# Random Lévy Looptrees and Lévy Maps

Igor Kortchemski & Cyril Marzouk Centre de Mathématiques Appliquées (CMAP),
CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau,
France<EMAIL_ADDRESS>de Mathématiques Appliquées
(CMAP), CNRS, École polytechnique, Institut Polytechnique de Paris, 91120
Palaiseau, France<EMAIL_ADDRESS>
###### Abstract

Motivated by scaling limits of random planar maps in random geometry, we
introduce and study Lévy looptrees and Lévy maps. They are defined using
excursions of general Lévy processes with no negative jump and extend the
known stable looptrees and stable maps, associated with stable processes. We
compute in particular their fractal dimensions in terms of the upper and lower
Blumenthal–Getoor exponents of the coding Lévy process. In a second part, we
consider excursions of stable processes with a drift and prove that the
corresponding looptrees interpolate continuously as the drift varies from
$-\infty$ to $\infty$, or as the self-similarity index varies from $1$ to $2$,
between a circle and the Brownian tree, whereas the corresponding maps
interpolate between the Brownian tree and the Brownian sphere.

Figure 1: Left: two looptrees coded by (approximations of) an excursion of a
stable process with index $3/2$ and with a drift $+7$ (top) and $-20$
(bottom); the colours represent Gaussian labels. Right: the associated plane
maps embedded in three-dimensional space; the colours represent the degree of
the faces.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation and literature
    2. 1.2 Main results
    3. 1.3 Open questions and plan
  2. 2 Planar maps and labelled looptrees
    1. 2.1 The discrete setting
    2. 2.2 Lévy looptrees and Gaussian labels
    3. 2.3 Invariance principles
  3. 3 A spinal decomposition and some volume bounds
    1. 3.1 A spinal decomposition
    2. 3.2 A volume estimate in the looptree
    3. 3.3 On distances in the map
    4. 3.4 A technical lemma with Brownian motions
    5. 3.5 Proof the technical lemma
  4. 4 Computation of the dimensions
    1. 4.1 Hölder continuity of the distances
    2. 4.2 Fractal dimensions
  5. 5 Approximations in the continuum world
    1. 5.1 Convergence of looptrees and labels
    2. 5.2 Convergence of maps
  6. 6 Looptrees and maps driven by stable excursions with a drift
    1. 6.1 Main convergence results
    2. 6.2 Exponential tilting
    3. 6.3 The Brownian regime
    4. 6.4 Technical estimates on stable laws
    5. 6.5 The condensation regime
    6. 6.6 Remarks on the regime $\vartheta=-1$ and $\alpha\downarrow 1$
    7. 6.7 Proof the technical estimates on stable laws

##  1. Introduction

In short, our main purpose is to extend the construction of stable looptrees
and stable maps, which are random metric spaces built using stable Lévy
processes, to the setting of general Lévy processes with no negative jumps. We
calculate their fractal dimensions and study their behaviour as the coding
Lévy processes vary. Figure 1 presents some simulations of such non stable
objects.

###  1.1. Motivation and literature

In random geometry, the _Brownian sphere_ (also called the Brownian map) is a
random metric space almost surely homeomorphic to the two dimensional sphere,
whose study has attracted a lot of attention in the past twenty years. It has
in particular been shown to be the universal scaling limit of many different
models of random planar maps, see e.g. [LG13, Mie13, BLG13, BJM14, ABA17,
ABA21, Abr16, CLG19, Mar22a], where we recall that a planar map is the
embedding of a finite, connected multigraph on the two-dimensional sphere,
viewed up to orientation-preserving homeomorphisms. In two recent
breakthroughs, it was also shown to appear at the scaling limit of models of
random, non embedded, planar graphs [AFL23, Stu22]. The models of maps in the
previous references share the common feature that no face has a size which
dominates the other ones, so they vanish in the limit after applying a
suitable rescaling. In this sense, the Brownian sphere is the canonical model
of spherical random surfaces. It bears intimate connections with the Brownian
excursion, the Brownian tree, and the Brownian snake, which are also important
objects in probability. It also relates to the theory of Liouville Quantum
Gravity [MS20, MS21a, MS21b, MS21c]; let us refer to the expository papers
[Mil18, Gwy20, GHS23, She23] for a gentle introduction to this topic.

Parallel to this, quite some effort has been put into escaping the
universality class of the Brownian sphere, motivated in part by statistical
physics models on maps. In this direction, Le Gall & Miermont [LGM11]
constructed a one-parameter family of continuum models, called _stable maps_ ,
which are the scaling limits (along subsequences) of random planar maps with
large degree faces. They are built using decorated excursions of stable Lévy
processes with no negative jumps. Interestingly, the so-called _stable
looptrees_ can also be used to construct stable maps. Stable looptrees are
random metric spaces which were introduced in [CK14], also motivated by the
study of percolation clusters on random maps [CK15]. Looptrees have been used
in relation with maps [DMS21, MS21a, KR19, KR20, LG18, Ric18, SS19, BHS23],
but also for their own interest or in relation with other models [Arc20,
Arc21, AS23, BS15, CDKM15, CHK15, Kha22]. Very roughly speaking stable
looptrees are obtained by replacing branchpoints of the stable trees of
[Duq03, DLG05] by “loops” and then gluing these loops along the tree
structure. See Figure 2 for a representation of discrete looptrees. Formally,
stable looptrees are built from an excursion of a stable Lévy process with no
negative jumps.

Figure 2: Left: a plane tree in dotted lines and its looptree version. Right:
a slight variation, used here, obtained by merging each internal vertex of the
tree with its right-most offspring, as shown by the dashed lines in the
middle.

It is then natural to try to define maps and looptrees using more general Lévy
processes, still with no negative jumps, and to study their properties. In the
present work we construct them following [Mar22b]. Our motivation mainly comes
from the fact that non stable maps appear as scaling limits of natural models
of random maps. Indeed, in [KM23] we have considered random planar maps with
large faces conditioned to have fixed large numbers of vertices, edges, and
faces at the same time. By appropriately tuning these values, new scalings
limits related to non stable Lévy processes appear, see precisely Section 6
below. We also believe that such objects can appear as scaling limits of
block-weighted random maps very recently considered in [FS23]. We will see
that the framework of Lévy maps allows in some sense to interpolate between
the Brownian sphere, stable maps, and the Brownian tree. More generally, Lévy
maps describe the possible scaling limits of the so-called Boltzmann random
maps, which are sampled from a weight sequence on the face degrees which may
vary with the size of the map.

Our first main contribution is to compute the fractal dimensions (Hausdorff,
Minkowski, and packing dimensions) of Lévy looptrees and Lévy maps.
Intuitively, these dimensions quantify the “roughness” of these random metric
spaces, and their identification is an important question in random geometry.
Fractal dimensions of stable trees, stable maps, and stable looptrees have
been computed respectively in [HM04, LGM11, CK14], and dimensions of general
Lévy trees have been computed in [DLG05]. In another direction, Blanc-Renaudie
[BR23, BR22] has computed the dimensions of the so-called inhomogeneous
continuum random trees and their associated looptrees. His proofs use very
different ideas, more geometric rather than appealing to a coding function and
we believe that both points of view shed lights on these objects. In addition,
if the Lévy setting can intuitively be seen as a mixture of the inhomogeneous
setting, it is not clear how one can transfer the results of [BR23, BR22] to
the Lévy-driven objects.

###  1.2. Main results

Let us state right away our main theorems on the dimensions of Lévy looptrees
and maps, while deferring the precise definition of these objects to the next
section.

#### Background on Lévy processes

Throughout this work, we let $X$ be a Lévy process with no negative jump and
with paths of infinite variation. We refer to [Ber96, DLG02] for details on
Lévy processes and further discussions related to this work. Let us denote by
$\psi\colon\lambda\in[0,\infty)\mapsto\log\mathop{}\\!\mathbb{E}[\operatorname{e}^{-\lambda
X_{1}}]$ its Laplace exponent; according to the Lévy–Khintchine formula, the
latter takes the form:

$\psi(\lambda)=-d\lambda+\beta\lambda^{2}+\int_{0}^{\infty}(\operatorname{e}^{-\lambda
r}-1+\lambda r)\pi(\mathop{}\\!\mathrm{d}r),$

where $d\in\mathbb{R}$ is the drift coefficient, $\beta\geq 0$ is the Gaussian
parameter, and the Lévy measure $\pi$ satisfies $\int_{0}^{\infty}(r\wedge
r^{2})\pi(\mathop{}\\!\mathrm{d}r)<\infty$. From the condition of infinite
variation paths, we have either $\beta>0$ or both $\beta=0$ and
$\int_{0}^{1}r\pi(\mathop{}\\!\mathrm{d}r)=\infty$. The case of stable Lévy
processes corresponds to $\psi(\lambda)=\lambda^{\alpha}$ with $1<\alpha<2$,
for which $d=\beta=0$ and
$\pi(\mathop{}\\!\mathrm{d}r)=\alpha(\alpha-1)\Gamma(2-\alpha)^{-1}r^{-\alpha-1}\mathop{}\\!\mathrm{d}r$.

An idea to deal with general Lévy processes is to compare them with stable
processes. To this end, recall the lower and upper exponents of $\psi$ at
infinity introduced by Blumenthal & Getoor [BG61]:

$\boldsymbol{\gamma}\coloneqq\sup\mathopen{}\mathclose{{}\left\\{c\geq
0\colon\lim_{\lambda\to\infty}\lambda^{-c}\psi(\lambda)=\infty}\right\\}\qquad\text{and}\qquad\boldsymbol{\eta}\coloneqq\inf\mathopen{}\mathclose{{}\left\\{c\geq
0\colon\lim_{\lambda\to\infty}\lambda^{-c}\psi(\lambda)=0}\right\\}.$ (1.1)

We use in this work the notation from [DLG05] and the exponents
$\boldsymbol{\gamma}$ and $\boldsymbol{\eta}$ actually correspond to
$\beta^{\prime\prime}$ and $\beta$ respectively in [BG61]. Let us mention that
$\boldsymbol{\eta}$ relates to the Lévy measure $\pi$ by
$\boldsymbol{\eta}=\inf\\{c\geq
0\colon\int_{0}^{1}x^{c}\pi(\mathop{}\\!\mathrm{d}x)<\infty\\}$. We always
have $\lim_{\lambda\to\infty}\lambda^{-2}\psi(\lambda)=\beta$, and since we
assume that $\psi$ has paths with infinite variation, then
$\lim_{\lambda\to\infty}\lambda^{-1}\psi(\lambda)=\infty$. In particular
$1\leq\boldsymbol{\gamma}\leq\boldsymbol{\eta}\leq 2$; the two exponents
coincide (with $\alpha$) for $\alpha$-stable processes and more generally when
$\psi$ varies regularly at infinity, but they differ in general.

We shall assume throughout this paper the integrability condition:

$\int_{\mathbb{R}}|\mathop{}\\!\mathbb{E}[\operatorname{e}^{iuX_{t}}]|\mathop{}\\!\mathrm{d}u<\infty$
(1.2)

for every $t>0$. Then by inverse Fourier transform $X_{t}$ admits a continuous
density for every $t>0$ (even jointly continuous in time and space). These
transition densities can then be used to define a regular version
$X^{\mathrm{br}}$ of $(X_{t})_{t\in[0,1]}$ conditioned on $X_{0}=X_{1}=0$,
which we call the bridge version of $X$. Next, by exchanging the parts prior
and after the first minimum of $X^{\mathrm{br}}$, the so-called Vervaat
transform allows to define an excursion $X^{\mathrm{ex}}$, which is informally
a version of $(X_{t})_{t\in[0,1]}$ conditioned on $X_{0}=X_{1}=0$ and
$X_{t}>0$ for every $t\in(0,1)$. Let us refer to Section 2.2 and references
therein for a few details.

We can construct a looptree $\mathscr{L}(X^{\mathrm{ex}})$ from the excursion
path $X^{\mathrm{ex}}$, in a way which somehow extends the construction of a
tree from a continuous excursion via its contour exploration. Very informally,
we see each positive jump, say $\Delta X^{\mathrm{ex}}_{t}>0$, as a vertical
line segment with this length, turned into a cycle by gluing together its two
extremities. We then glue these cycles together by attaching the bottom
$X^{\mathrm{ex}}_{t-}$ of such a segment to the first point we meet on another
vertical segment (the one corresponding to the last time $s<t$ such that
$X^{\mathrm{ex}}_{s-}<X^{\mathrm{ex}}_{t-}<X^{\mathrm{ex}}_{s}$) when going
horizontally to the left, starting from $(t,X^{\mathrm{ex}}_{t-})$. In
reality, due to the infinite variation paths, two macroscopic cycles never
touch each other, and when the Gaussian parameter $\beta$ is nonzero, the
looptree contains infinitesimal “tree parts”. In the case when
$X^{\mathrm{ex}}$ is the Brownian excursion, the looptree
$\mathscr{L}(X^{\mathrm{ex}})$ actually has no cycle and reduces to the usual
Brownian tree. The formal definition of $\mathscr{L}(X^{\mathrm{ex}})$ is
given in Section 2.2, let us only mention that it is a metric measured space,
which is the quotient of the interval $[0,1]$ by a continuous pseudo-distance
defined from $X^{\mathrm{ex}}$.

#### Fractal dimensions

Let us denote by $\dim_{H}$ the Hausdorff dimension, by $\dim_{p}$ the packing
dimension, and by $\operatorname{\underline{\dim}}$ and
$\operatorname{\overline{\dim}}$ respectively the lower and upper Minkowski
dimensions (sometimes also called box counting dimensions). We refer to
[Mat95, Chapter 4 and 5] for definitions and basic properties of these
dimensions. Recall in particular that the Minkowski dimensions of a metric
space $(E,d)$ are defined as:

$\operatorname{\underline{\dim}}E=\liminf_{\varepsilon\downarrow 0}\frac{\log
N(\varepsilon)}{\log
1/\varepsilon}\qquad\text{and}\qquad\operatorname{\overline{\dim}}E=\limsup_{\varepsilon\downarrow
0}\frac{\log N(\varepsilon)}{\log 1/\varepsilon},$

where $N(\varepsilon)\coloneqq\min\\{k\geq 1\colon\exists x_{1},\dots,x_{k}\in
E\text{ such that }E\subset\bigcap_{i=1}^{k}B(x_{i},\varepsilon)\\}$ is the
minimal number of $\varepsilon$-balls required to cover the space.

######  Theorem 1.1.

Almost surely, it holds

$\dim_{H}\mathscr{L}(X^{\mathrm{ex}})=\operatorname{\underline{\dim}}\mathscr{L}(X^{\mathrm{ex}})=\boldsymbol{\gamma}\qquad\text{and}\qquad\dim_{p}\mathscr{L}(X^{\mathrm{ex}})=\operatorname{\overline{\dim}}\mathscr{L}(X^{\mathrm{ex}})=\boldsymbol{\eta},$

where $1\leq\boldsymbol{\gamma}\leq\boldsymbol{\eta}\leq 2$ are the exponents
defined in (1.1).

This theorem extends [CK14, Theorem 1.1] in the case of $\alpha$-stable Lévy
processes for which $\boldsymbol{\gamma}=\boldsymbol{\eta}=\alpha$ for each
$\alpha\in(1,2)$. In the Brownian case $\alpha=2$, we also recover the
dimension $2=\boldsymbol{\gamma}=\boldsymbol{\eta}$ of the Brownian tree. It
is also consistent with [BR22, Theorem 2.3].

In the discrete setting, looptrees equipped with suitable integer labels on
the vertices encode bipartite planar maps, see Figure 3 for an example. This
corresponds to a reformulation of the celebrated bijection from [BDFG04] which
is written in terms of the sort of dual two-type tree of the looptree,
obtained for every cycle by joining every vertex on this cycle to an extra
vertex inside. It also extends the famous Cori–Vauquelin–Schaeffer bijection
for quadrangulations, in which case the looptree is simply a tree in which
each edge is doubled into a cycle of length $2$.

Figure 3: Left: a well-labelled looptree. Right: a bipartite plane map with
vertices labelled by their graph distance to a distinguished vertex (labelled
$0$). Middle: the bijection linking them.

We shall prove in Theorem 2.1 that when discrete excursions which encode
discrete looptrees converge in distribution after suitable scaling to
$X^{\mathrm{ex}}$, then the rescaled labelled looptrees converge to the
looptree $\mathscr{L}(X^{\mathrm{ex}})$ equipped with a random Gaussian field
of labels. This implies, by now classical arguments originally due to Le Gall
[LG07], that the associated rescaled discrete maps are tight, and thus
converge _along subsequences_ to some limit metric space, see Theorem 2.4
below. We shall denote by $M$ any such subsequential limit, which we call a
_Lévy map_ associated with $X^{\mathrm{ex}}$.

######  Theorem 1.2.

Almost surely, it holds

$\dim_{H}M=\operatorname{\underline{\dim}}M=2\boldsymbol{\gamma}\qquad\text{and}\qquad\dim_{p}M=\operatorname{\overline{\dim}}M=2\boldsymbol{\eta},$

where $1\leq\boldsymbol{\gamma}\leq\boldsymbol{\eta}\leq 2$ are the exponents
defined in (1.1).

Again, this extends the calculation of the dimension of the stable maps in
[LGM11] as well as the Brownian sphere [LG07].

Let us say a few words on the main techniques of the proofs. The lower bounds
on the dimensions follow from an upper bound on the rate of decay of the
volume of balls centred at a uniform random point as the radius tends to $0$.
The latter is established for both looptrees and maps in Section 3. In the
case of looptrees, it relies on a spinal decomposition, which rephrases the
usual spinal decomposition of a tree and is formally described in terms of the
coding Lévy excursion. A geometric argument, summarised in Figure 7, allows to
include the ball in a simpler set defined using hitting times of Lévy
processes. The core of the argument is the control of such quantities. By
local absolute continuity of the excursion near an independent uniform random
time with a bi-infinite Lévy process $(X_{t})_{t\in\mathbb{R}}$, we are able
to transfer estimates for that set obtained for the unconditioned process to
its excursion. The volume of balls in the map is controlled similarly, with a
close geometric argument, now depicted in Figure 10. The estimates are much
more challenging in the case of maps, since here one needs to control the
Gaussian labels on the looptree.

Upper bounds on the dimensions rely on Hölder continuity estimates, obtained
in Section 4. Recall that both the looptree and the map are a quotient of the
interval $[0,1]$ by some pseudo-distance; we prove there that the canonical
projections are Hölder continuous. For looptrees, this is obtained by cutting
the excursion path into small pieces, whenever it makes a jump larger than a
small threshold, and controlling the variation between two such jumps; again
we rely on local absolute continuity with respect to an unconditioned Lévy
process. Hölder continuity estimates for the map are then easily derived from
this, using the representation with labels on the looptree; by their Gaussian
nature, the regularity of the labels is nearly half that of the looptree
(which leads to the factor two in the dimensions).

#### Convergence of Lévy-driven objects

We next discuss the convergence of these continuum labelled looptrees and maps
when the driving Lévy excursion varies. We will in particular identify natural
families of random compact metric spaces which interpolate between the
Brownian tree and the Brownian sphere. To keep this introduction as short as
possible, we state here our results in the particular case where the driving
Lévy process is a spectrally positive Lévy process _with a drift_. As was
already mentioned, this case naturally appears when considering large discrete
stable random planar maps, conditioned on having a fixed number of edges,
vertices, and faces at the same time in a near-critical regime (see Section 6
for a precise statement).

Fix $\alpha\in(1,2)$ and $\vartheta\in(-\infty,\infty)$ and let
$X^{\alpha,\vartheta,\mathrm{ex}}$ denote an excursion with duration $1$ of
the $\alpha$-stable Lévy process with drift $-\vartheta$, characterised by the
Laplace exponent $\psi(\lambda)=\lambda^{\alpha}-\vartheta\lambda$. We choose
to let $-\vartheta$ denote the drift instead of $\vartheta$ in order to see
the bridge version as describing the fluctuations of the stable process
without drift and conditioned to end at position $\vartheta$ at time $1$.
Further, let $\mathscr{L}(X^{\alpha,\vartheta,\mathrm{ex}})$ and
$M^{\alpha,\vartheta}$ denote respectively the looptree coded by
$X^{\alpha,\vartheta,\mathrm{ex}}$ and the corresponding Lévy map. By
application of Theorem 1.1 and Theorem 1.2, all the four dimensions are equal
in this case, to $\alpha$ and $2\alpha$ respectively.

The next result describes the behaviour of $X^{\alpha,\vartheta,\mathrm{ex}}$,
$\mathscr{L}(X^{\alpha,\vartheta,\mathrm{ex}})$, and $M^{\alpha,\vartheta}$
for any fixed $\alpha\in(1,2)$ when $\vartheta\to\pm\infty$. It shows in
particular that appropriately rescaled, the map $M^{\alpha,\vartheta}$
interpolates between the Brownian sphere and the Brownian tree as $\vartheta$
varies from $-\infty$ to $\infty$. See Figure 4 for simulations of
$X^{\alpha,\vartheta,\mathrm{ex}}$, and Figure 1 for simulations of
$\mathscr{L}(X^{\alpha,\vartheta,\mathrm{ex}})$ and of $M^{\alpha,\vartheta}$.

######  Theorem 1.3.

Fix $\alpha\in(1,2)$. The following convergences in distribution hold for the
Skorokhod $J_{1}$ topology for paths and the Gromov–Hausdorff–Prokhorov
topology for metric spaces.

  1. (i)

As $\vartheta\to-\infty$, we have:

$s_{\alpha,\vartheta}^{1/2}\cdot
X^{\alpha,\vartheta,\mathrm{ex}}\enskip\mathop{\longrightarrow}^{(d)}_{\vartheta\to-\infty}\enskip\boldsymbol{e}\qquad\text{with}\qquad
s_{\alpha,\vartheta}=\frac{|\vartheta|^{(2-\alpha)/(\alpha-1)}}{(\alpha-1)\alpha^{1/(\alpha-1)}},$

where $\boldsymbol{e}$ is the standard Brownian excursion, and further:

$(4s_{\alpha,\vartheta})^{1/2}\cdot\mathscr{L}(X^{\alpha,\vartheta,\mathrm{ex}})\enskip\mathop{\longrightarrow}^{(d)}_{\vartheta\to-\infty}\enskip\mathscr{T}\qquad\text{and}\qquad(9s_{\alpha,\vartheta})^{1/4}\cdot
M^{\alpha,\vartheta}\enskip\mathop{\longrightarrow}^{(d)}_{\vartheta\to-\infty}\enskip\mathscr{S},$

where $\mathscr{T}$ is the standard Brownian tree and $\mathscr{S}$ the
standard Brownian sphere.

  2. (ii)

As $\vartheta\to\infty$, we have:

$(\vartheta^{-1}\cdot
X^{\alpha,\vartheta,\mathrm{ex}}_{(1-t)-})_{t\in[0,1]}\enskip\mathop{\longrightarrow}^{(d)}_{\vartheta\to\infty}\enskip(t\,\mathbf{1}_{\\{t\neq
1\\}})_{t\in[0,1]}$

as well as

$\vartheta^{-1}\cdot\mathscr{L}(X^{\alpha,\vartheta,\mathrm{ex}})\enskip\mathop{\longrightarrow}^{(d)}_{\vartheta\to\infty}\enskip\mathscr{C}\qquad\text{and}\qquad\vartheta^{-1/2}\cdot
M^{\alpha,\vartheta}\enskip\mathop{\longrightarrow}^{(d)}_{\vartheta\to\infty}\enskip\mathscr{T},$

where $\mathscr{C}$ is the circle with unit perimeter and $\mathscr{T}$ is
again the standard Brownian tree.

We interpret the convergence of the path in the second regime as
$\vartheta^{-1}X^{\alpha,\vartheta,\mathrm{ex}}_{t}\to(1-t)\mathbf{1}_{\\{0<t\leq
1\\}}$; the latter is not right-continuous at $0$, which is the reason why we
reverse time in the statement.

Figure 4: Simulations of excursions of $\alpha$-stable processes with a drift
$-\vartheta$ with a fixed duration. Top: $(\alpha,\vartheta)=(1.5,-8)$ and
$(\alpha,\vartheta)=(1.5,0)$. Bottom: $(\alpha,\vartheta)=(1.5,10)$ and
$(\alpha,\vartheta)=(1.9,20)$.

Let us give some intuition concerning the asymptotic behaviour of
$X^{\alpha,\vartheta,\mathrm{ex}}$. When $\vartheta\to-\infty$, any positive
jump for the stable process conditioned to end at $\vartheta$ becomes very
costly, and in the limit it does not jump at all, so one must expand the
fluctuations (and indeed $s_{\alpha,\vartheta}\to\infty)$, which in the limit
are naturally described by a Brownian bridge, hence a Brownian excursion after
applying the Vervaat transform. On the other hand, the regime when
$\vartheta\to\infty$ exhibits a new instance of the so-called “one big jump”
phenomenon [DDS08, AL11, BBY23, KR02, DJ12], in that the easiest way for the
stable process to end at a large $\vartheta$ is to make a single jump of size
approximately $\vartheta$, and behave microscopically compared to $\vartheta$
otherwise. It results in a so-called “condensation phenomenon” already
observed in several models of (loop)trees and maps [JS15, Kor15, KR19,
Mar22a].

Let us note that when $\vartheta=0$, the behaviour of stable excursions and
stable looptrees as $\alpha\to 1$ and $\alpha\to 2$ have been obtained in
[CK14]. Our general results actually allow $\alpha$ and $\vartheta$ to vary at
the same time, see precisely Theorem 6.1 and Corollary 6.2 below.

In a general setting, the convergence of looptrees and maps when the driving
Lévy excursion varies is discussed in Theorem 5.2 and Theorem 5.3. It is
essentially established that if the transition densities of a sequence of Lévy
processes converge uniformly to those of another Lévy process, then the
associated labelled looptrees converge in law, and further the corresponding
Lévy maps converge in law after extracting a subsequence.

Let us say a few words on the techniques used to derive these general results.
First, tightness of the looptree distances is easy but that of the labels is
not: we argue by adapting the proof the usual Kolmogorov criterion and using
some general concentration bounds for Gaussian processes. We then turn to the
finite-dimensional marginals, that is the pairwise distances and joint labels
of finitely many points in the looptree. The contribution of the large cycles
can easily be controlled, and the contribution of the microscopic parts is
proved by similar arguments to [CK14], by proving a weak law of large numbers.
Indeed, roughly speaking, the Lévy excursion codes the distances on the
looptree when one always follows the cycles on their right, whereas the true
distance minimises left and right lengths. We show that this roughly divides
the lengths by two. We actually argue on the unconditioned process first and
then use a local absolute continuity argument for the excursion; the
convergence of the densities is used at this point. Once the convergence of
labelled looptrees is proved, the convergence of maps along a subsequence, and
the identification of the Brownian sphere and tree in the extreme regimes, is
routine as the usual arguments to control discrete maps can be adapted.

Finally, the proof Theorem 1.3, and its actual more general version in Theorem
6.1, uses different arguments according as wether we aim for a Brownian
excursion in the limit or a one-big jump. In the former case we prove the
convergence of the transition densities and use the preceding paragraph. A key
idea is to use an exponential change of measure to make the imposed drift
$\vartheta$ “typical” for the stable Lévy process. The one-big jump regime is
treated very differently. We consider a large threshold, but small compared to
$\vartheta$, and we show that the trajectory obtained by removing all the
jumps larger than this threshold, under the bridge law and rescaled by
$\vartheta^{-1}$, converges to the linear drift $-t$. This is done via
Hoeffding-type concentration bounds available since the jumps are bounded.
Then we show that the bridge can only make at most two jumps larger than our
threshold with high probability, and then actually only one by an
exchangeability argument. This shows that our rescaled bridge converges to
$(\mathbf{1}_{\\{U\leq t\\}}-t)_{t\in[0,1]}$ where $U$ has the uniform
distribution on $[0,1]$; the claim follows from the Vervaat transform. Again
here, we argue first on the unconditioned process and then transfer the
results to the bridge by absolute continuity. This requires to understand the
asymptotic behaviour of the stable density, and we provide some new results
about it, as $\alpha\to 1$, in Section 6.4.

###  1.3. Open questions and plan

This work leaves open several natural questions. First, here the Lévy map $M$
is only defined as a subsequential limit of discrete maps, associated with
discrete excursions which converge to $X^{\mathrm{ex}}$. This was the case of
the Brownian sphere until uniqueness was finally proved [LG13, Mie13] and this
is still the case for stable maps, for which uniqueness is under current
investigation [CMR]. We believe that several geometric ideas of [CMR] can
apply to prove uniqueness in the more general Lévy setting when the Gaussian
parameter $\beta$ vanishes, although the technical inputs are more involved.
When the Gaussian parameter is nonzero, it is likely that other ideas are
needed.

In another direction, after computing the fractal dimensions of Lévy trees
[DLG05], Duquesne and Le Gall [DLG06] investigated the existence and the
computation of a gauge function associated with a nontrivial Hausdorff or
packing measure for the Brownian tree. Duquesne [Duq12] expressed more
generally the exact packing measure of Lévy trees, and proved on the contrary
that there is no (regular) Hausdorff measure, even for the stable trees
[Duq10]. More recently, Le Gall [LG22] expressed the Hausdorff measure of the
Brownian sphere. It is then natural to investigate these questions on the Lévy
looptrees and maps.

In a last direction, in Section 6 we consider the asymptotic behaviour of the
models related to $\alpha$-stable processes with a drift $-\vartheta$ as the
two parameters vary. Our results cover all regimes except the two limits
$(\alpha,\vartheta)\to(1,-1)$ and $(\alpha,\vartheta)\to(2,\infty)$, for which
nothing can be said in general and the behaviour depends more precisely on the
speed of each convergence. It is however interesting to look further at these
two points, which may provide other interpolations between the two extreme
behaviours displayed in Theorem 1.3.

The rest of this paper is organised as follow. In Section 2 we first define
formally discrete and continuous looptrees and maps. We stress again that Lévy
looptrees are defined directly from the excursion $X^{\mathrm{ex}}$, whereas
Lévy maps are defined as subsequential limits of discrete maps, in Theorem
2.4. Then in Section 3 we state and prove technical bounds on the volume of
small balls in looptrees and maps, which provide lower bounds for the fractal
dimensions. In Section 4 we first state and prove Hölder continuity estimates
for the looptree and map distances, which provide upper bounds for the fractal
dimensions. We then prove Theorem 1.1 and Theorem 1.2. Then Section 5
discusses the convergence of Lévy looptrees and maps when the associated
processes converge. Finally Section 6 applies these results to the model of
stable processes with a drift; there we state and prove extensions of Theorem
1.3.

##  2. Planar maps and labelled looptrees

We first define in Section 2.1 the discrete labelled looptrees and plane maps
as well as their coding by discrete paths. Then we construct analogously
labelled looptrees associated with (excursions of) Lévy processes in Section
2.2. We shall stay brief and refer to [Mar22b] and references therein for
details. See also [Kha22] for more results on looptrees coded by a function.
Finally in Section 2.3 we state and prove invariance principles, showing that
the discrete objects converge to the continuum ones. We define at this
occasion the Lévy maps as subsequential limits of discrete maps.

###  2.1. The discrete setting

Recall that a plane map is the embedding of a planar graph on the sphere and
viewed up to direct homeomorphisms. In order to break symmetries, we shall
always distinguish an oriented edge, called the root edge of the map. The
faces of the map are defined as the connected components of the complement of
the edges. The face to the left of the root edge is called the boundary of the
map and can be seen as an outer face. The degree of a face is defined as the
number of edges surrounding it, counted with multiplicity; a plane map is
bipartite if and only if all faces have even degree.

#### Looptrees

A looptree, as represented on the right of Figure 2, is by definition a plane
map with the property that every edge has exactly one side incident to the
outer face. This necessarily implies that all the other faces are simple
cycles and are edge-disjoint; also no edge is pending inside the outer face.
One can naturally order the edges of a looptree, oriented to keep the boundary
to their left, by following its contour: we start from the root edge $e_{0}$,
then recursively, the edge $e_{k+1}$ is the leftmost edge originating from the
tip of $e_{k}$. If the looptree has $n\geq 1$ edges, then the recursion ends
at $e_{n}=e_{0}$, once the tour of the looptree is complete. One can then
define a path $W^{n}=(W^{n}_{k};0\leq k\leq n+1)$ by setting $W^{n}_{0}=0$ and
letting for each $0\leq k\leq n$ the increment $W^{n}_{k+1}-W^{n}_{k}$ be as
follows:

  1. (i)

Let $W^{n}_{k+1}-W^{n}_{k}$ be equal to the length minus $1$ of the cycle to
the right of the edge $e_{k}$ _if_ no previously listed edge
$e_{0},\dots,e_{k-1}$ belongs to this cycle;

  2. (ii)

Otherwise let $W^{n}_{k+1}-W^{n}_{k}=-1$.

One can check that $W^{n}$ is a _Łukasiewicz path_ with length $n+1$, i.e. a
path whose increments all belong to
$\mathbb{Z}_{\geq-1}=\\{-1,0,1,2,\dots\\}$, started from $W^{n}_{0}=0$, with
$W^{n}_{n+1}=-1$ and $W^{n}_{k}\geq 0$ for every $1\leq k\leq n$. We extend it
to a càdlàg path defined on the real interval $[0,n+1]$ by letting it drift at
speed $-1$ between two integer times. We refer to Figure 5 for an example.
This construction is invertible, we shall describe below the converse
construction directly in the continuum setting. We denote by
$\mathsf{L}(W^{n})$ the looptree coded by the path $W^{n}$. Let us mention
that there is a well-known bijection between Łukasiewicz paths and plane
trees, in which the increments of $W^{n}$ code the offspring numbers minus $1$
when performing a depth first search exploration of the tree; the tree
associated with $W^{n}$ is the dotted tree on the left of Figure 2.

Figure 5: _Left:_ A looptree equipped with a good labelling. _Right:_ Its
Łukasiewicz path on top and its label process on the bottom.

#### Labels and maps

Let us further equip a looptree with a _good labelling_ on the vertices. For
every edge, oriented to keep the boundary on its left, we require that the
difference of labels between the tip and the origin is an integer larger than
or equal to $-1$. This fixes uniquely the labels of the vertices up to a
global additive constant, which we prescribe by requiring the origin of the
root edge to have label $0$. One can again encode the labels into a discrete
path $Z^{n}=(Z^{n}_{0},\dots,Z^{n}_{n})$ by following the contour of the
looptree. We further extend this path continuously by linear interpolation.
See again Figure 5 for an example. One can notice that the labels of the
vertices of any cycle of the looptree, when read clockwise, form a bridge with
increments larger than or equal to $-1$; if the cycle has length $\ell\geq 1$,
there are $\binom{2\ell-1}{\ell}$ such bridges.

The bijection from [BDFG04, Section 2] once reformulated in terms of looptrees
shows that looptrees equipped with a good labelling are in one-to-two
correspondence with bipartite plane maps equipped with a distinguished vertex,
called _pointed maps_. Moreover the bijection can be constructed by a simple
algorithm in both directions (in addition to [BDFG04], let us refer to
[Mar22b, Section 2.2]) and enjoys the following properties:

  1. (i)

The map and the looptree have the same amount of edges;

  2. (ii)

The cycles of the looptree correspond to the faces of the map, and the length
of a cycle is half the degree of the associated face;

  3. (iii)

The vertices of the looptree correspond to the non-distinguished vertices of
the map, and the labels, shifted so the minimum is $1$, equal the graph
distances to the distinguished vertex in the map.

  4. (iv)

The factor $2$ in the correspondence only comes from the loss of the
orientation of the root edge in the map.

Let us refer to Figure 3 for a graphical representation of this bijection. The
reader acquainted with the Schaeffer bijection between labelled trees and
quadrangulations can observe that the latter is a particular case of the
present bijection, when every cycle of the looptree has length $2$, so the
looptree is really just a tree in which every edge is doubled. In words, the
construction from a labelled looptree to a map works as follows. First, shift
all the labels so the minimum is $1$. Then assign to every corner of the outer
face a successor, which is the first one which carries a smaller label when
following the contour of the looptree, in the sense defined at the beginning
of this section but extended by periodicity. The fact that the labelling is
good implies that the difference of label between a corner and its successor
is always exactly $-1$. The successor of the corners which carry the label $1$
is an extra vertex labelled $0$ in the outer face. Then the map is obtained by
removing the edges of the looptree and instead linking every corner to its
successor.

#### Boltzmann distributions

A fairly general way to sample random maps is by using a Boltzmann
distribution [MM07]. This is also motivated by the study of statistical physic
models on maps, as argued e.g. in [LGM11, Section 8]. Given a sequence
$(q_{k})_{k\geq 1}$, one assigns to any finite bipartite pointed map a weight
given by:

$\prod_{f\text{ face}}q_{\deg(f)/2},$

where $\deg(f)$ is the degree of the face $f$. One can then sample a pointed
map with $n$ edges proportionally to its weight (assuming that the total
weight of $n$-edged maps, which is a finite set, is non-zero). By the previous
bijection, this transfers to sampling a looptree with $n$ edges and equipped
with a good labelling proportionally to the weight $\prod_{c\text{
cycle}}q_{\text{length}(c)}$. Recall that each cycle $c$ with length $\ell\geq
1$ in a looptree can be labelled in $\binom{2\ell-1}{\ell}$ different ways.
Then the looptree, without the labels, is sampled proportionally to the
weight:

$\prod_{c\text{
cycle}}\binom{2\text{length}(c)-1}{\text{length}(c)}q_{\text{length}(c)},$

and then conditionally on it, the labels on each cycle are uniform random
bridges with increments larger than or equal to $-1$, independent of each
other. Recall that the length of the cycles of the looptree equal the size
plus one of the nonnegative increments of the associated Łukasiewicz path.
This Łukasiewicz path, with duration $n+1$, is thus sampled proportionally to
the weight:

$\prod_{k=1}^{n+1}\widetilde{q}_{x_{k}}\qquad\text{where}\qquad\widetilde{q}_{j}=\binom{2j+1}{j+1}q_{j+1}\enskip\text{for}\enskip
j\geq 0\enskip\text{and}\enskip\widetilde{q}_{-1}=1,$

and where $x_{k}$ is the $k$’th increment of the path. By the so-called cyclic
lemma [Pit06, Lemma 6.1] one can realise such a random excursion by first
sampling a _bridge_ with duration $n+1$, starting from $0$ and ending at $-1$,
proportionally to the previous weight, and then cyclicly exchanging the
increments at the first time this bridge achieves its overall minimum to turn
it into an excursion.

Let us mention that this paper focuses on the continuum objects, so we
consider pointed maps with $n$ edges which is the simplest model to study.
However one may consider maps with $n$ vertices or $n$ faces instead, or even
more general notions of size, see e.g. [Mar22a, Section 6.4] and references
therein, or combine them and condition the map by its number of vertices,
edges, and faces at the same time [KM23]. In another direction, one can
consider maps without any distinguished vertex. Note that when we do not
condition on the number of vertices, distinguishing a vertex in the map leads
to a size-biasing of the latter. However standard arguments in the theory
allow to show that the effect of this size-biasing disappears in the limit,
provided some technical inputs, see e.g. [Mar22a, Proposition 6.4] and
references therein. Finally one can consider maps with a boundary [Bet15,
BM17] by prescribing the degree of the face to the right of the root edge in
the map, and hence the length of the cycle to the right of the root edge in
the looptree, so finally the size of the first increment of the Łukasiewicz
path. By removing this first jump, one obtains a so-called first-passage
bridge, and all this present work extends readily to this setting, see
[Mar22a, Mar22b]. We simply consider maps without boundary to ease the
notation.

###  2.2. Lévy looptrees and Gaussian labels

Recall from Section 1.2 the model of Lévy processes that we consider: $X$ has
no negative jump, infinite variation paths, and its characteristic function is
integrable (1.2). Let us mention that, this integrability condition holds as
soon as $\beta>0$ and also when $u^{-2}|\log
u|^{-1}\int_{0}^{u}r^{2}\pi(\mathop{}\\!\mathrm{d}r)\to\infty$ as $u\to 0$,
where $\pi$ is the Lévy measure of $X$, see [Kni96, Section 2], which is taken
from [Kal81, Section 5]. By inverse Fourier transform, the process $X$ then
admits transition densities, say $(p_{t}(x))_{t>0,x\in\mathbb{R}}$, which are
jointly continuous in time and space as shown e.g. in [Kni96, Lemma 2.4]. In
addition, they never vanish, and precisely Sharpe [Sha69] shows that if one of
them vanishes somewhere, then the process $X$ is, up to a drift, either a
subordinator or the negative of one.

Following [Kni96, Section 2], thanks to the positivity and joint continuity of
the transition densities, we may use them to define a bridge with unit
duration from $0$ to any $\vartheta\in\mathbb{R}$. Precisely, the conditional
law $\mathop{}\\!\mathbb{P}(\,\cdot\mid X_{1}=\vartheta)$ is characterised by
the following absolute continuity relation with respect to the unconditioned
path: for any $\varepsilon\in(0,1)$ and any continuous and bounded function
$F$, it holds

$\mathop{}\\!\mathbb{E}\bigl{[}F\bigl{(}(X_{t})_{t\in[0,1-\varepsilon]}\bigr{)}\mid
X_{1}=\vartheta\bigr{]}=\mathop{}\\!\mathbb{E}\Bigl{[}F\bigl{(}(X_{t})_{t\in[0,1-\varepsilon]}\bigr{)}\,\frac{p_{\varepsilon}(\vartheta-
X_{1-\varepsilon})}{p_{1}(\vartheta)}\Bigr{]}.$ (2.1)

One can actually start from this identity and show that the family of
processes it defines extends consistently to a process on the whole interval
$[0,1]$, see again [Kni96]. We shall denote by $X^{\mathrm{br}}$ the bridge
from $0$ to $0$, that is the path $(X_{t})_{t\in[0,1]}$ under
$\mathop{}\\!\mathbb{P}(\,\cdot\mid X_{1}=0)$. This bridge achieves its
overall minimum at a unique time $U$ almost surely, which has the uniform
distribution on $[0,1]$ by [Kni96, Lemma 2.1 and Theorem 2.1]. We then define
the Vervaat transformation $\mathcal{V}$, which exchanges the pre and post
minimum, namely for every $t\in[0,1]$ we set:

$X^{\mathrm{ex}}_{t}=(\mathcal{V}X^{\mathrm{br}})_{t}=X^{\mathrm{br}}_{U+t\bmod
1}-X^{\mathrm{br}}_{U}.$ (2.2)

Then $X^{\mathrm{ex}}$ is a unit duration excursion version of the process
$X$, see also [UB14, Theorem 4].

#### Lévy looptrees

Let $Y$ be any càdlàg function on $[0,\infty)$ with no negative jump, which
can be e.g. the Lévy process $X$ or its excursion $X^{\mathrm{ex}}$ extended
to $0$ on $[1,\infty)$. Let us recall from [Mar22b] the construction of the
looptree coded by $Y$ as well as random Gaussian labels on it. This originates
from [LGM11, CK14] which applies in a pure jump case, where the continuous
part $C$ below vanishes. First define from $Y$ a partial order $\preceq$ by
setting for every $s,t\geq 0$:

$s\prec t\qquad\text{when both}\qquad s<t\quad\text{and}\quad
Y_{s-}\leq\inf_{[s,t]}Y.$

We then set $s\preceq t$ when either $s\prec t$ or $s=t$. In analogy to the
discrete setting, when $Y$ is the Łukasiewicz path associated with a plane
tree, we interpret $s=\sup\\{r\geq 0\colon r\prec t\\}$ as the parent of $t$
and $\Delta Y_{s}$ as the degree of $s$. Then for any $s,t\geq 0$, we let

$s\wedge t\coloneqq\sup\\{r\geq 0\colon r\prec s\text{ and }r\prec t\\}$

denote their last common ancestor. For every pair $s,t\geq 0$, let us set

$R^{t}_{s}\coloneqq\inf_{r\in[s,t]}Y_{r}-Y_{s-}\qquad\text{if}\enskip s\preceq
t,$ (2.3)

and let $R^{t}_{s}=0$ otherwise, which is the case as soon as $Y$ does not
jump at time $s$ for example. Still in analogy to the discrete setting this
corresponds to the number of siblings of $t$ that lie to its right. Another
way to visualise this quantity is to consider for $t$ fixed the dual path
$Y^{t}=(Y_{t-}-Y_{(t-s)-})_{s\in[0,t]}$ and its running supremum
$\overline{Y}\vphantom{Y}^{t}=(\sup_{r\in[0,s]}Y^{t}_{r})_{s\in[0,t]}$, then
$R^{t}_{t-s}=\Delta\overline{Y}\vphantom{Y}^{t}_{s}$ is the overshoot of
$Y^{t}$ when it jumps to a new record at time $s$. In general, the path
$Y^{t}$ can also make records in a continuous way, thus we also define a
continuous process by:

$C_{t}=|\\{\overline{Y}\vphantom{Y}^{t}_{r};r\in[0,t]\\}|=|\\{\inf_{[s,t]}Y;s\in[0,t]\\}|,$
(2.4)

where $|\cdot|$ stands for the Lebesgue measure. It is standard that the
function defined for every $0\leq s\leq t\leq 1$ by:

$d_{C}(s,t)=d_{C}(t,s)=C_{s}+C_{t-}-2\inf_{[s,t]}C$ (2.5)

is a pseudo-distance and the quotient space $T_{C}$ obtained by identifying
all pairs of times at $d_{C}$-distance $0$ is a compact rooted real tree
[LG05].

Let $Y=X$ be our Lévy process. In this case the dual path $X^{t}$ has the same
law as $X$, so $C_{t}$ has the same law as the Lebesgue measure of the range
of the supremum process. The latter coincides with the measure of the range of
the ladder process, obtained by time-changing this supremum process by the
inverse local time at the supremum. In turn the ladder process is well-known
to be a subordinator with Laplace exponent $\psi(\lambda)/\lambda$, see e.g.
[DLG02, Lemma 1.1.2], whose drift coincides with the Gaussian parameter
$\beta$ of $X$. Since the Lebesgue measure of the range of a subordinator
equals its drift, then we conclude that for Lévy processes, we have:

$C\text{ vanishes}\quad\text{if and only if}\quad\beta=0.$

By the Vervaat transform and absolute continuity of the bridge, this holds
also for $Y=X^{\mathrm{ex}}$.

If one wants again to draw an analogy with the discrete setting, viewing $X$
or $X^{\mathrm{ex}}$ as the Łukasiewicz path of a tree, then when $\beta>0$,
the nontrivial process $C$ coincides with $\beta H$ where $H$ is the so-called
height process [DLG02, Equation 14], so $T_{C}=\beta T_{H}$ is the so-called
Lévy tree coded by $X$ or $X^{\mathrm{ex}}$, up to a multiplicative constant.
However when $\beta=0$, the process $C$ vanishes so the tree $T_{C}$ is
reduced to a single point, whereas it is shown in [DLG02] that one can still
define a nontrivial height process $H$, and thus a nontrivial Lévy tree
$T_{H}$. We shall not need the height process in this work.

We may now define the looptree coded by $Y$. We associate with any jump time
$t\geq 0$ a cycle $[0,\Delta Y_{t}]$ with $\Delta Y_{t}$ identified with $0$,
equipped with the metric $\delta_{t}(a,b)=\min\\{|a-b|,\Delta Y_{t}-|a-b|\\}$
for all $a,b\in[0,\Delta Y_{t}]$. For definiteness, if $\Delta Y_{t}=0$, then
we set $\delta_{t}(0,0)=0$. Define the looptree distance with parameter $a\geq
0$ between any $s,t\geq 0$ by:

$d^{a}_{\mathscr{L}(Y)}(s,t)\coloneqq\sum_{s\wedge t\prec r\prec
s}\delta_{r}(0,R^{s}_{r})+\sum_{s\wedge t\prec r\prec
t}\delta_{r}(0,R^{t}_{r})+\delta_{s\wedge t}(R^{s}_{s\wedge t},R^{t}_{s\wedge
t})+a\,d_{C}(s,t),$ (2.6)

where here and below, a sum over an empty set is null and where $R^{t}_{r}$ is
defined in (2.3) and $d_{C}$ is the tree distance defined in (2.5). Let us
refer to Figure 6 for a schematic representation of this distance.

Figure 6: A schematic representation of the looptree distance between two
times $s$ and $t$ and the coding excursion. In different colours are
highlighted the four components of the distance in (2.6): from left to right
in the latter in red, orange, blue, and green. Each portion of the geodesic
from $s$ to $t$ is numbered and associated with the corresponding part of the
excursion.

It can be checked that $d^{a}_{\mathscr{L}(Y)}$ is indeed a continuous pseudo-
distance on $[0,1]$ for every $a$, which moreover satisfies:

$d^{1}_{\mathscr{L}(Y)}\leq d_{Y},$ (2.7)

where the right-hand side is defined in (2.5), see [Mar22b, Proposition 3.2]
which extends [CK14, Lemma 2.1] when $C=0$. Henceforth we denote by
$\mathscr{L}^{a}(Y)$ the quotient space obtained by identifying all pairs of
times at pseudo-distance $0$. We shall simply write $\mathscr{L}(Y)$ and
$d_{\mathscr{L}(Y)}$ for $\mathscr{L}^{1}(Y)$ and $d^{1}_{\mathscr{L}(Y)}$.
The value of $a>0$ does not affect the fractal dimensions of the looptree, and
Theorem 1.1 extends to $\mathscr{L}^{a}(X^{\mathrm{ex}})$. This constant only
appears in the invariance principles in Section 2.3. In the case where $Y$ is
a Brownian excursion, we have $C=Y$ so $\mathscr{L}^{a}(Y)=a\cdot T_{C}=a\cdot
T_{Y}$ is a scaled Brownian tree coded by $Y$.

Observe that a discrete looptree as defined in Section 2.1 is a map, while a
continuous looptree $\mathscr{L}$ is a compact metric space. Let us note
however that the two definitions are very close: specifically if $Y$ is a
Łukasiewicz path, extended to the real line by a drift $-1$ between the jumps,
as depicted in Figure 5, then the continuum looptree $\mathscr{L}(Y)$ is
obtained from the discrete one $\mathsf{L}(Y)$ by replacing each edge by a
segment of length $1$, and also attaching an extra such segment to the root.

#### Gaussian labels on looptrees

Recall that $Y$ is any càdlàg function on $[0,\infty)$ with no negative jump
and that $C$ is defined by (2.4). One may define a random Gaussian field on
the tree $T_{C}$, known as the _Brownian snake_ driven by $C$, see [DLG02,
Chapter 4] for details on such processes in a broader setting. Formally,
conditionally given $C$, there exists a centred Gaussian process
$Z^{C}=(Z^{C}_{t})_{t\in[0,1]}$ with covariance function

$\mathop{}\\!\mathbb{E}[Z^{C}_{s}Z^{C}_{t}]=\min_{r\in[s,t]}C_{r},\qquad
s,t\in[0,1].$

Equivalently, we have
$\mathop{}\\!\mathbb{E}[(Z^{C}_{s}-Z^{C}_{t})^{2}]=d_{C}(s,t)$. On an
intuitive level, after associating a time $t$ with its projection in $T_{C}$,
the values of $Z^{C}$ evolve along the branches of the tree like a Brownian
motion, and these Brownian motions split at branchpoints and afterwards evolve
independently.

We construct a Gaussian field on the looptree $\mathscr{L}^{a}(Y)$ by placing
on each cycle an independent Brownian bridge, with duration given by the
length of the cycle, which describes the increments of the field along the
cycle. Formally, recall that the standard Brownian bridge
$b=(b_{t})_{t\in[0,1]}$ is a centred Gaussian process with covariance

$\mathop{}\\!\mathbb{E}[b_{s}b_{t}]=\min(s,t)-st,\qquad s,t\in[0,1].$

One can consider a bridge of any given duration using a diffusive scaling. Now
recall the partial order $\prec$ as well as the notation
$R^{t}_{s}=\inf_{r\in[s,t]}Y_{r}-Y_{s-}$ from (2.3). Let $C$ be the continuous
part defined in (2.4) and let $Z^{C}$ be associated Brownian snake.
Independently let $(b_{i})_{i\geq 1}$ denote a sequence of i.i.d. standard
Brownian bridges, and define for every $a\geq 0$ and $t\in[0,1]$,

$Z^{a}_{t}\coloneqq\sqrt{a}\,Z^{C}_{t}+\sum_{t_{i}\prec t}\sqrt{\Delta
Y_{t_{i}}}\,b_{i}\bigl{(}(\Delta Y_{t_{i}})^{-1}R^{t}_{t_{i}}\bigr{)},$ (2.8)

where the $t_{i}$’s are the jump times of $Y$ up to time $t$. Given $Y$, the
summands in (2.8) are independent zero-mean Gaussian random variables with
variance

$\displaystyle\mathop{}\\!\mathbb{E}\biggl{[}\Bigl{(}\sqrt{\Delta
Y_{t_{i}}}b_{i}\bigl{(}(\Delta
Y_{t_{i}})^{-1}R^{t}_{t_{i}}\bigr{)}\Bigr{)}^{2}\biggr{]}=\frac{R^{t}_{t_{i}}(\Delta
Y_{t_{i}}-R^{t}_{t_{i}})}{\Delta Y_{t_{i}}}\leq\min\\{R^{t}_{t_{i}},\Delta
Y_{t_{i}}-R^{t}_{t_{i}}\\}=\delta_{t_{i}}(0,R^{t}_{t_{i}}).$

Arguing as in the proof Proposition 6 in [LGM11] (recast in the formalism of
looptrees), one deduces that for any $q>0$, there exists $K_{q}>0$ such that
for every $s,t\in[0,1]$, it holds that

$\mathop{}\\!\mathbb{E}[|Z^{a}_{t}-Z^{a}_{s}|^{q}]\leq K_{q}\cdot
d^{a}_{\mathscr{L}(Y)}(s,t)^{q/2}.$ (2.9)

Taking $s=0$, this shows that the random variable $Z^{a}_{t}$ in (2.8) is
well-defined for any fixed $t$, which in particular implies that the series in
(2.8) converges. Further, given $Y$, we see that any pair $s,t\geq 0$ that has
$d^{a}_{\mathscr{L}(Y)}(s,t)=0$ also has $Z^{a}_{s}=Z^{a}_{t}$ almost surely
so $Z^{a}$ can actually be seen as a random process indexed by the looptree.
Note that by the scaling property of Gaussian variables the process $cZ^{a}$
constructed from $Y$ has the same law as the process $Z^{a}$ but constructed
from $c^{2}Y$. The value $a=1/3$ shall play a predominant role in relation
with maps and we simply write $Z=Z^{1/3}$. We shall also denote by
$Z^{\mathrm{ex}}$ this process $Z=Z^{1/3}$ constructed from

###  2.3. Invariance principles

We have not defined Lévy maps yet; they will shortly be defined as
subsequential limits of rescaled finite maps. Before that, let us start by
discussing limits of labelled looptrees built from random bridges with
exchangeable increments. Typical examples of applications include the
Boltzmann distributions recalled earlier, which relate to random walk
excursions of duration $n$, with a step distribution which may depend on $n$.
We let $X$ be a Lévy process with Laplace exponent $\psi$, with Gaussian
parameter $\beta\geq 0$, and we let $X^{\mathrm{br}}$ and $X^{\mathrm{ex}}$
denote the unit duration bridge and excursion versions of $X$ respectively.
For $a\geq 0$, we also let $\mathscr{L}^{a}(X^{\mathrm{ex}})$ denote the
looptree defined as in (2.6) and the lines below. Finally, let
$Z^{\mathrm{ex}}$ denote the label process $Z^{a}$ defined as in (2.8) with
$a=1/3$ and $Y=X^{\mathrm{ex}}$. The next result is an easy consequence of
[Mar22b, Theorems 7.4 and 7.9] by conditioning on the increment sizes.

######  Theorem 2.1.

For every $n\geq 1$, let $B^{n}=(B^{n}_{k})_{0\leq k\leq n+1}$ be a random
bridge, with $B^{n}_{n+1}=-1$, with increments $\Delta
B^{n}_{i}\in\mathbb{Z}_{\geq-1}$ and which are exchangeable. Let
$W^{n}=(W^{n}_{k})_{0\leq k\leq n+1}$ denote the excursion obtained by
cyclicly shifted $B^{n}$ at its first minimum. Let further $Z^{n}$ denote the
label process associated with a uniform random good labelling of the looptree
coded by $W^{n}$.

  1. (i)

Suppose that there exists a sequence $r_{n}\to\infty$ such that

$(r_{n}^{-1}B^{n}_{\lfloor
nt\rfloor})_{t\in[0,1]}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip
X^{\mathrm{br}}.$

for the Skorokhod $J_{1}$ topology. Then the following convergences in
distribution hold jointly:

$(r_{n}^{-1}W^{n}_{\lfloor
nt\rfloor})_{t\in[0,1]}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip
X^{\mathrm{ex}}\qquad\text{and}\qquad((2r_{n})^{-1/2}Z^{n}_{nt})_{t\in[0,1]}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip
Z^{\mathrm{ex}}$

for the Skorokhod $J_{1}$ topology.

  2. (ii)

Assume in addition that there exists $a\geq 0$ such that:

$r_{n}^{-2}\\#\\{i\leq n+1\colon\Delta B^{n}_{i}\in
2\mathbb{Z}-1\\}\enskip\mathop{\longrightarrow}^{\mathop{}\\!\mathbb{P}}_{n\to\infty}\enskip
a\beta^{2},$ (2.10)

where $a=0$ by convention when $\beta=0$. Then we have the convergence in
distribution:

$r_{n}^{-1}\,\mathsf{L}(W^{n})\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip\mathscr{L}^{(a+1)/2}(X^{\mathrm{ex}})$

for the Gromov–Hausdorff–Prokhorov topology where $\mathsf{L}(W^{n})$ is
equipped either with the uniform probability on the vertices or the uniform
probability on the corners.

######  Remark 2.2.

Given a Lévy bridge $X^{\mathrm{br}}$ and $a\geq 0$, one can always construct
a sequence of discrete bridges $B^{n}$ of duration $n+1$ respectively which
satisfies the assumptions of Theorem 2.1. Therefore the Lévy looptree
$\mathscr{L}^{(a+1)/2}(X^{\mathrm{ex}})$ can always be realised as the
Gromov–Hausdorff limit of rescaled finite looptrees. As such, it is a length
space, and since it is compact, a geodesic space.

Let us comment on Assumption (2.10). When following a geodesic path in the
looptree and traversing a cycle, one takes the shortest between the left and
right length. Roughly speaking, typically half of the cycles are traversed on
the right and half on the left, thus the microscopic cycles contributes to
roughly half the continuous part of the coding path, leading to
$\mathscr{L}^{1/2}(X^{\mathrm{ex}})$ in the limit. However, at the discrete
level, the fact that the cycle has even or odd length influences the law of
the length of the smallest part between left and right, namely
$\sum_{j=1}^{k}\min(j,k-j)$ depends on the parity of $k$. This contribution is
precisely captured in (2.10) and adds an extra factor to the continuous part
in the looptree distance in the limit. Such a phenomenon was already observed
in [KR20, Theorem 1.2] (see however [Mar22b, Remark 7.7] on the extra
constant). We note that this on the other hand has no influence on the labels.
The parameter $1/3$ appears in the limit due to our choice of labels in the
discrete model, see [Mar22b, Theorem 7.9] for more general labels.

###### Proof.

Using Skorokhod’s representation theorem, let us work on a probability space
where the assumptions hold almost surely. For every $i\geq 1$, let $\Delta
X^{\shortdownarrow}_{i}\geq 0$ denote the $i$’th largest increment of
$X^{\mathrm{br}}$ and then set

$\theta_{i}=\Bigl{(}\beta^{2}+\sum_{i\geq 1}(\Delta
X^{\shortdownarrow}_{i})^{2}\Bigr{)}^{-1/2}\Delta
X^{\shortdownarrow}_{i}\qquad\text{and then}\qquad\theta_{0}^{2}=1-\sum_{i\geq
1}\theta_{i}^{2}=\frac{\beta^{2}}{\beta^{2}+\sum_{i\geq 1}(\Delta
X^{\shortdownarrow}_{i})^{2}}.$

Finally let

$Y=\Bigl{(}\beta^{2}+\sum_{i\geq 1}(\Delta
X^{\shortdownarrow}_{i})^{2}\Bigr{)}^{-1/2}X^{\mathrm{br}},$

which is the exchangeable increment process with Gaussian parameter
$\theta_{0}$ and jump sizes $(\theta_{i})_{i\geq 1}$. Then the assumptions of
[Mar22b, Theorem 7.4] are fulfilled, namely let
$\sigma^{2}_{n}=\sum_{i=1}^{n}\Delta B^{n}_{i}(\Delta
B^{n}_{i}+1)=\sum_{i=1}^{n}(\Delta B^{n}_{i})^{2}-1$, then according to
[Kal02, Theorem 16.23] the convergence of $r_{n}^{-1}B^{n}$ to
$X^{\mathrm{br}}$ is equivalent to the convergence for every $i\geq 1$ of the
$i$’th largest increment of $r_{n}^{-1}B^{n}$ to $\Delta
X^{\shortdownarrow}_{i}$ and the convergence
$r_{n}^{-2}\sigma_{n}^{2}\to\beta^{2}+\sum_{i\geq 1}(\Delta
X^{\shortdownarrow}_{i})^{2}$. Thus if we instead rescale $B^{n}$ by
$\sigma_{n}^{-1}$, then for every $i\geq 1$, the $i$’th largest increment of
$\sigma_{n}^{-1}B^{n}$ converges to $(\beta^{2}+\sum_{i\geq 1}(\Delta
X^{\shortdownarrow}_{i})^{2})^{-1/2}\Delta
X^{\shortdownarrow}_{i}=\theta_{i}$, while

$\sigma_{n}^{-2}\\#\\{i\leq n\colon\Delta B^{n}_{i}\in
2\mathbb{Z}-1\\}\enskip\mathop{\longrightarrow}_{n\to\infty}\enskip\frac{a\beta^{2}}{\beta^{2}+\sum_{i\geq
1}(\Delta X^{\shortdownarrow}_{i})^{2}}=a\theta_{0}^{2}$

by our assumption (2.10). Denote by $\mathcal{V}Y$ the Vervaat transform of
the bridge $Y$, as in (2.2), so

$\mathcal{V}Y=\Bigl{(}\beta^{2}+\sum_{i\geq 1}(\Delta
X^{\shortdownarrow}_{i})^{2}\Bigr{)}^{-1/2}X^{\mathrm{ex}}.$

We can then apply [Mar22b, Theorem 7.4] and deduce that.

$r_{n}^{-1}\,\mathsf{L}(W^{n})\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip\Bigl{(}\beta^{2}+\sum_{i\geq
1}(\Delta
X^{\shortdownarrow}_{i})^{2}\Bigr{)}^{1/2}\mathscr{L}^{(a+1)/2}(\mathcal{V}Y)=\mathscr{L}^{(a+1)/2}(X^{\mathrm{ex}}),$

where the equality simply follows from the identity
$c\cdot\mathscr{L}^{(a+1)/2}(\mathcal{V}Y)=\mathscr{L}^{(a+1)/2}(c\mathcal{V}Y)$
for every $c>0$, which is plain from the construction of the looptree distance
(2.6).

As for the labels, we apply [Mar22b, Theorem 7.9] in the particular case of
the labels that code maps (see the beginning of [Mar22b, Section 7.4]) and
deduce the convergence in distribution

$((2r_{n})^{-1/2}Z^{n}_{nt})_{t\in[0,1]}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip\Bigl{(}\beta^{2}+\sum_{i\geq
1}(\Delta X^{\shortdownarrow}_{i})^{2}\Bigr{)}^{1/4}\zeta^{1/3},$

where the process $\zeta^{1/3}$ is defined as in (2.8) using $\mathcal{V}Y$
instead of $X^{\mathrm{ex}}$. As above, by the scaling property of Gaussian
variables, multiplying the label process by some $c>0$ amounts to multiplying
the underlying excursion with no negative jump by $c^{2}$, namely the limit
above equals precisely $Z^{\mathrm{ex}}$. ∎

######  Remark 2.3.

Let us mention that behind the abstract Gromov–Hausdorff–Prokhorov convergence
of the looptrees lies a very explicit convergence. Indeed, let us denote by
$(v^{n}_{k})_{0\leq k\leq n}$ the vertices of $\mathsf{L}(W^{n})$ encountered
when following its contour. Formally, recall from Section 2.1 the sequence
$(e_{0},\dots,e_{n})$ of oriented edges (with $e_{n}=e_{0}$), then set
$v^{n}_{k}$ to be the origin of $e_{k}$. One can then define a pseudo-distance
$d^{n}$ on $\\{0,\dots,n\\}$ by letting $d^{n}(i,j)$ be the looptree distance
between $v^{n}_{i}$ and $v^{n}_{j}$. It is shown in [Mar22b] that

$(r_{n}^{-1}d^{n}(\lfloor ns\rfloor,\lfloor
nt\rfloor))_{s,t\in[0,1]}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip(d^{(a+1)/2}_{\mathscr{L}(X^{\mathrm{ex}})}(s,t))_{s,t\in[0,1]}$

for the uniform topology. This immediately implies the
Gromov–Hausdorff–Prokhorov convergence for the uniform law on the corners. In
order to control the uniform law on the vertices, one needs to get rid of the
redundancies in the list $(v^{n}_{0},\dots,v^{n}_{n})$, and the argument is
provided by [Mar22a, Lemma 2.2] with $A=\\{-1\\}$.

Theorem 2.1 finds applications to scaling limits of random planar maps as in
[Mar22b, Theorem 7.12]. Indeed, recall that a pointed bipartite map
$(M^{n},v_{\star}^{n})$ with $n$ edges is coded by a labelled looptree such
that the labels, shifted to have minimum $1$, code the graph distances to
$v^{n}_{\star}$ in $M^{n}$. Let us denote by $(v^{n}_{k})_{0\leq k\leq n}$ the
vertices of the looptree encountered when following its contour as in the
previous remark. They can naturally be seen through the bijection as the
vertices of the map $M^{n}$ different from $v_{\star}^{n}$; in particular
$v^{n}_{0}$ is an extremity of the root edge of $M^{n}$ (the one farther away
from $v_{\star}^{n}$). Finally let $d_{M^{n}}$ denote the graph distance in
the map $M^{n}$.

######  Theorem 2.4.

Let $(M^{n},v_{\star}^{n})$ be a pointed map with $n$ edges and assume that
the Łukasiewicz path $W^{n}$ of its associated looptree satisfies the
assumptions of Theorem 2.1. Then the following holds jointly with this
theorem.

  1. (i)

The convergence in distribution

$\Bigl{(}\frac{1}{\sqrt{2r_{n}}}d_{M^{n}}(v^{n}_{\star},v^{n}_{\lfloor
nt\rfloor})\Bigr{)}_{t\in[0,1]}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip(Z^{\mathrm{ex}}_{t}-\min
Z^{\mathrm{ex}})_{t\in[0,1]},$

holds for the uniform topology. Consequently, if $V(M^{n})$ denotes the set of
vertices of $M^{n}$, then

$\frac{1}{\sqrt{2r_{n}}}d_{M^{n}}(v^{n}_{\star},v^{n}_{0})\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip-\min
Z^{\mathrm{ex}}\quad\text{and}\quad\frac{1}{\sqrt{2r_{n}}}\max_{v\in
V(M^{n})}d_{M^{n}}(v^{n}_{\star},v)\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip\max
Z^{\mathrm{ex}}-\min Z^{\mathrm{ex}},$

and for every continuous and bounded function $F$, we have:

$\frac{1}{\\#V(M^{n})}\sum_{v\in
V(M^{n})}F\Bigl{(}\frac{1}{\sqrt{2r_{n}}}d_{M^{n}}(v^{n}_{\star},v)\Bigr{)}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip\int_{0}^{1}F(Z^{\mathrm{ex}}_{t}-\min
Z^{\mathrm{ex}})\mathop{}\\!\mathrm{d}t.$

  2. (ii)

From every increasing sequence of integers, one can extract a subsequence
along which the convergence in distribution

$\Bigl{(}\frac{1}{\sqrt{2r_{n}}}d_{M^{n}}(v^{n}_{\lfloor
ns\rfloor},v^{n}_{\lfloor
nt\rfloor})\Bigr{)}_{s,t\in[0,1]}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip(D(s,t))_{s,t\in[0,1]},$

holds for the uniform topology, where $D$ is a random continuous pseudo-
distance which satisfies

$D(U,\cdot)\enskip\mathop{=}^{(d)}\enskip Z^{\mathrm{ex}}-\min
Z^{\mathrm{ex}},$

where $U$ is sampled uniformly at random on $[0,1]$ and independently of the
rest. Finally along this subsequence, we have

$\frac{1}{\sqrt{2r_{n}}}M^{n}\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip{}M=[0,1]/\\{D=0\\}$

in the Gromov–Hausdorff–Prokhorov topology.

  3. (iii)

If $X^{\mathrm{ex}}$ has no jump, so it reduces to $\beta$ times the Brownian
excursion, then these subsequential limits agree with $\sqrt{\beta/3}$ times
the standard Brownian sphere.

###### Proof.

The first claim is a straightforward consequence of Theorem 2.1, see e.g.
[Mar22b, Lemma 2.5]. The second claim is not immediate but is very standard
since the original work of Le Gall [LG07], see also [Mar22b, Lemma 2.6] for a
recent account of the argument is a similar context. As explained in Remark
2.3, this lemma requires an extra condition in order to remove the
redundancies in the list $(v^{n}_{0},\dots,v^{n}_{n})$ of vertices, see
[Mar22b, Equation 4]. As recalled there, the latter is provided by [Mar22a,
Lemma 2.2] with $A=\\{-1\\}$. The last claim is a consequence of the
identification of the Brownian sphere and the argument can be found in [LG13],
see precisely the proof Equation 59 there. ∎

In the rest of this paper, we call a _Lévy map_ any subsequential limit $M$ in
the previous theorem. We equip it with the metric $d_{M}$, the image of $D$ by
the canonical projection, and the probability $p_{M}$, given by the push-
forward of the Lebesgue measure on $[0,1]$. Similarly to looptrees in Remark
2.2, Lévy maps are geodesic spaces.

##  3. A spinal decomposition and some volume bounds

Throughout this section we let $X$ denote a Lévy process with Laplace exponent
$\psi$ and $X^{\mathrm{ex}}$ its excursion version. The main results of this
section are the following volume bounds. They will be the key to establish the
lower bounds on the dimensions of looptrees and maps in the subsequent
section. Recall that both the looptrees and maps are given by quotient of the
interval $[0,1]$ by a pseudo-distance so we naturally identify times in this
interval with their projection in the looptrees and maps.

######  Proposition 3.1.

Fix $\delta>0$ and let $U$ be a random time with the uniform distribution on
$[0,1]$ and independent of $X^{\mathrm{ex}}$. Then almost surely, for every
$n$ large enough, the ball of radius $2^{-n}$ centred at the image of $U$ in
the looptree $\mathscr{L}(X^{\mathrm{ex}})$ has volume less than
$\psi(2^{n(1-\delta)})^{-1}$.

We then turn to maps. Recall that we let $(M,d_{M},p_{M})$ denote a
subsequential limit of discrete random maps, related to a pair
$(X^{\mathrm{ex}},Z^{\mathrm{ex}})$, from Theorem 2.1 and Theorem 2.4. In
addition, thanks to Skorokhod’s representation theorem, we assume that all
convergences in these theorems hold in the almost sure sense. Notice the
additional square inside $\psi$ in the next statement compared to the first
one.

######  Proposition 3.2.

Fix $\delta>0$ and let $U$ be a random time with the uniform distribution on
$[0,1]$ and independent of $X^{\mathrm{ex}}$. Then almost surely, for every
$n$ large enough, the ball of radius $2^{-n}$ centred at the image of $U$ in
the map $M$ has volume less than $\psi(2^{2n(1-\delta)})^{-1}$.

In order to prove these results, we shall first describe Section 3.1 the path
from the root to a uniformly random point in the Lévy looptree, as well as the
labels on it. Then Proposition 3.1 is proved in Section 3.2, whereas
Proposition 3.2 is proved in Section 3.3, relying on a technical estimates on
Brownian paths stated in Section 3.4 and proved in Section 3.5.

###  3.1. A spinal decomposition

Let us first describe the equivalence classes in the looptree.

######  Lemma 3.3.

Fix $a\geq 0$. Almost surely, for every $0\leq s<t\leq 1$, we have:

$d^{a}_{\mathscr{L}(X^{\mathrm{ex}})}(s,t)=0\iff\begin{cases}\text{either}\enskip&\Delta
X^{\mathrm{ex}}_{s}>0\enskip\text{and}\enskip t=\inf\\{r>s\colon
X^{\mathrm{ex}}_{r}=X^{\mathrm{ex}}_{s-}\\},\\\ \enskip\text{or}\enskip&\Delta
X^{\mathrm{ex}}_{s}=0\enskip\text{and}\enskip
X^{\mathrm{ex}}_{s}=X^{\mathrm{ex}}_{t}=\inf_{[s,t]}X^{\mathrm{ex}}.\end{cases}$

Moreover, almost surely for any $s\in[0,1]$, the time $t$ is unique in the
first case, and there are either one or two such times $t$ in the second case.

###### Proof.

Let us first consider a general càdlàg function $Y$ and recall that the
looptree distance $d^{a}_{\mathscr{L}(Y)}$ is defined in (2.6). Then
$d^{a}_{\mathscr{L}(Y)}(s,t)=0$ when all terms on the right-hand side of (2.6)
vanish, namely:

  1. (i)

For every $r$ such that $s\wedge t\prec r\prec s$, either
$\inf_{[r,s]}Y=Y_{r}$ or $\inf_{[r,s]}Y=Y_{r-}$;

  2. (ii)

For every $r$ such that $s\wedge t\prec r\prec t$, either
$\inf_{[r,t]}Y=Y_{r}$ or $\inf_{[r,t]}Y=Y_{r-}$;

  3. (iii)

Either $\inf_{[s\wedge t,s]}Y=\inf_{[s\wedge t,t]}Y$ or both $Y_{s\wedge
t}=\inf_{[s\wedge t,s]}Y$ and $Y_{(s\wedge t)-}=\inf_{[s\wedge t,t]}Y$;

  4. (iv)

Finally $C_{s}=C_{t}=\inf_{[s,t]}C$.

It is straightforward to check that these properties hold in each case on the
right of the claim, let us prove the direct implication when
$Y=X^{\mathrm{ex}}$ by relying on some properties of such a path.

Consider $X$ the unconditioned Lévy process. First almost surely if $\Delta
X_{r}>0$, then for every $\varepsilon>0$, we have both
$\inf_{[r-\varepsilon,r]}X<X_{r-}$ and $\inf_{[r,r+\varepsilon]}X<X_{r}$. The
latter follows from the Markov property and the fact that $0$ is regular for
$(-\infty,0)$ and the former by invariance under time reversal and the fact
that $0$ is regular for $(0,\infty)$, see Theorem VII.1 and Corollary VII.5 in
[Ber96]. Consequently almost surely if $X$ achieves a local minimum, then it
does not jump at this time. Next, the strong Markov property also entails that
the local minima are unique. Finally, by [Ber96, Proposition III.2], we know
that almost surely every weak record of $X$ obtained by a jump is actually a
strict record. Applied to the dual process at a time just after $t$, it shows
that almost surely if $X$ realises a local minimum at time $t$ and if
$r=\sup\\{u<t\colon X_{u}<X_{t}\\}$ has $\Delta X_{r}>0$, then
$X_{r-}<X_{t}<X_{r}$.

All these properties transfer to the bridge $X^{\mathrm{br}}$ by absolute
continuity and then to the excursion $X^{\mathrm{ex}}$ by the Vervaat
transform. Therefore on an event of probability one, the four conditions above
simplify as:

  1. (i)

For every $r$ such that $s\wedge t\prec r\prec s$, it holds
$s=\inf\\{u>r\colon X^{\mathrm{ex}}_{u}=X^{\mathrm{ex}}_{r-}\\}$;

  2. (ii)

For every $r$ such that $s\wedge t\prec r\prec t$, it holds
$t=\inf\\{u>r\colon X^{\mathrm{ex}}_{u}=X^{\mathrm{ex}}_{r-}\\}$;

  3. (iii)

It holds $\inf_{[s\wedge t,s]}X^{\mathrm{ex}}=\inf_{[s\wedge
t,t]}X^{\mathrm{ex}}$;

  4. (iv)

Finally $C_{s}=C_{t}=\inf_{[s,t]}C$.

If $s\not\prec t$, then almost surely there exists $u\in(s,t)$ such that
$X^{\mathrm{ex}}_{u}=\inf_{[s,t]}X^{\mathrm{ex}}<X^{\mathrm{ex}}_{s-}$. Let
$r=s\wedge t$. Since $X^{\mathrm{ex}}$ realises a local minimum at time $u$,
then almost surely we must have $\Delta X^{\mathrm{ex}}_{r}>0$ and further
$X^{\mathrm{ex}}_{r-}<X^{\mathrm{ex}}_{u}<X^{\mathrm{ex}}_{s}<X^{\mathrm{ex}}_{r}$,
which contradicts the third condition above. Therefore almost surely we have
$s\prec t$, namely $X^{\mathrm{ex}}_{s-}\leq\inf_{[s,t]}X^{\mathrm{ex}}$. If
$\Delta X^{\mathrm{ex}}_{s}>0$ then indeed the second condition implies that
$t=\inf\\{u>r\colon X^{\mathrm{ex}}_{u}=X^{\mathrm{ex}}_{r-}\\}$. Otherwise if
$\Delta X^{\mathrm{ex}}_{s}=0$, then there is no $r\in(s,t)$ such that
$X^{\mathrm{ex}}$ achieves a local minimum at time $r$ with
$X^{\mathrm{ex}}_{r}=X^{\mathrm{ex}}_{s}$, nor
$X^{\mathrm{ex}}_{s}=X^{\mathrm{ex}}_{r-}<X^{\mathrm{ex}}_{r}$. Therefore if
$X^{\mathrm{ex}}_{t-}>X^{\mathrm{ex}}_{s}$, and since $d_{C}(s,t)=0$, then
there exists $r\in(s,t)$ such that $\Delta X^{\mathrm{ex}}_{r}>0$ and
$X^{\mathrm{ex}}_{s}<X^{\mathrm{ex}}_{r-}$ and
$X^{\mathrm{ex}}_{t-}<X^{\mathrm{ex}}_{r}$, which contributes to the second
term in (2.6). We conclude that in this case we must have
$X^{\mathrm{ex}}_{t-}=X^{\mathrm{ex}}_{s}$. Furthermore, the time $t$ cannot
be a jump time, for otherwise $X^{\mathrm{ex}}$ would reach smaller values
just before, hence
$X^{\mathrm{ex}}_{t}=X^{\mathrm{ex}}_{s}=\inf_{[s,t]}X^{\mathrm{ex}}$. Finally
there can only be at most two such times $t$, if $\inf\\{t>s\colon
X^{\mathrm{ex}}_{t}=X^{\mathrm{ex}}_{s}\\}$ is a time of a local minimum since
then it is unique at this height. ∎

Let us now describe the “chain of loops” from the root $0$ to a given point
$u\in[0,1]$ in the looptree, see Figure 7 for a graphical representation.
First define the following set of pairs of times:

$A_{u}=\\{(a,b)\in[0,u]\times[u,1]\colon\text{either
}X^{\mathrm{ex}}_{a}=X^{\mathrm{ex}}_{b}=\min_{[a,b]}X^{\mathrm{ex}}\text{ or
}b=\inf\\{t>a\colon X^{\mathrm{ex}}_{t}=X^{\mathrm{ex}}_{a-}\\}\\}.$ (3.1)

By Lemma 3.3, pairs $(a,b)\in A_{u}$ are identified in the looptree. These
points intuitively correspond to the ancestors of $u$ in the tree. Define then
the set of pairs of points that belong to the same cycle on the geodesic from
$u$ to the root, with one point on the left and one on the right. Formally,
set:

$B_{u}=\\{(a,b)\in[0,u]\times[u,1]\colon\exists s<a\text{ such that
}X^{\mathrm{ex}}_{s}>X^{\mathrm{ex}}_{a-}=\min_{[s,a-]}X^{\mathrm{ex}}>X^{\mathrm{ex}}_{b-}=\min_{[s,b-]}X^{\mathrm{ex}}>X^{\mathrm{ex}}_{s-}\\}.$
(3.2)

Pairs in $(a,b)\in B_{u}$ are not identified in the looptree (still by Lemma
3.3). Instead, the time $s$ in (3.2) together with $t=\inf\\{v>s\colon
X^{\mathrm{ex}}_{v}=X^{\mathrm{ex}}_{s-}\\}>u$ belong to $A_{u}$, so they
correspond to an ancestor of $u$. This ancestor has a large offspring number
$\Delta X^{\mathrm{ex}}_{s}$ which creates a cycle in the looptree. One point
on this cycle corresponds to the next ancestor of $u$ and splits the cycle
into a left part, to which $a$ belongs, and a right part, to which $b$
belongs. One can easily prove that almost surely, for any $u\in[0,1]$ the set
$A_{u}$ is closed and $A_{u}\cup B_{u}$ is the closure of $B_{u}$.

For any given $\varepsilon>0$ fixed, let us consider the set:

$B_{u}^{\varepsilon}=\\{(s,t)\in A_{u}\cup B_{u}\colon
d_{\mathscr{L}(X^{\mathrm{ex}})}(s,u)\leq\varepsilon\enskip\text{and}\enskip
d_{\mathscr{L}(X^{\mathrm{ex}})}(t,u)\leq\varepsilon\\}.$

Let us construct an extremal pair
$(u^{-}_{\varepsilon},u^{+}_{\varepsilon})\in B_{u}^{\varepsilon}$, see again
Figure 7 for a graphical representation. First let
$s_{\varepsilon}=\sup\\{s<u\colon
X^{\mathrm{ex}}_{s-}\leq\min_{[s,u]}X^{\mathrm{ex}}\text{ and
}d_{\mathscr{L}(X^{\mathrm{ex}})}(s,u)\geq\varepsilon\\}$, which codes the
last ancestor of $u$ (starting from the root) at
$d_{\mathscr{L}(X^{\mathrm{ex}})}$-distance at least $\varepsilon$ from $u$.

  1. (i)

If $d_{\mathscr{L}(X^{\mathrm{ex}})}(s_{\varepsilon},u)=\varepsilon$, then
define

$u^{-}_{\varepsilon}=\inf\\{s<u\colon
X^{\mathrm{ex}}_{s-}=X^{\mathrm{ex}}_{s_{\varepsilon}-}=\min_{[s,u]}X^{\mathrm{ex}}\\}\qquad\text{and}\qquad
u^{+}_{\varepsilon}=\inf\\{t>u\colon
X^{\mathrm{ex}}_{t}<X^{\mathrm{ex}}_{s-}\\},$ (3.3)

so $u^{-}_{\varepsilon}\leq s_{\varepsilon}$ with a strict inequality when
$X^{\mathrm{ex}}$ realizes a local minimum at time $s_{\varepsilon}$ and that
it previously crossed continuously this level. Then
$(u^{-}_{\varepsilon},u^{+}_{\varepsilon})\in A_{u}$ satisfies
$d_{\mathscr{L}(X^{\mathrm{ex}})}(u,u^{-}_{\varepsilon})=d_{\mathscr{L}(X^{\mathrm{ex}})}(u,u^{+}_{\varepsilon})=\varepsilon$.

  2. (ii)

Otherwise if
$d_{\mathscr{L}(X^{\mathrm{ex}})}(s_{\varepsilon},u)>\varepsilon$, then
necessarily $\Delta X^{\mathrm{ex}}_{s_{\varepsilon}}>0$. Let
$t_{\varepsilon}=\inf\\{t>s_{\varepsilon}\colon
X^{\mathrm{ex}}_{t-}\leq\min_{[t,u]}X^{\mathrm{ex}}\\}$ be the next ancestor
of $u$, then $d_{\mathscr{L}(X^{\mathrm{ex}})}(t_{\varepsilon},u)<\varepsilon$
and every time $a\in[s_{\varepsilon},t_{\varepsilon}]$ such that
$X^{\mathrm{ex}}_{a}=\min_{[s_{\varepsilon},a]}X^{\mathrm{ex}}$ has
$d_{\mathscr{L}(X^{\mathrm{ex}})}(a,u)=d_{\mathscr{L}(X^{\mathrm{ex}})}(t_{\varepsilon},u)+(X^{\mathrm{ex}}_{a}-X^{\mathrm{ex}}_{t_{\varepsilon}-})$.
We then define:

$\displaystyle u^{-}_{\varepsilon}$
$\displaystyle=\inf\\{a>s_{\varepsilon}\colon X^{\mathrm{ex}}_{a}\leq
X^{\mathrm{ex}}_{s_{\varepsilon}}-(d_{\mathscr{L}(X^{\mathrm{ex}})}(s_{\varepsilon},u)-\varepsilon)\\}$
$\displaystyle=\inf\\{a>s_{\varepsilon}\colon X^{\mathrm{ex}}_{a}\leq
X^{\mathrm{ex}}_{t_{\varepsilon}-}+(\varepsilon-
d_{\mathscr{L}(X^{\mathrm{ex}})}(t_{\varepsilon},u))\\},$

which is the first point in the looptree when following the left part of the
cycle from $s_{\varepsilon}$ to $t_{\varepsilon}$ which lies at looptree
distance $\varepsilon$ from $u$. Finally we let

$u^{+}_{\varepsilon}=\inf\\{b>u\colon
X^{\mathrm{ex}}_{b}<X^{\mathrm{ex}}_{t_{\varepsilon}-}+(\varepsilon-
d_{\mathscr{L}(X^{\mathrm{ex}})}(t_{\varepsilon},u))\\},$ (3.4)

which is now the first point in the looptree when following the right part of
the cycle from $s_{\varepsilon}$ to $t_{\varepsilon}$ which lies at looptree
distance $\varepsilon$ from $u$ (equivalently, the last one when going from
$t_{\varepsilon}$ to $s_{\varepsilon}$). Then
$(u^{-}_{\varepsilon},u^{+}_{\varepsilon})\in B_{u}$ satisfies
$d_{\mathscr{L}(X^{\mathrm{ex}})}(u^{-}_{\varepsilon})=d_{\mathscr{L}(X^{\mathrm{ex}})}(u^{+}_{\varepsilon})=\varepsilon$.

Figure 7: Left: A schematic representation of the spine from a time $u$ to the
root $0$ (a very big loop on the left has been cropped). The black dots
correspond to ancestors of $u$, i.e. points in $A_{u}$, whereas the cycles
correspond to the points in $B_{u}$. The two white points
$u^{-}_{\varepsilon}$ and $u^{+}_{\varepsilon}$ belong to the same cycle and
lie at distance precisely $\varepsilon$ from $u$. Right: The geodesic from $v$
to $u$ and from $w$ to $u$ are indicated in fat coloured paths (red and blue,
as well as purple for their common part). Since $v$ and $w$ are visited before
$u^{-}_{\varepsilon}$ then one must necessarily go through either
$u^{-}_{\varepsilon}$ or $u^{+}_{\varepsilon}$ in order to reach $u$, so they
lie outside the ball centred at $u$ and with radius $\varepsilon$.

The following geometric lemma should be clear on a picture, see e.g. Figure 7
right.

######  Lemma 3.4.

Almost surely, the ball of radius $\varepsilon$ centred at $u$ in the looptree
is contained in the interval $[u^{-}_{\varepsilon},u^{+}_{\varepsilon}]$.

It should also be clear from Figure 7 that the inclusion is strict in general;
but it turns out that this rough inclusion will give us the correct upper
bound for computing the dimensions.

###### Proof.

In the first case above, when $(u^{-}_{\varepsilon},u^{+}_{\varepsilon})\in
A_{u}$, the path $X^{\mathrm{ex}}$ achieves values smaller than
$X^{\mathrm{ex}}_{u^{-}_{\varepsilon}-}=X^{\mathrm{ex}}_{u^{+}_{\varepsilon}}$
immediately before $u^{-}_{\varepsilon}$ and immediately after
$u^{+}_{\varepsilon}$. We infer from the definition of $d_{\mathscr{L}(X)}$
that any time $t<u^{-}_{\varepsilon}$ or $t>u^{+}_{\varepsilon}$ has
$d_{\mathscr{L}(X)}(t,u)>d_{\mathscr{L}(X)}(u^{-}_{\varepsilon},u)=d_{\mathscr{L}(X)}(u^{+}_{\varepsilon},u)=\varepsilon$.
For the same reason, in the second case, any time $t>u^{+}_{\varepsilon}$ has
$d_{\mathscr{L}(X)}(t,u)>d_{\mathscr{L}(X)}(u^{+}_{\varepsilon},u)=\varepsilon$
and any time $t<s_{\varepsilon}$ has
$d_{\mathscr{L}(X)}(t,u)>d_{\mathscr{L}(X)}(s_{\varepsilon},u)>\varepsilon$.
Finally, times $t\in(s_{\varepsilon},u^{-}_{\varepsilon})$ have
$d_{\mathscr{L}(X)}(t,u)>\min\\{d_{\mathscr{L}(X)}(s_{\varepsilon},u),d_{\mathscr{L}(X)}(u^{-}_{\varepsilon},u)\\}=d_{\mathscr{L}(X)}(u^{-}_{\varepsilon},u)=\varepsilon$.
This shows that the complement of the interval
$[u^{-}_{\varepsilon},u^{+}_{\varepsilon}]$ is contained in the complement of
the ball of radius $\varepsilon$ centred at $u$ in the looptree. ∎

###  3.2. A volume estimate in the looptree

Recall the claim of Proposition 3.1 on the volume of balls in the looptree.
Given Lemma 3.4, it suffices to upper bound the length of the interval
$[U^{-}_{\varepsilon},U^{+}_{\varepsilon}]$ where $U$ is a uniform random time
independent of $X^{\mathrm{ex}}$. In order to make calculations, we shall rely
on the local absolutely continuity between the excursion and the unconditioned
process, which simply comes from the construction of the excursion. Precisely,
let $(\overleftarrow{X}_{s})_{s\geq 0}$ and $(\overrightarrow{X}_{t})_{t\geq
0}$ be two independent copies of the Lévy process $X$, then by (2.1) and
(2.2), the pair
$(X^{\mathrm{ex}}_{U}-X^{\mathrm{ex}}_{(U-s)-},X^{\mathrm{ex}}_{t+U})_{s\in[0,U],t\in[0,1-U]}$
is locally absolutely continuous with respect to
$(\overleftarrow{X}_{s},\overrightarrow{X}_{t})_{s,t\geq 0}$ with bounded
density.

We shall keep the same notation as in the preceding subsection and consider
the sets $A_{0}$, $B_{0}$, and $B^{\varepsilon}_{0}$ defined as above for
$(-\overleftarrow{X}_{s-},\overrightarrow{X}_{t})_{s,t\geq 0}$. In particular,
the first elements of pairs in $A_{0}$ are given by the negative of the set of
times at which $(\overleftarrow{X}_{s})_{s\geq 0}$ realises a weak record. Let
us thus consider the running supremum process $(S_{t})_{t\geq 0}$ and the
local time at the supremum $(L_{t})_{t\geq 0}$ of the process
$(\overleftarrow{X}_{t})_{t\geq 0}$. We henceforth drop the arrow for better
readability. According to [DLG02, Equation 96] the random point measure
$\mathscr{N}(\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}u)=\sum_{i\in
I}\delta_{(t_{i},x_{i},u_{i})}$ defined by:

$\mathscr{N}(\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}u)=\sum_{s:S_{s}>S_{s-}}\delta_{(L_{s},S_{s}-X_{s-},(S_{s-}-X_{s-})/(S_{s}-X_{s-}))}(\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}u)$
(3.5)

is a Poisson random measure on $[0,\infty)\times[0,\infty)\times[0,1]$ with
intensity $\mathop{}\\!\mathrm{d}t\otimes
x\pi(\mathop{}\\!\mathrm{d}x)\otimes\mathop{}\\!\mathrm{d}u$.

Let us next define four subordinators by $X^{\prime}$, $X^{R}$,
$\widetilde{X}$, and $\sigma$ by setting for every $t>0$:

$X^{\prime}_{t}=\beta t+\sum_{t_{i}\leq t}x_{i}\qquad\text{and}\qquad
X^{R}_{t}=\beta t+\sum_{t_{i}\leq t}u_{i}x_{i},$

as well as

$\widetilde{X}_{t}=\beta t+\sum_{t_{i}\leq
t}\min\\{u_{i},1-u_{i}\\}x_{i}\qquad\text{and}\qquad\sigma_{t}=\beta
t+\sum_{t_{i}\leq t}u_{i}(1-u_{i})x_{i}.$

Notice that we give them all the same drift $\beta$ for the continuous part.
Informally, in the infinite looptree coded by
$(\overleftarrow{X},\overrightarrow{X})$ with a distinguished path of loops by
coded $A_{0}\cup B_{0}$, when reading this path starting from $0$, the jumps
of process $X^{\prime}$ code the total length of the loops on this path, those
of $X^{R}$ code their right length, those of $\widetilde{X}$ the shortest
length among left and right, so related to the looptree distance, and finally
those of $\sigma$ the product of the left and right lengths, which is related
to a Brownian bridge on the loop.

######  Lemma 3.5.

The Laplace exponents of these processes are given by:

$\psi^{\prime}(\lambda)-\beta\lambda\quad\text{for
}X^{\prime},\qquad\widetilde{\psi}(\lambda)\quad\text{for
}X^{R},\qquad\widetilde{\psi}(\lambda/2)+\beta\lambda/2\quad\text{for
}\widetilde{X},\quad\text{and}\quad\phi(\lambda)+\beta\lambda\quad\text{for
}\sigma,$

where $\psi^{\prime}$ simply denotes the derivative of $\psi$ and
$\widetilde{\psi}(\lambda)=\psi(\lambda)/\lambda$ and finally
$\phi(\lambda)=\int_{0}^{\infty}x\pi(\mathop{}\\!\mathrm{d}x)\int_{0}^{1}\mathop{}\\!\mathrm{d}u(1-\exp(-\lambda
u(1-u)x))$.

###### Proof.

Recall the Poisson random measure $\mathscr{N}$ from (3.5), then for every
$t,\lambda>0$, by exchanging the differentiation and integral, we get:

$-\frac{1}{t}\log\mathop{}\\!\mathbb{E}\Bigl{[}\exp\Bigl{(}-\sum_{t_{i}\leq
t}x_{i}\Bigr{)}\Bigr{]}=\int_{0}^{\infty}(1-\operatorname{e}^{-\lambda
x})x\pi(\mathop{}\\!\mathrm{d}x)=\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}\int_{0}^{\infty}(\operatorname{e}^{-\lambda
x}-1+\lambda
x)\pi(\mathop{}\\!\mathrm{d}x)=\psi^{\prime}(\lambda)-2\beta\lambda.$

For $X^{R}$, we have:

$-\frac{1}{t}\log\mathop{}\\!\mathbb{E}\Bigl{[}\exp\Bigl{(}-\sum_{t_{i}\leq
t}u_{i}x_{i}\Bigr{)}\Bigr{]}=\int_{0}^{\infty}x\pi(\mathop{}\\!\mathrm{d}x)\int_{0}^{1}(1-\operatorname{e}^{-\lambda
ux})\mathop{}\\!\mathrm{d}u=\frac{1}{\lambda}\int_{0}^{\infty}\pi(\mathop{}\\!\mathrm{d}x)(\lambda
x-1+\operatorname{e}^{-\lambda x})=\widetilde{\psi}(\lambda)-\beta\lambda.$

Noticing that the random variables $\min\\{u_{i},1-u_{i}\\}$ have density
$2\,\mathbf{1}_{[0,1/2]}$, and that
$\int_{0}^{1/2}2\mathop{}\\!\mathrm{d}v(1-\operatorname{e}^{-\lambda
vx})=1+\frac{2}{\lambda x}(\operatorname{e}^{-\lambda
x/2}-1)=\frac{2}{\lambda}(\operatorname{e}^{-\lambda x/2}-1+\frac{\lambda
x}{2})$, we have similarly:

$-\frac{1}{t}\log\mathop{}\\!\mathbb{E}\Bigl{[}\exp\Bigl{(}-\sum_{t_{i}\leq
t}x_{i}\min\\{u_{i},1-u_{i}\\}\Bigr{)}\Bigr{]}=\int_{0}^{\infty}x\pi(\mathop{}\\!\mathrm{d}x)\int_{0}^{1/2}2\mathop{}\\!\mathrm{d}v(1-\operatorname{e}^{-\lambda
vx})=\widetilde{\psi}\Bigl{(}\frac{\lambda}{2}\Bigr{)}-\frac{\beta\lambda}{2}.$

Finally

$-\frac{1}{t}\log\mathop{}\\!\mathbb{E}\Bigl{[}\exp\Bigl{(}-\sum_{t_{i}\leq
t}x_{i}u_{i}(1-u_{i})\Bigr{)}\Bigr{]}=\int_{0}^{\infty}x\pi(\mathop{}\\!\mathrm{d}x)\int_{0}^{1}(1-\operatorname{e}^{-\lambda
u(1-u)x})\mathop{}\\!\mathrm{d}u=\phi(\lambda).$

This characterises the law of these subordinators. ∎

The following result claims that in the infinite model, on the distinguished
chain of loops, the sum of the lengths of the loops we encounter is close to
the sum of the shortest between their left and right length.

######  Lemma 3.6.

Let $\tau_{\varepsilon}=\inf\\{t\geq 0\colon\widetilde{X}_{t}>\varepsilon\\}$
for every $\varepsilon>0$. Fix $\delta\in(0,1)$, then almost surely, for every
$\varepsilon>0$ small enough, it holds $X^{R}_{\tau_{\varepsilon}-}\leq
X^{\prime}_{\tau_{\varepsilon}-}\leq\varepsilon^{1-\delta}$.

###### Proof.

Clearly $X^{R}\leq X^{\prime}$ so we focus on the second bound. Also, it is
enough to prove the claim assuming that $\beta=0$. Indeed, when $\beta>0$,
since $\beta$ is the common drift coefficient to both $X^{\prime}$ and
$\widetilde{X}$, then by adding this contribution plus that of the jumps of
$\widetilde{X}$, we infer from the $\beta=0$ case that
$X^{\prime}_{\tau_{\varepsilon}-}\leq\varepsilon^{1-\delta}+\widetilde{X}_{\tau_{\varepsilon}-}\leq\varepsilon^{1-\delta}+\varepsilon$.
We now assume that $\beta=0$. To simplify notation, we set
$v_{i}=\min(u_{i},1-u_{i})$ and $y_{i}=v_{i}x_{i}$. For $K>1$, let us
decompose the process $X^{\prime}$ by introducing a cutoff for small values of
the $v_{i}$’s as follows: for every $t>0$, write

$X^{\prime}_{t}=\sum_{t_{i}\leq t}\frac{y_{i}}{v_{i}}\mathbf{1}_{\\{v_{i}\geq
1/(2K)\\}}+\sum_{t_{i}\leq
t}\frac{y_{i}}{v_{i}}\mathbf{1}_{\\{v_{i}<1/(2K)\\}}.$

Also, since the $v_{i}$’s have density $2\,\mathbf{1}_{[0,1/2]}$, we have

$\mathop{}\\!\mathbb{P}\Bigl{(}v_{i}\leq\frac{1}{2K}\Bigr{)}=\frac{1}{K}\qquad\text{and}\qquad\mathop{}\\!\mathbb{E}\Bigl{[}\frac{1}{v_{i}}\mathbf{1}_{\\{v_{i}\geq
1/(2K)\\}}\Bigr{]}=\int_{1/(2K)}^{1/2}\frac{2}{x}\mathop{}\\!\mathrm{d}x=2\log
K.$

We claim that for $K$ well-chosen, the first sum in the first display will be
close to the sum of the $y_{i}$’s, whereas in the second sum, there are too
few terms and this sum will be small.

Fix $\delta>\delta^{\prime}>0$ and let
$K_{\varepsilon}=\exp(\varepsilon^{-\delta^{\prime}})$. Then the Markov
inequality implies:

$\displaystyle\mathop{}\\!\mathbb{P}\Bigl{(}\sum_{t_{i}\leq\tau_{\varepsilon}-}x_{i}\mathbf{1}_{\\{v_{i}\geq\frac{1}{2K_{\varepsilon}}\\}}\geq\varepsilon^{1-\delta}\bigm{|}(t_{i},y_{i})_{i\in
I}\Bigr{)}$
$\displaystyle\leq\mathop{}\\!\mathbb{P}\Bigl{(}\sum_{t_{i}\leq\tau_{\varepsilon}-}\frac{1}{v_{i}}y_{i}\mathbf{1}_{\\{v_{i}\geq\frac{1}{2K_{\varepsilon}}\\}}\geq\varepsilon^{-\delta}\sum_{t_{i}\leq\tau_{\varepsilon}-}y_{i}\bigm{|}(t_{i},y_{i})_{i\in
I}\Bigr{)}$ $\displaystyle\leq 2\varepsilon^{\delta}\log K_{\varepsilon}$
$\displaystyle\leq 2\varepsilon^{\delta-\delta^{\prime}}.$

Note that the sum of the last quantity over all values of $\varepsilon$ of the
form $2^{-n}$ with $n\in\mathbb{N}$ is finite.

We next consider the other sum in the decomposition above, when the $v_{i}$’s
are smaller than $1/(2K_{\varepsilon})$. Recall that we assume that $\beta=0$
and note in this case the identity in law (as processes):

$\sum_{t_{i}\leq
t}x_{i}\mathbf{1}_{\\{v_{i}<\frac{1}{2K_{\varepsilon}}\\}}\enskip\mathop{=}^{(d)}\enskip
X^{\prime}_{t\mathop{}\\!\mathbb{P}(v_{i}<\frac{1}{2K_{\varepsilon}})}=X^{\prime}_{t/K_{\varepsilon}}.$

Recall from Lemma 3.5 that $X^{\prime}$ has Laplace exponent $\psi^{\prime}$,
then for every $t,x>0$, we have by the Markov property and the upper bound
$1-\operatorname{e}^{-y}\leq y$:

$\mathop{}\\!\mathbb{P}\Bigl{(}\sum_{t_{i}\leq
t}x_{i}\mathbf{1}_{\\{v_{i}<\frac{1}{2K_{\varepsilon}}\\}}>x\Bigr{)}=\mathop{}\\!\mathbb{P}(1-\operatorname{e}^{X^{\prime}_{t/K_{\varepsilon}}/x}>1-\operatorname{e}^{-1})\leq\frac{\operatorname{e}}{\operatorname{e}-1}\bigl{(}1-\mathop{}\\!\mathbb{E}\bigl{[}\operatorname{e}^{-X^{\prime}_{t/K_{\varepsilon}}/x}\bigr{]}\bigr{)}\leq\frac{\operatorname{e}}{\operatorname{e}-1}\frac{t}{K_{\varepsilon}}\psi^{\prime}(1/x).$

On the other hand, since $\widetilde{X}$ has Laplace exponent
$\widetilde{\psi}(\cdot/2)$,

$\mathop{}\\!\mathbb{P}(\tau_{\varepsilon}\geq
t)=\mathop{}\\!\mathbb{P}(\widetilde{X}_{t}\leq\varepsilon)=\mathop{}\\!\mathbb{P}\bigl{(}\operatorname{e}^{-\widetilde{X}_{t}/\varepsilon}\geq\operatorname{e}^{-1}\bigr{)}\leq\operatorname{e}\cdot\mathop{}\\!\mathbb{E}\bigl{[}\operatorname{e}^{-\widetilde{X}_{t}/\varepsilon}\bigr{]}=\operatorname{e}\cdot\operatorname{e}^{-t\widetilde{\psi}(1/(2\varepsilon))}.$

We infer that for any $t>0$, we have:

$\mathop{}\\!\mathbb{P}\Bigl{(}\sum_{t_{i}\leq\tau_{\varepsilon}-}x_{i}\mathbf{1}_{\\{v_{i}<\frac{1}{2K_{\varepsilon}}\\}}>\varepsilon^{1-\delta}\Bigr{)}\leq\frac{\operatorname{e}}{\operatorname{e}-1}tK_{\varepsilon}^{-1}\psi^{\prime}(1/\varepsilon^{1-\delta})+\operatorname{e}\cdot\operatorname{e}^{-t\widetilde{\psi}(1/(2\varepsilon))}.$

For $t=\varepsilon^{-\delta}\widetilde{\psi}(1/(2\varepsilon))^{-1}$ with
$\delta>0$, we read:

$\mathop{}\\!\mathbb{P}\Bigl{(}\sum_{t_{i}\leq\tau_{\varepsilon}-}x_{i}\mathbf{1}_{\\{v_{i}<\frac{1}{2K_{\varepsilon}}\\}}>\varepsilon^{1-\delta}\Bigr{)}\leq\frac{\operatorname{e}}{\operatorname{e}-1}\frac{\psi^{\prime}(1/\varepsilon^{1-\delta})}{\varepsilon^{\delta}K_{\varepsilon}\widetilde{\psi}(1/(2\varepsilon))}+\operatorname{e}\cdot\operatorname{e}^{-\varepsilon^{-\delta}}.$

Since $\widetilde{\psi}$ is increasing and $\psi^{\prime}$ grows at most
polynomially (in fact, its upper exponent at infinity is
$\boldsymbol{\eta}-1$), whereas $K_{\varepsilon}$ grows exponentially fast as
$\varepsilon\downarrow 0$, then the sum of the upper bound when $\varepsilon$
ranges over the set $\\{2^{-n},n\in\mathbb{N}\\}$ is finite. We conclude from
our two bounds and the Borel–Cantelli lemma that almost surely for every $n$
large enough, we have $X^{\prime}_{\tau_{2^{-n}}-}\leq 2^{-n(1-\delta)}$. We
extend this to all values of $\varepsilon>0$ small enough by monotonicity. ∎

Recall that Proposition 3.1 claims that almost surely, the volume of the ball
in the looptree centred at an independent uniform random point $U$ and with
radius $2^{-n}$ is less than $\psi(2^{n(1-\delta)})^{-1}$ for $n$ large
enough.

###### Proof Proposition 3.1.

As we have already observed, thanks to Lemma 3.4, it suffices to prove that if
$U$ is an independent random time with the uniform distribution on $[0,1]$,
then almost surely, for every $n$ large enough, we have both
$U-U^{-}_{2^{-n}}\leq\psi(2^{n(1-\delta)})^{-1}$ and
$U^{+}_{2^{-n}}-U\leq\psi(2^{n(1-\delta)})^{-1}$. Let us first focus on the
second bound, we shall then deduce the first one by a symmetry argument. By
the local absolute continuity relation which follows from (2.1) and (2.2),
since the density is bounded and we aim for an almost sure result, it is
sufficient to prove the claim after replacing
$(X^{\mathrm{ex}}_{U}-X^{\mathrm{ex}}_{(U-s)-},X^{\mathrm{ex}}_{t+U})_{s\in[0,U],t\in[0,1-U]}$
by $(\overleftarrow{X}_{s},\overrightarrow{X}_{t})_{s,t\geq 0}$. In
particular, we replace $U$ by $0$ and write $0^{+}_{\varepsilon}$ for the time
$u^{+}_{\varepsilon}$ defined from
$(\overleftarrow{X}_{s},\overrightarrow{X}_{t})_{s,t\geq 0}$ for $u=0$.
Relying on the Borel–Cantelli lemma, it suffices to prove that
$\sum_{n}\mathop{}\\!\mathbb{P}(0^{+}_{2^{-n}}\geq\psi(2^{n(1-\delta)})^{-1})<\infty$
for this infinite model.

Let us rewrite the definition of $0^{+}_{\varepsilon}$ (recall (3.3) and
(3.4)) in a more convenient way and let us refer to Figure 8 for a figurative
representation. Let $\widetilde{X}$ and $X^{R}$ be defined as previously from
$\overleftarrow{X}$ instead of $X$, let
$\tau_{\varepsilon}=\inf\\{s>0\colon\widetilde{X}_{s}>\varepsilon\\}$, and
$T(x)=\inf\\{t>0\colon\overrightarrow{X}_{t}\leq-x\\}$. Then we have
$0^{+}_{\varepsilon}=T(X^{R}_{\tau_{\varepsilon}-}+(\varepsilon-\widetilde{X}_{\tau_{\varepsilon}-}))$.

Fix $\delta>0$, then according to Lemma 3.6, almost surely for every
$\varepsilon$ small enough, it holds
$X^{R}_{\tau_{\varepsilon}-}\leq\varepsilon^{1-\delta/4}$. We then implicitly
work under the event that
$X^{R}_{\tau_{\varepsilon}-}+(\varepsilon-\widetilde{X}_{\tau_{\varepsilon}-})\leq\varepsilon^{1-\delta/2}$,
so it remains to upper bound $T(\varepsilon^{1-\delta/2})$. First, for every
$x>0$, we have by the Markov inequality:

$\mathop{}\\!\mathbb{P}(T(\varepsilon^{1-\delta/2})\geq
x)=\mathop{}\\!\mathbb{P}\bigl{(}1-\operatorname{e}^{-T(\varepsilon^{1-\delta/2})/x}\geq
1-\operatorname{e}^{-1}\bigr{)}\leq\frac{\operatorname{e}}{\operatorname{e}-1}\bigl{(}1-\mathop{}\\!\mathbb{E}\bigl{[}\operatorname{e}^{-x^{-1}T(\varepsilon^{1-\delta/2})}\bigr{]}\bigr{)}.$

It is well-known that $T$ is a subordinator with Laplace exponent
$\Phi=\psi^{-1}$ the inverse of $\psi$, see e.g. [Ber96, Theorem VII.1]. Then,
using also that $1-\operatorname{e}^{-y}\leq y$, we infer that

$\mathop{}\\!\mathbb{P}(T(\varepsilon^{1-\delta/2})\geq
x)\leq\frac{\operatorname{e}}{\operatorname{e}-1}\bigl{(}1-\operatorname{e}^{-\varepsilon^{1-\delta/2}\Phi(x^{-1})}\bigr{)}\leq\frac{\operatorname{e}}{\operatorname{e}-1}\varepsilon^{1-\delta/2}\Phi(x^{-1}).$

Consequently, for $\varepsilon=2^{-n}$ and
$x=\psi(\varepsilon^{-(1-\delta)})^{-1}=\psi(2^{n(1-\delta)})^{-1}$, we infer
that

$\mathop{}\\!\mathbb{P}(T(2^{-n(1-\delta/2)})\geq\psi(2^{n(1-\delta)})^{-1})\leq\frac{\operatorname{e}}{\operatorname{e}-1}2^{-n(1-\delta/2)+n(1-\delta)}=\frac{\operatorname{e}}{\operatorname{e}-1}2^{-n\delta/2},$

which is a convergent series as we wanted. This implies that almost surely,
for every $n$ large enough, we have
$0^{+}_{2^{-n}}\geq\psi(2^{n(1-\delta)})^{-1}$ in the infinite volume model.
By local absolute continuity, we deduce that almost surely, for every $n$
large enough, we have $U^{+}_{2^{-n}}-U\leq\psi(2^{n(1-\delta)})^{-1}$ in the
excursion model.

Figure 8: Schematic representation of the times $u^{-}_{\varepsilon}$ and
$u^{+}_{\varepsilon}$ in terms of the Lévy excursion. On the excursion, the
blue and green vertical lines on the very left have the same length. The sum
of this length and that of all the red vertical lines (which correspond to the
geodesic from $u$ to the root) equals precisely $\varepsilon$. This is less
than $X^{\mathrm{ex}}_{u}-X^{\mathrm{ex}}_{u^{+}_{\varepsilon}}$, which codes
the length of the path from $u$ to $u^{+}_{\varepsilon}$ in the looptree if
one always follows the cycles on the spine on their right, and which is at
most of order $\varepsilon^{1-\delta}$ by Lemma 3.6.

Let $B(U,r)$ denote the closed ball centred at $U$ and with radius $r$ in
$\mathscr{L}(X^{\mathrm{ex}})$ and let us denote by $|\cdot|$ the volume
measure. We just proved that almost surely, for every $n$ large enough, we
have $|B(U,2^{-n})\cap[U,1]|\leq\psi(2^{n(1-\delta)})^{-1}$. Lemma 3.7 below,
together with the symmetry of $U$, implies that the two sequences
$(|B(U,2^{-n})\cap[0,U]|)_{n\geq 1}$ and $(|B(U,2^{-n})\cap[U,1]|)_{n\geq 1}$
have the same distribution, which completes the proof. ∎

We need the following lemma to conclude the proof Proposition 3.1. It roughly
says that the looptree $\mathscr{L}(X^{\mathrm{ex}})$ is invariant under
mirror symmetry. This property is not immediate from the construction from the
excursion path $X^{\mathrm{ex}}$, although we believe that a suitable path
construction could map $X^{\mathrm{ex}}$ to a process which has the same law
as its time-reversal. Also this mirror symmetry relates informally to that of
the Lévy trees from [DLG02, Corollary 3.1.6]. One could rely on this result
(beware that it is written under the $\sigma$-finite Itō excursion measure),
and indeed the exploration process $\rho$ and its dual $\eta$, defined in the
form of Equations 89 and 90 in [DLG02], relate explicitly to the looptree
distances. However we prefer a self-contained soft argument using an
approximation with discrete looptrees which are (almost) symmetric.

Recall from the previous proof that $B(u,r)$ denotes the closed ball centred
at $U$ and with radius $r$ in $\mathscr{L}(X^{\mathrm{ex}})$, seen as a subset
of $[0,1]$ and that $|\cdot|$ denotes the Lebesgue measure.

######  Lemma 3.7.

For every $u\in[0,1]$, the processes $(|B(u,r)\cap[0,u]|)_{r\geq 0}$ and
$(|B(1-u,r)\cap[1-u,1]|)_{r\geq 0}$ have the same distribution.

###### Proof.

Fix $u\in[0,1]$. Combining Remark 2.2 and Remark 2.3 we infer the following:
there exist discrete Łukasiewicz paths, obtained by cyclicly shifting a bridge
with exchangeable increments, which converge in distribution to
$X^{\mathrm{ex}}$ once properly rescaled, such that if $d^{n}$ denotes the
corresponding looptree distance, normalised both in time and space, then
$(d^{n}(u,t))_{t\in[0,1]}$ converges to
$(d_{\mathscr{L}(X^{\mathrm{ex}})}(u,t))_{t\in[0,1]}$ for the uniform
topology. In particular, the processes $(|B(u,r)\cap[0,u]|)_{r\geq 0}$ and
$(|B(1-u,r)\cap[1-u,1]|)_{r\geq 0}$ are the limits in law of the analogous
processes for these rescaled discrete looptrees. We claim that these discrete
volume processes for any given $n$ have the same distribution as in fact the
discrete looptrees are almost invariant under mirror symmetry.

Figure 9: Middle: a looptree obtained from a dotted plane tree by replacing
the edges from each vertex to its parent by cycles. Left: a looptree as we
consider in this work, obtained from that in the middle by merging each
internal vertex of the tree with its right-most offspring. Right: a looptree
obtained by first taking the mirror image of that in the middle and then again
merging each internal vertex of the tree with its right-most offspring. The
looptrees on the left and on the right are the mirror image of each other,
except for the loop of length $1$.

Recall that the looptrees we consider here are those on the left of Figure 9,
where each internal node in the associated tree (in the middle of the figure)
is merged with its right-most offspring. Let us call such a looptree $L_{n}$.
By exchangeability of the coding bridge, the law of the looptree, say
$\Lambda_{n}$, in the middle of Figure 9, without this merging operation, is
invariant under mirror symmetry. Hence, if one starts from this looptree
$\Lambda_{n}$, takes its mirror image $\widehat{\Lambda}_{n}$, and constructs
a further looptree $L_{n}^{\prime}$ by merging each internal vertex with its
right-most offspring, then $L_{n}^{\prime}$ has the same law as $L_{n}$.
Beware that the looptree $L_{n}^{\prime}$ is not exactly the mirror image
$\widehat{L}_{n}$ of $L_{n}$. The only difference however is caused by loops
of length $1$, which always pend to the left in $L_{n}$ and $L_{n}^{\prime}$,
but get moved instead to the right in $\widehat{L}_{n}$. Recall that the
random looptrees we are considering are obtained from a Łukasiewicz path which
is a discrete approximation of $X^{\mathrm{ex}}$; we can always assume that
this path has no null increment, and thus the looptree $L_{n}$ has no such
loop of length $1$, in which case $\widehat{L}_{n}=L_{n}^{\prime}$ has indeed
the same law as $L_{n}$. We conclude by letting $n\to\infty$ as in the first
part of the proof. ∎

###  3.3. On distances in the map

Let us now turn to the proof Proposition 3.2 on the volume of balls in the map
$(M,d_{M},p_{M})$. Let us start by noticing that, similarly to the discrete
setting, the distance in the map can be seen as a pseudo-distance on the
looptree. This will be useful later. Recall from Theorem 2.1 and Theorem 2.4
that $M$ is defined as a subsequential limit of rescaled discrete random maps
$M^{n}$, where $M^{n}$ is coded by a labelled looptree, itself coded by a pair
of discrete paths $(W^{n},Z^{n})$. We shall denote by
$X^{n}=r_{n}^{-1}W^{n}_{\lfloor n\rfloor\cdot}$ the rescaled Łukasiewicz path
which converges to $X^{\mathrm{ex}}$ almost surely by combining Theorem 2.1
with Skorokhod’s representation theorem.

######  Lemma 3.8.

Almost surely, for every $s,t\in[0,1]$, if
$d_{\mathscr{L}(X^{\mathrm{ex}})}(s,t)=0$, then $d_{M}(s,t)=0$.

###### Proof.

Fix $s\leq t$ such that $d_{\mathscr{L}(X^{\mathrm{ex}})}(s,t)=0$, which means
by Lemma 3.3 that

$\text{(i)}\enskip\text{either}\enskip
X^{\mathrm{ex}}_{s}=X^{\mathrm{ex}}_{t}=\inf_{[s,t]}X^{\mathrm{ex}}\qquad\text{(ii)}\enskip\text{or}\enskip\Delta
X^{\mathrm{ex}}_{s}>0\enskip\text{and}\enskip t=\inf\\{r>s\colon
X^{\mathrm{ex}}_{r}=X^{\mathrm{ex}}_{s-}\\}.$ (3.6)

Suppose first that we are in the first case and in addition that
$X^{\mathrm{ex}}_{r}>X^{\mathrm{ex}}_{s}=X^{\mathrm{ex}}_{t}$ for all
$r\in(s,t)$. Fix some small $\varepsilon>0$, then on the one hand we have
$\inf_{[s+\varepsilon,t-\varepsilon]}X^{\mathrm{ex}}>X^{\mathrm{ex}}_{t}+2\delta$
for some $\delta>0$, and on the other hand by right-continuity there exists
$s^{\prime}\in(s,s+\varepsilon)$ such that
$X^{\mathrm{ex}}_{s}<X^{\mathrm{ex}}_{s^{\prime}}\leq
X^{\mathrm{ex}}_{s}+\delta$; finally since $X^{\mathrm{ex}}$ has no negative
jumps, then all values in $(X^{\mathrm{ex}}_{t},X^{\mathrm{ex}}_{t}+2\delta)$
are achieved by some times in $(t-\varepsilon,t)$. Recall that
$X^{\mathrm{ex}}$ is the almost sure limit in the Skorokhod $J_{1}$ topology
of some rescaled discrete Łukasiewicz path $X^{n}$. Then for every $n$ large
enough, there exist $s_{n}\in(s,s+\varepsilon)$ and
$t_{n}\in(t-\varepsilon,t)$ with
$X^{n}_{s_{n}}=X^{n}_{t_{n}}=\inf_{[s_{n},t_{n}]}X^{n}$, namely the two times
correspond to the same vertex in the discrete looptree. This implies that they
also correspond to the same vertex in the discrete map, i.e. that
$d_{M^{n}}(s_{n},t_{n})=0$. When passing to the limit along the corresponding
subsequence and letting then $\varepsilon\downarrow 0$, we infer that indeed
$d_{M}(s,t)=0$.

Suppose next that we are in the first case in (3.6), but now that there exists
$r\in(s,t)$ with
$X^{\mathrm{ex}}_{r}=X^{\mathrm{ex}}_{s}=X^{\mathrm{ex}}_{t}$. Then by
uniqueness of the local minima, neither $s$ nor $t$ are local minima. In
addition, as recalled in the proof Lemma 3.3, we have $\Delta
X^{\mathrm{ex}}_{s}=0$ in this case since otherwise the reversed process at
time $r$ would may a weak record by a jump. This implies that for every
$\varepsilon>0$, there exist $s^{\prime}\in(s-\varepsilon,s)$ and
$t^{\prime}\in(t,t+\varepsilon)$ which satisfy
$X^{\mathrm{ex}}_{s^{\prime}}=X^{\mathrm{ex}}_{t^{\prime}}<X^{\mathrm{ex}}_{r}$
for all $r\in(s^{\prime},t^{\prime})$ and we conclude from the preceding case.

Similarly, if we fall in the second case in (3.6), then the previous argument
shows that $t$ is not a time of local minimum for otherwise the reversed
process started just after would reach this record again by a jump. Using the
left-limit at $s$, we infer that for every $\varepsilon>0$, there exist times
$s^{\prime}\in(s-\varepsilon,s)$ and $t^{\prime}\in(t,t+\varepsilon)$ with
$X^{\mathrm{ex}}_{s^{\prime}}=X^{\mathrm{ex}}_{t^{\prime}}<X^{\mathrm{ex}}_{r}$
for every $r\in(s^{\prime},t^{\prime})$ and we conclude again from the first
case. ∎

Let us next continue with a geometric lemma inspired by [LGM11, Lemma 14]
which will allow us to include balls in the map in some well-chosen intervals
analogously to Lemma 3.4 for looptrees. Recall the sets $A_{u}$ and $B_{u}$
defined in (3.1) and (3.2) respectively.

######  Lemma 3.9.

Almost surely for every $u\in(0,1)$ and for every pair $(a,b)\in A_{u}\cup
B_{u}$, we have

$d_{M}(u,t)\geq\min\\{Z_{u}-Z_{a},Z_{u}-Z_{b}\\},$

for every $t\in[0,a)\cup(b,1]$.

We shall assume throughout the proof that both $Z_{a}<Z_{u}$ and $Z_{b}<Z_{u}$
otherwise the claim is trivial.

###### Proof.

Let us first consider the analogue in the discrete setting of a pointed map
$(M^{n},v^{n}_{\star})$ coded by a pair $(W^{n},Z^{n})$ and the associate
labelled looptree. Here times are integers and we use the same notation for a
vertex in the map different from the distinguished one, the corresponding
vertex in the looptree, and a time of visit of one of its corners when
following the contour of the looptree. The assumption that $(a,b)\in A_{u}\cup
B_{u}$ means that either $a$ and $b$ are the same vertex, which is an ancestor
of $u$, or they both belong to the same cycle between two consecutive
ancestors of $u$, with $a$ on the left and $b$ on the right.

Then take a vertex $v$ outside the interval $[a,b]$ and consider a geodesic
_in the map_ from $v$ to $u$. This geodesic necessarily contains an edge of
the map with one extremity $x\in[a,b]$ and the other one $y\notin[a,b]$. From
the construction of the bijection briefly recalled in Section 2.1, this means
that either $x$ is the successor of $y$ or vice versa. Now if $x$ is the
successor of $y$, this means that $x$ is the first vertex after $y$ when
following the contour of the looptree (extended by periodicity) which has a
label smaller than that of $y$. Since one necessarily goes through $a$ when
going from $y$ to $x$ by following the contour of the looptree, then
$Z^{n}_{a}\geq Z^{n}_{y}\geq Z^{n}_{x}+1$, see e.g. Figure 10. Symmetrically,
if $y$ is the successor of $x$, then $Z^{n}_{x}\leq Z^{n}_{b}$. Hence
$Z^{n}_{x}\leq\max\\{Z^{n}_{a},Z^{n}_{b}\\}$ in any case. On the one hand this
vertex $x$ lies on the geodesic in the map between $u$ and $v$, so
$d_{M^{n}}(u,x)<d_{M^{n}}(u,v)$. On the other hand the labels encode the graph
distance to the reference point $v^{n}_{\star}$ in the map, so by the triangle
inequality, we have $d_{M^{n}}(u,x)\geq Z^{n}_{u}-Z^{n}_{x}$, which leads to
our claim in the discrete setting.

Figure 10: A portion of a discrete labelled looptree in plain lines. In dashed
lines is represented a geodesic path between $t$ and $u$ in the associated
map. Each dashed edge thus links a time to its successor. The pictures display
the two situations: when $x$ is the successor of $y$ on the left and vice
versa on the right. Since the edge between $x$ and $y$ steps over $a$ or $b$,
then the label of $x$ is necessarily smaller than or equal to that of $a$ or
$b$ in each case.

In the continuum setting, we argue by approximation. Recall that we assume
$(X^{\mathrm{ex}},Z^{\mathrm{ex}})$ to arise as the almost sure limit of a
pair $(X^{n},Z^{n})$, which is already rescaled in time and space to lighten
the notation. Let us first consider the case when $(a,b)\in B_{u}$, namely
that there exists $s<a$ such that

$X^{\mathrm{ex}}_{s}>X^{\mathrm{ex}}_{a}=\min_{[s,a]}X^{\mathrm{ex}}>X^{\mathrm{ex}}_{b}=\min_{[s,b]}X^{\mathrm{ex}}>X^{\mathrm{ex}}_{s-}.$

We infer from the convergence $X^{n}\to X^{\mathrm{ex}}$ in the Skorokhod
$J_{1}$ topology that there exist times $s_{n}<a_{n}<u_{n}<b_{n}$ in
$\mathbb{N}/n$ which converge to $s$, $a$, $u$, $b$ respectively and such that
$X^{n}_{a_{n}}=\min_{[s_{n},a_{n}]}X^{n}>X^{n}_{b_{n}}=\min_{[s_{n},b_{n}]}X^{n}>X^{n}_{s_{n}-1/n}$.
This implies that $a_{n}$ and $b_{n}$ both belong to the same cycle between
two consecutive ancestors of $u_{n}$, with $a_{n}$ on the left and $b_{n}$ on
the right and we infer from the first part of the proof that the graph
distance in the rescaled map between $u_{n}$ and any vertex $t_{n}$ which lies
outside $[a_{n},b_{n}]$ is larger than or equal to
$\min\\{Z^{n}_{u_{n}}-Z^{n}_{a_{n}},Z^{n}_{u_{n}}-Z^{n}_{b_{n}}\\}$. We
finally can pass to the limit along the corresponding subsequence, using that
$Z^{n}\to Z$ for the uniform topology.

Suppose now that $(a,b)\in A_{u}$, which means that either
$X^{\mathrm{ex}}_{a}=X^{\mathrm{ex}}_{b}=\min_{[a,b]}X^{\mathrm{ex}}$ or
$b=\inf\\{t>a\colon X^{\mathrm{ex}}_{t}=X^{\mathrm{ex}}_{a-}\\}$. As shown in
the proof Lemma 3.8, in each case we can approximate this pair by pairs that
satisfy

$X^{\mathrm{ex}}_{t}>X^{\mathrm{ex}}_{b}=X^{\mathrm{ex}}_{a}\qquad\text{for
every }t\in(a,b).$

We thus henceforth assume that the latter holds. In particular, for any
$\varepsilon>0$ small, the values of $X^{\mathrm{ex}}_{t}$ for
$t\in[a+\varepsilon,b-\varepsilon]$ stay bounded away from
$X^{\mathrm{ex}}_{b}=X^{\mathrm{ex}}_{a}$. From the fact that
$X^{\mathrm{ex}}$ is right-continuous at $s$ and makes no negative jump, we
infer that every value larger than but close enough to
$X^{\mathrm{ex}}_{b}=X^{\mathrm{ex}}_{a}$ is achieved both by times
arbitrarily close to $a$ and by times arbitrarily close to $b$. Then the
convergence $X^{n}\to X^{\mathrm{ex}}$ in the Skorokhod $J_{1}$ topology
yields the existence of times $a_{n}<u_{n}<b_{n}$ which converge to $a$, $u$,
$b$ respectively and such that for $n$ large enough, we have $X^{n}_{t}\geq
X^{n}_{a_{n}}=X^{n}_{b_{n}}$ for every $t\in(a_{n},b_{n})\cap\mathbb{N}/n$. It
follows that $a_{n}$ and $b_{n}$ correspond to the same vertex in the
looptree, and it is an ancestor of $u_{n}$. We conclude from the discrete case
again. ∎

Recall that Proposition 3.2 claims that almost surely, the volume of the ball
in the map centred at an independent uniform random point $U$ and with radius
$2^{-n}$ is less than $\psi(2^{2n(1-\delta)})^{-1}$ for $n$ large enough. This
now easily follows from Lemma 3.9 and a technical result that we state and
prove in the next subsection.

###### Proof Proposition 3.2.

According to Lemma 3.9, almost surely the ball of radius $r$ centred at the
image $U$ in the map is entirely contained in the image of any interval
$[a_{r},b_{r}]$ such that the pair $(a_{r},b_{r})\in A_{U}\cup B_{U}$
satisfies $\min\\{Z_{U}-Z_{a_{r}},Z_{U}-Z_{b_{r}}\\}\geq r$. Consider the
infinite volume model, obtained by replacing the excursion $X^{\mathrm{ex}}$
by the two-sided unconditioned Lévy process and the time $U$ by $0$.
Proposition 3.10 stated below shows that almost surely, for every $r>0$ small
enough, there exists such a pair $(a_{r},b_{r})\in A_{0}\cup B_{0}$ such that
$\min\\{Z_{0}-Z_{a_{r}},Z_{0}-Z_{b_{r}}\\}\geq r$ and both $a_{r}$ and $b_{r}$
lie at looptree distance at most $r^{2(1-\delta)}$ from $0$. Take $r=2^{-n}$.
We infer from the proof Proposition 3.1 that almost surely, for every $n$
large enough we have both $a_{2^{-n}}\geq U^{-}_{2^{-2n(1-\delta)}}\geq
U-\psi(2^{2n(1-\delta)^{2}})^{-1}$ and $b_{2^{-n}}\leq
U^{+}_{2^{-2n(1-\delta)}}\leq U+\psi(2^{2n(1-\delta)^{2}})^{-1}$. The claim
follows by local absolute continuity. ∎

###  3.4. A technical lemma with Brownian motions

In order to complete the proof Proposition 3.2, it remains to establish the
following.

######  Proposition 3.10.

For every $\varepsilon>0$, almost surely, for every $r>0$ small enough, in the
infinite volume labelled looptree coded by the pair of paths
$X=(\overleftarrow{X}_{s},\overrightarrow{X}_{t})_{s,t\geq 0}$, there exists a
pair $(a_{r},b_{r})\in A_{0}\cup B_{0}$ which satisfies both $-Z_{a_{r}}\geq
r$ and $-Z_{b_{r}}\geq r$ as well as $d_{\mathscr{L}(X)}(a_{r},0)\leq
r^{2(1-\varepsilon)}$ and $d_{\mathscr{L}(X)}(b_{r},0)\leq
r^{2(1-\varepsilon)}$.

In the case of stable Lévy processes, Le Gall & Miermont [LGM11, Lemma 15]
were able to find such a pair $(a_{r},b_{r})\in A_{0}$, namely that codes an
ancestor of the distinguished point $0$. Their argument can be extended when
the upper and lower exponents $\boldsymbol{\gamma}$ and $\boldsymbol{\eta}$
coincide, or when the exponent $\boldsymbol{\delta}\in[1,\boldsymbol{\gamma}]$
introduced Duquesne [Duq12] as
$\boldsymbol{\delta}\coloneqq\sup\\{c>0\colon\exists K\in(0,\infty)\text{ such
that }K\,\mu^{-c}\psi(\mu)\leq\lambda^{-c}\psi(\lambda),\text{ for all
}1\leq\mu\leq\lambda\\}$ is not equal to $1$. However we were not able to
extend it when $\boldsymbol{\delta}=1$. We therefore argue in a general
setting, conditionally on the underlying looptree, and relying only on the
properties of the Brownian labels.

Let us first introduce the setting in terms of Brownian motion and Brownian
bridges. Let $\mathcal{C}$ be a random closed subset of $[0,\infty)$
containing $0$, its complement is a countable union of disjoint open
intervals, say, $\mathcal{C}^{c}=\bigcup_{i\geq 1}(g_{i},d_{i})$ with
$d_{i}>g_{i}$. For every $i\geq 1$, let us denote by $\delta_{i}=d_{i}-g_{i}$
the length of these intervals and suppose we are given random lengths
$\delta^{L}_{i},\delta^{R}_{i}\geq\delta_{i}$. Independently, let
$W=(W_{t},t\geq 0)$ be a standard Brownian motion started from $0$. Consider
its trace on $\mathcal{C}$, then independently for every $i\geq 1$,
conditionally on the values $W_{g_{i}}$ and $W_{d_{i}}$ and on the lengths
$\delta^{L}_{i}$ and $\delta^{R}_{i}$, let us sample two independent Brownian
bridges $W^{L,i}=(W^{L,i}_{t};t\in[0,\delta^{L}_{i}])$ and
$W^{R,i}=(W^{R,i}_{t};t\in[0,\delta^{R}_{i}])$ with duration $\delta^{L}_{i}$
and $\delta^{R}_{i}$ respectively, both starting from
$W^{L,i}_{0}=W^{R,i}_{0}=W_{g_{i}}$ and ending at
$W^{L,i}_{\delta^{L}_{i}}=W^{R,i}_{\delta^{R}_{i}}=W_{d_{i}}$ and independent
of the rest.

The connection with our problem is the following: the successive ancestors of
$0$ in the infinite looptree, correspond to the times of weak records of
$\overleftarrow{X}$, and their labels form a subordinate Brownian motion,
precisely $W_{\sigma(\cdot)}$, where $W$ is a standard Brownian motion and
$\sigma$ is the last subordinator in Lemma 3.5. We shall thus take
$\mathcal{C}$ to be the range of $\sigma$, and the labels on each side of the
successive cycles between these ancestors roughly correspond to the collection
$\\{W^{L,i},W^{R,i}\\}$.

######  Lemma 3.11.

For every $\varepsilon>0$, almost surely, for every $r>0$ small enough, the
following alternative holds:

  * •

either there exists $t\in\mathcal{C}\cap[0,r^{2(1-\varepsilon)}]$ such that
$W_{t}\leq-r$,

  * •

or there exist $i\geq 1$ with $g_{i}<r^{2(1-\varepsilon)}$ and
$s\in[0,\min\\{\delta^{L}_{i},r^{2(1-\varepsilon)}\\}]$ and
$t\in[0,\min\\{\delta^{R}_{i},r^{2(1-\varepsilon)}\\}]$ such that both
$W^{L,i}_{s}\leq-r$ and $W^{R,i}_{t}\leq-r$.

The key point in this statement is that in the second case, both bridges reach
$-r$ in the same interval of $\mathcal{C}^{c}\cap[0,r^{2(1-\varepsilon)}]$.
The idea of the proof is quite intuitive: first we know by the law of iterated
logarithm that $W$ reaches values much less than $-r$ on
$[0,r^{2(1-\varepsilon)}]$ so, if the first alternative fails, then there
exists an interval $(g_{i},d_{i})$ on which $W$ goes much below $-r$, while
starting and ending above this value. Then this interval cannot be too small
for the Brownian motion to fluctuate enough. Therefore, if one resamples two
Brownian bridges with the same duration $\delta_{i}$ and the same endpoints
$W_{g_{i}}$ and $W_{d_{i}}$, then they have a reasonable chance to both go
below $-r$ in this interval. Actually the Brownian bridges we sample have a
longer duration, so intuitively, they fluctuate even more and have an even
greater chance to reach $-r$. However turning this discussion into a rigorous
proof is quite challenging, and the proof Lemma 3.11 is deferred to Section
3.5.

Let us finish this section by relying on this result in order to establish
Proposition 3.10, thereby finishing the proof Proposition 3.2.

###### Proof Proposition 3.10.

Recall the Poisson random measure $\mathscr{N}$ in (3.5) and the four
subordinators defined from it. By construction of the looptree, the successive
ancestors of $0$ in the infinite looptree, started from $0$, are coded by the
negative of the first element of pairs in $A_{0}$ (defined in (3.1)) and
correspond to the times of weak records of $\overleftarrow{X}$. Further, by
construction of the label process, the labels of these ancestors form a
subordinate Brownian motion, precisely $W_{\sigma(\cdot)}$, where $W$ is a
standard Brownian motion and $\sigma$ is the last subordinator in Lemma 3.5.
We shall apply Lemma 3.11 to the closed set $\mathcal{C}$ given by the range
of $\sigma$.

Every jump of this subordinator, and more precisely every atom of the Poisson
random measure $\mathscr{N}$, codes two consecutive ancestors separated by a
cycle in the looptree, whose left and right length is encoded in the atom of
$\mathscr{N}$. In Lemma 3.11, these two lengths correspond to $\delta^{L}_{i}$
and $\delta^{R}_{i}$, whereas the length $\delta_{i}=d_{i}-g_{i}$ equals
$\delta_{i}=\delta^{L}_{i}\delta^{R}_{i}/(\delta^{L}_{i}+\delta^{R}_{i})\leq\min(\delta^{L}_{i},\delta^{R}_{i})$
and corresponds indeed to the variance of a Brownian bridge of duration
$\delta^{L}_{i}+\delta^{R}_{i}$ and evaluated at $\delta^{L}_{i}$ or
$\delta^{R}_{i}$. By standard properties of the Brownian bridge, conditionally
on these lengths and on the labels of the two ancestors $W_{g_{i}}$ and
$W_{d_{i}}$, the labels on each side of the cycle indeed evolve like two
independent Brownian bridges, given precisely by $W^{L,i}$ and $W^{R,i}$.

Lemma 3.11 thus shows the following alternative: either there exists a value
$t_{r}\leq r^{2(1-\varepsilon)}$ in the range of $\sigma$, say
$t_{r}=\sigma(s_{r})$, and such that $W_{\sigma(s_{r})}\leq-r$. In this case,
the time $s_{r}$, once time-changed by the local time at the supremum of
$\overleftarrow{X}$, corresponds to the first element of a pair
$(-a_{r},b_{r})\in A_{0}$. Otherwise there exists a jump of $\sigma$, say
$g_{r}=\sigma(s_{r}-)<\sigma(s_{r})=d_{r}$, with $g_{r}<r^{2(1-\varepsilon)}$,
which corresponds to a cycle in the looptree, and then there exists one point
on each side of this cycle, both at distance (on the cycle) at most
$r^{2(1-\varepsilon)}$ from $g_{r}$ and with label less than $-r$.

Notice the easy bound $\min\\{u,1-u\\}\leq 2u(1-u)$ for any $0\leq u\leq 1$
which shows that the subordinators $\sigma$ and $\widetilde{X}$ from Lemma 3.5
are related by $\widetilde{X}\leq 2\sigma$. Observe that $\widetilde{X}$
encodes the looptree distance to $0$ of the successive ancestors. Therefore,
in the first case of the alternative, the time $-a_{r}$ lies at looptree
distance at most $2r^{2(1-\varepsilon)}$ from $0$. So does the time
corresponding to $g_{r}$ in the second case, and thus the two points on each
side of the cycle lie at looptree distance at most $3r^{2(1-\varepsilon)}$
from $0$. One can then get rid of this factor $3$ by replacing $\varepsilon$
by a smaller value. ∎

###  3.5. Proof the technical lemma

Let us end this section with the proof of Lemma 3.11. The statement can be
rephrased as follows: if the Brownian motion $W$ is such that $W_{t}>-r$ for
all $t\in\mathcal{C}\cap[0,r^{2(1-\varepsilon)}]$, then necessarily the two
Brownian bridges we resample both reach $-r$ in the same interval in
$\mathcal{C}^{c}\cap[0,r^{2(1-\varepsilon)}]$.

We start with an easy intermediate result. We shall denote by
$\tau_{z}=\inf\\{t\geq 0\colon W_{t}=z\\}$ the first hitting time of
$z\in\mathbb{R}$ by the Brownian motion $W$. For every $r\in(0,1)$ small
enough, we let $K_{r}$ be the largest integer such that

$K_{r}\leq\frac{\log(1/r)}{(\log\log(1/r))^{3}}.$ (3.7)

######  Lemma 3.12.

Almost surely, for every $r\in\\{2^{-n},n\in\mathbb{N}\\}$ small enough, the
following assertions hold:

  1. (i)

The path $W$ reaches $-r\log^{10K_{r}}(1/r)$ before time
$r^{2(1-1/\log\log(1/r))}$,

  2. (ii)

For every $k\leq K_{r}$, the path $W$ reaches $-r\log^{10k}(1/r)$ before
$r\log^{10k+3}(1/r)$,

  3. (iii)

There exists a universal constant $c\in(0,1)$ such that for at least $cK_{r}$
different indices $k\leq K_{r}$ the following holds: at the first time the
path $W$ reaches $-r\log^{10k}(1/r)$, it was below $-r$ for a duration at
least $(r\log^{10k}(1/r))^{2}$ and it previously never reached
$r\log^{10k}(1/r)$.

Let us note that by comparing any $r\in(0,1)$ to a negative power of $2$, the
claims extend by monotonicity to any $r>0$ small enough, provided a minor
change in the exponents.

###### Proof.

Let us prove that for each event, the probability to fail is bounded above by
a quantity which depends on $r$ and such that the sum over all values of
$r\in\\{2^{-n},n\geq 1\\}$ is finite. The claims then follow from the
Borel–Cantelli lemma. For the first assertion, the reflection principle and
the scaling property show that:

$\displaystyle\mathop{}\\!\mathbb{P}(\tau_{-r\log^{10K_{r}}(1/r)}>r^{2(1-1/\log\log(1/r))})$
$\displaystyle=\mathop{}\\!\mathbb{P}(\inf\\{W_{t},t\in[0,r^{2(1-1/\log\log(1/r))}]\\}\geq-r\log^{10K_{r}}(1/r))$
$\displaystyle=\mathop{}\\!\mathbb{P}(|W_{r^{2(1-1/\log\log(1/r))}}|\leq
r\log^{10K_{r}}(1/r))$ $\displaystyle=\mathop{}\\!\mathbb{P}(|W_{1}|\leq
r^{1/\log\log(1/r)}\log^{10K_{r}}(1/r)).$

Since $K_{r}=o(\log(1/r)/(\log\log(1/r))^{2})$, then for $r$ small enough, we
have

$r^{1/\log\log(1/r)}\log^{10K_{r}}(1/r)\leq\exp\Bigl{(}-\frac{\log(1/r)}{2\log\log(1/r)}\Bigr{)},$

whose sum of over all values of $r\in\\{2^{-n},n\geq 1\\}$ is finite. By a
Taylor expansion of the last probability, written as the integral of the
density, we obtain:

$\mathop{}\\!\mathbb{P}(\tau_{-r\log^{10K_{r}}(1/r)}>r^{2(1-1/\log\log(1/r))})=O\Bigl{(}\exp\Bigl{(}-\frac{\log(1/r)}{2\log\log(1/r)}\Bigr{)}\Bigr{)},$

hence the sum of the left-hand side over all values of $r\in\\{2^{-n},n\geq
1\\}$ is finite as we wanted.

Next, by the classical ruin problem, we have:

$\mathop{}\\!\mathbb{P}(\tau_{r\log^{10k+3}(1/r)}<\tau_{-r\log^{10k}(1/r)})=\frac{r\log^{10k}(1/r)}{r\log^{10k}(1/r)+r\log^{10k+3}(1/r)}=\frac{1}{1+\log^{3}(1/r)}.$

Since $K_{r}=o(\log(1/r))$, then we infer by a union bound that

$\mathop{}\\!\mathbb{P}(\exists k\leq
K_{r}\colon\tau_{r\log^{10k+3}(1/r)}<\tau_{-r\log^{10k}(1/r)})=o(\log^{-2}(1/r)),$

and again the right-hand side forms a convergent series when summing over all
values $r\in\\{2^{-n},n\in\mathbb{N}\\}$.

For the last claim, by symmetry, for any given $k$, the probability that $W$
reaches $-r\log^{10k}(1/r)$ before $r\log^{10k}(1/r)$ equals $1/2$ so if these
events were independent, then this would follow e.g. from Hoeffding’s
inequality. They are not independent, however by the previous bound, we may
assume that at the first time $W$ reaches $-r\log^{10k}(1/r)$, it has not
reached $r\log^{10(k+1)}(1/r)$ yet, and by the strong Markov property, the
probability starting from this position to reach $-r\log^{10(k+1)}(1/r)$
before $r\log^{10(k+1)}(1/r)$ is (slightly) larger than $1/2$. Formally, one
can e.g. apply the Azuma inequality to the martingale whose increments are

$\Delta
M_{k}=\mathbf{1}_{\\{\tau_{-r\log^{10(k+1)}(1/r)}<\tau_{r\log^{10(k+1)}(1/r)}\\}}-\mathop{}\\!\mathbb{P}(\tau_{-r\log^{10(k+1)}(1/r)}<\tau_{r\log^{10(k+1)}(1/r)}\mid(W_{t},t\leq\tau_{-r\log^{10k}(1/r)})),$

and since the conditional probability on the right is larger than $1/2$, then

$\displaystyle\mathop{}\\!\mathbb{P}\Bigl{(}\\#\\{k\leq
K_{r}\colon\tau_{-r\log^{10(k+1)}(1/r)}<\tau_{r\log^{10(k+1)}(1/r)}\\}<\frac{K_{r}}{4}\Bigr{)}$
$\displaystyle=\mathop{}\\!\mathbb{P}\Bigl{(}\sum_{k=1}^{K_{r}}\Bigl{(}\mathbf{1}_{\\{\tau_{-r\log^{10(k+1)}(1/r)}<\tau_{r\log^{10(k+1)}(1/r)}\\}}-\frac{1}{2}\Bigr{)}<-\frac{K_{r}}{4}\Bigr{)}$
$\displaystyle\leq\mathop{}\\!\mathbb{P}\Bigl{(}\sum_{k=1}^{K_{r}}\Delta
M_{k}<-\frac{K_{r}}{4}\Bigr{)}$ $\displaystyle\leq\exp(-cK_{r}),$

where $c>0$ is some universal constant. The sum of these exponentials over all
values $r\in\\{2^{-n},n\in\mathbb{N}\\}$ is finite again.

Similarly, for any $z>0$, the law of the time-reversed path
$(z+W_{(\tau_{-z}-t)^{+}})_{t\geq 0}$ is that of a three-dimensional Bessel
process started from $0$, or equivalently the Euclidean norm of a three-
dimensional Brownian motion, stopped at the last passage at $z$. The
probability that a Bessel process does not reach $z/2$ before time $z^{2}$ is
bounded below by some constant $c>0$ independent of $z$. We conclude as above,
that the proportion of the number of indices $k\leq K_{r}$ such that this
occurs for $z=r\log^{10k}(1/r)$ is bounded away from $0$. ∎

Let us finally prove Lemma 3.11.

###### Proof Lemma 3.11.

Let us work on the event that $W_{t}>-r$ for all
$t\in\mathcal{C}\cap[0,r^{2(1-\varepsilon)}]$ and let us prove that
necessarily the second case of the alternative occurs. By monotonicity, it is
sufficient to prove the claim when $r$ is restricted to the set
$\\{2^{-n},n\in\mathbb{N}\\}$. Let us implicitly assume also that the three
assertions from Lemma 3.12 hold, and let us prove that there exists $k\leq
K_{r}$, the integer from (3.7), such that on the interval of $\mathcal{C}^{c}$
on which $W$ reaches $-r\log^{10k}(1/r)$ for the first time, the two bridges
that we sample also reach $-r$.

Let us introduce some notation. For every $k\geq 1$, denote by $G_{k}$ and
$D_{k}$ the boundaries $g_{i}$ and $d_{i}$ of the unique interval
$(g_{i},d_{i})\subset\mathcal{C}^{c}$ such that $W$ reaches the level
$-r\log^{10k}(1/r)$ for the first time in the interval $(g_{i},d_{i})$. Note
that these intervals are not necessarily distinct from each others and that
when $r$ is small enough, by Lemma 3.12, they all intersect
$(0,r^{2(1-\varepsilon)})$ for $k\leq K_{r}$, and only $D_{K_{r}}$ may exceed
$r^{2(1-\varepsilon)}$. Let $\Delta_{k}=D_{k}-G_{k}$ and let
$\Delta^{L}_{k},\Delta^{R}_{k}\geq\Delta_{k}$ denote the corresponding
extensions; let us change the notation and write $W^{L,k}$ and $W^{R,k}$ for
the Brownian bridges with duration $\Delta^{L}_{k}$ and $\Delta^{R}_{k}$
respectively, with initial and terminal value $W_{G_{k}}$ and $W_{D_{k}}$
respectively. We shall upper bound the probability of the event

$A_{r}=\bigcap_{k\leq
K_{r}}\Bigl{(}\Bigl{\\{}\inf_{s\in[0,\Delta^{L}_{k}]\cap(0,r^{2(1-\varepsilon)})}W^{L,k}_{s}>-r\Bigr{\\}}\cup\Bigl{\\{}\inf_{t\in[0,\Delta^{R}_{k}]\cap(0,r^{2(1-\varepsilon)})}W^{R,k}_{t}>-r\Bigr{\\}}\Bigr{)}$

by a quantity that only depends on $r$ and whose sum over all values of
$r\in\\{2^{-n},n\geq 1\\}$ is finite, such as $\log^{-2}(1/r)$. We then
conclude as in the previous proof from the Borel–Cantelli lemma.

Step 1: Proof for Brownian motions. Let us first prove the claim when the
Brownian bridges $W^{L}$ and $W^{R}$ are replaced by Brownian motions.
Precisely, for every $k\leq K_{r}$, conditionally on $\Delta_{k}$,
$\Delta_{k}^{L}$, $\Delta_{k}^{R}$, and $W_{G_{k}}$, let
$X^{L,k}=(X^{L,k}_{t};t\in[0,\Delta_{k}^{L}])$ and
$X^{R,k}=(X^{R,k}_{t};t\in[0,\Delta_{k}^{L}])$ be two independent Brownian
motions started from $X^{L,k}_{0}=X^{R,k}_{0}=W_{G_{k}}>-r$.

We first claim that if the interval $[G_{k},D_{k}]$ is too long, then the
Brownian motions $X^{L,k}$ and $X^{R,k}$ on it cannot reasonably stay above
$-r$. Specifically, if we denote by $X$ a generic Brownian motion independent
of the rest, then by the same reasoning as in the previous proof, we have for
every $k$:

$\displaystyle\mathop{}\\!\mathbb{P}\Bigl{(}\Delta_{k}>2(r+W_{G_{k}})^{2}\log^{6}(1/r)\enskip\text{and}\enskip\max\Bigl{\\{}\inf_{t\in[0,\Delta_{k}/2]}X^{L,k}_{t},\inf_{t\in[0,\Delta_{k}/2]}X^{R,k}_{t}\Bigr{\\}}>-r\Bigr{)}$
$\displaystyle\leq 2\mathop{}\\!\mathbb{P}\bigl{(}\inf\\{X_{s},0\leq
s\leq(r+W_{G_{k}})^{2}\log^{6}(1/r)\\}>-(W_{G_{k}}+r)\bigr{)}$