\int_{0}^{\sigma}\ell^{a}(ds)\,\phi(\rho_{s},{\hat{\rho}}_{s},W_{s})\right]=E^{0}\otimes\Pi_{x}\left(\phi(J_{a},\hat{J}_{a},(\xi_{s})_{0\leq
s\leq a})\right).$

Note that this also implies that $N(\langle\ell^{a},1\rangle)=1$ for all
$a>0$.

##  3 Trees, maps and scaling limits

###  3.1 The pointed Gromov-Hausdorff-Prokhorov topology

To study the asymptotic metric-measure structure of a large random map, we
will make use of the pointed Gromov-Hausdorff-Prokhorov distance, defined
between (isometry classes of) compact rooted metric-measure spaces. In
particular, we let $\mathbb{F}^{c}$ denote the set of quadruples
$(F,R,\mu,\rho)$ such that $(F,R)$ is a compact metric space, $\mu$ is a
locally finite Borel measure of full support on $F$, and $\rho$ is a
distinguished point of $F$, which we call the root, considered up to root-
preserving, measure-preserving isometries.

Let $(E_{1},d_{1},\nu_{1},\rho_{1})$ and $(E_{2},d_{2},\nu_{2},\rho_{2})$ be
two compact rooted metric-measure spaces in $\mathbb{F}^{c}$. The pointed
Gromov-Hausdorff-Prokhorov distance between them is defined as

$d_{GHP}(E_{1},E_{2})=\inf\left\\{d_{H}(\varphi_{1}(E_{1}),\varphi_{2}(E_{2}))+d(\varphi_{1}(\rho_{1}),\varphi_{2}(\rho_{2}))+d_{P}(\varphi_{1}^{*}\nu_{1},\varphi_{2}^{*}\nu_{2})\right\\},$
(29)

where the infimum is over all isometric embeddings $\varphi_{i}:E_{i}\to E$ of
$E_{1}$ and $E_{2}$ into the same metric space $(E,d)$, $d_{H}$ is the
Hausdorff distance between compact subsets of $E$, and $d_{P}$ denotes the
Prokhorov distance between measures on $E$, as defined in [Bil68, Chapter 1].
It is well-known (for example, see [ADH13, Theorem 2.3]) that this defines a
metric on the space of equivalence classes of $\mathbb{F}^{c}$, where we say
that two spaces $(F,R,\mu,\rho)$ and
$(F^{\prime},R^{\prime},\mu^{\prime},\rho^{\prime})$ are equivalent if there
is a measure and root-preserving isometry between them.

We can also consider the pointed Gromov-Hausdorff distance
$d_{GH}(\cdot,\cdot)$ between two spaces in $\mathbb{F}^{c}$; this is defined
by removing the term involving the Prokhorov distance from (29) above. A
useful characterisation of the pointed Gromov-Hausdorff distance can be made
via _correspondences_.

######  Definition 3.1.

If $(E_{1},d_{1})$ and $(E_{2},d_{2})$ are two compact metric spaces, a
correspondence between $E_{1}$ and $E_{2}$ is a subset $R\subseteq E_{1}\times
E_{2}$ such that, for every $x_{1}\in E_{1}$, there exists $x_{2}\in E_{2}$
such that $(x_{1},x_{2})\in R$, and for every $x_{2}\in E_{2}$, there exists
$x_{1}\in E_{1}$ such that $(x_{1},x_{2})\in R$.

If $R$ is a correspondence between $(E_{1},d_{1})$ and $(E_{2},d_{2})$, the
distortion of $R$ is given by

$\textnormal{dis}_{R}:=\sup\\{\lvert
d_{1}(x_{1},y_{1})-d_{2}(x_{2},y_{2})\rvert;\,(x_{1},x_{2}),(y_{1},y_{2})\in
R\\}.$

As was done in the Brownian case, we will use the following theorem to
establish tightness in the Gromov-Hausdorff topology.

######  Theorem 3.2 (Theorem 7.3.25 in [BBI01]).

Let $(E_{1},d_{1},\nu_{1},\rho_{1})$ and $(E_{2},d_{2},\nu_{2},\rho_{2})$ be
in $\mathbb{F}^{c}$. Then

$d_{GH}((E_{1},d_{1},\nu_{1},\rho_{1}),(E_{2},d_{2},\nu_{2},\rho_{2}))=\frac{1}{2}\inf\textnormal{dis}_{R},$

where the infimum is taken over the set of correspondences between $E_{1}$ and
$E_{2}$ containing the pair $(\rho_{1},\rho_{2})$.

In this article, we will prove (subsequential) pointed Gromov-Hausdorff-
Prokhorov convergence by first proving pointed Gromov-Hausdorff convergence
using correspondences, and then show Prokhorov convergence of the measures on
the appropriate metric space.

###  3.2 Discrete trees

A tree is a connected graph (i.e. with vertices and edges) with no cycles. A
plane tree is a tree that is embedded in the sphere (it can be equivalently
embedded in the plane). In this paper we will deal mainly with rooted plane
trees, meaning that there is a distinguished oriented edge known as the root
edge. The initial vertex of the root edge is called the root vertex. This
induces the notion of genealogy: we say that $u$ is the parent of $v$, or $v$
is a child of $u$ if $u$ and $v$ are neighbours and $u$ is strictly closer to
the root vertex.

In what follows, we denote by $\mathbf{T}$ the set of rooted plane trees, and
by $\mathbf{T}_{n}$ the set of rooted plane trees with $n$ edges.

To describe rooted plane trees, we will make use of the Ulam-Harris encoding
(see [Nev86] for further background), in which vertices of a tree
$\mathfrak{t}\in\mathbf{T}$ are labelled by elements of $\bigcup_{n\geq
0}\mathbb{N}^{n}$, where $\mathbb{N}=\\{1,2,\ldots\\}$ and where we set
$\mathbb{N}^{0}=\varnothing$. The root of $\mathfrak{t}$ receives Ulam-Harris
label $\varnothing$, and the other labels are determined recursively, with the
children of vertex $v=v_{1}v_{2}\dotsm v_{h}\in\mathbb{N}^{h}$ receiving Ulam-
Harris labels $(vi,1\leq i\leq c_{\mathfrak{t}}(v))$ (where
$c_{\mathfrak{t}}(v)$ is the number of children of $v$ in $\mathfrak{t}$), in
the left-to-right ordering given by the embedding of $\mathfrak{t}$ in the
plane, in which the outer endpoint of the root edge is designated to be the
leftmost child of the root. For $v=v_{1}v_{2}\dotsm v_{h}$, we denote by
$\lvert v\rvert=h$ the generation of $v$.

Let $\mathfrak{t}\in\mathbf{T}$, and $u,v\in V(\mathfrak{t})$. We denote by
$[[u,v]]$ the set of vertices of $\mathfrak{t}$ that lie on the unique
shortest path from $u$ to $v$ in $\mathfrak{t}$, and
$]]u,v[[=[[u,v]]\setminus\\{u,v\\}$. We also denote by $u\wedge v$ the most
recent common ancestor of $u$ and $v$ in $\mathfrak{t}$, which may be defined
by the equality $[[\varnothing,u\wedge
v]]=[[\varnothing,u]]\cap[[\varnothing,v]]$. This also leads to the notion of
lexicographical order: $u$ comes before $v$ in this ordering, denoted $u\leq
v$, if and only if $u_{u\wedge v+1}\leq v_{u\wedge v+1}$ in the Ulam-Harris
encoding.

######  Remark 3.3.

To avoid confusion, we point out that elsewhere in the literature that we
cite, rooted plane trees are often referred to as “rooted ordered plane trees”
where the root is a vertex and the ordering is equivalent to the choice of the
leftmost child of the root which, together with the planar embedding, fixes
the Ulam-Harris/lexicographical ordering of the vertices. Since we started
with a root edge, rather than a root vertex, we have already fixed this
ordering. Sometimes they are also just referred to as “plane trees” and the
root vertex and ordering is automatically included in this definition.

We denote by $\preceq$ the genealogical relation on $\mathfrak{t}$: we write
$u\preceq v$ and say $u$ is an ancestor of $v$, or $v$ is a descendant of $u$,
if $u\in[[\varnothing,v]]$. We use $\leq$ for the lexicographical order on
$\mathfrak{t}$, and $\ll$ for the _reverse_ lexicographical order: $u\ll v$
means that the image of $v$ comes before the image of $u$ in the
lexicographical order of the mirror tree. More precisely, if neither $u\prec
v$ nor $v\prec u$ holds, then $u\ll v$ iff $u\leq v$; if $u\prec v$, then
$v\ll u$ (and $u\leq v$). We also use the lexicographical order to define a
projection $p_{\mathfrak{t}}$ from $\\{0,\ldots,|V(\mathfrak{t})|-1\\}$ to
$\mathfrak{t}$: $p_{\mathfrak{t}}(i)$ denotes the $(i+1)^{th}$ vertex in the
lexicographical ordering of $\mathfrak{t}$.

A tree $\mathfrak{t}\in\mathbf{T}_{n}$ can be coded by three different
functions as illustrated in Figure 3. We define the contour exploration of
$\mathfrak{t}$ as the sequence $e_{0},e_{1},\dots,e_{2n-1}$ of oriented edges
bounding the unique face of $\mathfrak{t}$, starting with the root edge, and
ordered counterclockwise around this face (it therefore traces the “outer
boundary” of the tree in “left-to-right” fashion). For $0\leq i\leq 2n-1$, the
$i^{th}$ visited vertex in this contour exploration is the initial vertex
$e^{-}_{i}$ of $e_{i}$, and we set $C_{\mathfrak{t}}(i)=|e^{-}_{i}|$. Note
that the contour ordering is such that the ordering of the vertices determined
by the times of their first visit by the contour function is the same as their
lexicographical ordering.

To define the height function, we now list the $n+1$ vertices of
$\mathfrak{t}$ in lexicographical order as $u_{0},u_{1},\ldots,u_{n}$. We
define the height process $H_{\mathfrak{t}}=(H_{\mathfrak{t}}(j))_{0\leq j\leq
n}$ by

$H_{\mathfrak{t}}(j)=\lvert u_{j}\rvert,\,0\leq j\leq n.$ (30)

We also define the Łukasiewicz path $(X_{\mathfrak{t}}(m))_{0\leq m\leq n}$ by
setting $X_{\mathfrak{t}}(0)=0$, then by considering the vertices
$u_{0},u_{1},\ldots,u_{n}$ in lexicographical order and setting
$X_{\mathfrak{t}}(m+1)=X_{\mathfrak{t}}(m)+c_{\mathfrak{t}}(u_{m})-1$ (recall
that $c_{\mathfrak{t}}(u)$ denotes the number of children of $u$ in
$\mathfrak{t}$). The height function and Łukasiewicz path are connected by the
following relation.

$H_{\mathfrak{t}}(m)=|\\{j\in\\{0,1,\ldots,m-1\\}:X_{\mathfrak{t}}(j)=\inf_{j\leq
k\leq m}X_{\mathfrak{t}}(k)\\}|.$ (31)

The intuition behind this is that the times $j\in\\{0,1,\ldots,m-1\\}$
satisfying $X_{\mathfrak{t}}(j)=\inf_{j\leq k\leq m}X_{\mathfrak{t}}(k)$
correspond exactly to the ancestors of the vertex $u_{m}$.

Although the height function and the contour function directly encode the
geometry of the tree, we will see below that in the setting of random trees,
the Łukasiewicz path is often more convenient to work with.

Figure 3: Example of contour function, height function and Łukasiewicz path
for the given tree.

Let $\mathfrak{t}$ be a random element of $\mathbf{T}$. We say it has the
branching property if, for any $n\in\mathbb{N}$, conditionally on
$\\{c_{\mathfrak{t}}(\varnothing)=n\\}$, the subtrees
$(\mathfrak{t}_{1},\dotsm,\mathfrak{t}_{n})$ of descendants of the $n$
children of the root are independent and each distributed as the original tree
$\mathfrak{t}$. The law of such a tree is characterised by its offspring
distribution: for $\mu_{\alpha}$ a probability measure on $\mathbb{Z}_{\geq
0}$, a random rooted plane tree $\mathfrak{t}$ is a Bienaymé–Galton–Watson
tree with offspring distribution $\mu_{\alpha}$ if it has the branching
property and the distribution $c_{\mathfrak{t}}(\varnothing)$ of the number of
children of the root is $\mu_{\alpha}$. We say an offspring distribution is
critical if it has mean $1$. Note that Bienaymé–Galton–Watson trees thus come
equipped with a root edge and a lexicographical order.

Duquesne [Duq03, Theorem 3.1] showed that when $T_{n}^{(\alpha)}$ is a
Bienaymé–Galton–Watson tree with critical offspring distribution satisfying
$\mu_{\alpha}[x,\infty)\sim cx^{-\alpha}$ for some $\alpha\in(1,2)$ and
$c\in(0,\infty)$, conditioned to have $n$ edges, and $X_{n},H_{n}$ and $C_{n}$
respectively denote its Łukasiewicz path, height function and contour
function, then there exists $c^{\prime}\in(0,\infty)$ (explicitly depending on
$c$) such that the following joint convergence holds with respect to the
Skorokhod-$J_{1}$ product topology:

$\Bigg{(}\frac{1}{c^{\prime}}n^{-\frac{1}{\alpha}}X_{n}(nt),c^{\prime}n^{-(1-\frac{1}{\alpha})}H_{n}(nt),c^{\prime}n^{-(1-\frac{1}{\alpha})}C_{n}(2nt)\Bigg{)}_{t\in[0,1]}\xrightarrow[n\to\infty]{(d)}\Big{(}X^{\text{exc}}_{t},H_{t},H_{t}\Big{)}_{t\in[0,1]},$
(32)

where $(H_{t})_{t\in[0,1]}$ denotes the height function defined from
$X^{\text{exc}}$ as in (8). (The joint convergence including the Łukasiewicz
path is not specifically stated in [Duq03, Theorem 3.1], but is implicit in
the proof.) A straightforward consequence is that

$(T_{n}^{(\alpha)},c^{\prime}n^{-(1-\frac{1}{\alpha})}d_{n},\rho_{n})\xrightarrow[n\to\infty]{(d)}(\mathcal{T}_{\alpha},d_{\alpha},\rho)$

with respect to the pointed Gromov-Hausdorff topology, where $d_{n}$ denotes
the graph metric on $T_{n}^{(\alpha)}$, $\rho_{n}$ its root vertex, and
$(\mathcal{T}_{\alpha},d_{\alpha},\rho)$ is the stable tree defined in Section
2.1.3. This can also be straightforwardly extended to include convergence of
the uniform measure on $T_{n}^{(\alpha)}$ to the measure $\upsilon$ introduced
in Section 2.1.3 and therefore give convergence in the pointed Gromov-
Hausdorff-Prokhorov topology.

###  3.3 Random planar maps

We now give some basic definitions related to maps.

A planar map is an embedding of a graph $G=(V,E)$ into the sphere in such a
way no two edges cross, considered up to orientation-preserving homeomorphism.
In what follows we will sometimes drop the adjective “planar”, but all maps
should be assumed to planar maps in this paper. A rooted planar map is a
planar map equipped with a distinguished oriented edge, called its root edge.
The starting vertex of the root edge is called the root vertex.

A pointed rooted planar map is a rooted planar map that is equipped with an
additional distinguished vertex (this may or may not coincide with the root
vertex).

We will also need the notion of corners: if we imagine all edges to be
replaced by two (oppositely) directed edges, a corner in a map is an angular
sector $c$ between an incoming edge and an outgoing edge that follow each
other in the clockwise cyclic order around a vertex $v$. The corner is
canonically associated with this latter outgoing edge. If $v$ is has degree 1,
it is associated with one corner.

######  Definition 3.4.

A quadrangulation is a planar map whose faces all have degree four. We denote
by $\mathcal{Q}_{n}$ the set of rooted planar quadrangulations with $n$ faces,
and by $\mathcal{Q}_{n}^{\bullet}$ the set of pointed, rooted planar
quadrangulations with $n$ faces.

A map $\mathfrak{m}$ induces a metric space by considering the set of its
vertices $V(\mathfrak{m})$ equipped with the graph distance; the latter is
denoted by $d_{\mathfrak{m}}$. For $u\in V(\mathfrak{m})$ and $r\geq 0$, we
denote by

$B_{\mathfrak{m}}(u,r)=\\{v\in
V(\mathfrak{m})\,\lvert\,d_{\mathfrak{m}}(u,v)\leq r\\}$

the ball of radius $r$ around $u$ in $\mathfrak{m}$. A geodesic in
$\mathfrak{m}$ is a discrete path $\gamma=(\gamma(i),0\leq i\leq k)$ taking
values in $V(\mathfrak{m})$, such that
$d_{\mathfrak{m}}(\gamma(i),\gamma(j))=\lvert i-j\rvert$ for every
$i,j\in\\{0,\dotsm,k\\}$.

###  3.4 The Cori–Vauquelin–Schaeffer bijection

Figure 4: The CVS bijection: an admissible labelled tree with 6 edges (in
black), and the associated planar quadrangulation with 6 faces (in red), with
its pointed vertex $v_{*}$.

Let us now get to the bijection between rooted planar quadrangulations with
$n$ faces, and a family of decorated trees with $n$ edges, following very
closely the presentation of [Mie14, Section 2.3]. This bijection was first
introduced in the work of Cori and Vauquelin [CV81] and further developed in
the thesis of Schaeffer [Sch98]. We will actually only use the inverse of the
pointed version of this bijection, from trees to quadrangulations, which was
constructed explicitly by Chassaing and Schaeffer [CS04] and refer to it as
the CVS bijection.

Let $\mathfrak{t}$ be a rooted plane tree with $n$ edges, with root edge
$e_{0}$ and root vertex $u_{0}$. An admissible label function on
$\mathfrak{t}$ is a function $\ell:V(\mathfrak{t})\to\mathbb{Z}$ such that
$\ell(u_{0})=0$, and, for every adjacent $u,v\in V(\mathfrak{t})$,
$|\ell(u)-\ell(v)|\leq 1$. If $\mathfrak{t}$ is a rooted plane tree, and
$\ell$ is an admissible label function on $\mathfrak{t}$, we say that the pair
$(\mathfrak{t},\ell)$ is a admissible labelled tree. We denote by
$\mathbb{T}_{n}$ the set of admissible labelled trees with $n$ edges.

Consider a labelled tree $(\mathfrak{t},\ell)\in\mathbb{T}_{n}$. Let
$e_{0},e_{1},\dots,e_{2n-1}$ be the contour exploration of the oriented edges
of $\mathfrak{t}$, as defined in Section 3.2. Recall that $e^{-}_{i}$ is the
initial vertex of $e_{i}$. It will be useful extend the sequences
$(e_{i})_{i}$ and $(e^{-}_{i})_{i}$ to $\mathbb{N}$ by periodicity. In what
follows we will identify the oriented edge $e_{i}$ with the associated corner,
and thus use the notation $\ell(e_{i})$ for $\ell(e^{-}_{i})$.

For every $i\geq 0$, we define the successor of $i$ by

$s(i):=\inf\\{j>i\,|\,\ell(e_{j})=\ell(e_{i})-1\\},$ (33)

with the convention that $\inf\varnothing=\infty$. Note that $s(i)=\infty$ if
and only if $\ell(e_{i})=\min_{j}\ell(e_{j})$, as the values of $\ell$
decrease only by unit steps. We introduce a new vertex, corresponding to a
corner labelled $e_{\infty}$, that will play the role of successor for all
such $i$. To do so, recall that $\mathfrak{t}$ is embedded in the sphere, and
consider a point $v_{*}$ in the sphere that does not belong to the support of
$\mathfrak{t}$. Denote by $e_{\infty}$ a small neighbourhood of $v_{*}$ not
intersecting $\mathfrak{t}$ or any of its corners. We will also refer to
$e_{\infty}$ as a corner in what follows. We set

$\ell(v_{*})=\ell(e_{\infty})=\min_{j}\ell(e_{j})-1.$

We can then define, for every $0\leq i<\infty$, the successor of the corner
$e_{i}$ as $s(e_{i})=e_{s(i)}$, where we consider $e_{s(i)}$ to be its
associated corner.

The reverse CVS construction involves drawing, for every
$i\in\\{0,1,...,2n-1\\}$, an edge, that we will call an arc, from the corner
$e_{i}$ to the corner $s(e_{i})$, without crossing $\mathfrak{t}$, $v_{*}$, or
another arc (see Figure 4). This is always possible; see [Mie14, Lemma 2.3.2].
We denote by $\mathfrak{q}$ the embedded graph with vertex set
$V(\mathfrak{t})\cup{v_{*}}$, and edge set formed by the arcs.

A first property of $\mathfrak{q}$, which is contained in the stronger
statement Theorem 3.5, is that it is a quadrangulation with $n$ faces. Note
that it comes equipped with a distinguished vertex $v_{*}$, but for now it is
not a rooted quadrangulation. To fix this root, we use an extra parameter
$\varepsilon\in\\{-1,1\\}$. If $\varepsilon=1$ we let the root edge of
$\mathfrak{q}$ be the arc linking $e_{0}$ with $s(e_{0})$, and oriented from
$s(e_{0})$ to $e_{0}$. If $\varepsilon=-1$, the root edge is this same arc,
but oriented from $e_{0}$ to $s(e_{0})$. We have thus defined a mapping $\Phi$
from $\mathbb{T}_{n}\times\\{-1,1\\}$ to the set $\mathcal{Q}^{\bullet}_{n}$
of pointed rooted planar quadrangulations $(\mathfrak{q},v_{*})$.

######  Theorem 3.5 (Proposition 2 and Theorem 4 in [CS04]).

For every $n\geq 1$, $\Phi$ is a bijection from
$\mathbb{T}_{n}\times\\{-1,1\\}$ onto $\mathcal{Q}^{\bullet}_{n}$.

A very important aspect of the CVS bijection is that the labels of the tree
correspond to some of the distances in the quadrangulation. Indeed, if
$(\mathfrak{q},v_{*})=\Phi((\mathfrak{t},\ell),\varepsilon)$, then, for every
$v\in V(\mathfrak{q})$,

$d_{\mathfrak{q}}(v,v_{*})=\ell(v)-\ell(v_{*}).$ (34)

######  Remark 3.6.

Another property of the CVS bijection is that every edge in $\mathfrak{t}$ is
inscribed inside a face of $\mathfrak{q}$. Hence, any path of adjacent
vertices in $\gamma_{0},\gamma_{1},\ldots,\gamma_{m}$ in $\mathfrak{t}$ can be
extended (not necessarily uniquely) to a path in $\mathfrak{q}$ by adding at
most one extra vertex between each pair of the form $\gamma_{i}$ and
$\gamma_{i+1}$. Hence, if the extended path intersects some set $A\subset
V(\mathfrak{q})$, then (the image of) the original path is at distance at most
$1$ from $A$.

For $(\mathfrak{t},\ell)\in\mathbb{T}_{n}$, recall that
$u_{0},u_{1},\ldots,u_{n}$ denotes the lexicographical ordering of the tree
vertices. Let us define for $0\leq i,j\leq n$

$d^{\circ}_{\mathfrak{t},\ell}(i,j):=\ell(u_{i})+\ell(u_{j})-2\inf_{i\wedge
j\leq k\leq i\vee j}\ell(u_{k})+2.$ (35)

We also write $d_{\mathfrak{q}}(i,j)$ for $d_{\mathfrak{q}}(u_{i},u_{j})$. We
then have the following bound.

######  Lemma 3.7 (Lemma 3.1 in [LG07]).

For every $i,j\in\\{0,1,\dotsm,n\\}$,

$d_{\mathfrak{q}}(i,j)\leq d^{\circ}_{\mathfrak{t},\ell}(i,j).$

In the rest of this paper, we will consider random admissible labelled trees
$(T_{n}^{(\alpha)},\ell_{n})$, constructed as follows. Let $\mu_{\alpha}$ be a
probability measure on $\mathbb{Z}_{\geq 0}$ such that $\mu_{\alpha}(0)>0$ and
with expectation $1$ and let $Y$ be a real random variable. We assume that

$\exists\,c\in(0,\infty)\,:\,\mu_{\alpha}[x,\infty)\sim
cx^{-\alpha},\,p_{Y}(0)<1\textnormal{ and
}p_{Y}(-1)=p_{Y}(1)=\frac{1-p_{Y}(0)}{2}.$ (36)

We anticipate that all of our results are adaptable to any law $\mu_{\alpha}$
in the domain of attraction of an $\alpha$-stable law, but we focus on the
aforementioned case for ease of notation.

For each $n\geq 2$, let $T_{n}^{(\alpha)}$ be a Bienaymé–Galton–Watson tree
with offspring distribution $\mu_{\alpha}$, conditioned to have $n$ edges
(this will be well-defined for all sufficiently large $n$ under our
assumptions). Recall that with our conventions, $T_{n}^{(\alpha)}$ comes pre-
equipped with a root edge, going from the root vertex to its first child in
the lexicographical ordering.

Conditionally on $T_{n}^{(\alpha)}$, we endow each vertex $u$ of
$T_{n}^{(\alpha)}$ with a random label $\ell_{n}(u)\in\mathbb{R}$, setting
$\ell_{n}(\varnothing)=0$ for the root $\varnothing$ of the tree, and, for
$v\neq\varnothing$,

$\ell_{n}(v)=\sum_{u\preceq v}Y(u),$ (37)

where $(Y(u))_{u\in T_{n}^{(\alpha)}\setminus\\{\varnothing\\}}$ are i.i.d.
copies of $Y$. Note that $(T_{n}^{(\alpha)},\ell_{n})\in\mathbb{T}_{n}$.

We moreover choose $\varepsilon_{n}$ uniformly in $\\{-1,1\\}$, independently
for each $n$. The quadrangulation $M_{n}^{(\alpha)}$ is defined as the image
of the triple $(T_{n}^{(\alpha)},\ell_{n},\varepsilon_{n})$ under the reverse
CVS bijection, i.e. $\Phi(T_{n}^{(\alpha)},\ell_{n},\varepsilon_{n})$. We
moreover let $d_{n}$ denote the graph metric on $M_{n}^{(\alpha)}$, let
$\partial_{n}$ denote the extra vertex added when constructing
$M_{n}^{(\alpha)}$, let $\rho_{n}$ denote its root vertex, and let $\pi_{n}$
denote the canonical projection from $\\{0,1,\ldots,n\\}$ to
$V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\}$ (via the vertex with that
lexicographical index in $T_{n}^{(\alpha)}$: it is essentially the identity
map). We also let $\nu_{n}$ denote the uniform probability measure on the
vertices of $M_{n}^{(\alpha)}$, and let $d^{\circ}_{n}$ denote
$d^{\circ}_{T_{n},\ell_{n}}$ as defined in (35).

######  Remark 3.8.

By construction, each vertex in
$V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\}$ is incident to at least one
edge per corner $c$ that it had in $T_{n}^{(\alpha)}$ (this edge is the arc
that connects $c$ to its successor), so that the vertex degrees in
$M_{n}^{(\alpha)}$ are at least as heavy-tailed as in $T_{n}^{(\alpha)}$. In
that regard, our stable quadrangulations are similar to the images by Tutte’s
bijection of Boltzmann stable maps, that are planar quadrangulations with
stably-decaying vertex degrees, and that should have the same scaling limit as
the duals of these Boltzmann maps. In our model however, it is not clear yet
if typical vertex degrees are indeed stably-decaying, since each corner of
$T_{n}^{(\alpha)}$ is not only the origin of a successor arc, but potentially
the recipient of many. We plan to address this question in future works.

###  3.5 Subsequential scaling limits of random quadrangulations

The aim of this subsection is to show that the sequence
$(M_{n}^{(\alpha)},n^{-\frac{\alpha-1}{2\alpha}}d_{n},\nu_{n})$ is tight in
the Gromov-Hausdorff-Prokhorov topology. This follows by the same arguments as
in the Brownian case, using the main result of [Mar20].

Let $(T_{n}^{(\alpha)},\ell_{n},\varepsilon_{n})$ be defined as in Section 3.4
(around (37)), and recall that
$M_{n}^{(\alpha)}=\Phi(T_{n}^{(\alpha)},\ell_{n},\varepsilon_{n})$, with root
vertex $\rho_{n}$. On $M_{n}^{(\alpha)}$ are defined a distance $d_{n}$ and a
pseudo-distance $d_{n}^{\circ}$, and we define these on $\\{0,\ldots,n\\}^{2}$
as well using the pushforward by the lexicographical ordering. Let $X_{n}$ be
the Łukasiewicz path of $T_{n}^{(\alpha)}$, let $H_{n}$ be the height process
of $T_{n}^{(\alpha)}$, both as defined in and below (30), and extend $H_{n}$,
$X_{n}$, $\ell_{n}$, $d_{n}^{\circ}$ and $d_{n}$ to the entire time interval
$[0,n]$ and $[0,n]^{2}$ by linear interpolation. Also let $X^{\text{exc}}$ be
as in Section 2.1.1. We denote by $H$ the height function defined from
$X^{\text{exc}}$ as in (8), and by $W$ the Brownian snake driven by $H$ as
defined in Section 2.2.1, with tip $\widehat{W}$. The key result is the
following.

######  Theorem 3.9 (Special case of Theorem 1.1 in [Mar20]).

Assume that (36) holds, set $\sigma^{2}_{Y}$ to be the variance of $Y$ and set
$c^{\prime}=\left(\frac{c}{C_{\alpha}}\right)^{1/\alpha}$ (recall (5)).

Then the following convergence holds jointly in distribution with respect to
the Skorokhod-$J_{1}$ topology.

$\left(\frac{1}{c^{\prime}}n^{-\frac{1}{\alpha}}X_{n}(nt),c^{\prime}n^{-(1-\frac{1}{\alpha})}H_{n}(nt),\left(\frac{c^{\prime}n^{-(1-\frac{1}{\alpha})}}{\sigma_{Y}^{2}}\right)^{1/2}\ell_{n}(nt)\right)_{t\in[0,1]}\xrightarrow[n\to\infty]{(d)}\Big{(}X^{\text{exc}}_{t},H_{t},\widehat{W}_{t}\Big{)}_{t\in[0,1]}.$

Although not stated in [Mar20, Theorem 1.1], it is implicit in the proof that
we can add the convergence of the Łukasiewicz paths to the above statement
(Marzouk works on a probability space where (32) holds and shows that the
snakes converge too).

From the convergence result of Theorem 3.9 we derive, similarly to [LG07],
some preliminary results on subsequential limits for the distances in
$M_{n}^{(\alpha)}$. Like in the Brownian case, a first straightforward
consequence of Theorem 3.9 is that

$\left(\left(\frac{c^{\prime}n^{-(1-\frac{1}{\alpha})}}{\sigma_{Y}^{2}}\right)^{1/2}d^{\circ}_{n}(ns,nt)\right)_{0\leq
s,t\leq 1}\xrightarrow[n\to\infty]{(d)}\left(D^{\circ}(s,t)\right)_{0\leq
s,t\leq 1},$ (38)

where $D^{\circ}$ is defined as in (1). Together with Lemma 3.7, this
immediately implies the following result, similarly to Proposition 3.2 and its
corollary (11) in [LG07] for the Brownian case.

######  Proposition 3.10.

The sequence of the laws of the processes
$\left(n^{-\frac{\alpha-1}{2\alpha}}d_{n}(ns,nt)\right)_{0\leq s,t\leq 1}$ is
tight in the space of probability measures on the continuous functions from
$[0,1]^{2}$ to $\mathbb{R}$.

In particular, subsequential limits exist: for any sequence $(n_{k})_{k\geq
1}$ there exists a subsequence $(k_{m})_{m\geq 1}$ for which there is a
limiting function $D$ from $[0,1]^{2}$ to $\mathbb{R}_{+}$ along which we have
the following joint convergence in distribution:

$\displaystyle\begin{split}&\left(\frac{1}{c^{\prime}}n^{-\frac{1}{\alpha}}X_{n}(nt),c^{\prime}n^{-\frac{\alpha-1}{\alpha}}H_{n}(nt),\left(\frac{c^{\prime}n^{-(1-\frac{1}{\alpha})}}{\sigma_{Y}^{2}}\right)^{1/2}\ell_{n}(nt),\left(\frac{c^{\prime}n^{-(1-\frac{1}{\alpha})}}{\sigma_{Y}^{2}}\right)^{1/2}d_{n}(ns,nt)\right)_{s,t\in[0,1]}\\\
&\qquad\xrightarrow[n\to\infty]{(d)}\left(X^{\text{exc}}_{t},H_{t},\widehat{W}_{t},D(s,t)\right)_{s,t\in[0,1]}.\end{split}$
(39)

###### Proof.

The first statement has the same proof as [LG07, Proposition 3.2]. The second
statement is the analogue of [LG07, Equation (11)] and comes from combining
the first statement with Theorem 3.9. ∎

It is important to note that a priori, the limiting function $D$ may depend on
the subsequence. In what follows, we only consider limits along such a
subsequence $(n_{k})_{k\geq 1}$. By the Skorokhod representation theorem, we
can further assume that the convergence (LABEL:eqn:subsequential_limits) holds
almost surely. This has the following consequence, that is similar the first
assertion of [LG07, Theorem 3.4] in the Brownian case.

######  Corollary 3.11.

The sequence of the laws of the metric spaces
$(V(M_{n}^{(\alpha)}),n^{-\frac{\alpha-1}{2\alpha}}d_{n},\nu_{n},\rho_{n})$ is
tight in the space of probability measures on the space $\mathbb{F}^{c}$ of
isometry classes of compact rooted metric spaces equipped with a finite Borel
measure, endowed with the pointed Gromov-Hausdorff-Prokhorov topology. In
particular, for any sequence $(n_{k})_{k\geq 1}$ there exists a subsequence
$(k_{m})_{m\geq 1}$ for which there exists a limiting compact rooted metric-
measure space $(M,D,\nu,\rho)$ such that

$\displaystyle\begin{split}\left(V(M^{(\alpha)}_{n_{k_{m}}}),\frac{\sqrt{c^{\prime}}}{\sigma_{Y}}n_{k_{m}}^{-\frac{\alpha-1}{2\alpha}}d_{n_{k_{m}}},\nu_{n_{k_{m}}},\rho_{n_{k_{m}}}\right)\xrightarrow[n\to\infty]{(d)}(M,D,\nu,\rho).\end{split}$
(40)

Moreover, $M$ is a quotient space of
$\mathbf{m}_{\alpha}=\mathcal{T}/\approx$, as defined after (2), and in
particular there is a canonical projection $\pi:[0,1]\to M$. The limiting
metric is the pushforward of the function $D$ appearing in
(LABEL:eqn:subsequential_limits) under $\pi$, the limiting measure $\nu$ is
the pushforward of Lebesgue measure under $\pi$, and the root $\rho$ is the
pushforward of $0$ under $\pi$.

###### Proof.

Let us define $C^{\prime}=\sqrt{c^{\prime}}/\sigma_{Y}$ for ease of notation.
Throughout the proof we will consider the space
$(V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\},C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n},\nu_{n},\rho_{n})$
in place of
$(V(M_{n}^{(\alpha)}),C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n},\nu_{n},\rho_{n})$
and for the rest of the proof let $\nu_{n}$ denote uniform measure on
$V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\}$; by the triangle inequality,
the two spaces necessarily have the same scaling limit.

Firstly, note that (LABEL:eqn:subsequential_limits) implies that the sequence
$(V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\},C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n})$
is tight in the Gromov-Hausdorff topology by [BBI01, Theorem 7.4.15]. By the
definitions of the various Gromov-Hausdorff and Gromov-Hausdorff-Prokhorov
distances, it immediately follows that the spaces are also compact with
respect to the pointed Gromov-Hausdorff-Prokhorov topology on adding the roots
and measures. Since the space $\mathbb{F}^{c}$ is Polish (this follows from a
very minor extension of [Mie09, Proposition 8]), this implies that there are
subsequential limits taking values in $\mathbb{F}^{c}$.

We now work on a probability space and along a subsequence where
(LABEL:eqn:subsequential_limits) holds almost surely. Our candidate for the
limiting metric space is $([0,1]/\sim_{D},D)$, where $s\sim_{D}t\iff
D(s,t)=0$. Note that $([0,1]/\sim_{D},D)$ is indeed a compact metric space
since all of the relevant properties are inherited directly from
(LABEL:eqn:subsequential_limits). We write $M=[0,1]/\sim_{D}$, with root
$\\{0\\}$, and let $\pi$ denote the canonical projection from $[0,1]$ to $M$.

We first show that the pointed Gromov-Hausdorff distance between
$(V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\},C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n},\rho_{n})$
and $(M,D,\\{0\\})$ vanishes almost surely using Theorem 3.2. In particular,
since $M_{n}^{(\alpha)}$ is defined as the image of
$(T_{n}^{(\alpha)},\ell_{n},\varepsilon_{n})$ under $\Phi$, it follows that
$(V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\},C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n},\rho_{n})$
is isometric to

$\left([0,1]/\sim_{n},C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n}(\lfloor
sn\rfloor,\lfloor tn\rfloor),0\right),$

where $\sim_{n}$ denotes the equivalence relation $s\sim_{n}t\iff
d_{n}\left(\lfloor sn\rfloor,\lfloor tn\rfloor\right)=0$. Recall from Section
3.4 that the two spaces are actually related via the projection
$s\mapsto\pi_{n}(\lfloor sn\rfloor)\in
V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\}$.

We construct a correspondence $\mathcal{R}_{n}$ between
$(V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\},C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n},\rho_{n})$
and $(M,D,\\{0\\})$ by including all points of the form $(\pi_{n}(\lfloor
ns\rfloor),\pi(s))$ where $s\in[0,1]$. We denote its distortion by $r_{n}$; it
then follows directly from (LABEL:eqn:subsequential_limits) that $r_{n}\to 0$
along the same subsequence.

To prove that the measures also converge on this space, we take the Gromov-
Hausdorff embedding
$F_{n}=(V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\})\sqcup M$ endowed with
the metric

$D_{F_{n}}(x,y)=\begin{cases}C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n}(x,y)&\text{
if }x,y\in V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\}\\\ D(x,y)&\text{ if
}x,y\in M\\\
\inf_{u,v\in\mathcal{R}_{n}}(C^{\prime}n^{-\frac{\alpha-1}{2\alpha}}d_{n}(x,u)+D(y,v)+\frac{1}{2}r_{n})&\text{
if }x\in V(M_{n}^{(\alpha)})\setminus\\{\partial_{n}\\},y\in M.\end{cases}$

We now claim that $d^{F_{n}}_{P}(\nu_{n},\nu)\rightarrow 0$ as
$n\rightarrow\infty$. To prove this, for each $0\leq i\leq n$ we define the
interval $I_{n,i}=[\frac{i}{n},\frac{i}{n}+\frac{1}{n})$. Now take a set
$A_{n}$ of vertices in $M_{n}^{(\alpha)}\setminus\\{\partial_{n}\\}$, and set

$A^{\prime}_{n}=\bigcup_{i:\pi_{n}(i)\in A_{n}}I_{n,i}.$

Let $A_{n}^{\prime\prime}=\pi(A^{\prime}_{n})$. We will show that
$A_{n}^{\prime\prime}\subset A_{n}^{r_{n}}$, where $E^{\varepsilon}$ denotes
the $\varepsilon$-neighborhood of a set $E$. To this end, take any $v\in
A_{n}^{\prime\prime}$. By our definitions, there exists $s\in A^{\prime}_{n}$
with $v=\pi(s)$ and $s\in I_{n,i}$ for some $i$ with $\pi_{n}(i)\in A_{n}$.
Consequently, we must have that $i=\lfloor ns\rfloor$, and hence
$(u_{i},v)\in\mathcal{R}_{n}$ and $D_{F}(u_{i},v)=\frac{1}{2}r_{n}$. It
follows that $v\in A_{n}^{r_{n}}$ and hence that $A_{n}^{\prime\prime}\subset
A_{n}^{r_{n}}$.

Also note that $\nu_{n}(A_{n})=\nu(A_{n}^{\prime\prime})$ by construction, and
so $\nu_{n}(A_{n})\leq\nu(A_{n}^{r_{n}})$.

Similarly, take any set $B\subset M$. We use the same argument to show that
$\nu(B)\leq\nu_{n}(B^{r_{n}})$. Let $B^{\prime}=\pi^{-1}(B)$, and

$B_{n}=\\{\pi_{n}(i)\colon\exists\ s\in B^{\prime}\text{ with }s\in
I_{n,i}\\}.$

Clearly $B^{\prime}\subset\bigcup_{i:\pi_{n}(i)\in B_{n}}I_{n,i}$ and so

$\nu(B)=Leb(B^{\prime})\leq Leb\left(\bigcup_{i:\pi_{n}(i)\in
B_{n}}I_{n,i}\right)=\frac{|B_{n}|}{n}=\nu_{n}(B_{n}).$ (41)

To conclude, note that if $v\in B_{n}$ and $v=\pi_{n}(i)$, then (by our
definitions) there necessarily exists $s\in B^{\prime}$ with $s\in I_{n,i}$,
hence $i=\lfloor sn\rfloor$ and therefore $(v,\pi(s))\in\mathcal{R}_{n}$.
Moreover $\pi(s)\in B$. Hence $B_{n}\subset B^{r_{n}}$, so
$\nu_{n}(B_{n})\leq\nu_{n}(B^{r_{n}})$, so combining with (41) gives
$\nu(B)\leq\nu_{n}(B^{r_{n}})$.

It follows that $d^{F_{n}}_{P}(\nu_{n},\nu)\leq r_{n}$, and hence converges to
zero as $n\rightarrow\infty$.

Moreover, it follows from Theorem 3.9 and from the construction of
$D_{\alpha}^{*}$ that $D\leq D_{\alpha}^{*}$, and therefore that
$(M,D,\\{0\\})$ is a quotient space of
$(\mathcal{T}/\approx,D_{\alpha}^{*},\rho)$. Since the roots are the same we
will henceforth identify them. ∎

######  Remark 3.12.

Note that each limiting space $(M,D,\nu,\rho)$ is naturally defined under
$\mathbb{N}^{(1)}_{0}$. However, thanks to points (i), (ii) and (iii) just
above (6), it can equally be defined under $\mathbb{N}^{(s)}$ for any $s>0$ or
simply under $\mathbb{N}_{0}$.

We stress again that the limiting space $(M,D,\nu,\rho)$ a priori depends on
the subsequence. Similarly to the Brownian case, the construction allows us to
immediately deduce some properties for $D$. This follows by the same proof as
[LG07, Proposition 3.3].

######  Proposition 3.13.

The following properties hold almost surely.

  1. (1)

The function $D$ is a pseudo-distance on $[0,1]$, i.e., for every
$s,t,u\in[0,1]$

     * •

$D(s,s)=0$

     * •

$D(s,t)=D(t,s)$

     * •

$D(s,u)\leq D(s,t)+D(t,u)$.

  2. (2)

For every $s,t\in[0,1]$, $D(s,t)\leq D_{\alpha}^{*}(s,t)\leq
D_{\alpha}^{\circ}(s,t)$ (recall the definitions from (1) and (2)).

  3. (3)

Let $s^{*}$ denote the location of the unique (see Proposition 2.2) infimum of
$\widehat{W}$. Then for every $s\in[0,1]$,
$D(s^{*},s)=\widehat{W}_{s}-\widehat{W}_{s^{*}}$. Consequently, for any $r\geq
0$, $\nu(B(\pi(s^{*}),r))=\overline{\mathcal{I}}[0,r]$ (recall the definition
of $\overline{\mathcal{I}}$ in (21)).

We end with a useful fact. Recall that $[[u,v]]$ is the direct tree path from
$u$ to $v$; the following is morally due to the fact that the geodesic cannot
take “shortcuts” to cut out the direct path.

######  Lemma 3.14.

For any $u,v\in M$,

$\displaystyle D(u,v)\geq Z_{u}+Z_{v}-2\min_{x\in[[u,v]]}Z_{x}.$

This can be proved formally using [Mie14, Proposition 2.3.8(ii)], which gives
the discrete analogue of this statement (and is deterministic), and taking
appropriate limits. Alternatively, one could also repeat the proof leading up
to [LG10, Corollary 3.2], which gives the analogous result in the Brownian
case, and for which the key input is the fact that the tree and the snake do
not have common increase points (in the stable case, this latter statement was
established in [ACM24, Theorem 1.4]). There is also an another viewpoint in
terms of cactuses: see [CLGM13, Remark 3.3] and the associated discussion.

##  4 Large volumes

The purpose of this section is to obtain some estimates on the probability of
having unusually large volumes at typical points in $(M,D,\nu,\rho)$.
Throughout this section, $U$ will denote a uniform random variable on either
$[0,\sigma]$ or $[0,1]$ (under $\mathbb{N}_{0}$ or $\mathbb{N}^{(1)}_{0}$ as
appropriate), chosen independently of everything else, $p(U)$ will denote its
image in $\mathcal{T}_{\alpha}$, and $\pi(U)$ will denote its image in the
limiting space $M$.

The upper bounds on the corresponding tail probabilities are very
straightforward; the lower bounds are more involved.

###  4.1 Upper tail bounds for large volumes

In this subsection we fix an $a>0$ and consider the law of
$M_{\varepsilon}(U_{a})$, where
$M_{\varepsilon}(s):=\varepsilon^{-\frac{2\alpha}{\alpha-1}}\nu(B(s,\varepsilon))$
and where $U_{a}$ is chosen “uniformly at level $a$ in the tree”. Formally, we
sample a pointed rooted tree and snake $(\mathcal{T}_{\alpha},U_{a},Z)$
according to $\mathbb{N}_{0}\times\ell^{a}$, where the local time measure
$\ell^{a}$ is as in (28). Note that this defines a probability measure on
pointed rooted trees since $\langle\ell^{a},1\rangle=1$ by Proposition 2.6.

######  Proposition 4.1.

There exists $c<\infty$ such that for all $a\in(0,\infty)$, $r>0$ and
$\lambda>1$,

$\mathbb{N}_{0}\left(\nu(B(U_{a},r))\geq\lambda
r^{\frac{2\alpha}{\alpha-1}}\right)\leq c\lambda^{-\frac{\alpha-1}{2\alpha}}.$

######  Remark 4.2.

We do not believe that the exponent for the tail decay is optimal.

###### Proof.

The proof is similar to that of [DLG06, Proposition 5.1]. We apply Proposition
2.6 and use the same notation, except that it will be more convenient to
consider the linear Brownian motion $(\xi_{t})_{t\geq 0}$ as starting at
$U_{a}$ rather than at the root. Let $\tau_{x}$ denote the hitting time of $x$
by $(\xi_{t})_{t\geq 0}$. It follows from Proposition 2.5, Proposition 2.6 and
Lemma 3.14 that, for any $\delta>0$,

$\displaystyle\mathbb{N}_{0}\left(\int\ell^{a}(ds)e^{-\delta
M_{\varepsilon}(s)}\right)$
$\displaystyle\geq\Pi_{0}\left(\exp\left\\{-\alpha\int_{0}^{\tau_{-\varepsilon}\wedge
a}\mathbb{N}_{0}(1-e^{-\delta\varepsilon^{-\frac{2\alpha}{\alpha-1}}\sigma})^{\alpha-1}dt\right\\}\right)$
$\displaystyle\geq\Pi_{0}\left(\exp\left\\{-\alpha\int_{0}^{\varepsilon^{-2}\tau_{-\varepsilon}}\varepsilon^{2}\mathbb{N}_{0}(1-e^{-\delta\varepsilon^{-\frac{2\alpha}{\alpha-1}}\sigma})^{\alpha-1}dt\right\\}\right)$
$\displaystyle=\Pi_{0}\left(\exp\left\\{-\alpha\int_{0}^{\tau_{-1}}\mathbb{N}_{0}(1-e^{-\delta\sigma})^{\alpha-1}dt\right\\}\right).$

Here in the second and third line we used Brownian scaling and the Itô scaling
property (18).

Note that $\tau_{-1}$ has density function

$\frac{1}{\sqrt{4\pi x^{3}}}e^{-1/4x}dx,$

and so a Tauberian theorem implies that there exists $c>0$ such that
$1-\Pi_{0}\left(\exp\left\\{-\theta\tau_{-1}\right\\}\right)\sim
c\theta^{1/2}$ as $\theta\downarrow 0$. Hence by (19) we can write for all
sufficiently small $\delta>0$:

$\displaystyle\Pi_{0}\left(\exp\left\\{-\alpha\int_{0}^{\tau_{-1}}\mathbb{N}_{0}(1-e^{-\delta\sigma})^{\alpha-1}dt\right\\}\right)$
$\displaystyle=\Pi_{0}\left(\exp\left\\{-\alpha\delta^{\frac{\alpha-1}{\alpha}}\cdot{\tau_{-1}}\right\\}\right)\geq
1-c\delta^{\frac{\alpha-1}{2\alpha}}.$

Recall from below Proposition 2.6 that
$\mathbb{N}_{0}\left(\int\ell^{a}(ds)\right)=1$. Hence applying Markov’s
inequality we deduce that

$\displaystyle\mathbb{N}_{0}\left(\int\ell^{a}(ds)\mathbbm{1}\\{M_{\varepsilon}(s)\geq\lambda\\}\right)$
$\displaystyle\leq(1-e^{-1})^{-1}\mathbb{N}_{0}\left(\int\ell^{a}(ds)(1-e^{-\lambda^{-1}M_{\varepsilon}(s)})\right)$
$\displaystyle\leq c(1-e^{-1})^{-1}\lambda^{-\frac{\alpha-1}{2\alpha}}.\qed$

This allows us to deduce the following.

######  Corollary 4.3.

Take any $\varepsilon>0$. Then $\mathbb{N}_{0}$-almost surely, it holds for
$\nu$-almost every $x$ in $M$ that

$\limsup_{r\downarrow 0}\frac{\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log
r^{-1})^{\frac{2\alpha+\varepsilon}{\alpha-1}}}=0.$

###### Proof.

We start by fixing an arbitrary $a>0$ and proving the result for
$\ell^{a}$-almost every $x$; this then extends straightforwardly to
$\nu$-almost every $x$ using (28). By Fubini’s theorem, it is sufficient to
prove the claim for $B(U_{a},r)$, where $U_{a}$ is chosen according to
$\ell^{a}$. This latter statement follows quite straightforwardly from
Proposition 4.1. In particular, combining the tail estimate of Proposition 4.1
with Borel-Cantelli along the subsequence $r_{k}=2^{-k}$ and using
monotonicity of volumes, we deduce that, $\mathbb{N}_{0}$-almost everywhere

$\limsup_{r\downarrow 0}\frac{\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log
r^{-1})^{\frac{2\alpha+\varepsilon}{\alpha-1}}}\leq
2^{\frac{2\alpha}{\alpha-1}}\limsup_{k\to\infty}\frac{\nu(B(x,r_{k}))}{r_{k}^{\frac{2\alpha}{\alpha-1}}(\log
r_{k}^{-1})^{\frac{2\alpha+\varepsilon}{\alpha-1}}}=0,\text{ for
}\ell^{a}\text{-almost every }x.$

Moreover by considering the restriction to $\\{1<\sigma<2\\}$ and rescaling,
we deduce the same result under $\mathbb{N}^{(1)}_{0}$. ∎

###  4.2 Lower tail bounds for large volumes

We now turn to establishing a similar lower bound. We anticipate that the
exponent $\alpha-1$ appearing here is sharp.

######  Proposition 4.4.

  1. (a)

There exists $c>0$ such that for all
$r>0,1<\lambda<r^{-\frac{\alpha}{\alpha-1}}$,

$\displaystyle\mathbb{N}^{(1)}_{0}\left(\nu(B(\pi(U),r))\geq\lambda
r^{\frac{2\alpha}{\alpha-1}}\right)$ $\displaystyle\geq
c\lambda^{-(\alpha-1)}.$

  2. (b)

There exists $c>0$ such that for all $r>0,1<\lambda<\infty$,

$\displaystyle\mathbb{N}_{0}\left(\nu(B(\pi(U),r))\in[\lambda
r^{\frac{2\alpha}{\alpha-1}},2\lambda r^{\frac{2\alpha}{\alpha-1}}]\right)$
$\displaystyle\geq c\lambda^{-(\alpha-1)}r^{-\frac{2}{\alpha-1}}.$

######  Remark 4.5.

The bound in part (a) should really hold for all $1<\lambda\ll
r^{-\frac{2\alpha}{\alpha-1}}$. The reason for the considering the interval
$[\lambda r^{\frac{2\alpha}{\alpha-1}},2\lambda r^{\frac{2\alpha}{\alpha-1}}]$
in part (b) will become apparent in the proof of Corollary 4.7, where it will
enable us to consider disjoint events for a decreasing sequence of radii.

Comparing with the following proposition for the volume of the ball around the
minimum of the snake (recall this is denoted $\pi(s^{*})$) leads to a proof of
Corollary 1.6. This contrasts with the Brownian case, where $\pi(s^{*})$ has
the same law as a vertex chosen according to $\nu$.

######  Proposition 4.6.

It holds that

$\limsup_{\varepsilon\downarrow
0}\varepsilon^{-\frac{2\alpha}{\alpha-1}}\mathbb{N}^{(1)}_{0}\left(\nu(B(\pi(s^{*}),\varepsilon))\right)<\infty.$
(42)

###### Proof.

This is a straightforward consequence of [ACM24, Theorem 1.3] and Proposition
3.13 (3) (to obtain the result under $\mathbb{N}^{(1)}_{0}$ we can restrict to
the event $\\{1<\sigma<2\\}$ and apply rescaling). ∎

###### Proof of Corollary 1.6.

If $\pi(s^{*})$ did have the same law as $\pi(U)$ for an independently chosen
$U$, then Markov’s inequality combined with Proposition 4.6 would imply that
there exists a constant $C<\infty$ such that for all
$r>0,1<\lambda<r^{-\frac{\alpha}{\alpha-1}}$,

$\mathbb{N}^{(1)}_{0}\left(\nu(B(\pi(U),r))\geq\lambda
r^{\frac{2\alpha}{\alpha-1}}\right)\leq C\lambda^{-1}.$

Taking for example $\lambda=\log r^{-1}$ we find that this result is
incompatible with Proposition 4.4(a) for all sufficiently small $r$. ∎

Similarly to the previous subsection, this has consequences for volume
fluctuations around typical points in $M$.

######  Corollary 4.7.

Take any $\varepsilon>0$. Then $\mathbb{N}_{0}$-almost everywhere, it holds
for $\nu$-almost every $x$ in $M$ that

$\limsup_{r\downarrow 0}\frac{\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log
r^{-1})^{\frac{1-\varepsilon}{\alpha-1}}}=\infty.$

We now turn to the proof of Proposition 4.4, which will proceed by a series of
lemmas. The strategy is to consider an event on which $U$ is close to a large
hub, which in turn has a correspondingly large volume.

Recall that $U$ is a uniform point in $[0,\sigma]$; we let $H_{U}$ denote the
height of $p(U)$ in $\mathcal{T}_{\alpha}$. We consider the spinal
decomposition along the spine from $\rho$ to $p(U)$ as in Section 2.2.4 and
recall that the sizes of hubs along this spine are determined by two
subordinators $U^{(1)}$ and $U^{(2)}$: one for each side of the path. We let
$I_{r}$ denote the part of this spine that is within tree distance $r^{2}$ of
$p(U)$, and consider the sizes of the parts of the hubs that are on the right
hand side of this part of the spine (i.e. the side that comes after $p(U)$ in
the contour exploration). Wlog we suppose that $U^{(1)}$ corresponds to the
right hand side; each jump of $U^{(1)}$ then corresponds to a “half-hub” of
the same size on the spine. We consider the event $A_{r}=A_{r,\lambda}$ (we
drop the $\lambda$ to lighten notation) that in $I_{r}$ there exists a half-
hub of size exceeding $\lambda r^{\frac{2}{\alpha-1}}$, and that the sum of
the sizes of all of the rest of the half-hubs in this interval is at most
$r^{\frac{2}{\alpha-1}}$.

Figure 5: Schematic representation of the sizes of hubs (and half-hubs) on
each side of the spine from $\rho$ to $p(U)$. There are subtrees attached
according to the Poisson process described in Section 2.2.4. The red and blue
parts can be treated independently.

We also define $\tilde{\sigma}_{r}$ to be the sum of the lifetimes of all the
subtrees grafted to the spine outside of the interval $I_{r}$ (i.e. the sum of
the volumes of all of the subtrees in blue in Figure 5). The quantity
$\tilde{\sigma}_{r}$ will be useful because it is a good approximation for
$\sigma$, but it can be treated independently of the red subtrees in
calculations.

######  Lemma 4.8.

There exists $c>0$ such that for all $r>0,1<\lambda\ll
r^{-\frac{2}{\alpha-1}}$,

$\mathbb{N}_{0}(A_{r},1<\tilde{\sigma}_{r}<2)\geq c\lambda^{-(\alpha-1)}.$

###### Proof.

We extend and abuse our notation slightly by writing $A_{r}(s)$ for the event
$A_{r}$ conditionally on the event $U=s$, and by writing
$A_{r}(U^{(1)}_{[0,r^{2}]})$ for the relevant event in terms of $U^{(1)}$ on
the interval $[0,r^{2}]$ where $U^{(1)}$ is the subordinator discussed above.

We will apply Proposition 2.4 and lower bound the right hand side by
considering only the contribution from $h\in[1,2]$. We take the setup
described above this lemma, and for convenience we will index everything by
distance from $p(U)$, so that a jump of $U^{(1)}$ at time $t$ corresponds to a
half-hub on the right hand side of the path at distance $t$ from $p(U)$.

In addition, we will let $\mathcal{P}_{1}$ denote the Poisson process of trees
grafted to the spine within distance $r^{2}$ of $p(U)$ (in red in Figure 5),
and $\mathcal{P}_{2}$ denote the Poisson process of trees grafted everywhere
else (in blue in Figure 5). These are independent. Similarly the jumps of
$U^{(1)}$ in $I_{r}$ are independent of the jumps of $U^{(1)}+U^{(2)}$ outside
of $I_{r}$. By Proposition 2.4, this allows us to factorise as follows:

$\displaystyle\mathbb{N}_{x}\left[\frac{\mathbbm{1}\\{1<\tilde{\sigma}_{r}<2\\}}{\sigma}\int_{0}^{\sigma}A_{r}(s)ds\right]$
$\displaystyle\qquad\geq\frac{1}{2}\int_{1}^{2}dhE^{0}\left[A_{r}(U^{(1)}_{[0,r^{2}]}),\sigma<2,1<\tilde{\sigma}_{r}<2\right]$
$\displaystyle\qquad\geq\frac{1}{2}\int_{1}^{2}dhE^{0}\left[A_{r}(U^{(1)}_{[0,r^{2}]}),\sigma-\tilde{\sigma}_{r}\leq\frac{1}{2}\right]E^{0}\left[\mathbb{P}_{\mathcal{P}_{2}}\left(1<\tilde{\sigma}_{r}<\frac{3}{2}\right)\right]$
$\displaystyle\qquad\geq\frac{1}{2}\int_{1}^{2}dh\left\\{P^{0}\left(A_{r}(U^{(1)}_{[0,r^{2}]})\right)-E^{0}\left[\mathbb{P}_{\mathcal{P}_{1}}\left(\sigma-\tilde{\sigma}_{r}>\frac{1}{2}\right)\right]\right\\}E^{0}\left[\mathbb{P}_{\mathcal{P}_{2}}\left(1<\tilde{\sigma}_{r}<\frac{3}{2}\right)\right].$

We start by bounding $P^{0}\left(A_{r}(U^{(1)}_{[0,r^{2}]})\right)$ (this
takes up most of the proof). We assume for now that $\lambda>2$ and separate
the jumps of $U^{(1)}_{[0,r^{2}]}$ into three groups:

  * •

those of size less than $r^{\frac{2}{\alpha-1}}$,

  * •

those of size more than $\lambda r^{\frac{2}{\alpha-1}}$,

  * •

those of size in $[r^{\frac{2}{\alpha-1}},\lambda r^{\frac{2}{\alpha-1}}]$.

By standard properties of subordinators and Poisson processes, these three
collections of jumps can be considered independently. The sum of the jumps of
$U^{(1)}$ in $I_{r}$ with size at most $r^{\frac{2}{\alpha-1}}$ is
stochastically dominated by an independent copy of $U^{(1)}_{r^{2}}$; hence
there is a constant probability that the sum is at most
$r^{\frac{2}{\alpha-1}}$ using scaling invariance of $U^{(1)}$ (which implies
that the probability is the same for all $r$). The number of jumps of size
exceeding $xr^{\frac{2}{\alpha-1}}$ is Poisson($C_{\alpha}x^{-(\alpha-1)}$),
where $C_{\alpha}$ is as in (5); hence for all $\lambda>2$ the probability
that there are zero with size in $[r^{\frac{2}{\alpha-1}},\lambda
r^{\frac{2}{\alpha-1}}]$ and exactly one with size exceeding $\lambda
r^{\frac{2}{\alpha-1}}$ is lower bounded by
$e^{-2C_{\alpha}}C_{\alpha}\lambda^{-(\alpha-1)}$. We obtain the desired
result by multiplying these three probabilities. This implies that
$P^{0}\left(A_{r}(U^{(1)}_{[0,r^{2}]})\right)\geq c\lambda^{-(\alpha-1)}$ for
all $\lambda>2$ and extends to all $\lambda>1$ by monotonicity.

It is then straightforward to verify that
$E^{0}\left[\mathbb{P}_{\mathcal{P}_{2}}\left(1<\tilde{\sigma}_{r}<\frac{3}{2}\right)\right]$
is lower bounded by positive quantities which don’t depend on $r$ (whenever
$r$ is sufficiently small), and that (by (7), standard results on tails of
stable processes and our assumption that $\lambda\ll r^{-\frac{2}{\alpha-1}}$)
there exists $c>0$ such that

$E^{0}\left[\mathbb{P}_{\mathcal{P}_{1}}\left(\sigma-\tilde{\sigma}_{r}>\frac{1}{2}\right)\right]\leq
cr^{-2}\ll c\lambda^{-(\alpha-1)},$

thus concluding the proof of the lemma. ∎

According to the formula in Proposition 2.4, conditionally on $U^{(1)}$, the
subtrees grafted to the right hand side of the spine in the interval $I_{r}$
have the law of a Poisson process with intensity

$\mathbbm{1}\\{t\in[0,r^{2}]\\}dU^{(1)}_{t}\otimes\mathbb{N}_{\xi_{t}}(dw),$
(43)

where $(\xi_{t})_{t\geq 0}$ is the Brownian motion appearing in Proposition
2.4; in other words the Brownian motion on the spine indexed by distance from
$\pi(U)$.

Conditionally on the event $A_{r}$, we now consider the following event
$B_{r}=B_{r,\lambda}$ (again we drop the dependence on $\lambda$ to ease
notation). Let $h=h_{r,\lambda}$ denote the hub of size at least $\lambda
r^{\frac{2}{\alpha-1}}$. Throughout the rest of the proof, when we consider
the values of the snake restricted to a subtree, we will retain its original
values (i.e. we will not re-root the snake to start at $0$).

The event $B_{r}$ is then defined as follows (recall that we parametrised time
so that $\xi_{0}$ is the value of the snake at $p(U)$, and that $\mathcal{R}$
denotes the range of the snake):

  * •

The values of the snake in $I_{r}$ are all in the interval
$[\xi_{0}-\frac{r}{16},\xi_{0}+\frac{r}{16}]$.

  * •

All subtrees grafted to a hub $h^{\prime}$ in $[0,r^{2}]$ that is not $h$
satisfy
$\mathcal{R}(Z)\subset[\xi_{h^{\prime}}-\frac{r}{16},\xi_{h^{\prime}}+\frac{r}{16}]$.

We then have the following.

######  Lemma 4.9.

There exists $c^{\prime}>0$ such that for all $r>0,1<\lambda<\infty$,

$\mathbb{N}_{0}(B_{r}|A_{r},1<\tilde{\sigma}_{r}<2)\geq c^{\prime}.$

###### Proof.

Note that the two events appearing in the bullet points above can be treated
independently of each other and also independently of the event
$\\{1<\tilde{\sigma}_{r}<2\\}$ since the Poisson process representation holds
conditionally on $\xi$, and since different parts of the Poisson process are
independent. The first event has uniformly positive probability by the scaling
property of Brownian motion. Since the total mass of all the other hubs
excluding $h$ is at most $r^{\frac{2}{\alpha-1}}$ on the event $A_{r}$, the
number of subtrees violating the second bullet point is stochastically
dominated by a Poisson random variable with mean
$2\left(16^{2}\frac{\alpha+1}{(\alpha-1)^{2}}\right)^{\frac{1}{\alpha-1}}$
(also using (20)), and hence is also zero with positive probability.

In summary, we obtained a uniformly positive probability for all both of the
desired events, so the result follows on multiplying these (and adjusting the
resulting constant lower bound to extend to all $\lambda>1$). ∎

On the event $A_{r}$ we now let $V_{r}=V_{r,\lambda}$ denote the collection of
subtrees grafted to $h$ in which the snake does not exit
$[\xi_{h}-r/8,\xi_{h}+r/8]$.

######  Lemma 4.10.

Let $c^{\prime}$ be the constant appearing in Lemma 4.9. Then there exists
$\tilde{c}>0$ such that, for all $r>0,2\tilde{c}<\lambda<\infty$,

$\mathbb{N}_{0}(\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}}|A_{r},1<\tilde{\sigma}_{r}<2)\geq
1-\frac{c^{\prime}}{2}.$

###### Proof.

First note that we can drop the dependence on $\tilde{\sigma}_{r}$ since this
is independent of all variables that only depend on the trees grafted to
$I_{r}$, and in particular $(V_{r},\mathbbm{1}_{\left\\{A_{r}\right\\}})$. By
Proposition 2.4, Proposition 2.5 and the scaling property (18), for any
$\kappa>0$ we can then write

$\displaystyle\mathbb{N}_{0}\left(e^{-\kappa V_{r}}|A_{r}\right)$
$\displaystyle\leq\exp\left\\{-\lambda
r^{\frac{2}{\alpha-1}}\mathbb{N}_{0}\left((1-e^{-\kappa\sigma})\mathbbm{1}\left\\{\frac{r}{8},\frac{-r}{8}\notin\mathcal{R}\right\\}\right)\right\\}$
$\displaystyle\leq\exp\left\\{-\lambda(\kappa
r^{\frac{2\alpha}{\alpha-1}}-\kappa^{2}r^{\frac{4\alpha}{\alpha-1}})\mathbb{N}_{0}\left(1\leq\sigma\leq
2,\frac{1}{8},\frac{-1}{8}\notin\mathcal{R}\right)\right\\}.$

Hence setting $\kappa=c\lambda^{-1}r^{-\frac{2\alpha}{\alpha-1}}$ and
$C=\mathbb{N}_{0}\left(1\leq\sigma\leq
2,\frac{1}{8},\frac{-1}{8}\notin\mathcal{R}\right)$ we see that

$\displaystyle\mathbb{N}_{0}\left(e^{-c\lambda^{-1}r^{-\frac{2\alpha}{\alpha-1}}V_{r}}|A_{r}\right)$
$\displaystyle\leq\exp\left\\{-C(c-c^{2}\lambda^{-1})\right\\}\leq\exp\left\\{-Cc/2\right\\}$

for all $\lambda>2c$. In particular, we can choose $c>0$ large enough so that
for all such $\lambda$,

$\displaystyle\mathbb{N}_{0}(V_{r}\leq c^{-1}\lambda
r^{-\frac{2\alpha}{\alpha-1}})\leq
e\exp\left\\{-Cc/2\right\\}\leq\frac{c^{\prime}}{2},$

where $c^{\prime}$ is as in Lemma 4.9. ∎

In particular the results of the last two lemmas imply the following.

######  Corollary 4.11.

There exist $c,\tilde{c}\in(0,\infty)$ such that:

  1. (a)

For all $r>0,1<\lambda<\infty$, it holds that $\mathbb{N}_{0}(B_{r}\text{ and
}\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}}|A_{r},1<\tilde{\sigma}_{r}<2)\geq c$.

  2. (b)

For all $r>0,1<\lambda<\infty$,

$\mathbb{N}_{0}(B_{r}\text{ and }\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}},A_{r},1<\tilde{\sigma}_{r}<2)\geq
c\lambda^{-(\alpha-1)}.$

  3. (c)

For all $r>0,1<\lambda<\infty$,

$\mathbb{N}_{0}(B_{r}\text{ and }\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}},A_{r},r^{\frac{2\alpha}{\alpha-1}}<\tilde{\sigma}_{r}<2r^{\frac{2\alpha}{\alpha-1}})\geq
c\lambda^{-(\alpha-1)}r^{-\frac{2}{\alpha-1}}.$

###### Proof.

Part (a) follows by subtracting the estimate of Lemma 4.10 (for the
complementary event) from that of Lemma 4.9. Note that we can remove the
restriction that $\lambda>2\tilde{c}$ using monotonicity and adapting the
constant $c$. Part (b) follows by combining (a) with Lemma 4.8. Part (c) then
follows by applying (b) with $r=1$, and then applying the scaling relation
(18). ∎

Part (a) in turn leads us to the following result.

######  Corollary 4.12.

For all $r>0,1<\lambda<r^{-\frac{\alpha}{\alpha-1}}$, there exists $c>0$ such
that $\mathbb{N}^{(1)}_{0}(B_{r}\text{ and }\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}}|A_{r})\geq c$.

###### Proof.

We first claim that Corollary 4.11 straightforwardly implies the following:
for all $\lambda\geq 1$ it holds that $\mathbb{N}_{0}(B_{r}\text{ and
}\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}}|A_{r},1<\sigma<3)\geq c$. For this we will see
that it is sufficient to show that

$\mathbb{N}_{0}(\sigma>3|A_{r},1<\tilde{\sigma}_{r}<2)\leq
1-\frac{1-c}{1-\frac{c}{2}}.$

By the Poisson process decomposition of Proposition 2.4 and [Ber96, Section
VIII.1, Proposition 4], we can apply Proposition 2.4 and (19) obtain that
there exists $C<\infty$ such that, for $\beta=\frac{2-\alpha}{\alpha}$,

$\displaystyle\mathbb{N}_{0}(\sigma>3|A_{r},1<\tilde{\sigma}_{r}<2)$
$\displaystyle\leq P^{0}\left((U^{(1)}+U^{(2)})_{r^{2}}\geq
r^{\beta}|U^{(1)}_{r^{2}}\geq\lambda
r^{\frac{2}{\alpha-1}}\right)+\mathbb{N}_{0}(\sigma>3|(U^{(1)}+U^{(2)})_{r^{2}}<r^{\beta}))$
$\displaystyle\leq
C\lambda^{\alpha-1}r^{2-\beta(\alpha-1)}+(1-e^{-1})(1-\exp\\{-r^{\beta}\mathbb{N}_{0}(1-e^{-\sigma})\\})$
$\displaystyle\leq
C\lambda^{\alpha-1}r^{2-\beta(\alpha-1)}+(1-e^{-1})(1-\exp\\{-r^{\beta}\\})\leq
Cr^{\frac{2-\alpha}{\alpha}}.$

In particular we use that $\lambda<r^{-\frac{\alpha}{\alpha-1}}\ll
r^{-\frac{2}{\alpha-1}}$ to give the final line. Hence

$\displaystyle\mathbb{N}_{0}(\sigma>3|A_{r},1<\tilde{\sigma}_{r}<2)\leq
Cr^{\frac{2-\alpha}{\alpha}}\leq 1-\frac{1-c}{1-\frac{c}{2}}$

for all sufficiently small $r>0$. Hence we obtain a lower bound for the
complementary event and deduce that

$\mathbb{N}_{0}(B_{r}\text{ and }\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}}|A_{r},1<\tilde{\sigma}_{r}<2,1<\sigma<3)\geq
1-(1-c)\frac{1-\frac{c}{2}}{1-c}=\frac{c}{2}.$

It is moreover straightforward to show that
$\mathbb{N}_{0}(1<\tilde{\sigma}_{r}<2|A_{r},1<\sigma<3)$ is lower bounded by
a constant using the same types of arguments (we omit the details) and hence
that there exists $c>0$ such that, for any
$r>0,1<\lambda<r^{-\frac{\alpha}{\alpha-1}}$,

$\displaystyle\mathbb{N}_{0}(B_{r}\text{ and }\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}}|A_{r},1<\sigma<3)$ $\displaystyle\geq
C\mathbb{N}_{0}(B_{r}\text{ and }\nu(V_{r})\geq\tilde{c}\lambda
r^{\frac{2\alpha}{\alpha-1}}|A_{r},1<\tilde{\sigma}_{r}<2,1<\sigma<3)$
$\displaystyle\geq C.$

This implies the stated result by rescaling and adjusting the constants
slightly. ∎

The final step is to connect $V_{r}$ to the ball of radius $r$ round $\pi(U)$
(recall that $V_{r}$ is the collection of subtrees grafted to $h$ in which the
snake does not exit $[\xi_{h}-r/8,\xi_{h}+r/8]$).

######  Lemma 4.13.

Almost surely under $\mathbb{N}^{(1)}_{0}(\cdot|A_{r},B_{r})$, it holds that
$V_{r}\subset B_{M}(\pi(U),r)$ for all sufficiently small $r$. The same is
true almost everywhere under $\mathbb{N}_{0}(\cdot|A_{r},B_{r})$,

###### Proof.

We start with the claim under $\mathbb{N}^{(1)}_{0}$ (it then transfers to
$\mathbb{N}_{0}$ using (6)) and wlog we work on a probability space where
(LABEL:eqn:subsequential_limits) holds almost surely.

On the event $A_{r}$, let $h$ be the large hub. Consider the subtrees grafted
to $h$ on the right hand side of the path from $\rho=p(0)$ to $p(U)$. We call
subtrees in which the snake exits the interval
$[\xi_{h}-\frac{r}{8},\xi_{h}+\frac{r}{8}]$ “bad”; the rest are “good”, see
Figure 6.

Figure 6: The setup around the hub $h$. Bad subtrees are in blue, and good
subtrees are in red.

Hence $V_{r}$ consists precisely of the collection of good subtrees. Note that
on the event $B_{r}$ this also implies that the snake does not exit
$[\xi_{0}-3r/16,\xi_{0}+3r/16]$ in the good subtrees.

On the event $B_{r}$, the snake does not exit the interval
$[Z_{\pi(U)}-\frac{r}{8},Z_{\pi(U)}+\frac{r}{8}]$ in any of the subtrees
grafted to the spine between $\pi(U)$ and $h$. It therefore follows from the
fact that $D\leq D_{\alpha}^{*}$ that $D(\pi(U),h)\leq\frac{r}{2}$.

In addition, each collection of consecutive good subtrees is coded by an
interval $[s,t]$ for which $p(s)=p(t)=h$. It therefore similarly follows from
the fact that $D\leq D_{\alpha}^{*}$ that any $x$ in a good subtree satisfies
$D(h,x)\leq\frac{r}{2}$. The claim therefore follows by the triangle
inequality. ∎

This allows us to tie up as follows.

###### Proof of Proposition 4.4.

The result in (a) follows from multiplying the probabilities in Lemma 4.8,
Corollary 4.12 and Lemma 4.13. Similarly, one can prove using Corollary
4.11(c) and Lemma 4.13 that there exists $c>0$ such that for all
$r>0,1<\lambda<\infty$,

$\displaystyle\mathbb{N}_{0}\left(\nu(B(\pi(U),r))\geq\lambda
r^{\frac{2\alpha}{\alpha-1}}\right)$ $\displaystyle\geq
c\lambda^{-(\alpha-1)}r^{-\frac{2}{\alpha-1}}.$

This is almost the same as the statement of part (b); to prove it as written,
one in fact needs to refine the estimate of Lemma 4.10 slightly to consider a
similar interval for the volumes there; this can be done straightforwardly by
counting subtrees in the Poisson process more precisely using the same
principles as in the rest of the proofs, so we have chosen to omit it. ∎

######  Remark 4.14.

The same logic, combined with the result of Miermont [Mie05, Proposition 2]
that for a hub $p(t)$

$\lim_{r\downarrow
0}\frac{\upsilon(B_{\mathcal{T}_{\alpha}}(p(t),r))}{r}=\Delta_{t},$ (44)

can be used to show that $\nu(B_{M}(\pi(t),r))$ is typically at least of order
$r^{2}$ when $p(t)$ is a hub of $\mathcal{T}_{\alpha}$.

####  4.2.1 Proof of Corollary 4.7

###### Proof of Corollary 4.7.

Again by Fubini’s theorem it is sufficient to prove this for (the projection
of) a uniform point $\pi(U)$. We set $r_{n}=2^{-n}$ and $\lambda_{n}=(\log
r_{n}^{-1})^{\frac{1-\varepsilon}{\alpha-1}}$. By Proposition 2.3, we can
further assume that $\pi(U)$ is in fact the root $\rho$ and we consider the
spine from $\rho$ to another independent uniform point as considered in
Proposition 2.4. Let $(\xi_{s})_{s\geq 0}$ be the linear Brownian motion
appearing there, and for each $n\geq 0$ set

$\tau_{n}=\inf\\{s\geq 0:|\xi_{s}|\geq\varepsilon r_{n}\\}.$

Note that it is indeed the case that $\tau_{n}<\infty$ for all sufficiently
large $n$, $\mathbb{N}_{0}$-almost everywhere.

By Proposition 2.4, the subtrees grafted to the right and left sides of the
spine in the interval $[\tau_{n+1},\tau_{n}]$ are coded by a Poisson process
and, conditionally on the sizes of the hubs on this part of the spine, the
Poisson processes on the right and left hand sides are independent. We will
consider two events, $A_{n}^{l}$ and $A_{n}^{r}$ on the left and right hand
sides respectively. The event $A_{n}^{l}$ will guarantee that all vertices
corresponding to time points in $[\tau_{n+1},\tau_{n}]$ along the spine are
within distance $r_{n}$ of the root. The event $A_{n}^{r}$ will ensure that
there is a subtree on the right rooted at point in $[\tau_{n+1},\tau_{n}]$
such that the volume of the ball of radius $r_{n}$ around the root in the
subtree is at least $\lambda_{n}r_{n}^{\frac{2\alpha}{\alpha-1}}$. These two
points combined will imply that the ball of radius $4r_{n}$ in the original
tree has volume at least $\lambda_{n}r_{n}^{\frac{2\alpha}{\alpha-1}}$, so it
is sufficient to show that $A_{n}^{l}\cap A_{n}^{r}$ occurs infinitely often,
almost surely.

Let us fix $\delta\in(0,1)$. It will be helpful to consider the event

$B_{n}:=\left\\{U^{(1)}_{\tau_{n}}-U^{(1)}_{\tau_{n+1}},U^{(2)}_{\tau_{n}}-U^{(2)}_{\tau_{n+1}}\in[(\delta
r_{n})^{\frac{2}{\alpha-1}},2(\delta r_{n})^{\frac{2}{\alpha-1}}]\right\\},$

where $U^{(1)}$ and $U^{(2)}$ are as in Section 2.2.4.

Note that the events $(B_{n})_{n\geq 0}$ are independent, and moreover it
follows by scaling invariance that there exists $c>0$ such that for all $n\geq
0$,

$P^{0}\otimes\Pi_{0}\left(B_{n}|\tau_{n}<\infty\right)>c.$

We then let $S^{l}_{n}$ denote the collection of subtrees that are grafted to
the spine to the left of $[0,\tau_{n}]$, and let $S^{r}_{n}$ denote the
collection of subtrees grafted to the right of $[\tau_{n+1},\tau_{n}]$. We let
(slightly abusing notation)

$A_{n}^{l}=\left\\{\sup_{x\in S_{n}^{l}}|Z_{x}|\leq r_{n}\right\\},\qquad
A^{r}_{n}=\left\\{\exists T\in S^{r}_{n}:\nu(B(\rho_{T},r_{n})\cap
T)\geq\lambda_{n}r_{n}^{\frac{2\alpha}{\alpha-1}}\right\\},$

where here $B(\rho_{T},r_{n})\cap T$ denotes the part of the ball intersecting
the tree $T$, measured with respect to the metric $D$ of $M$.

We first consider a bound for the probability of $A_{n}^{r}$. On the event
$B_{n}$ it follows from Proposition 4.4(b) that the number of such subtrees
stochastically dominates a Poisson random variable with parameter
$c_{\delta}\lambda_{n}^{-(\alpha-1)}$, and hence we deduce that there exists
$c_{\delta}^{\prime}>0$ such that, for all $n\geq 1$,

$\mathbb{P}\\!\left(A_{n}^{r}|\tau_{n}<\infty,B_{n}\right)\geq
c_{\delta}^{\prime}\lambda_{n}^{-(\alpha-1)}.$

Now note that, since the events $B_{n}\cap A_{n}^{r}$ refer to disjoint parts
of the spine for distinct $n$, they form a sequence of independent events.
Hence it follows from the second Borel-Cantelli lemma, the fact that
$\tau_{n}<\infty$ for all sufficiently large $n$ $\mathbb{N}_{0}$-almost
everywhere, our choice of $\lambda_{n}$ and the estimates above that
$B_{n}\cap A_{n}^{r}$ occurs infinitely often, $\mathbb{N}_{0}$-almost
everywhere.

We now suppose that this is indeed the case and condition on a subsequence
$n_{k}$ such that $B_{n_{k}}\cap A_{n_{k}}^{r}$ occurs for all $k\geq 1$. We
claim that there exists $c<\infty$ such that

$\mathbb{N}_{0}(A_{n}^{l}|B_{n}\cap A_{n}^{r},\tau_{n}<\infty)\geq
1-c\delta^{\frac{2}{\alpha-1}}.$

for all $n$. To see this, we apply Proposition 2.4 and (20) which imply that,
conditionally on $B_{n}$, the number of subtrees where the snake exits the
interval $[-r_{n},r_{n}]$ is stochastically dominated by a Poisson random
variable with parameter $c\varepsilon^{\frac{2}{\alpha-1}}$, for some
$c<\infty$, and hence is non-zero with probability less than
$c\varepsilon^{\frac{2}{\alpha-1}}$. In particular, it therefore follows from
Fatou’s lemma that, conditionally on the subsequence $(n_{k})_{k}$,

$\mathbb{N}_{0}((A_{n_{k}}^{l})^{c}\text{ for all }k\text{ large
enough}|B_{n_{k}}\cap A_{n_{k}}^{r}\text{ i.o.})\leq
c\delta^{\frac{2}{\alpha-1}}.$

Moreover, as discussed above, on the event $B_{n}\cap A_{n}^{l}\cap A_{n}^{r}$
it holds that
$\nu(B(\pi(U),4r_{n})\geq\lambda_{n}r_{n}^{\frac{2\alpha}{\alpha-1}}$. Since
$B_{n_{k}}\cap A_{n_{k}}^{r}\text{ i.o.}$ $\mathbb{N}_{0}$-almost everywhere,
we deduce that

$\mathbb{N}_{0}(\nu(B(\pi(U),4r_{n_{k}})\geq\lambda_{n_{k}}r_{n_{k}}^{\frac{2\alpha}{\alpha-1}}\text{
for all }k\text{ large enough})\geq 1-c\delta^{\frac{2}{\alpha-1}}.$

Since $\delta>0$ was arbitrary, this readily implies the statement of the
corollary.

∎

##  5 Small volumes

We now turn to looking at the occurrence of unusually small volumes at typical
points. As in Section 4.1, we fix an $a>0$ and consider the law of
$M_{\varepsilon}(U_{a})$, where
$M_{\varepsilon}(s):=\varepsilon^{-\frac{2\alpha}{\alpha-1}}\nu(B(s,\varepsilon))$
and where $U_{a}$ is as in the statement of Proposition 4.1.

###  5.1 Upper tail bounds for small volumes

######  Proposition 5.1.

There exist $\kappa>0,C<\infty,c>0$ such that for all $r>0$, $\lambda>1$ and
$a\in(r^{2},\infty)$,

$\mathbb{N}_{0}\left(\nu(B(U_{a},r))\leq\lambda^{-1}r^{\frac{2\alpha}{\alpha-1}}\right)\leq
Ce^{-c\lambda^{\kappa}}.$

In particular this holds for $\kappa=\frac{\alpha-1}{32\alpha}$.

###### Proof.

Outline. We will construct an event $E_{r,\lambda}$ on which
$\nu(B(U_{a},r))\geq\lambda^{-1}r^{\frac{2\alpha}{\alpha-1}}$ and show that
the required tail bound holds for $E_{r,\lambda}$. To this end, we take
$\beta,\beta^{\prime},\gamma,\gamma^{\prime}$ to be positive constants that
will be fixed later.

The setup and the argument is similar to that of the proof of Proposition 4.4.
We consider the spine from $U_{a}$ to $\rho$, and the associated decomposition
as in Proposition 2.6. For concreteness we say that the half of the spine
coming after $U_{a}$ in the contour order is the right hand side of the spine.
We also let $I_{r}$ denote the interval of the spine falling within distance
$r^{2}\lambda^{-2\gamma^{\prime}}$ of $U_{a}$. We also assume that the
Brownian motion appearing in Proposition 2.6 is indexed by its distance from
$U_{a}$ rather than from $\rho$. In addition we assume that the values of $Z$
are all shifted so that $Z_{U_{a}}=0$.

Figure 7: The setup between $U_{a}$ and $h$ on the event $E_{r,\lambda}$.
Between $U_{a}$ and $h$ the hubs are small and the snake has only small
fluctuations. The roots of larger subtrees where the snake exits the interval
$[-r\lambda^{-\gamma},r\lambda^{-\gamma}]$ are labelled in orange.

We then let $E_{r,\lambda}$ be the intersection of the following events (see
Figure 7):

  1. (1)

The Brownian motion $\xi$ does not exit the interval
$[-r\lambda^{-\gamma},r\lambda^{-\gamma}]$ on the interval $I_{r}$.

  2. (2)

There exists a hub of size at least $r^{\frac{2}{\alpha-1}}\lambda^{-\beta}$
on $I_{r}$. On this event we let $t_{\text{hub}}$ be the time index of the
first of these (i.e. closest to $U_{a}$).

  3. (3)

The sum of the sizes of the hubs appearing on the interval
$[0,t_{\text{hub}})$ is at most
$r^{\frac{2}{\alpha-1}}\lambda^{\beta^{\prime}-\beta}$.

  4. (4)

The number of subtrees grafted to the spine in the interval $I_{r}$ in which
the snake exits the interval $[-r\lambda^{-\gamma},r\lambda^{-\gamma}]$ is at
most $\frac{\lambda^{\gamma}}{8}$.

  5. (5)

The volume of the ball of radius $\frac{r}{2}$ centred at
$h:=\pi(t_{\text{hub}})$ is at least
$\lambda^{-1}r^{\frac{2\alpha}{\alpha-1}}$.

Construction. We first explain why
$\nu(B(U_{a},r))\geq\lambda^{-1}r^{\frac{2\alpha}{\alpha-1}}$ on the event
$E_{r,\lambda}$. The logic is very similar to that used to prove Proposition
4.4, and we will therefore omit some details. In particular, we let $N$ be the
number of subtrees appearing in point (4) above, we order them by their
(increasing) distance from the root, and for $i=1,\ldots,N$ we let $x_{i}$ be
the root of the $i^{th}$ subtree. In addition we let $x_{0}=U_{a}$ and
$x_{N+1}=h$. Hence each $x_{i}$ is on the spine from $U_{a}$ to $\rho$, and by
construction, it holds for each $i=0,\ldots,N$ that there exist
$t_{i},t_{i+1}\in[0,\sigma]$ such that $p(t_{i})=x_{i},p(t_{i+1})=x_{i+1}$,
and $\sup_{r,s\in[t_{i},t_{i+1}]}|Z_{r}-Z_{s}|\leq 4r\lambda^{-\gamma}$. Using
that $D\leq D_{\alpha}^{*}$ and combining with the triangle inequality
therefore gives that

$D(U_{a},h)\leq\frac{\lambda^{\gamma}}{8}\cdot
4r\lambda^{-\gamma}=\frac{r}{2}.$

Tail estimates. We now bound the probabilities of the complements of each of
the above events in turn. We will then deduce the result using a union bound.

  1. (1)

Using the Markov property of Brownian motion, we write

$\displaystyle\mathbb{P}\\!\left((\xi_{t})_{0\leq t\leq
r^{2}\lambda^{-2\gamma^{\prime}}}\not\subset[-r\lambda^{-\gamma},r\lambda^{-\gamma}]\right)=\mathbb{P}\\!\left((\xi_{t})_{0\leq
t\leq
1}\not\subset[-\lambda^{\gamma^{\prime}-\gamma},\lambda^{\gamma^{\prime}-\gamma}]\right)\leq
Ce^{-c\lambda^{\gamma^{\prime}-\gamma}}.$

  2. (2)

By Proposition 2.6, the number of such hubs is lower bounded by a Poisson
random variable with parameter $c\lambda^{\beta(\alpha-1)-2\gamma^{\prime}}$,
for an appropriate constant $c\in(0,\infty)$. Hence the probability that it is
zero is at most $e^{-c\lambda^{\beta(\alpha-1)-2\gamma^{\prime}}}$.

  3. (3)

Let $S_{r^{\frac{2}{\alpha-1}}\lambda^{-\beta}}$ denote the sum in question.
Then $S_{r^{\frac{2}{\alpha-1}}\lambda^{-\beta}}$ concerns the hubs that all
occur before $h$, and which are therefore all conditioned to have size at most
$r^{\frac{2}{\alpha-1}}\lambda^{-\beta}$. The number of such hubs moreover has
a geometric distribution with parameter $cr^{2}\lambda^{-\beta(\alpha-1)}$,
and by standard thinning properties of Poisson processes this geometric time
is independent of the sizes of the hubs. In addition applying the memoryless
property of the geometric distribution along with scaling invariance, it
therefore follows that for any $k>1$ (with the obvious extension of notation)

$\displaystyle\mathbb{P}\\!\left(S_{r^{\frac{2}{\alpha-1}}\lambda^{-\beta}}>kr^{\frac{2}{\alpha-1}}\lambda^{-\beta}\right)\leq\mathbb{P}\\!\left(S_{r^{\frac{2}{\alpha-1}}\lambda^{-\beta}}>r^{\frac{2}{\alpha-1}}\lambda^{-\beta}\right)^{k/2}=\mathbb{P}\\!\left(S_{1}>1\right)^{k/2},$

and so taking $k=\lambda^{\beta^{\prime}}$ we deduce that the probability in
question is bounded by $e^{-c\lambda^{\beta^{\prime}}}$.

  4. (4)

It will suffice to bound the probability of this event conditionally on the
high probability event in (3), since we can exclude the alternative using the
previous estimate. In this case the number of such subtrees is upper bounded
by a Poisson random variable with parameter
$\lambda^{\frac{2\gamma}{\alpha-1}+\beta^{\prime}-\beta}$. Hence, by a
Chernoff bound, the probability that it exceeds $\frac{\lambda^{\gamma}}{8}$
is upper bounded by
$C\exp\\{-c\lambda^{\gamma-\frac{2\gamma}{\alpha-1}-\beta^{\prime}+\beta}\\}$.

  5. (5)

The final estimate is the most involved. We condition on the event in (2), and
consider the collection of subtrees grafted to $h$ in which the snake does not
exit $[\xi_{h}-\frac{r}{8},\xi_{h}+\frac{r}{8}]$. Each of these subtrees are
contained within $B(h,\frac{r}{2})$ since $D\leq D_{\alpha}^{*}$ and since we
can bound fluctuations of the snake on the interval coding the subtree. It
will in fact suffice to consider only the subtrees with total volume in
$[r^{\frac{2\alpha}{\alpha-1}}\lambda^{-1},2r^{\frac{2}{\alpha-1}}\lambda^{-1}]$.
For a tree with total volume in this interval, the probability that the snake
does not exit the interval $[\xi_{h}-\frac{r}{8},\xi_{h}+\frac{r}{8}]$ is
uniformly bounded below by a positive constant by scaling invariance and since
$\lambda>1$. Hence the total number of trees satisfying both of these
properties is Poisson with parameter $c\lambda^{1/\alpha-\beta}$, for some
$c\in(0,\infty)$. Hence the probability that this number is zero is
$e^{-c\lambda^{1/\alpha-\beta}}$.

One can verify that it is possible to choose
$\beta,\beta^{\prime},\gamma,\gamma^{\prime}>0$ so that all of the
probabilities above decay as stretched exponentials in $\lambda$. One viable
option (which is not optimal) is to take

$\beta=\frac{3}{4\alpha},\beta^{\prime}=\frac{1}{4\alpha},\gamma=\frac{\alpha-1}{32\alpha},\gamma^{\prime}=\frac{\alpha-1}{32\alpha},$

which results in a final decay of $Ce^{-c\lambda^{\frac{\alpha-1}{32\alpha}}}$
via a union bound on the above events. ∎

Proposition 5.1 has the following consequences.

######  Corollary 5.2.

There exists $\kappa>0$ such that, $\mathbb{N}_{0}$-almost surely, it holds
for $\nu$-almost every $x$ in $M$ that

$\liminf_{r\downarrow
0}\frac{\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log\log
r^{-1})^{-\kappa}}>0.$

In particular this is true for $\kappa=\frac{32\alpha}{\alpha-1}$.

###### Proof.

This follows by the same proof as Corollary 4.3. In particular, we start by
fixing an arbitrary $a>0$ and proving the result for $\ell^{a}$-almost every
$x$; this then extends straightforwardly to $\nu$-almost every $x$ using (28).
By Fubini’s theorem, it is sufficient to prove the claim for $B(U_{a},r)$,
where $U_{a}$ is chosen according to $\ell^{a}$. We then take $r_{n}=2^{-n}$
and $\lambda_{n}=c^{\prime}\log\log r_{n}^{-1}$ where $c^{\prime}>2/c$ where
$c$ is the constant appearing in the statement of Proposition 5.1, and then
apply Borel-Cantelli, monotonicity and (28). ∎

We also note the following slight extension of Proposition 5.1. In what
follows, $\textsf{Diam}(\mathcal{T}_{\alpha})$ refers to the diameter of
$\mathcal{T}_{\alpha}$ measured with respect to the tree metric.

######  Proposition 5.3.

Let $U$ be Uniform($[0,\sigma]$) under $\mathbb{N}_{0}$. Then there exist
$\kappa,c>0,C<\infty$ such that for any $a>r^{2}$,

$\mathbb{N}_{0}\left(\nu(B(U,r))\leq\lambda^{-1}r^{\frac{2\alpha}{\alpha-1}}\middle|\textsf{Diam}(\mathcal{T}_{\alpha})>a\right)\leq
Ce^{-c\lambda^{\kappa}}.$

###### Proof.

We could have used the same argument as in Proposition 5.1 to obtain that
there exists $\kappa,c>0,C<\infty$ such that for any $a>r^{2}$,

$\mathbb{N}_{0}\left(\nu(B(\rho,r))\leq\lambda^{-1}r^{\frac{2\alpha}{\alpha-1}}\middle|H>a\right)\leq
Ce^{-c\lambda^{\kappa}}.$ (45)

(In particular, conditionally on $H>a$ we can consider the same decomposition
to the point $U_{a}$ and instead construct the same event in the vicinity of
$\rho$ rather than $U_{a}$.) Then note that, for all $a>0$,

$\\{H>2a\\}\subset\\{\textsf{Diam}(\mathcal{T}_{\alpha})>2a\\}\subset\\{H>a\\},$

and the tail estimate of [GH10, Proposition 5.6] shows that

$\frac{N(H>a)}{N(\textsf{Diam}(\mathcal{T}_{\alpha})>2a)}$

is bounded above by a constant, uniformly in $a>0$. Hence it follows from (45)
that we can modify the constant $C<\infty$ so that

$\mathbb{N}_{0}\left(\nu(B(\rho,r))\leq\lambda^{-1}r^{\frac{2\alpha}{\alpha-1}}\middle|\textsf{Diam}(\mathcal{T}_{\alpha})>2a\right)\leq
Ce^{-c\lambda^{\kappa}}.$

for all $a>0$. The claim then follows by uniform re-rooting invariance
(Proposition 2.3). ∎

######  Corollary 5.4.

There exists $\kappa>0$ such that, $\mathbb{N}^{(1)}_{0}$-almost surely,

$\liminf_{r\downarrow 0}\frac{\inf_{x\in
M}\nu(B(x,r))}{r^{\frac{2\alpha}{\alpha-1}}(\log r^{-1})^{-\kappa}}>0.$ (46)

In particular this is true for $\kappa=\frac{32\alpha}{\alpha-1}$.

###### Proof.

We will start by proving the result under
$\mathbb{N}_{0}(\cdot\cap\\{\textsf{Diam}(\mathcal{T}_{\alpha})>1,K<\sigma<2K\\})$
for an arbitrary $K>0$. Choose $\varepsilon>0$ and
$U\sim\textsf{Uniform}([0,\sigma])$ and for each $k\leq\lceil
r^{-\frac{2\alpha+2\varepsilon}{\alpha-1}}\rceil$ set
$U_{k}=(U+kr^{\frac{2\alpha+2\varepsilon}{\alpha-1}})\mod 1$. Note that it
follows from Lemma 2.1 that for any $\varepsilon>0$, the snake tip process
$(\widehat{W}_{t})_{t\in[0,1]}$ is almost everywhere
$\frac{\alpha-1}{2\alpha+\varepsilon}$ Hölder continuous. In particular, this
implies that for all sufficiently small $r>0$, we have for all $t\in[0,1]$
that

$\sup_{q,s\in[t,t+r^{\frac{2\alpha+2\varepsilon}{\alpha-1}}]}|\widehat{W}_{s}-\widehat{W}_{q}|\leq
r.$ (47)

Combined with the fact that $D\leq D_{\alpha}^{*}$, this implies that,
$\mathbb{N}_{0}$-almost surely, the collection of balls around the points
$(\pi(U_{k}))_{0\leq k\leq 2K\lceil
r^{-\frac{2\alpha+2\varepsilon}{\alpha-1}}\rceil}$ form an $\frac{r}{2}$-cover
of $M$ for all sufficiently small $r>0$. Then, by a union bound and
Proposition 5.3 (if $U_{k}+r^{\frac{2\alpha}{\alpha-1}}\lambda^{-1}<1$ and
$U_{k+1}+r^{\frac{2\alpha}{\alpha-1}}\lambda^{-1}>1$ we interpret the interval
below as the union of the appropriate two intervals),

$\displaystyle\mathbb{N}_{0}\left(\exists
k:\nu(B(U_{k},r))<r^{\frac{2\alpha}{\alpha-1}}\lambda^{-1},\textsf{Diam}(\mathcal{T}_{\alpha})>1,K<\sigma<2K\right)\leq
2K\lceil r^{-\frac{2\alpha+2\varepsilon}{\alpha-1}}\rceil
Ce^{-c\lambda^{\frac{\alpha-1}{32\alpha}}}.$

Hence setting $r_{n}=2^{-n}$ and $\lambda_{n}=c^{\prime}(\log
r_{n}^{-1})^{\frac{1}{\alpha-1}}$ for some sufficiently large $c^{\prime}>0$,
and then applying Borel-Cantelli as in the proof of Corollary 4.3, we obtain
that almost everywhere, $\nu(B(U_{k},\frac{r_{n}}{2}))\geq
r_{n}^{\frac{2\alpha}{\alpha-1}}\lambda_{n}^{-1}$ for all $k$ and all
sufficiently large $n$. In addition, on the event that the points
$(\pi(U_{k}))_{k}$ form an $\frac{r_{n}}{2}$-cover of $M$, we have that

$\inf_{u\in M}\nu(B(u,r_{n}))\geq\inf_{k}\nu(B(U_{k},\frac{r_{n}}{2}))\geq
r_{n}^{\frac{2\alpha}{\alpha-1}}\lambda_{n}^{-1}.$

This concludes the proof along the subsequence $r_{n}$ since both of the
required events hold for all sufficiently large $n$,
$\mathbb{N}_{0}(\cdot\cap\\{\textsf{Diam}(\mathcal{T}_{\alpha})>1,K<\sigma<2K\\})$
almost surely. The result extends to the limit $r\downarrow 0$ using
monotonicity, implying that (46) holds under the conditioning
$\mathbb{N}_{0}(\cdot\cap\\{\textsf{Diam}(\mathcal{T}_{\alpha})>1,K<\sigma<2K\\})$.