Scaling limits of random planar maps with large faces

# Scaling limits of random planar maps with large faces

[ [    [ [ Université Paris-Sud Département de Mathématiques
Université Paris-Sud
Centre d’Orsay
91405 ORSAY Cedex
France
\smonth7 \syear2009\smonth3 \syear2010
\smonth7 \syear2009\smonth3 \syear2010
\smonth7 \syear2009\smonth3 \syear2010
###### Abstract

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index . When the number of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative jumps and index . In particular, the profile of distances in the map, rescaled by the factor , converges to a random measure defined in terms of the distance process. With the same rescaling of distances, the vertex set viewed as a metric space converges in distribution as , at least along suitable subsequences, toward a limiting random compact metric space whose Hausdorff dimension is equal to .

[
\kwd
\doi

10.1214/10-AOP549 \volume39 \issue1 2011 \firstpage1 \lastpage69 \newproclaimRemarkRemark \newproclaimAssumptionAssumption

\runtitle

Scaling limits of planar maps

{aug}

A]\fnmsJean-François \snmLe Gall\correflabel=e1]jean-francois.legall@math.u-psud.fr and A]\fnmsGrégory \snmMiermontlabel=e2]gregory.miermont@math.u-psud.fr

class=AMS] \kwd05C80 \kwd60F17 \kwd60G51. Random planar map \kwdscaling limit \kwdgraph distance \kwdprofile of distances \kwdstable distribution \kwdstable tree \kwdGromov–Hausdorff convergence \kwdHausdorff dimension.

## 1 Introduction

The goal of the present work is to discuss the continuous limits of large random planar maps when the distribution of the degree of a typical face has a heavy tail. Recall that a planar map is a proper embedding of a finite connected graph in the two-dimensional sphere. For technical reasons, it is convenient to deal with rooted planar maps, meaning that there is a distinguished oriented edge called the root edge. One is interested in the “shape” of the graph and not in the particular embedding that is considered. More rigorously, two rooted planar maps are identified if they correspond via an orientation-preserving homeomorphism of the sphere. The faces of the map are the connected components of the complement of edges and the degree of a face counts the number of edges that are incident to it. Large random planar graphs are of particular interest in theoretical physics, where they serve as models of random geometry ADJ ().

A simple way to generate a large random planar map is to choose it uniformly at random from the set of all rooted -angulations with faces (a planar map is a -angulation if all faces have degree ). It is conjectured that the scaling limit of uniformly distributed -angulations with faces, when tends to infinity (or, equivalently, when the number of vertices tends to infinity), does not depend on the choice of and is given by the so-called Brownian map. Since the pioneering work of Chassaing and Schaeffer CSise (), there have been several results supporting this conjecture. Marckert and Mokkadem MaMo () introduced the Brownian map and proved a weak form of the convergence of rescaled uniform quadrangulations toward the Brownian map. A stronger version, involving convergence of the associated metric spaces in the sense of the Gromov–Hausdorff distance, was derived in Le Gall legall06 () in the case of uniformly distributed -angulations. Because the distribution of the Brownian map has not been fully characterized, the convergence results of legall06 () require the extraction of suitable subsequences. Proving the uniqueness of the distribution of the Brownian map is one of the key open problems in this area.

A more general way of choosing a large planar map at random is to use Boltzmann distributions. In this work, we restrict our attention to bipartite maps, where all face degrees are even. Given a sequence of nonnegative real numbers and a bipartite planar map , the associated Boltzmann weight is

 Wq(m)=∏f∈F(m)qdeg(f)/2, (1)

where denotes the set of all faces of and is the degree of the face . One can then generate a large planar map by choosing it at random from the set of all planar maps with vertices (or with faces) with probability weights that are proportional to . Such distributions arise naturally (possibly in slightly different forms) in problems involving statistical physics models on random maps. This is discussed in Section 8 below.

Assuming certain integrability conditions on the sequence of weights, Marckert and Miermont jfmgm05 () obtain a variety of limit theorems for large random bipartite planar maps chosen according to these Boltzmann distributions. These results are extended in Miermont Mi1 () and Miermont and Weill MW () to the nonbipartite case, including large triangulations. In all of these papers, limiting distributions are described in terms of the Brownian map. Therefore, these results strongly suggest that the Brownian map should be the universal limit of large random planar maps, under the condition that the distribution of the degrees of faces satisfies some integrability property. Note that, even though the distribution of the Brownian map has not been characterized, many of its properties can be investigated in detail and have interesting consequences for typical large planar maps; see, in particular, the recent papers legall08 () and Mi2 () (and Bettinelli betti (), for similar results, for random maps on surfaces of higher genus).

In the present work, we consider Boltzmann distributions such that, even for large , a random planar map with vertices will have “macroscopic” faces, which, in some sense, will remain present in the scaling limit. This leads to a (conjectured) scaling limit which is different from the Brownian map. In fact, our limit theorems involve new random processes that are closely related to the stable trees of duqleg02 (), in contrast to the construction of the Brownian map MaMo (), legall06 (), which is based on Aldous’ continuum random tree (CRT).

Let us informally describe our main results, referring to the following sections for more precise statements. For technical reasons, we consider planar maps that are both rooted and pointed (in addition to the root edge, there is a distinguished vertex, denoted by ). Roughly speaking, we choose the Boltzmann weights in (1) in such a way that the distribution of the degree of a (typical) face is in the domain of attraction of a stable distribution with index . This can be made more precise by using the Bouttier–Di Francesco–Guitter bijection BdFGmobiles () between bipartite planar maps and certain labeled trees called mobiles. A mobile is a (rooted) plane tree, where vertices at even distance (resp., odd distance) from the root are called white (resp., black) and white vertices are assigned integer labels that satisfy certain simple rules; see Section 3.1. In the Bouttier–Di Francesco–Guitter bijection, a (rooted and pointed) planar map corresponds to a mobile in such a way that each face of is associated with a black vertex of and each vertex of (with the exception of the distinguished vertex ) is associated with a white vertex of . Moreover, the degree of a face of is exactly twice the degree of the associated black vertex in the mobile (see Section 3.1 for more details).

Under appropriate conditions on the sequence of weights , formula (1) defines a finite measure on the set of all rooted and pointed planar maps. Moreover, if is the probability measure obtained by normalizing , then the mobile associated with a planar map distributed according to is a critical two-type Galton–Watson tree, with different offspring distributions and for white and black vertices, respectively, and labels chosen uniformly over all possible assignments (see jfmgm05 () and Proposition 4 below). The distribution is always geometric, whereas has a simple expression in terms of the weights .

We now come to our basic assumption. In the present work, we choose the weights in such a way that behaves like , when , for some . Recalling that the degree of a face of is equal to twice the degree of the associated black vertex in the mobile , we see that, in a certain sense, the face degrees of a planar map distributed according to are independent, with a common distribution that belongs to the domain of attraction of a stable law with index .

We equip the vertex set of a planar map with the graph distance and would like to investigate the properties of this metric space when is distributed according to and conditioned to be large. For every integer , denote by a random planar map distributed according to . Our goal is to get information about typical distances in the metric space when is large and, if at all possible, to prove that these (suitably rescaled) metric spaces converge in distribution as in the sense of the Gromov–Hausdorff distance. As a motivation for studying the particular conditioning , we note that our results will have immediate application to Boltzmann distributions on nonpointed rooted planar maps: simply observe that a given rooted planar map with vertices corresponds to exactly different rooted and pointed planar maps.

To achieve the preceding goal, we use another nice feature of the Bouttier–Di Francesco–Guitter bijection: up to an additive constant depending on , the distance between and an arbitrary vertex coincides with the label of the white vertex of associated with . Thus, in order to understand the asymptotic behavior of distances from in the map , it suffices to get information about labels in the mobile when is large. To this end, we first consider the tree obtained by ignoring the labels in . Under our basic assumption, the results of duqleg02 () can be applied to prove that the tree converges in distribution, modulo a rescaling of distances by the factor , toward the so-called stable tree with index . The stable tree can be defined by a suitable coding from the sample path of a centered stable Lévy process with no negative jumps and index , under an appropriate excursion measure. The preceding convergence to the stable tree is, however, not sufficient for our purposes since we are primarily interested in labels. Note that, under the assumptions made in jfmgm05 () on the weight sequence (and, in particular, in the case of uniformly distributed -angulations), the rescaled trees converge toward the CRT and the scaling limit of labels is described in jfmgm05 () as Brownian motion indexed by the CRT or, equivalently, as the Brownian snake driven by a normalized Brownian excursion. In our “heavy tail” setting, however, the scaling limit of the labels is not Brownian motion indexed by the stable tree, but is given by a new random process of independent interest, which we call the continuous distance process.

Let us give an informal presentation of the distance process—a rigorous definition can be found in Section 4 below. We view the stable tree as the genealogical tree for a continuous population and the distance of a vertex from the root is interpreted as its generation in the tree. Fix a vertex in the stable tree. Among the ancestors of , countably many of them, denoted by correspond to a sudden creation of mass in the population: each has a macroscopic number of “children” and one can also consider the quantity , which is the rank among these children of the one that is an ancestor of . The preceding description is informal in our continuous setting (there are no children), but can be made rigorous thanks to the ideas developed in duqleg02 () and, in particular, to the coding of the stable tree by a Lévy process. We then associate with each vertex a Brownian bridge (starting and ending at ) with duration , independently when varies, and we set

 D(a)=∞∑k=1Bk(rk).

The resulting process when varies in the stable tree is the continuous distance process. As a matter of fact, since vertices of the stable tree are parametrized by the interval (using the coding by a Lévy process), it is more convenient to define the continuous distance process as a process indexed by the interval (or even by when we consider a forest of trees).

Much of the technical work contained in this article is devoted to proving that the rescaled labels in the mobile converge in distribution to the continuous distance process. The proper rescaling of labels involves the multiplicative factor instead of , as in earlier work. This indicates that the typical diameter of our random planar maps is of order , rather than in the case of maps with faces of bounded degree. Because conditioning on the total number of vertices makes the proof more difficult, we first establish a version of the convergence of labels for a forest of independent mobiles having the distribution of under . The proof of this result (Theorem 1) is given in Section 5. We then derive the desired convergence for the conditioned objects in Section 6.

Finally, we obtain asymptotic results for the planar maps in Section 7. Theorem 4 gives precise information about the profile of distances from the distinguished vertex in . Precisely, let be the measure on defined by

 ∫ρ(n)Mn(dx)φ(x)=1n∑v∈V(Mn)φ(n−1/2αdgr(v∗,v)).

Then, the sequence of random measures converges in distribution toward the measure defined by

 ∫ρ(∞)(dx)φ(x)=∫10dtφ(c(Dt−D––)),

where is a constant depending on the sequence of weights and .

We also investigate the convergence of the suitably rescaled metric spaces in the Gromov–Hausdorff sense. Theorem 5 shows that, at least along a subsequence, the random metric spaces converge in distribution toward a limiting random compact metric space. Furthermore, the Hausdorff dimension of this limiting space is a.s. equal to , which should be compared with the value for the dimension of the Brownian map legall06 (). The fact that the Hausdorff dimension is bounded above by follows from Hölder continuity properties of the distance process that are established in Section 4. The proof of the corresponding lower bound is more involved and depends on some properties of the stable tree and its coding by Lévy processes, which are investigated in duqleg02 (). Similarly as in the case of the convergence to the Brownian map, the extraction of a subsequence in Theorem 5 is needed because the limiting distribution is not characterized.

The paper is organized as follows. Section 2 introduces Boltzmann distributions on planar maps and formulates our basic assumption on the sequence of weights. Section 3 recalls the Bouttier–Di Francesco–Guitter bijection and the key result giving the distribution of the random mobile associated with a planar map under the Boltzmann distribution (Proposition 4). Section 3 also introduces several discrete functions coding mobiles, in terms of which most of the subsequent limit theorems are stated. Section 4 is devoted to the definition of the continuous distance process and to its Hölder continuity properties. In Section 5, we address the problem of the convergence of the discrete label process of a forest of random mobiles toward the continuous distance process of Section 4. We then deduce a similar convergence for labels in a single random mobile conditioned to be large in Section 6. Section 7 deals with the existence of scaling limits of large random planar maps and the calculation of the Hausdorff dimension of limiting spaces. Finally, Section 8 discusses some motivation coming from theoretical physics.

### Notation

The symbols will stand for positive constants that may depend on the choice of the weight sequence , but, unless otherwise indicated, do not depend on other quantities. The value of these constants may vary from one proof to another. The notation stands for the space of all continuous functions from into and the notation stands for the Skorokhod space of all càdlàg functions from into . If is a process with càdlàg paths, denotes the left limit of at for every . We denote the set of all finite measures on by and this set is equipped with the usual weak topology. If and are two sequences of positive numbers, the notation (as ) means that the ratio tends to as . Unless otherwise indicated, all random variables and processes are defined on a probability space .

## 2 Critical Boltzmann laws on bipartite planar maps

### 2.1 Boltzmann distributions

A rooted and pointed bipartite map is a pair , where is a rooted bipartite planar map and is a distinguished vertex of . As in Section 1, the graph distance on the vertex set is denoted by and we let be, respectively, the origin and the target of the root edge of . By the bipartite nature of , the quantities differ. Moreover, this difference is at most in absolute value since and are linked by an edge. We say that is positive if

 dgr(e+,v∗)=dgr(e−,v∗)+1.

It is called negative otherwise, that is, if .

We let denote the set of all rooted and pointed bipartite planar maps that are positive. In the sequel, the mention of will usually be implicit, so we will simply denote the generic element of by . For our purposes, it is useful to add an element to , which can be seen roughly as the vertex map with no edge and a single vertex “bounding” a single face of degree .

Let be a sequence of nonnegative real numbers. For every , set

 Wq(m)=∏f∈F(m)qdeg(f)/2,

where denotes the set of all faces of . By convention, we set . This defines a -finite measure on , whose total mass is

 Zq=Wq(M∗+)∈[1,∞].

We say that is admissible if , in which case we can define as the probability measure obtained by normalizing . The measure is called the Boltzmann distribution on with weight sequence .

Following jfmgm05 (), we have the following simple criterion for the admissibility of . Introduce the function

 fq(x)=∞∑k=1N(k)qkxk−1,x≥0, (2)

where

 N(k)=(2k−1k−1).

Let be the radius of convergence of this power series. Note that by monotone convergence, the quantity exists, as well as .

###### Proposition 1 (jfmgm05 ())

The sequence is admissible if and only if the equation

 fq(x)=1−1x,x≥1, (3)

has a solution. If this holds, then the smallest such solution equals .

On the interval , the function is convex, so the equation (3) has at most two solutions. Let us now pause for a short informal discussion, inspired by jfmgm05 (). For a “typical” admissible sequence , the graphs of and of the function will cross at without being tangent. In this case, the law of the number of vertices of a -distributed random map will have an exponential tail. An admissible sequence is called critical if the graphs are tangent at , that is, if

 Z2qf′q(Zq)=1. (4)

For critical sequences, the law of the number of vertices of a -distributed random map may have a tail heavier than exponential. In the case where , jfmgm05 () shows that this tail follows a power law with exponent . However, the law of the degree of a typical face in such a random map will have an exponential tail.

In the present paper, we will be interested in the “extreme” cases where is a critical sequence such that . We will show that in a number of these cases, the degree of a typical face in a -distributed random map also has a heavy tail distribution.

### 2.2 Choosing the Boltzmann weights

We start from a sequence of nonnegative real numbers such that

 q∘k∼k→∞k−a (5)

for some real number . In agreement with (2), we set

 f∘(x)=fq∘(x)=∞∑k=1N(k)q∘kxk−1

for every . By Stirling’s formula, we have

 N(k)∼k→∞22k−1√πk

so that the radius of convergence of the series defining is . Furthermore, the condition guarantees that and are (well defined and) finite.

###### Proposition 2

Set

 c=44f∘(1/4)+f′∘(1/4),β=f′∘(1/4)4f∘(1/4)+f′∘(1/4)

and define a sequence by setting

 qk=c(β/4)k−1q∘k. (6)

Then, the sequence is both admissible and critical, and .

{Remark*}

As the proof will show, the choice given for the constants and is the only one for which the conclusion of the proposition holds. {pf*}Proof of Proposition 2 Consider a sequence defined as in the proposition, with an arbitrary choice of the positive constants and . If is defined as in (2), it is immediate that

 fq(x)=cf∘(βx/4).

Hence, . Assume, for the moment, that the sequence is admissible and . By Proposition 1, we have or, equivalently,

 cf∘(1/4)=1−β. (7)

Furthermore, the criticality of will hold if and only if or, equivalently,

 cf′∘(1/4)=4β. (8)

Conversely, if (7) and (8) both hold, then the sequence is admissible by Proposition 1, the curves and are tangent at and a simple convexity argument shows that is the unique solution of (3) so that , again by Proposition 1.

We conclude that the conditions (7) and (8) are necessary and sufficient for the conclusion of the proposition to hold. The desired result thus follows.

We now introduce our basic assumption, placing a further restriction on the value of the parameter . {Assumption} The sequence is of the form given in Proposition 2, with a sequence satisfying (5) for some . We set .

This assumption will be in force throughout the remainder of this work, with the exception of the beginning of Section 3.2 (including Proposition 4), where we consider a general admissible sequence .

Many of the subsequent asymptotic results will be written in terms of the constant , which lies in the interval , and the constant defined by

 c0=(2cΓ(2−α)α(α−1)β√π)1/α. (9)

The reason for introducing this other constant will become clearer in Section 3.2.

## 3 Coding maps with mobiles

### 3.1 The Bouttier–Di Francesco–Guitter bijection

Following BdFGmobiles (), we now recall how bipartite planar maps can be coded by certain labeled trees called mobiles.

By definition, a plane tree is a finite subset of the set

 U=∞⋃n≥0Nn (10)

of all finite sequences of positive integers (including the empty sequence ) which satisfies three obvious conditions. First, . Then, for every with , the sequence (the “parent” of ) also belongs to . Finally, for every , there exists an integer (the “number of children” of ) such that belongs to if and only if . The elements of are called vertices. The generation of a vertex is denoted by . The notions of an ancestor and a descendant in the tree are defined in an obvious way.

For our purposes, vertices such that is even will be called white vertices and vertices such that is odd will be called black vertices. We denote by (resp., ) the set of all white (resp., black) vertices of .

A (rooted) mobile is a pair that consists of a plane tree and a collection of integer labels assigned to the white vertices of such that the following properties hold:

1. [(a)]

2. .

3. Let , be the parent of , and , be the children of . Then, for every , , where, by convention, .

Condition (b) means that if one lists the white vertices adjacent to a given black vertex in clockwise order, then the labels of these vertices can decrease by at most 1 at each step. See Figure 1 for an example of a mobile.

We denote by the (countable) set of all mobiles. We will now describe the Bouttier–Di Francesco–Guitter (BDG) bijection between and . This bijection can be found in Section 2 of BdFGmobiles (), with the minor difference that BdFGmobiles () deals with maps that are pointed, but not rooted.

Let be a mobile with vertices. The contour sequence of is the sequence of vertices of which is obtained by induction as follows. First, and then, for every , is either the first child of that has not yet appeared in the sequence or the parent of if all children of already appear in the sequence . It is easy to verify that and that all vertices of appear in the sequence . In fact, a given vertex appears exactly times in the contour sequence and each appearance of corresponds to one “corner” associated with this vertex.

The vertex is white when is even and black when is odd. The contour sequence of , also called the white contour sequence of , is, by definition, the sequence defined by for every .

The image of under the BDG bijection is the element of that is defined as follows. First, if , meaning that , we set . Suppose that so that has at least one element. We extend the white contour sequence of to a sequence , , by periodicity, in such a way that for every . Then, suppose that the tree is embedded in the plane and add an extra vertex not belonging to the embedding. We construct a rooted planar map whose vertex set is equal to

 V(m)=T∘∪{v∗}

and whose edges are obtained by the following device. For , we let

 ϕ(i)=inf{j>i\dvtxℓ(v∘j)=ℓ(v∘i)−1}∈{i+1,i+2,…}∪{∞}.

We also set , by convention. Then, for every , we draw an edge between and . More precisely, the index corresponds to one specific “corner” of and the associated edge starts from this corner. The construction can then be made in such a way that edges do not cross (and do not cross the edges of the tree) so that one indeed gets a planar map. This planar map is rooted at the edge linking to , which is oriented from to . Furthermore, is pointed at the vertex , in agreement with our previous notation.

See Figure 2 for an example and Section 2 of BdFGmobiles () for a more detailed description.

###### Proposition 3 ((BDG bijection))

The preceding construction yields a bijection from onto . This bijection enjoys the following two properties:

1. each face of contains exactly one vertex of , with ;

2. the graph distances in to the distinguished vertex are linked to the labels of the mobile in the following way: for every ,

 dgr(v∗,v)=ℓ(v)−minv′∈T∘ℓ(v′)+1.

In our study of scaling limits of random planar maps, it will be important to derive asymptotics for the random mobiles associated with these maps via the BDG bijection. These asymptotics are more conveniently stated in terms of random processes coding the mobiles. Let us introduce such coding functions.

Let be a mobile with vertices (so that ) and let be, as previously, the white contour sequence of . We set

 Cθi=12|v∘i|for 0≤i≤n,Cθi=0for i>n. (11)

We call the contour process of the mobile . It is a simple exercise to check that the contour process determines the tree . Similarly, we set

 Λθi=ℓ(v∘i)for 0≤i≤n,Λθi=0for i>n (12)

and call the contour label process of . The pair determines the mobile .

For technical reasons, we introduce variants of the preceding contour functions. Let and let , be the list of vertices of in lexicographical order. The height process of is defined by

 Hθi=12|w∘i|for 0≤i≤n∘,Hθi=0for i>n∘.

Similarly, we introduce the label process, which is defined by

 Lθi=ℓ(w∘i)for 0≤i≤n∘,Lθi=0for i>n∘.

We will also need the Lukasiewicz path of . This is the sequence , defined as follows. First, . Then, for every , is the number of (white) grandchildren of in . Finally, for every . It is easy to see that for every so that

 #T∘=n∘+1=inf{i≥0\dvtxSθi=−1}.

Let us briefly comment on the reason for introducing these different processes. In our applications to random planar maps, asymptotics for the pair , which is directly linked to the white contour sequence of , turn out to be most useful. On the other hand, in order to derive these asymptotics, it will be more convenient to consider first the pair .

In the following, the generic element of will be denoted by as previously.

### 3.2 Boltzmann distributions and Galton–Watson trees

Let be an admissible sequence, in the sense of Section 2, and let be a random element of with distribution . Our goal is to describe the distribution of the random mobile associated with via the BDG bijection. We closely follow Section 2.2 of jfmgm05 ().

We first need the notion of an alternating two-type Galton–Watson tree. Recall that white vertices are those of even generation and black vertices are those of odd generation. Informally, an alternating two-type Galton–Watson tree is just a Galton–Watson tree where white and black vertices have a different offspring distribution. More precisely, if and are two probability distributions on the nonnegative integers, the associated (alternating) two-type Galton–Watson tree is the random plane tree whose distribution is specified by saying that the numbers of children of the different vertices are independent, the offspring distribution of each white vertex is and the offspring distribution of each black vertex is ; see jfmgm05 (), Section 2.2, for a more rigorous presentation.

We also need to introduce the notion of a discrete bridge. Consider an integer and the set

 Ep:={(x1,…,xp)∈{−1,0,1,2,…}p\dvtxp∑i=1xi=0}.

Note that is a finite set and, indeed, , with as in (2). Let be uniformly distributed over . The sequence defined by and

 Yj=j∑i=1Xi,1≤j≤p,

is called a discrete bridge of length .

###### Proposition 4 ((jfmgm05 (), Proposition 7))

Let be a random element of with distribution and let be the random mobile associated with via the BDG bijection. Then:

1. the random tree is an alternating two-type Galton–Watson tree with offspring distributions and given by

 μ0(k)=Z−1qfq(Zq)k,k≥0,

and

 μ1(k)=ZkqN(k+1)qk+1fq(Zq),k≥0;
2. conditionally given , the labels are distributed uniformly over all possible choices that satisfy the constraints (a) and (b) in the definition of a mobile; equivalently, for every , with the notation introduced in property (b) of the definition of a mobile, the sequence is a discrete bridge of length and these sequences are independent when varies over .

A random mobile having the distribution described in the proposition will be called a -mobile. The law of a -mobile is a probability distribution on .

Note that the respective means of and are

 m0:=∑k≥0kμ0(k)=Zqfq(Zq),m1:=∑k≥0kμ1(k)=Zqf′q(Zq)/fq(Zq)

so that is less than or equal to and equality holds if and only if is critical.

We now return to a weight sequence satisfying our basic Assumption 2.2. Recall that the sequence , which is both admissible and critical, is given in terms of the sequence by (6) and that we have as , with .

Then, is the geometric distribution with parameter and

 μ1(k)=c1−β4−kN(k+1)q∘k+1,k=0,1,….

From the asymptotic behavior of , we obtain

 μ1(k)∼k→∞2c(1−β)√πk−α−1.

In particular, if we set , this yields

 ¯¯¯μ1(k)∼k→∞2cα(1−β)√πk−α. (13)

Let be the probability distribution on the nonnegative integers which is the law of

 U∑i=1Vi,

where is distributed according to , are distributed according to and the variables are independent. Then, is critical, in the sense that

 ∞∑k=0kμ(k)=m0m1=1.

Notice that is just the distribution of the number of individuals at the second generation of a -mobile. It will be important to have information on the tail of . This follows easily from the estimate (13) and the definition of . First, note that

 ¯¯¯μ(k)=P[U∑i=1Vi≥k]≥P[∃i∈{1,…,U}\dvtxVi≥k]=1−E[(1−¯¯¯μ1(k))U].

Then,

 1−E[(1−¯¯¯μ1(k))U]=1−β1−(1−¯¯¯μ1(k))(1−β)∼k→∞1−ββ¯¯¯μ1(k).

Using (13), we get

 ¯¯¯μ(k)≥2cαβ√πk−α+o(k−α).

A corresponding upper bound is easily obtained by writing, for every ,

 ¯¯¯μ(k) ≤ P[∃i∈{1,…,U}\dvtxVi≥(1−ε)k]

and checking that the second term in the right-hand side is as . To see this, first note that the probability of the event is if the constant is chosen sufficiently large. If , then the event in the second term may hold only if there are two distinct values of such that . The desired estimate then follows from (13).

We have thus obtained

 ¯¯¯μ(k)∼k→∞2cαβ√πk−α,

which we can rewrite in the form

 ¯¯¯μ(k)∼k→∞α−1Γ(2−α)cα0k−α (14)

with the constant defined in (9). The reason for introducing the constant and writing the asymptotics (14) in this form becomes clear when discussing scaling limits. Recall that by our assumption that . By (13) or (14), is then in the domain of attraction of a stable law with index . Recalling that is critical, we have the following, more precise, result.

Let be the probability distribution on obtained by setting for every [and if ]. Let be a random walk on the integers with jump distribution . Then,

 (n−1/αS[nt])t≥0(d)⟶n→∞(c0Xt)t≥0, (15)

where the convergence holds in distribution in the Skorokhod sense and is a centered stable Lévy process with index and no negative jumps, with Laplace transform given by

 E[exp(−uXt)]=exp(tuα),t,u≥0. (16)

See, for instance, Chapter VII of Jacod and Shiryaev JS () for a thorough discussion of the convergence of rescaled random walks toward Lévy processes.

### 3.3 Discrete bridges

Recall from Proposition 4 that the sequence of labels of white vertices adjacent to a given black vertex in a -mobile is distributed as a discrete bridge. In this section, we collect some estimates for discrete bridges that will be used in the proofs of our main results.

We consider a random walk on starting from and with jump distribution

 ν∗(k)=2−k−2,k=−1,0,1,….

Fix an integer and let be a vector whose distribution is the conditional law of given that . Then, the process is a discrete bridge with length . Indeed, a simple calculation shows that

 (Y(p)1,Y(p)2−Y(p)1,…,Y(p)p−Y(p)p−1)

is uniformly distributed over the set .

###### Lemma 1

For every real , there exists a constant such that for every integer and ,

{pf}

We may, and will, assume that . Let us first suppose that . By the definition of and then the Markov property of , we have

where for every integer and . A standard local limit theorem (see, e.g., Section 7 of spitzer ()) shows that if , then we have

 √nπn(x)=g(x/√n)+εn(x)% where supx∈Z|εn(x)|→n→∞0.

Then,

 πp−k′(−Yk′)πp(0)≤√3√p−k′πp−k′(−Yk′)√pπp(0)≤K,

where

 K=√3(4π)−1/2+supn≥1supx∈Z|εn(x)|infn≥1√nπn(0)<∞.

It follows that

Then, the bound with a finite constant depending only on , is a consequence of Rosenthal’s inequality for i.i.d. centered random variables petrov75 (), Theorem 2.10. We have thus obtained the desired estimate under the restriction .

If , the same estimate is readily obtained by observing that has the same distribution as . Finally, in the case , we apply the preceding bounds successively to and to .

An immediate consequence of the lemma (applied with ) is the bound

 E[(Y(p)j)2]≤K(1)min{j,p−j}≤2K(1)j(p−j)p (17)

for every integer and .

Finally, a conditional version of Donsker’s theorem gives

 (1√2pY(p)[pt])0≤t≤1(d)⟶p→∞(γt)0≤t≤1, (18)

where is a standard Brownian bridge. Such results are part of the folklore of the subject; see Lemma 10 in betti () for a detailed proof of a more general statement.

## 4 The continuous distance process

Our goal in this section is to discuss the so-called continuous distance process, which will appear as the scaling limit of the label processes and of Section 3.1 when is a -mobile conditioned to be large in some sense.

### 4.1 Definition and basic properties

We consider the centered stable Lévy process with no negative jumps and index , and Laplace exponent as in (16). The canonical filtration associated with is defined, as usual, by

 Ft=σ{Xs,0≤s≤t}

for every . We let be a measurable enumeration of the jump times of and set for every . Then, the point measure

 ∑i∈Nδ(ti,xi)

is Poisson on with intensity

 α(α−1)Γ(2−α)dtdxxα+1.

For , we set

 Ist=infs≤r≤tXr

and . For every , we set

 Tx=inf{t≥0\dvtx−It>x}.

We recall that the process is a stable subordinator of index with Laplace transform

 E[exp(−uTx)]=exp(−xu1/α); (19)

see, for example, Theorem 1 in bertlev96 (), Chapter VII.

Suppose that, on the same probability space, we are given a sequence of independent (one-dimensional) standard Brownian bridges over the time interval starting and ending at the origin. Assume that the sequence is independent of the Lévy process . Then, for every , we introduce the rescaled bridge

 ˜bi(r)=x1/2ibi(r/xi),0≤r≤x<