Random stable looptrees

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.19(2014), no. 108, 1–35.

ISSN:1083-6489 DOI:10.1214/EJP.v19-2732

Random stable looptrees

Nicolas Curien

Igor Kortchemski

Abstract

We introduce a class of random compact metric spaces Lα indexed byα ∈ (1,2) and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can informally be viewed as dual graphs ofα-stable Lévy trees. We study their properties and prove in particular that the Hausdorff dimension ofLα is almost surely equal toα. We also show that stable looptrees are universal scaling limits, for the Gromov–Hausdorff topology, of various combinatorial models. In a companion paper, we prove that the stable looptree of parameter ³₂ is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations.

Keywords:Stable processes; random metric spaces; random non-crossing configurations.

AMS MSC 2010:Primary 60F17; 60G52, Secondary 05C80.

Submitted to EJP on April 9, 2013, final version accepted on November 10, 2014.

Figure 1: An α = 1.1 stable tree, and its associated looptree L1.1, embedded non isometrically in the plane (this embedding of L1.1 contains intersecting loops, even though they are disjoint in the metric space).

*This work is partially supported by the French “Agence Nationale de la Recherche” ANR-08-BLAN-0190.

†CNRS and Université Paris 6, France. E-mail:nicolas.curien@gmail.com

‡DMA, École Normale Supérieure, France. E-mail:igor.kortchemski@normalesup.org

1 Introduction

In this paper, we introduce and study a new family(Lα)1<α<2 of random compact metric spaces which we call stable looptrees (in short, looptrees). Informally, they are constructed from the stable tree of index αintroduced in [17, 26] by replacing each branch-point of the tree by a cycle of length proportional to the “width” of the branch-point and then gluing the cycles along the tree structure (see Theorem 2.3 below). We study their fractal properties and calculate in particular their Hausdorff dimension. We also prove that looptrees naturally appear as scaling limits for the Gromov–Hausdorff topology of various discrete random structures, such as Boltzmann- type random dissections which were introduced in [23].

Perhaps more unexpectedly, looptrees appear in the study of random maps decorated with statistical physics models. More precisely, in a companion paper [15], we prove that the stable looptree of parameter ³₂ is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations and on the uniform infinite planar triangulation of Angel & Schramm [2]. We also conjecture a more general statement for O(n)models on random planar maps.

In this paperα∈(1,2).

Stable looptrees as limits of discrete looptrees. In order to explain the intuition leading to the definition of stable looptrees, we first introduce them as limits of random discrete graphs (even though they will be defined later without any reference to discrete objects). To this end, with every rooted oriented tree (or plane tree)τ, we associate a graph denoted byLoop(τ)and constructed by replacing each vertexu∈τby a discrete cycle of length given by the degree ofuⁱⁿτ(i.e. number of neighbors ofu) and gluing all these cycles according to the tree structure provided byτ, see Figure 2 (by discrete cycle of lengthk, we mean a graph onkverticesv1, . . . , vk with edgesv1v2, . . . , v_k−1vk, vkv1).

We endowLoop(τ)with the graph distance (every edge has unit length).

Figure 2: A discrete treeτ and its associated discrete looptreeLoop(τ)^.

Fixα∈(1,2)and letτn be a Galton–Watson tree conditioned on havingnvertices, whose offspring distributionµis critical and satisfiesµ([k,∞))∼ |Γ(1−α)|⁻¹·k^−αas k → ∞. The stable looptreeLα then appears (Theorem 4.1) as the scaling limit in distribution for the Gromov–Hausdorff topology of discrete looptreesLoop(τn):

n^−1/α·Loop(τn) −−−−→_n→∞^(d) Lα, (1.1) wherec·M stands for the metric space obtained fromM by multiplying all distances

byc >0. Recall that the Gromov–Hausdorff topology gives a sense to convergence of (isometry classes) of compact metric spaces, see Section 3.2 below for the definition.

It is known that the random treesτn converge, after suitable scaling, towards the so-called stable treeT^{α of index}α(see [16, 17, 26]). It thus seems natural to try to defineLαdirectly fromT^αby mimicking the discrete setting (see Figure 1). However this construction is not straightforward since the countable collection of loops ofLα

does not form a compact metric space: one has to take its closure. In particular, two different cycles ofLαnever share a common point. To overcome these difficulties, we defineLαby using the excursionX^exc,(α)of anα-stable spectrally positive Lévy process (which also codesT^α).

Properties of stable looptrees. Stable looptrees possess a fractal structure whose dimension is identified by the following theorem:

Theorem 1.1(Dimension).For everyα∈(1,2), almost surely,Lαis a random compact metric space of Hausdorff dimensionα.

The proof of this theorem uses fine properties of the excursion X^exc,(α). We also prove that the family of stable looptrees interpolates between the circle of unit length C¹:= (2π)⁻¹·S1and the2-stable treeT²which is the Brownian Continuum Random Tree introduced by Aldous [1] (up to a constant multiplicative factor).

Theorem 1.2(Interpolation loop-tree).The following two convergences hold in distribu- tion for the Gromov–Hausdorff topology

(i) Lα

−−→_α(d)

↓1 C¹, (ii) Lα

−−→_α(d)

↑2

1 2· T².

See Figure 3 for an illustration. The proof of (i) relies on a new “one big-jump principle” for the normalized excursion of theα-stable spectrally positive Lévy process which is of independent interest: informally, as α ↓ 1, the random process X^exc,(α) converges towards the deterministic affine function on[0,1]which is equal to1at time 0 and0at time 1. We refer to Theorem 3.6 for a precise statement. Notice also the appearance of the factor ¹₂ in(ii).

1 INTRODUCTION 4

Figure 3: On the leftL1.01, on the rightL1.9. fig:1et2

wheredeg(f)is the degree of the facef, that is the number of edges in the boundary off, andZnis a normalizing constant. Under mild assumptions onµ, this definition makes sense for everynlarge enough. LetD^µ_nbe a random dissection sampled according toP^µn. In [18],whenµhas a heavy tail, the second author studied theasymptotic behavior ofD^µ_nviewed as a random closed subset of the unit disk whenn!1. In thiscase, the limiting object (the so called stable lamination of index

↵) is a random compact subset of the diskwhich isthe union of infinitely many non-intersecting chords and has faces of infinite degree. Its Hausdorff dimension is a.s.2-↵^-1.

In this paper, instead of consideringD^µ_nas a random compact subset of the unit disk, we view D^µ_nas a metric space by endowingD^µ_nwith the graph distanced_gr(every edge ofD^µ_nhas length one). From this perspective, it turns out the scaling limit of the random Boltzmann dissectionsD^µn

is a stable looptree:

cor:discretencstable Corollary 1. Fix↵2(1,2)and letµbe a probability measure supported on{0,2,3,. . .}of mean1such thatµ([k,1))⇠c·k^-↵ask!1. Then the following convergence holds in distribution for the Gromov–

Hausdorff topology

n^-1/↵·D^µ_{n n}^(d)!

!1 (| (1-↵)|·c·µ0)^-1/↵·L↵.

Figure 4: A large dissection and a representation of its metric space.

fig:dissec+looptree

Looptrees in random planar maps. Another area where looptrees appear is the theory of Figure 3: On the leftL1.01, on the rightL1.9.

Scaling limits of Boltzmann dissections. Our previously mentioned invariance prin- ciple (Theorem 4.1) also enables us to prove that stable looptrees are scaling limits of Boltzmann dissections of [23]. Before giving a precise statement, we need to introduce some notation. Forn≥3, letPn be the convex polygon inscribed in the unit disk of the complex plane whose vertices are then-th roots of unity. By definition, a dissection is the

EJP19(2014), paper 108.

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union of the sides ofPn and of a collection of diagonals that may intersect only at their endpoints, see Figure 11. The faces are the connected components of the complement of the dissection in the polygon. Following [23], ifµ= (µj)_j≥0is a probability distribution on{0,2,3,4, . . .}^{of mean}1, we define a Boltzmann–type probability measureP^µn on the set of all dissections ofPn+1by setting, for every dissectionωofPn+1:

P^µn(ω) = 1 Zn

Y

fface ofω

µdeg(f)−1,

wheredeg(f)is the degree of the facef, that is the number of edges in the boundary of f, andZnis a normalizing constant. Under mild assumptions onµ, this definition makes sense for everynlarge enough. LetDn^µbe a random dissection sampled according toP^µn. In [23], the second author studied the asymptotic behavior ofDn^µviewed as a random closed subset of the unit disk whenn→ ∞in the case whereµhas a heavy tail. Then the limiting object (the so-called stable lamination of indexα) is a random compact subset of the disk which is the union of infinitely many non-intersecting chords and has faces of infinite degree. Its Hausdorff dimension is a.s.2−α⁻¹.

In this paper, instead of consideringDn^µas a random compact subset of the unit disk, we viewDn^µas a metric space by endowing the vertices ofDn^µwith the graph distance (every edge ofD^µn has length one). From this perspective, the scaling limit of the random Boltzmann dissectionsDn^µis a stable looptree (see Figure 4):

Corollary 1.3.Fixα∈(1,2)and letµbe a probability measure supported on{0,2,3, . . .} of mean1such thatµ([k,∞))∼c·k^−αask→ ∞, for a certainc >0. Then the following convergence holds in distribution for the Gromov–Hausdorff topology

n^−1/α· D^µn

−−−−→_n→∞(d) (cµ0|Γ(1−α)|)^−1/α·Lα.

Figure 4: A large dissection and a representation of its metric space.

Looptrees in random planar maps. Another area where looptrees appear is the theory of random planar maps. The goal of this very active field is to understand large-scale properties of planar maps or graphs, chosen uniformly in a certain class (triangulations, quadrangulations, etc.), see [2, 11, 27, 25, 30]. In a companion paper [15], we prove that the scaling limit of cluster boundaries of critical site-percolation on large random triangulations and the UIPT introduced by Angel & Schramm [2] isL3/2

(byboundary of a cluster, we mean the graph formed by the edges and vertices of a connected component which are adjacent to its exterior; see [15] for a precise definition and statement). We also give a precise conjecture relating the whole family of looptrees (Lα)α∈(1,2) to cluster boundaries of criticalO(n)models on random planar maps. We

refer to [15] for details.

Looptrees in preferential attachment. As another motivation for introducing loop- trees, we mention the subsequential work [13], which studies looptrees associated with random trees built by linear preferential attachment, also known in the literature as Barabási–Albert trees or plane-oriented recursive trees. As the number of nodes grows, it is shown in [13] that these looptrees, appropriately rescaled, converge in the Gromov–

Hausdorff sense towards a random compact metric space called the Brownian looptree, which is a quotient space of Aldous’ Brownian Continuum Random Tree.

Finally, let us mention that stable looptrees implicitly appear in [27], where Le Gall and Miermont have considered scaling limits of random planar maps with large faces.

The limiting continuous objects (the so-called α-stable maps) are constructed via a distance process which is closely related to looptrees. Informally, the distance process of Le Gall and Miermont is formed by a looptreeLαwhere the cycles support independent Brownian bridges of the corresponding lengths. However, the definition and the study of the underlying looptree structure is interesting in itself and has various applications.

Even though we do not rely explicitly on the article of Le Gall and Miermont, this work would not have been possible without it.

Outline. The paper is organized as follows. In Section 2, we give a precise definition ofLαusing the normalized excursion of theα-stable spectrally positive Lévy process.

Section 3 is then devoted to the study of stable looptrees, and in particular to the proofs of Theorems 1.1 and 1.2. In the last section, we establish a general invariance principle concerning discrete looptrees from which Theorem 1.3 will follow.

2 Defining stable looptrees

This section is devoted to the construction of stable looptrees using the normalized excursion of a stable Lévy process, and to the study of their properties. In this section, α∈(1,2)is a fixed parameter.

2.1 The normalized excursion of a stable Lévy process

We follow the presentation of [16] and refer to [5] for the proof of the results mentioned here. By α-stable Lévy process we will always mean a stable spectrally positive Lévy processX of indexα, normalized so that for everyλ >0

E[exp(−λXt)] = exp(tλ^α).

The processX takes values in the Skorokhod spaceD(R+,R)of right-continuous with left limits (càdlàg) real-valued functions, endowed with the Skorokhod topology (see [8, Chap. 3]). The dependence ofXinαwill be implicit in this section. Recall thatX enjoys the following scaling property: For everyc > 0, the process(c^−1/αXct, t ≥0) has the same law asX. Also recall that the Lévy measureΠofX is

Π(dr) = α(α−1)

Γ(2−α)r^−α−11(0,∞)dr. (2.1) Following Chaumont [12] we define the normalized excursion ofX above its infimum as the re-normalized excursion ofX above its infimum straddling time1. More precisely, set

g₁= sup{s≤1;Xs= inf

[0,s]X} ^and d₁= inf{s >1;Xs= inf

[0,s]X}. Note thatXd₁ =Xg

1 since a.s.X has no jump at timeg₁andX has no negative jumps . Then the normalized excursionX^excofX above its infimum is defined by

X_s^exc = (d₁−g₁)^−1/α(Xg

1+s(d₁−g

1)−Xg

1) for everys∈[0,1]. (2.2)

We shall see later in Section 3.1.2 another useful description of X^exc using the Itô excursion measure ofX above its infimum. Notice thatX^exc is a.s. a random càdlàg function on[0,1]such thatX₀^exc =X₁^exc= 0andX_s^exc >0for everys∈(0,1). IfY is a càdlàg function, we set∆Yt=Yt−Y_t−, and to simplify notation, for0< t≤1, we write

∆t=X_t^exc−X_t^exc₋

and set∆0= 0by convention.

2.2 The stable Lévy tree

We now discuss the construction of theα-stable treeT^α, which is closely related to theα-stable looptree. Even though it possible to defineLαwithout mentioningT^{α, this} sheds some light on the intuition hiding behind the formal definition of looptrees.

2.2.1 The stable height process

By the work of Le Gall & Le Jan [26] and Duquesne & Le Gall [17, 18], it is known that the random excursionX^exc encodes a random compactR^-treeT^{α called the}α-stable tree. To defineT^α, we need to introduce theheight process associated withX^{exc. We} refer to [17] and [18] for details and proofs of the assertions contained in this section.

First, for0≤s≤t≤1, set

I_s^t= inf

[s,t]X^exc.

The height processH^exc associated withX^exc is defined by the approximation formula H_t^exc = lim

ε→0

1 ε

Z t 0

ds1_{X_s^exc<I_s^t+ε}, t∈[0,1],

where the limit exists in probability. The process(H_t^exc)0≤t≤1has a continuous modifica- tion, which we consider from now on. ThenH^{exc satisfies}H₀^exc=H₁^exc= 0^andH_t^exc>0 fort ∈ (0,1). It is standard to define theR-tree coded byH^exc as follows. For every h: [0,1]→R+and0≤s, t≤1, we set

dh(s, t) =h(s) +h(t)−2 inf

[min(s,t),max(s,t)]h. ^(2.3)

Recall that a pseudo-distancedon a setXis a mapd:X×X →R+such thatd(x, x) = 0 andd(x, y)≤d(x, z)+d(z, y)for everyx, y, z∈X(it is a distance if, in addition,d(x, y)>0 ifx6=y). It is simple to check thatdhis a pseudo-distance on [0,1]. In the caseh=H^exc, forx, y∈[0,1], setx'yifdH^exc(x, y) = 0. The random stable treeT^αis then defined as the quotient metric space [0,1]/',dH^exc

, which indeed is a random compactR^-tree [18, Theorem 2.1]. Letπ : [0,1] → T^α be the canonical projection. The treeT^{α has a} distinguished pointρ=π(0), called theroot or the ancestor of the tree. Ifu, v ∈ T^α, we denote by[[u, v]]the unique geodesic betweenuandv. This allows us to define a genealogical order onT^α: For everyu, v ∈ T^{α, set}u4vifu∈[[ρ, v]]. Ifu, v∈ T^{α, there} exists a uniquez∈ T^{αsuch that}[[ρ, u]]∩[[ρ, v]] = [[ρ, z]], called themost recent common ancestortouandv, and is denoted byz=u∧v.

2.2.2 Genealogy ofT^{α and}X^exc

The genealogical order ofT^αcan be easily recovered fromX^exc as follows. We define a partial order on [0,1], still denoted by 4, which is compatible with the projection π: [0,1]→ T^αby setting, for everys, t∈[0,1],

s4t if s≤t and X_s^exc₋ ≤I_s^t,

where by conventionX₀₋^exc= 0. It is a simple matter to check that4is indeed a partial order which is compatible with the genealogical order onT^α, meaning that a pointa∈ T^α is an ancestor ofbif and only if there exists4t∈[0,1]witha=π(s)andb=π(t). For everys, t∈[0,1], lets∧tbe the most recent common ancestor (for the relation4^on [0,1]) ofsandt. Thenπ(s∧t)also is the most recent common ancestor ofπ(s)anπ(t)in the treeT^α.

We now recall several well-known properties ofT^α. By definition, themultiplicity (or degree) of a vertex u ∈ T^α is the number of connected components of T^α\{u}^. Vertices ofT^α\{ρ}which have multiplicity1are calledleaves, and those with multiplicity at least 3 are called branch-points. By [18, Theorem 4.6], the multiplicity of every vertex ofT^{αbelongs to}{1,2,∞}. In addition, the branch-points ofT^αare in one-to-one correspondence with the jumps ofX^exc [29, Proposition 2]. More precisely, a vertex u∈ T^αis a branch-point if and only if there exists a uniques∈[0,1]^{such that}u=π(s) and∆X_s^exc= ∆s>0. In this case∆sintuitively corresponds to the “number of children”

(although this does not formally make sense) orwidth ofπ(s).

We finally introduce a last notation, which will be crucial in the definition of stable looptrees in the next section. Ifs, t∈[0,1]ands4t, set

x^t_s=I_s^t−X_s−^exc ∈[0,∆s].

Roughly speaking,x^t_sis the “position” of the ancestor ofπ(t)among the∆s“children” of π(s).

2.3 Definition of stable looptrees

Informally, the stable looptreeLαis obtained from the treeT^αby replacing every branch-point of widthxby a metric cycle of lengthx, and then gluing all these cycles along the tree structure of T^α (in a very similar way to the construction of discrete looptrees from discrete trees explained in the Introduction, see Figures 1 and 2). But making this construction rigorous is not so easy because there are countably many loops (non of them being adjacent).

Recall that the dependence in α is implicit through the processX^exc. For every t∈[0,1]we equip the segment[0,∆t]with the pseudo-distanceδtdefined by

δt(a, b) = min

|a−b|,∆t− |a−b| , a, b∈[0,∆t].

Note that if∆t>0,([0,∆t), δt)is isometric to a metric cycle of length∆t(this cycle will be associated with the branch-pointπ(t)in the looptreeLα, as promised in the previous paragraph).

Fors≤t∈[0,1], we writes≺tifs4tands6=t. It is important to keep in mind that

≺^{does not}correspond to the strict genealogical order inT^αsince there exists≺twith π(s) =π(t). The stable looptreeLαwill be defined as the quotient of[0,1]by a certain pseudo-distancedinvolvingX^exc, which we now define. First, ifs4t, set

d0(s, t) = X

s≺r4t

δr(0, x^t_r). (2.4)

In the last sum, only jump times give a positive contribution, sinceδr(0, x^t_r) = 0when

∆r= 0. Note that even iftis a jump time, its contribution in (2.4) is null sinceδt(0, x^t_t) = 0 and we could have summed overs≺r≺t. Deliberately, we do not allowr=sin (2.4).

Also, it could happen that there is nor∈(s, t]such that boths≺randr4t(e.g. when s=t) in which case the sum (2.4) is equal to zero. Heuristically, ifs≺r4t, the term δr(0, x^t_r)represents the length of the portion of the path going from (the images in the

looptree of)s^totbelonging to the loop coded by the branch-pointr(see Figure 5). Then, for everys, t∈[0,1], set

d(s, t) = δs∧t x^s_s∧t, x^t_s∧t

+d0(s∧t, s) +d0(s∧t, t). (2.5)

Loop of length ∆r

t s

δ_s∧t(x_s∧t^s , x^t_s∧t)

Loop corresponding to the branch point s∧t δr(0, x_r^t)

corresponding to the branch pointπ(r)

π( )

Figure 5: Illustration of the definition ofd. The geodesic between the images ofsandt in the looptree is in bold. Here,s∧t≺r≺t. This is a simplified picture since in stable looptrees no loops are adjacent.

Let us give an intuitive meaning to this definition. The distance d(s, t) contains contributions given by loops which correspond to branch-points belonging to the geodesic [[π(s), π(t)]]in the tree: the third (respectively second) term of the right-hand side of (2.5) measures the contributions from branch-points belonging to the interior of[[π(s∧t), π(t)]]

(respectively[[π(s∧t), π(s)]]), while the termδ_s∧t(x^s_s∧t, x^t_s∧t)represents the length of the portion of the path going from (the images in the looptree of)stot belonging to the (possibly degenerate) loop coded byπ(s∧t)(this term is equal to 0ifπ(s∧t)is not a branch-point), see Figure 5.

In particular, ifs4t, note that

d(s, t) = δs(0, x^t_s) +d0(s, t) = X

s4r4t

δr(0, x^t_r). (2.6)

Lemma 2.1(Bounds ond).Letr, s, t∈[0,1]. Then:

(i) (Lower bound) Ifs≺r≺t, we haved(s, t)≥min(x^t_r,∆r−x^t_r). (ii) (Upper bound) Ifs < t, we haved(s, t)≤X_s^exc+X_t^exc₋ −2I_s^t. Proof. The first assertion is obvious from the definition ofd:

d(s, t)≥δr(0, x^t_r)≥min(x^t_r,∆r−x^t_r)

For(ii), let us first prove that ifs≺tthen

d0(s, t)≤X_t^exc₋ −I_s^t. (2.7) (Note that X_t^exc₋ −I_s^t ≥ 0 becauses 6= t.) To this end, remark that if s 4 r 4 t and s4r⁰ 4t, thenr4r⁰ orr⁰ 4r. It follows that ifs≺r0≺r1 ≺ · · · ≺rn =t, using the

fact thatI_r^r_iⁿ=Ir^riⁱ⁺¹ for0≤i≤n−1^{, we have}

n

X

i=0

δri(0, x^r_rⁿ_i) ≤

n−1

X

i=0

x^r_rⁿ_i +δrn(0, x^r_rⁿ_n)

=

n−1

X

i=0

I_r^r_iⁱ⁺¹−X_r^exc_i− + 0

≤

n−1

X

i=0

X_r^exc_i+1−−X_r^exc_i−

=X_r^exc_n−−X_r^exc₀₋≤X_t^exc₋ −I_s^t,

where for the last inequality we have used the fact thatI_s^t≤X_r^exc₀₋sinces < r0< t. Since d0(s, t) =P

s≺r4tδr(0, x^t_r), this gives (2.7).

Let us return to the proof of(ii). Lets < t. Ifs≺t, then by (2.6) and treating the jump atsseparately we can use (2.7) to get

d(s, t) = δs(0, x^t_s) +d0(s, t)

≤ (∆s−x^t_s) + (X_t−^exc−I_s^t)

= (X_s^exc₋ + ∆s−I_s^t) + (X_t^exc₋ −I_s^t) =X_s^exc+X_t^exc₋ −2I_s^t.

Otherwises∧t < s. It is then easy to check thatI_s^t=I_s^t_∧t. In addition,δ_s∧t(x^t_s∧t, x^s_s∧t)≤ x^s_s∧t−x^t_s∧t=I_s^s_∧t−I_s^t_∧t=I_s^s_∧t−I_s^t. Then by (2.5) and (2.7) we have

d(s, t)≤I_s^s_∧t−I_s^t+ (X_t^exc₋ −I_s^t_∧t) + (X_s^exc₋ −I_s^s_∧t) =X_s^exc₋ +X_t^exc₋ −2I_s^t.

This completes the proof.

Proposition 2.2.Almost surely, the functiond(·,·) : [0,1]×[0,1]→R+is a continuous pseudo-distance.

Proof. By definition of d and Theorem 2.1, for every s, t ∈ [0,1], we have d(s, t) ≤ 2 supX^exc<∞. The fact thatdsatisfies the triangular inequality is a straightforward but cumbersome consequence of its definition (2.5). We leave the details to the reader.

Let us now show that the functiond(·,·) : [0,1]×[0,1]→R+is continuous. To this end, fix(s, t)∈[0,1]²and letsn, tn(n≥1)be real numbers in[0,1]such that(sn, tn)→(s, t) asn→ ∞. The triangular inequality entails

|d(s, t)−d(sn, tn)| ≤d(s, sn) +d(t, tn).

By symmetry, it is sufficient to show thatd(s, sn)→0asn→ ∞. Suppose for a moment thatsn↑sandsn < s, then by Theorem 2.1(ii)we have

d(sn, s)≤X_s^exc_n +X_s−^exc−2I_s^s_{n n}−→

→∞ X_s−^exc+X_s−^exc−2X_s−^exc= 0.

The other case whensn↓sandsn< sis treated similarly. This proves the proposition.

We are finally ready to define the looptree coded byX^exc.

Definition 2.3.Forx, y∈[0,1], setx∼y ifd(x, y) = 0. The random stable looptree of indexαis defined as the quotient metric space

Lα = [0,1]/∼, d .

We will denote bypthe canonical projectionp: [0,1]→Lα. Sinced: [0,1]×[0,1]→ R+ is a.s. continuous by Theorem 2.2, it immediately follows that p : [0,1] → Lα is a.s. continuous. The metric spaceLαis thus a.s. compact, as the image of a compact metric space by an a.s. continuous map.

With this definition, it is maybe not clear whyLαcontains loops. For sake of clarity, let us give an explicit description of these. Fixs∈[0,1]with∆s>0, and foru∈[0,∆s] letsu= inf{t≥s:X_t^exc=X_s^exc−u}. It is easy to check that the image of{su}u∈[0,∆s] by pinLαis isometric to a circle of length∆s, which corresponds to the loop attached to the branch-pointπ(s)in the treeT^α.

To conclude this section, let us mention that it is possible to constructLαdirectly from the stable treeT^α in a measurable fashion. For instance, ifu = π(s)^{, one can} recover the jump∆sas follows (see [29, Eq. (1)]):

∆s

a.s.= lim

ε→0

1

εMass{v∈ T^α;d_Tα(u, v)< ε}, (2.8) where Mass is the push-forward of the Lebesgue measure on[0,1]by the projection π: [0,1]→ T^α. However, we believe that our definition ofLαusing Lévy processes is simpler and more amenable to computations (recall also that the stable tree is itself defined by the height processH^excassociated withX^exc).

3 Properties of stable looptrees

The goal of this section is to prove Theorems 1.1 and 1.2. Before doing so, we introduce some more background on spectrally positive stable Lévy processes. This will be our toolbox for studying fine properties of looptrees. The interested reader should consult [4, 5, 12] for additional details.

Let us stress that, to our knowledge, the limiting behavior of the normalized excursion ofα-stable spectrally positive Lévy processes asα↓1(Theorem 3.6) seems to be new.

3.1 More on stable processes 3.1.1 Excursions above the infimum

In Section 2.1, the normalized excursion process X^exc has been introduced as the normalized excursion ofXabove its infimum straddling time1. Let us present another definitionX^excusing Itô’s excursion theory (we refer to [5, Chapter IV] for details).

IfXis anα-stable spectrally positive Lévy process, denote byX_t= inf{Xs: 0≤s≤t} its running infimum process. Note thatX is continuous sinceX has no negative jumps.

The process X −X is strong Markov and 0 is regular for itself, allowing the use of excursion theory. We may and will choose−Xas the local time ofX−X at level0. Let (gj, dj), j∈ Ibe the excursion intervals ofX−X away from0. For everyj∈ I ^ands≥0, setω_s^j=X(gj+s)∧dj −Xg_j. We viewω^j as an element of the excursion spaceE^{, defined} by:

E ={ω∈D(R⁺,R⁺);ω(0) = 0andζ(ω) := sup{s >0;ω(s)>0} ∈(0,∞)}. Ifω∈ E^{, we call}ζ(ω)the lifetime of the excursionω. From Itô’s excursion theory, the point measure

N(dtdω) =X

j∈I

δ(−X_gj,ω^j)

is a Poisson measure with intensitydtn(dω), wheren(dω)is aσ-finite measure on the set Ecalled the Itô excursion measure. This measure admits the following scaling property.

For everyλ >0, defineS^(λ):E → E ^byS^(λ)(ω) = λ^1/αω(s/λ), s≥0

. Then (see [12] or

[5, Chapter VIII.4] for details) there exists a unique collection of probability measures (n_(a), a >0)on the set of excursions such that the following properties hold:

(i) For everya >0,n_(a)(ζ=a) = 1.

(ii) For everyλ >0anda >0, we haveS^(λ)(n_(a)) =n_(λa). (iii) For every measurable subsetAof the set of all excursions:

n(A) = Z ∞

0

n_(a)(A) da

αΓ(1−1/α)a^1/α+1.

In addition, the probability distributionn₍₁₎, which is supported on the càdlàg paths with unit lifetime, coincides with the law ofX^exc as defined in Section 2.1, and is also denoted byn(·|ζ= 1). Thus, informally,n(·|ζ= 1)is the law of an excursion under the Itô measure conditioned to have unit lifetime.

3.1.2 Absolute continuity relation forX^exc

We will use a path transformation due to Chaumont [12] relating the bridge of a stable Lévy process to its normalized excursion, which generalizes the Vervaat transformation in the Brownian case. IfU is a uniform variable over[0,1]independent ofX^exc, then the processX^brdefined by

X_t^br=

X_U+t^exc ifU+t≤1,

X_t+U^exc₋₁ ifU+t >1. ^fort∈[0,1]

is distributed according to the bridge of the stable processX, which can informally be seen as the process(Xt; 0≤t≤1)conditioned to be at level zero at time one. See [5, Chapter VIII] for definitions. In the other direction, to getX^excfromX^brwe just re-root X^br by performing a cyclic shift at the (a.s. unique) timeu?(X^br)where it attains its minimum.

We finally state an absolute continuity property betweenX^brandX^exc. Fixa∈(0,1). LetF : D([0, a],R)→Rbe a bounded continuous function. We have (see [5, Chapter VIII.3, Formula (8)]):

E

F X_t^br; 0≤t≤a

=E

F(Xt; 0≤t≤a)p1−a(−Xa) p1(0)

,

whereptis the density ofXt. Note that by time reversal, the law of(X1−X_(1−t)−)0≤t≤1

satisfies the same property.

The previous two results will be used in order to reduce the proof of a statement concerningX^excto a similar statement involvingX (which is usually easier to obtain).

More precisely, a property concerningX will be first transferred to X^brby absolute continuity, and then toX^excby using the Vervaat transformation.

3.1.3 Descents

LetY :R→Rbe càdlàg function. For every s, t∈R^{, we write}s4Y tif and only ifs≤t andY_s−≤inf[s,t]Y, and in this case we set

x^t_s(Y) = inf

[s,t]Y −Ys−≥0, and u^t_s(Y) =x^t_s(Y)

∆Ys ∈[0,1].

We writes≺^{Y if}s4Y tands6=t. When there is no ambiguity, we writex^t_sinstead of x^t_s(Y), etc. Fort∈R, the collection{(x^t_s(Y), u^t_s(Y)) :s4t}is called thedescent oftinY.

As the reader may have noticed, this concept is crucial in the definition of the distance involved in the definition of stable looptrees.

We will describe the law of the descents (from a typical point) in anα-stable Lévy process by using excursion theory. To this end, denoteXt = sup{Xs : 0≤s ≤t} ^the running supremum process ofX. The processX−X is strong Markov and0is regular for itself. Let(Lt, t≥0)denote a local time ofX−X at level0, normalized in such a way thatE

exp(−λXL⁻¹(t))

= exp(−tλ^α−1). Note that by [5, Chapter VIII, Lemma 1],L⁻¹is a stable subordinator of index1−1/α. Finally, to simplify notation, setxs=Xs−Xs−

andus = ^X_X^s−Xs−

s−Xs− for everys≥0such thatXs> Xs−. In order to describe the law of descents from a fixed point in anα-stable process we need to introduce the two-sided stable process. IfX¹andX²are two independent stable processes onR^{+, set}Xt=X_t¹ fort≥0andXt=−X_(−t)−² fort <0.

Proposition 3.1.The following assertions hold.

(i) Let (Xt : t ∈ R)be a two-sided spectrally positive α-stable process. Then the collection

{(−s, x⁰_s(X), u⁰_s(X)) :s40} has the same distribution as

(s,xs,us);s≥0s.t.Xs> Xs− . (ii) The point measure

X

Xs>Xs−

δ_(Ls,^xs

us,us) (3.1)

is a Poisson point measure with intensitydl·xΠ(dx)·1[0,1](r)dr.

Proof. The first assertion follows from the fact that the dual process Xˆ, defined by Xˆs=−X(−s)−fors≥0, has the same distribution asX and that

(x⁰_−s(X), u⁰_−s(X)) = Xˆs−Xˆs−,Xˆs−Xˆs−

Xˆs−Xˆ_s−

!

for everys≥0such that−s40, or equivalentlyXˆs>Xˆs−.

For(ii), denote by(gj, dj)j∈Jthe excursion intervals ofX−X above0. It is known (see [4, Corollary 1]) that the point measure

X

j∈J

(Lgj,∆Xdj,∆Xdj)

is a Poisson point measure with intensitydl·Π(dx)·1[0,x](r)dr. The conclusion follows.

We now state a technical but useful consequence of the previous proposition, which will be required in the proof of the lower bound of the Hausdorff dimension of stable looptrees.

Corollary 3.2.Fixη >0. Let(Xt:t∈R)be a two-sidedα-stable process. For >0, set

Aε=







∃s∈[−ε,0]withs40 :

x⁰_s(X)≥ε^1/α+η and

∆Xs−x⁰_s(X)≥ε^1/α+η





 .

ThenP(A^c_ε)≤C^γ for certain constantsC, γ >0(depending onαandη).

Proof. SetBε={∃s∈[0, ε] :Xs≥ε^1/α+ηand∆Xs−xs≥ε^1/α+η}. By Theorem 3.1(i)^, it is sufficient to establish the existence of two constantsC, γ >0such thatP(B_ε^c)≤C^γ. To simplify notation, setα= 1−1/αandc=^η(α−1)/2. Then write:

P(B_ε^c) ≤ P(B_ε^c, Lε> c^α) +P(Lε< c^α)

≤ P(∀ss.t. Ls≤c^α:xs< ε^1/α+η or∆Xs−xs< ε^1/α+η) +P(Lε< c^α(3.2)).

Using the fact that (3.1) is a Poisson point measure with intensitydl·xΠ(dx)·1[0,1](r)dr, it follows that the first term of (3.2) is equal to

exp

−c^α Z 1

0

dr Z _∞

−∞

xΠ(dx)1_{rx≥ε1/α+ηandx(1−r)≥ε^1/α+η}

= exp

−c^α·c^{−η(α−1)} for a certain constantc >0. In addition,

P(Lε< c^α)≤P

L⁻₁¹> ε (c^α)^1/α

≤P(L⁻₁¹> ^−ηα/2).

The conclusion follows sinceP(L⁻₁¹> u) =O(u^−α¯)asu→ ∞^.

We conclude this section by a lemma which will be useful in the proof of Theorem 4.1.

See also [27, Proof of Proposition 7] for a similar statement.

Lemma 3.3.Almost surely, for everyt≥0^{we have} Xt−inf

[0,t]X = X

s4t s≥0

x^t_s(X). (3.3)

Proof. The left-hand side of the equality appearing in the statement of the lemma is clearly a càdlàg function. It also simple, but tedious, to check that the right-hand side is a càdlàg function as well. It thus suffices to prove that (3.3) holds almost surely for every fixedt≥0.

SetXˆs=X(t−s)−−X_t− for0≤s≤t, and to simplify notation setSu= sup_[0,u]Xˆ. In particular,(Xs,0≤s≤t)and( ˆXs,0≤s≤t)have the same distribution. Hence

St, X

0≤s≤t

∆Ss ^(d)

=

Xt−inf

[0,t]X,X

s4t s≥0

x^t_s(X)

. (3.4)

Then notice that ladder height process(S_L⁻¹

t , t≥0)is a subordinator without drift [5, Chapter VIII, Lemma 1], hence a pure jump-process. This implies thatStis the sum of its jumps, i.e. a.sSt=P

0≤s≤t∆Ss. This completes the proof of the lemma.

The following result is the analog statement for the normalized excursion.

Corollary 3.4.Almost surely, for everyt∈[0,1]we have X_t^exc= X

04s4t

x^t_s(X^exc).

Proof. This follows from the previous lemma and the construction ofX^excas the normal- ized excursion above the infimum ofX straddling time1in Section 2.1. We leave details to the reader.

In particular Theorem 3.4 implies that almost surely, for every0≤t≤1, X_t^exc= X

04s4t

∆X_s^exc·u^t_s(X^exc). (3.5) By (2.6), a similar equality, which will be useful later, holds almost surely for every 0≤t≤1:

d(0, t) = X

04s4t

∆X_s^exc·min u^t_s(X^exc),1−u^t_s(X^exc)

. (3.6)

3.1.4 Limiting behavior of the normalized excursion asα↓1^andα↑2

In this section we study the behavior ofX^excasα→1orα→2. In order to stress the dependence inα, we add an additional superscript^(α), e.g. X^(α), X^br,(α), X^exc,(α)will respectively denote theα-stable spectrally positive process, its bridge and normalized excursion, andΠ^(α), n^(α)will respectively denote the Lévy measure and the excursion measure above the infimum ofX^(α).

Limiting caseα↑2. We prove thatX^exc,(α)converges, asα↑2, towards a multiple of the normalized Brownian excursion, denoted bye(see Figure 6 for an illustration). This is standard and should not be surprising, since theα= 2stable Lévy process is just√

2 times Brownian motion.

Proposition 3.5.The following convergence holds in distribution for the topology of uniform convergence on every compact subset ofR+

X^exc,(α) −−→_α^(d)

↑2

√2·e. (3.7)

Proof. We first establish an unconditioned version of this convergence. Specifically, ifB is a standard Brownian motion, we show that

X^(α) −→^(d)

α↑2

√2·B, ^(3.8)

where the convergence holds in distribution for the uniform topology on D([0,1],R). SinceBis almost surely continuous, by [33, Theorems V.19, V.23] it is sufficient to check that the following three conditions hold asα↑2:

(a) The convergenceX₀^(α)−→√

2·B0holds in distribution, (b) For every 0 < s < t, the convergence X_t^(α)−X_s^(α)−→√

2·(Bt−Bs) holds in distribution,

(c) For everyδ >0, there existη, >0such that for0≤s≤t≤1:

|t−s|< η =⇒ P

|X_t^(α)−X_s^(α)| ≤δ/2

≥.

It is clear that Condition (a) holds. The scaling property ofX^(α)entails thatX_t^(α)−Xs^(α)

has the same law as(t−s)^1/α·X₁^(α). On the other hand, for everyu∈R^{, we have} Eh

exp(iuX₁^(α))i

−−→α↑2 exp(−u²) =Eh

exp(iu√ 2B1)i

.

Condition (b) thus holds. The same argument gives Condition (c). This establishes (3.8).

The convergence (3.7) is then a consequence of the construction of X^exc,(α) from the excursion ofX^(α)above its infimum straddling time1(see Section 2.1). Indeed, by Skorokhod’s representation theorem, we may assume that the convergence (3.8) holds almost surely. Then set

g^(α)₁ = sup{s≤1 :X_s^(α)= inf

[0,s]X^(α)} ^and d^(α)₁ = inf{s >1 :X_s^(α)= inf

[0,s]X^(α)}. Similarly, defineg⁽²⁾₁ , d⁽²⁾₁ whenX^(α)is replaced by√

2·B. Since local minima of Brownian motion are almost surely distinct, we get thatg^(α)₁ → g⁽²⁾₁ a.s. as α↑ 2. On the other side, since for everyα∈(1,2], a.s.d⁽²⁾₁ is not a local minimum ofB(this follows from the Markov property applied at the stopping timed⁽²⁾₁ )we get thatd^(α)₁ →d⁽²⁾₁ in distribution asα↑2. The desired convergence (3.7) then follows from (2.2).

Limiting case α ↓ 1^. The limiting behavior of the normalized excursion X^{exc,(α) as} α ↓ 1 is very different from the case α↑ 2. Informally, we will see that in this case, X^exc,(α)converges towards the deterministic affine function on[0,1]which is equal to1 at time0and0at time1. Some care is needed in the formulation of this statement, since the functionx7→10<x≤1(1−x)is not càdlàg. To cope up with this technical issue, we reverse time:

Proposition 3.6.The following convergence holds in distribution inD([0,1],R): X₍₁^exc,(α)_−t)−,0≤t≤1 _(d)

−→_α

↓1 t1_{t6=1},0≤t≤1 .

Remark 3.7.Let us mention here that the caseα↓1is not (directly) related to Neveu’s branching process [32] which is often considered as the limit of a stable branching process when α → 1. Indeed, contrary to the latter, the limit of X^exc,α when α ↓ 1 is deterministic. The reason is that Neveu’s branching process has Lévy measure r⁻²1(0,∞)dr, but recalling our normalization (2.1), in the limitα↓1, the Lévy measure Π^(α)doesnot converge tor⁻²1(0,∞)dr.

Theorem 3.6 is thus a new “one-big jump principle” (see Figure 6 for an illustration), which is a well-known phenomenon in the context of subexponential distributions (see [20] and references therein). See also [3, 19] for similar one-big jump principles.

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 6: Simulations ofX^exc,(α)for respectivelyα= 1.00001andα= 1.99.

The strategy to prove Theorem 3.6 is first to establish the convergence ofX^exc,(α)on every fixed interval of the form[ε,1]with∈(0,1)and then to study the behavior near0. Lemma 3.8.For everyε∈(0,1),

X_t^exc,(α);ε≤t≤1 ⁽P)

−−→α↓1 (1−t;ε≤t≤1),

where the convergence holds in probability for the uniform norm.

Proof of Theorem 3.8. Following the spirit of the proof of Theorem 3.5, we first establish an analog statement for the unconditioned processX^(α)by proving that

X^(α) −→^(d)

α↓1 (−t;t≥0), (3.9)

where the convergence holds in distribution for the uniform convergence on every compact subset ofR+. To establish (3.9), we also rely on [33, Theorems V.19, V.23]

and easily check that Conditions(a),(b)and(c)hold, giving (3.9). Fixε∈(0,1/10). We shall use the notation[a±b] := [a−b, a+b]fora∈R^andb >0. We also introduce the

functions`(s) = 1−s<sup>and</sup>`ε(s) = 1−ε−s^fors∈[0,1]. To prove the lemma, we show that for everyε >0we have

n^(α) {ωt∈[1−t±ε], ∀t∈[ε,1]} |ζ= 1

−−→α↓1 1.

By the scaling property of the measuren^(α) (see property (iii) in Section 3.1.1), it is sufficient to show that

n^(α) sup

t∈[ε,ζ]|ωt−`(t)| ≤10ε|ζ∈[1±ε]

−−→α↓1 1. (3.10) Fort ≥0, denote by q_t^(α)(dx)the entrance measure at time t under n^(α), defined by relation

n^(α) f(ωt)1_{ζ>t}

= Z _∞

0

f(x)q_t^(α)(dx)

for every measurable function f : R+ → R+ Then, using the fact that, for every t >0, under the conditional probability measuren^(α)(· |ζ > t), the process (ωt+s)_s≥0 is Markovian with entrance lawq_t^(α)(dx)and transition kernels ofX^(α)stopped upon hitting0, we get

n^(α) sup

t∈[ε,ζ]|ωt−`(t)| ≤10ε|ζ∈[1±ε]

= 1

n^(α)(ζ∈[1±ε]) Z ∞

0

q^(α)_ε (dx)P_x^(α) sup

t∈[0,τ]|X_t^(α)−`ε(t)| ≤10ε and τ∈[1−ε±ε]

, (3.11) wherePx^(α)denotes the distribution of a standardα-stable processX^(α)started fromx and stopped at the first timeτ when it touches0. From (3.9) it follows that for every δ∈(0, ε)the convergence

P_x^(α) sup

[0,τ]|X^(α)−`ε| ≤10εandτ∈[1−ε±ε]

−−→α↓1 1 holds uniformly inx∈[1−ε±(ε−δ)].Consequently

lim inf

α↓1

Z ∞ 0

q_ε^(α)(dx)P_x^(α) sup

[0,τ]||X^(α)−`ε|| ≤10εand τ∈[1−ε±ε]

Z ∞ 0

q_ε^(α)(dx)1x∈[1−ε±(ε−δ)]

≥1. ^(3.12)

On the other hand, we can write provided that2δ < ε(notice that1−ε+ 2δ > ε) n^(α)(ζ∈[1±(ε−2δ)]) =

Z ∞ 0

q_ε^(α)(dx)P_x^(α)

τ∈[1−ε±(ε−2δ)]

.

Convergence (3.9) then entails thatg(x, α) := Px^(α)(τ ∈ [1−ε±(ε−2δ)])also tends towards 0 as α ↓ 1, uniformly for x ∈ R+\[1−ε±(ε−δ)]. Since the total mass R∞

0 q^(α)ε (dx) =n^(α)(ζ > ε)is finite, the dominated convergence theorem implies that Z

R+\[1−ε±(ε−δ)]

q^(α)_ε (dx)g(x, α) −−→

α↓1 0.

Finally, asg(x, α)is bounded by1we get by dominated convergence and the last display that

lim inf

α↓1

Z ∞ 0

q_ε^(α)(dx)1x∈[1−ε±(ε−δ)]

n^(α)(ζ∈[1±(ε−2δ)]) = lim inf

α↓1

Z ∞ 0

q_ε^(α)(dx)1x∈[1−ε±(ε−δ)]

Z ∞ 0

q_ε^(α)(dx)g(x, α)

≥ 1.(3.13)