Galton-Watson trees: the condensation case

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.19(2014), no. 56, 1–29.

ISSN:1083-6489 DOI:10.1214/EJP.v19-3164

Local limits of conditioned

Galton-Watson trees: the condensation case

Romain Abraham

Jean-François Delmas

Abstract

We provide a complete picture of the local convergence of critical or sub-critical Galton-Watson trees conditioned on having a large number of individuals with out- degree in a given set. The generic case, where the limit is a random tree with an infinite spine has been treated in a previous paper. We focus here on the non-generic case, where the local limit is a random tree with a node with infinite out-degree. This case corresponds to the so-called condensation phenomenon.

Keywords:Galton-Watson ; random tree ; local-limit ; non-extinction, branching process.

AMS MSC 2010:60J80 ; 60B10.

Submitted to EJP on November 26, 2013, final version accepted on June 20, 2014.

SupersedesHAL:hal-00909604.

1 Introduction

Conditioning critical or sub-critical Galton-Watson (GW) trees comes from the semi- nal work of Kesten, [12]. Letp= (p(n), n∈N)be an offspring distribution such that:

p(0)>0, p(0) +p(1)<1. (1.1) Letµ(p) =P+∞

n=0np(n)be its mean. Ifµ(p)<1(resp. µ(p) = 1, µ(p)>1), we say that the offspring distribution and the associated GW tree are sub-critical (resp. critical, super-critical). In the critical and sub-critical cases, the tree is a.s. finite, but Kesten considered in [12] the local limit of a sub-critical or critical tree conditioned to have height greater thann. Whenngoes to infinity, this conditioned tree converges in dis- tribution to the so-called size-biased GW tree. This random tree has an infinite spine on which are grafted a random number of independent GW trees with the same offspring distributionp. This limit tree can be seen as the GW tree conditioned on non-extinction.

Since then, other conditionings have been considered for critical GW trees: large total progeny see Kennedy [11] and Geiger and Kaufmann [6], large number of leaves

∗Laboratoire MAPMO, CNRS, UMR 7349, Fédération Denis Poisson, FR 2964, Université d’Orléans, Or- léans, FRANCE. E-mail:romain.abraham@univ-orleans.fr

†Université Paris-Est, École des Ponts, CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, Marne-la- Vallée, FRANCE. E-mail:delmas@cermics.enpc.fr

see Curien and Kortchemski [4]. We are interested in this paper in the conditioning on having a large number of individuals with out-degree belonging to a given set A which has been introduced in Rizzolo [18] and also appears in Kortchemski [13]. In Abraham and Delmas [1], it is proven that, in the critical case, the local limit of a GW tree conditioned to have a large number of individuals with out degree in a given setA is still the GW tree conditioned on non-extinction.

However, the results are different in the sub-critical case. LetA ⊂N^{. We set} p(A) =X

k∈A

p(k).

We first define for an offspring distributionpthat satisfies (1.1) and a set Asuch that p(A)>0a modified offspring distributionp_A,θ by:

∀k≥0, pA,θ(k) =

(c_A(θ)θ^kp(k) ifk∈ A,

θ^k−1p(k) ifk∈ A^c, ^(1.2) where the normalizing constantc_A(θ)is given by:

cA(θ) =θ−E

θ^X1_{X∈Ac}

θE

θ^X1_{X∈A} , (1.3)

whereXis a random variable distributed according top. LetI_Abe the set of positiveθ for whichp_A,θ is a probability distribution. Ifpis sub-critical, according to Lemma 5.2, either there exists (a unique)θ^c_A∈ IA such thatpA,θ^c_A is critical orθ_A^∗ := maxIA ∈IA

andpA,θ^∗_A is sub-critical. We shall say, see Definition 5.3, that pisgenericfor the set Ain the former case and that pis non-genericfor the set Ain the latter case. See Lemma 5.4 and Remark 5.5 on the non-generic property.

For a treet, letL_A(t)be the set of nodes oftwhose number of offspring belongs toAandL_A(t)be its cardinal (see definition in Section 6). It is proven in [1] that, for everyθ ∈I_A, ifτ is a GW tree with offspring distributionpandτ_A,θ is a GW tree with offspring distributionpA,θ, then the conditional distributions ofτgiven{LA(τ) =n}and that ofτ_A,θgiven{LA(τ_A,θ) =n}are the same. Therefore, ifpis generic for the setA, that is there exists aθ_A^c ∈I_Asuch thatp_A,θ^c

A is critical, then the GW treeτ conditioned onL_A(τ)being large converges to the size-biased GW tree associated withp_A,θ^c

A. When the sub-critical offspring distribution is non-generic for N, a condensation phenomenon has been observed when conditioning with respect to the total population size, see Jonnsson and Stefansson [8] and Janson [7]: the limiting tree is no more the size-biased tree but a tree that contains a single node with infinitely many offspring.

The goal of this paper is to give a short proof of this result and to show that such a condensation also appears when pis non-generic for A and conditioning by L_A(τ) being large. This and [1] give a complete description of the limit in distribution of a critical or sub-critical GW treeτ conditioned on{L_A(τ) =n}asngoes to infinity.

We summarize this complete description following Janson ([7], Section 5). Let p be an offspring distribution that satisfies (1.1) which is critical or sub-critical (that is µ(p)≤1). Letτ^∗(p)denote the random tree which is defined by:

i) There are two types of nodes: normalandspecial.

ii) The root is special.

iii) Normal nodes have offspring distributionp.

iv) Special nodes have offspring distribution the biased distributionp˜onN∪ {+∞}

defined by:

˜ p(k) =

(k p(k) ifk∈N, 1−µ ifk= +∞.

v) The offsprings of all the nodes are independent of each others.

vi) All the children of a normal node are normal.

vii) When a special node gets a finite number of children, one of them is selected uniformly at random and is special while the others are normal.

viii) When a special node gets an infinite number of children, all of them are normal.

Notice that:

• Ifpis critical, then a.s. τ^∗(p) has one infinite spine and all its nodes have finite degrees. This is the size-biased GW tree considered in [12].

• Ifµ(p)<1then a.s. τ^∗(p)has exactly one node of infinite degree and no infinite spine. This tree has been considered in [8, 7].

Definition 1.1. LetA ⊂N^{such that}p(A)>0. We definep^∗_Aas:

- critical case(µ(p) = 1):

p^∗_A=p.

- sub-critical and generic for A(µ(p) < 1 and there exists (a unique) θ^c_A ∈ I_A such thatµ(p_A,θ^c

A) = 1):

p^∗_A=p_A,θA^c.

- sub-critical and non-generic forA(µ(p)<1andµ(p_A,θA^∗)<1):

p^∗_A=p_A,θ^∗

A, withθ^∗_A= maxI_A. (1.4) Remark 1.2. The uniqueness ofθ_A^c is not immediate and follows from Lemma 5.2.

We state our main result (the convergence of random discrete trees is precisely defined in Section 2 and GW trees are presented in Section 3).

Theorem 1.3. Let τ be a GW tree with offspring distribution pwhich satisfies (1.1) and µ(p) ≤1. Let A ⊂ N^{such that} p(A) >0. We have the following convergence in distribution:

dist (τ

L_A(τ) =n) −→

n→+∞dist (τ^∗(p^∗_A)), (1.5) where the limit is understood along the infinite sub-sequence{n∈N^∗;P(LA(τ) =n)>

0}, as well as:

dist (τ

L_A(τ)≥n) −→

n→+∞dist (τ^∗(p^∗_A)). (1.6) The theorem has already been proven in the critical case and the sub-critical generic case in [1]. We concentrate here on the case of the sub-critical non-generic case. The non-generic case forA=N^,0∈ A,06∈ Aare respectively proven in Sections 4, 6 and 7.

Let us add that a sub-critical offspring distributionpis either generic for allA ⊂N^such thatp(A)>0, or non-generic at least for{0}and maybe for some other sets and generic for other setsAsuch thatp(A)>0, see Lemma 5.4. It is not possible for a sub-critical offspring distributionpto be non-generic for allA ⊂N ^{such that}p(A)>0 unless the radius of convergence of its generating function is 1, see Remark 5.5. By considering the last example of Remark 5.5, we exhibit a distribution pwhich is non-generic for {0} but generic forN. Thus the associated GW tree conditioned on havingnvertices

converges in distribution (asngoes to infinity) to a tree with an infinite spine whereas the same tree conditioned on havingnleaves converges in distribution to a tree with a node of infinite degree.

Let us add that we only focus here on local limits. Global limits of conditioned Galton- Watson trees have also been studied in the generic case first by Aldous [2] (conditioning on the total population size) and then by Rizzolo [18] for the conditioning studied here.

Global limits on non-generic trees have also been studied by Kortchemski [14] when conditioned to the total population size. Let us add that global limits require regularity for the offspring distribution (such as belonging to the domain of attraction of stable laws) whereas we only require here a first moment to obtain a local limit.

In Section 2, we recall the setting of the discrete trees (which is close to [1], but has to include discrete trees with nodes of infinite degree). We also give in Lemma 2.2, in the same spirit of Lemma 2.1 in [1], a convergence determining class which is the key result to prove the convergence in the non-generic case. Section 3 is devoted to some remarks on GW trees. We study in detail the distributionp_A,θdefined by (1.2) in Section 5. The proof of Theorem 1.3 is given in the following three sections. More precisely, the case A = N is presented in Section 4. This provides a short and self-contained proof of the results from [8, 7]. The case0∈ Acan be handled in the same spirit, see Section 6, using that the setL_A(τ)can be encoded into a GW treeτ^A, see Mimami [15]

or [18]. Notice that if06∈ A, thenL_A(τ), when non empty, can also be encoded into a GW treeτ^A, see [18]. However, we didn’t use this result, but rather use in Section 7 a more technical version of the previous proofs to treat the case0 6∈ A. We prove in the appendix, Section 8, consequences of the strong ratio limit property we used in the previous sections.

2 The set of discrete trees

We recall Neveu’s formalism [17] for ordered rooted trees. We set U = [

n≥0

(N^∗)ⁿ

the set of finite sequences of positive integers with the convention (N^∗)⁰ = {∅}. For n≥0andu= (u₁, . . . , u_n)∈ U, let|u|=nbe the length ofuand:

|u|_∞= max(|u|, u1, u2, . . . , u_|u|)

with the convention|∅|=|∅|∞ = 0. We will call|u|∞the norm ofualthough it is not a norm sinceU is not even a vector space. Ifuandv are two sequences ofU, we denote byuvthe concatenation of the two sequences, with the convention thatuv=uifv=∅ anduv=vifu=∅. The set of ancestors of∅isA_∅={∅}and ofu6=∅is:

A_u={v∈ U;there existsw∈ U,w6=∅, such thatu=vw}. (2.1) The most recent common ancestor of a subset s of U, denoted by MRCA(s), is the unique elementvofT

u∈sAu with maximal length|v|. Foru, v∈ U, we denote byu < v the lexicographic order onU i.e. u < vif eitheru∈Av or, if we setw=MRCA({u, v}), thenu=wiu⁰ andv=wjv⁰for somei, j∈N^{∗ with}i < j.

A treetis a subset ofU that satisfies:

• ∅ ∈t,

• Ifu∈t, thenAu⊂t.

• For everyu∈t, there existsk_u(t)∈N∪ {+∞}such that, for every positive integer i,ui∈tiff1≤i≤k_u(t).

The integer k_u(t)represents the number of offsprings of the vertexu∈ t. (Notice that k_u(t) has to be finite in [1], whereask_u(t) might take the value +∞ here.) The vertex u ∈ t is called a leaf if ku(t) = 0 and it is said infinite if ku(t) = +∞. By convention, we shall setku(t) =−1ifu6∈t. The vertex∅is called the root oft. We set:

|t|= Card (t).

Lettbe a tree. The set of its leaves isL₀(t) ={u∈t;k_u(t) = 0}. Its height and its

“norm” are resp. defined by

H(t) = sup{|u|, u∈t} and H_∞(t) = sup{|u|_∞, u∈t}= max(H(t),sup{ku(t), u∈t});

they can be infinite. Foru∈t, we define the sub-treeSu(t)oft“above”uas:

S_u(t) ={v∈ U, uv∈t}.

Foru∈t\ L0(t), we also define the forestFu(t)“above”uas the following sequence of trees:

Fu(t) = (Sui(t);i∈N^∗, i≤ku(t)).

Foru∈t\ {∅}, we also define the sub-treeS^u(t)oft“below”uas:

S^u(t) ={v∈t;u6∈Av}.

Notice thatu∈ S^u(t).

Forv= (v_k, k∈N^∗)∈(N^∗)^N∗, we setv¯_n= (v₁, . . . , v_n)forn∈N, with the convention thatv¯₀ =∅andv¯ ={¯v_n, n∈N} defines an infinite spine or branch. We denote byT∞

the set of trees. We denote byT⁰the subset of finite trees, T⁰={t∈T^∞; |t|<+∞}, byT^(h)∞ the subset of trees with norm less thanh,

T^(h)∞ ={t∈T∞; H_∞(t)≤h}, byT^∗0the subset of trees with no infinite branch,

T^∗0={t∈T^∞;∀v∈(N^∗)^N,v¯ 6⊂t},

and byT2the subset of trees with no infinite branch and with exactly one infinite vertex, T²={t∈T∞; Card{u∈t;ku(t) = +∞}= 1} ∩T^∗0.

Notice thatT⁰is countable andT²is uncountable.

Forh∈N, the restriction functionr_h,∞fromT∞toT∞is defined by:

r_h,∞(t) ={u∈t, |u|_∞≤h}.

Sets r_h,∞(t)are called “left-balls” in [8]. We endow the set T∞ with the ultra-metric distance

d∞(t,t⁰) = 2^{−max{h∈N, rh,∞(t)=rh,∞(t0)}}.

A sequence(tn, n∈N)of trees converges to a treetwith respect to the distanced_∞ if and only if, for everyh∈N^,

rh,∞(tn) =rh,∞(t) fornlarge enough,

that is for allu∈ U,lim_n→+∞k_u(t_n) = k_u(t)∈ N∪ {−1,+∞}. The Borelσ-field asso- ciated with the distanced_∞ is the smallestσ-field containing the singletons for which the restrictions functions(r_h,∞, h∈N)are measurable. With this distance, the restric- tion functions are contractant. SinceT⁰ is dense inT∞and(T∞, d_∞)is complete and compact, we get that(T^∞, d∞)is a compact Polish metric space.

Remark 2.1. In [1], we considered

T={t∈T∞;k_u(t)<+∞ ∀u∈t}

the subset of trees with no infinite vertex. OnT, we defined the distance:

d(t,t⁰) = 2^{−max{h∈N, rh(t)=rh(t0)}},

withrh(t) ={u∈t, |u| ≤h}. Notice that(T, d)is Polish but not compact and thatT^is not closed in(T^∞, d∞). If a sequence(tn, n∈N^∗)converges in(T, d)then it converges in(T∞, d_∞). And if a sequence(t_n, n∈N^∗)of elements ofTconverges in(T∞, d_∞)to a limit inTthen it converges to the same limit in(T, d).

Consider the closed ball B_∞(t,2^−h) ={t⁰ ∈ T∞;d_∞(t,t⁰)≤ 2^−h} for somet ∈T∞

andh∈Nand notice that:

B∞(t,2^−h) =r_h,∞⁻¹ ({rh,∞(t)}).

Since the distance is ultra-metric, the closed balls are open and the open balls are closed, and the intersection of two balls is either empty or one of them. We deduce that the family ((r_h,∞⁻¹ ({t}),t ∈ T^(h)∞), h ∈ N) is a π-system, and Theorem 2.3 in [3]

implies that this family is convergence determining for the convergence in distribution.

Let (Tn, n ∈ N^∗) and T be T∞-valued random variables. We denote by dist (T) the distribution of the random variableT (which is uniquely determined by the sequence of distributions ofr_h,∞(T)for everyh≥0), and we denote:

dist (T_n) −→

n→+∞dist (T)

for the convergence in distribution of the sequence (Tn, n ∈ N^∗) to T. Notice that this convergence in distribution is equivalent to the finite dimensional convergence in distribution of(k_u(T_n), u∈ U)to(k_u(T), u∈ U)asngoes to infinity.

We deduce from the portmanteau theorem that the sequence(T_n, n∈N^∗)converges in distribution toT if and only if for allh∈N^,t∈T^(h)∞:

n→+∞lim P(r_h,∞(Tn) =t) =P(r_h,∞(T) =t).

As we shall only considerT0-valued random variables that converge in distribution to a T2-valued random variable, we give an other characterization of convergence in distribution that holds for this restriction. To present this result, we introduce some notations. Ifv= (v1, . . . , vn)∈ U, withn >0, andk∈N, we define the shift ofvbykas θ(v, k) = (v1+k, v2, . . . , vn). Ift∈T^0,s∈T∞andx∈twe denote by:

t~(s, x) =t∪ {xθ(v, kx(t)), v∈s\ {∅}}

the tree obtained by grafting the treesat xon “the right” of the treet, with the con- vention thatt~(s, x) =tifs={∅}is the tree reduced to its root. Notice that ifxis a leaf oftands∈T, then this definition coincides with the one given in [1].

For everyt∈T^{0and every}x∈t, we consider the set of trees obtained by grafting a tree atxon “the right” oft:

T(t, x) ={t~(s, x), s∈T^∞}

as well as fork∈N^:

T(t, x, k) ={s∈T(t, x);k_x(s) =k} and T+(t, x, k) ={s∈T(t, x);k_x(s)≥k}

the subsets ofT(t, x)such that the number of offspring ofxare resp. kandkor more.

It is easy to see thatT+(t, x, k)is closed. It is also open, as for all s ∈ T+(t, x, k) we have thatB_∞(s,2^{−max(k,H∞(t))−1})⊂T⁺(t, x, k).

Moreover, notice that the setT2is a Borel subset of the setT. The next lemma gives another criterion for the convergence in distribution inT0∪T2. Its proof is very similar to the proof of Lemma 2.1 in [1].

Lemma 2.2. Let(Tn, n∈N^∗)andT beT^∞-valued random variables which belong a.s.

toT0∪T2. The sequence(T_n, n∈ N^∗)converges in distribution toT if and only if for everyt∈T0,x∈tandk∈N^{, we have:}

n→+∞lim P(T_n∈T+(t, x, k)) =P(T ∈T+(t, x, k)) and lim

n→+∞P(T_n=t) =P(T =t).

(2.2) Remark 2.3. Let

T¹={t∈T;∃!v∈(N^∗)^N∗s.t.v¯ ⊂t},

be the subset of trees with only one infinite spine (or branch). We give in [1] a charac- terization of the convergence inT0∪T1as follows. Let(T_n, n∈N^∗)andT beT^-valued random variables which belong a.s. toT⁰∪T¹. The sequence(Tn, n∈N^∗)converges in distribution toT if and only if (2.2) holds for everyt∈T^0,x∈ L0(t)andk= 0. In a sense, the convergence inT⁰∪T¹is thus easier to check.

Proof. The subclassF ={T⁺(t, x, k)T (T⁰S

T²), t∈T⁰, x∈t, k∈N} ∪ {{t}, t∈T⁰} of Borel sets onT⁰S

T^{2forms a}π-system since we have

T⁺(t1, x1, k1)∩T⁺(t2, x2, k2) =















T⁺(t1, x1, k1) ift1∈T(t2, x2)andx2∈Ax₁, T+(t₁, x₁, k₁∨k₂) ift₁∈T(t₂, x₂)andx₁=x₂,

{t1} ift1∈T(t2, x2)andx26∈Ax₁∪ {x1},

∅ in the other (non-symmetric) cases. For everyh ∈ N ^{and every} t ∈ T^(h)∞, we have that t⁰ belongs to r⁻¹_h,∞({t})T

T^{2 if and} only ift⁰ belongs to some T+(s, x, k)T

T2withx∈tsuch that|x|∞ =handsbelongs tor⁻¹_h,∞({t})T

T0 withx ∈ s. SinceT0 is countable, we deduce that F generates the Borelσ-field onT0∪T2. In particularF is a separating class inT0∪T2. SinceA∈ F is closed and open as well, according to Theorem 2.3 of [3], to prove that the familyF is a convergence determining class, it is enough to check that, for allt∈ T⁰∪T^{2 and} h∈N, there existsA∈ F such that:

t∈A⊂B∞(t,2^−h). (2.3)

Ift∈T⁰, this is clear as{t}=B∞(t,2^−h)for allh > H∞(t). Ift∈T^{2, for all}s∈T^0and x∈ssuch thatt∈T+(s, x, k), withk=k_x(s), we havet∈T+(s, x, k)⊂B_∞(t,2^−|x|∞). Since we can find such asandxsuch that|x|_∞is arbitrary large, we deduce that (2.3) is satisfied. This proves that the familyFis a convergence determining class inT⁰∪T^2. Since, fort∈T^0,x∈tandk∈N^{, the sets}T⁺(t, x, k)and{t}are open and closed, we deduce from the portmanteau theorem that if(Tn, n∈N^∗)converges in distribution to T, then (2.2) holds for everyt∈T^0,x∈tandk∈N^.

3 GW trees

3.1 Definition

Let p = (p(n), n ∈ N) be a probability distribution on the set of the non-negative integers. We assume thatpsatisfies (1.1). Let g(z) =P

k∈Np(k)z^k be the generating function ofp. We denote byρ(p)its radius of convergence and we will writeρforρ(p) when it is clear from the context. We say thatpis aperiodic if{k;p(k)>0} ⊂dN^implies d= 1.

AT-valued random variableτ is a Galton-Watson (GW) tree with offspring distribu- tionpif the distribution ofk_∅(τ)ispand forn ∈N^∗, conditionally on{k_∅(τ) =n}, the sub-trees(S1(τ),S2(τ), . . . ,Sn(τ))are independent and distributed as the original tree τ. Equivalently, for everyh∈N^∗andt∈T^{(h)∞ , we have:}

P(r_h,∞(τ) =t) = Y

u∈rh−1,∞(t)

p(ku(t)).

In particular, the restriction of the distribution ofτon the setT0is given by:

∀t∈T⁰, P(τ =t) =Y

u∈t

p(ku(t)). (3.1)

The GW tree is called critical (resp. sub-critical, super-critical) ifµ(p) = 1(resp. µ(p)<

1,µ(p)>1). In the critical and sub-critical case, we have that a.s. τ belongs toT^0. Let P^k be the distribution of the forest τ^(k) = (τ1, . . . , τk) of i.i.d. GW trees with offspring distributionp. We set:

|τ^(k)|=

k

X

j=1

|τj|.

When there is no confusion, we shall writeτforτ^(k). 3.2 Condensation tree

Assume thatpsatisfies (1.1) withµ(p)<1. Recall the definition of the treeτ^∗(p)in the introduction. Remark that, asµ(p)<1, the treeτ^∗(p)belongs a.s. toT^2.

Fort∈T^0,x∈t, we set:

D(t, x) =P(τ=S^x(t))

p(0) Pkx(t)(τ=Fx(t)).

Forz∈R^{, we set}z+ = max(z,0). LetX be a random variable with distributionp. The following lemma is elementary.

Lemma 3.1. Assume thatpsatisfies (1.1) andµ(p)<1. The distribution ofτ^∗(p)is also characterized by: a.s.τ^∗(p)∈T2and fort∈T0,x∈t,k∈N^,

P(τ^∗(p)∈T⁺(t, x, k)) =D(t, x) 1−µ(p) +E

(X−kx(t))+1_{X≥k}

. (3.2) In particular, we have that ifx∈ L0(t):

P(τ^∗(p)∈T(t, x), kx(τ^∗(p)) = +∞) = (1−µ(p))P(τ=t) p(0)

and

P(τ^∗(p)∈T(t, x)) = P(τ =t)

p(0) · (3.3)

Remark 3.2. Assume that psatisfies (1.1) and µ(p) = 1. Then, according to [1], the distribution ofτ^∗(p)is characterized byτ^∗(p)∈T1a.s. and (3.3) fort∈T0,x∈ L₀(t). Remark 3.3. Letτ^S(p)denote the limit (in distribution) of a critical or sub-critical GW treeτ conditionally on{H(τ) =n}or{H(τ)≥n} asngoes to infinity (see [12, 1] for the existence of this limit). The distribution ofτ^S(p)is characterized by the properties i) to vii) withp˜in iv) replaced by the size-biased distributionp^◦:

p^◦(k) = k p(k)

µ ^fork∈N.

Remark that, when pis critical, the definitions of τ^∗(p)and τ^S(p) coincide. We have that a.s. τ^S(p)belongs toT¹. Following [1], we notice that the distribution ofτ^S(p)is characterized by: a.s.τ^S(p)∈T1and for allt∈T0,x∈ L0(t),

P(τ^S(p)∈T(t, x)) = P(τ =t)

µ(p)^|x|p(0)· (3.4)

4 Conditioning on the total population size ( A = N ⁾

We prove Theorem 1.3 for the special case A = N and concentrate on the case p non-generic forN. The results of this section appear already in [7] see also [8]. We provide here an elementary proof relying on the strong ratio limit property of random walks on the integers.

Recall the definition ofp_N,θin (1.2) and thatI_Nis the set of positiveθfor whichp_N,θ is a probability distribution. It is easy to check thatµ(p_N,θ)is increasing inθ. Following [7], we define a non-generic distribution forNas follows.

Definition 4.1. Letpbe a distribution onN satisfying (1.1). We say pis non-generic forN^iflim_θ↑ρ(p)µ(p_N,θ)<1.

4.1 The non-generic case withρ(p) = 1

Notice that if psatisfies (1.1) and ifρ(p) = 1 and µ(p) < 1, thenpis in particular non-generic forN^.

Theorem 4.2. Assume thatpsatisfies (1.1),ρ(p) = 1andµ(p)<1. We have that:

dist (τ

|τ|=n) −→

n→+∞dist (τ^∗(p)), (4.1)

where the limit is understood along the infinite sub-sequence{n∈N^∗;P(|τ|=n)>0}, and:

dist (τ

|τ| ≥n) −→

n→+∞dist (τ^∗(p)). (4.2)

Proof. For simplicity, we shall assume thatpis aperiodic, that isP(|τ|=n)>0for alln large enough. The adaptation to the periodic case is left to the reader.

Recall thatρ(p) = 1. Letk∈N^,t∈T0,x∈t,`=k_x(t)andm=|t|. We have:

P(τ∈T⁺(t, x, k),|τ|=n) =D(t, x) X

j≥max(`+1,k)

p(j)Pj−`(|τ|=n−m).

Let (X_n, n ∈ N^∗) be a sequence of independent random variables taking values in Nwith distributionpand setSn=Pn

k=1Xk. Let us recall Dwass formula (see [5]): for everyk∈N^{∗and every}n≥k, we have

P^k(|τ|=n) = k

nP(Sn=n−k). (4.3)

Letτ_n be distributed asτ conditionally on{|τ|=n}. Using Dwass formula (4.3), we have

P(τn∈T⁺(t, x, k)) = P(τ ∈T⁺(t, x, k), |τ|=n) P(|τ|=n)

=D(t, x) X

j≥max(`+1,k)

p(j)Pj−`(|τ|=n−m) P(|τ|=n)

=D(t, x) X

j≥max(`+1,k)

p(j)n j−` n−m

P(S_n−m=n−m−j+`) P(S_n=n−1) · We then set

δ_n⁰(k, `) = 1 P(Sn=n)

X

j≥k

p(j)P(Sn =n+`−j) (4.4) and

δ¹_n(k, `) = 1 P(Sn =n)

X

j≥k

jp(j)P(Sn=n+`−j). (4.5) We get:

P(τn∈T⁺(t, x, k)) =D(t, x) n n−m

P(Sn−m=n−m) P(Sn=n−1)

δ_n−m¹ (max(`+ 1, k), `)−`δ<sub>n−m</sub><sup>0</sup> (max(`+ 1, k), `) .

Then use the strong ratio limit property (8.2) as well as its consequences (8.3) and (8.4), to get that:

n→+∞lim P(τ_n ∈T+(t, x, k)) =D(t, x)



1−µ(p) + X

j≥max(`+1,k)

(j−`)p(j)



. (4.6) Thanks to (3.2), we get:

n→+∞lim P(τn ∈T⁺(t, x, k)) =P(τ^∗(p)∈T⁺(t, x, k)).

Then use Lemma 2.2 to get (4.1). Sincedist (τ

|τ| ≥n)is a mixture ofdist (τ

|τ|=k) fork≥n, we deduce that (4.2) holds.

Remark4.3. Assumeµ(p) = 1. Then the proof of (4.6) still holds and we get in particular that for allt∈T^0andx∈ L0(t):

n→+∞lim P(τ_n∈T(t, x)) = P(τ=t) p(0) ·

Then the applicationT(t, x)7→P(τ =t)/p(0)can be extended into a probability distri- bution onT1which is given by the distribution ofτ^∗(p)(also equal to the distribution of τ^S defined in Remark 3.3). Then use Remark 2.3 to get thatdist (τ| |τ|=n)converges todist (τ^∗(p)).

4.2 The non-generic case withρ(p)>1

We consider the caseρ(p)>1. The offspring distributionp_N,θof (1.2) has generating function:

gθ(z) = g(θz) g(θ) ·

Notice that if pis non-generic for N^{, we have} I_N = (0, ρ(p)] and p^∗_N defined by (1.4) is p^∗_N = p_N,ρ(p). According to [11] (see also Proposition 5.5 in [1] for a more general setting), ifτ_N,θ denotes a GW tree with offspring distributionp_N,θ, then the distribution ofτ_N,θ conditionally on|τ_N,θ|does not depend onθ∈I_N.

Corollary 4.4. Assume thatpsatisfies (1.1) and is non-generic forN. We have that:

dist (τ

|τ|=n) −→

n→+∞dist (τ^∗(p^∗_N)),

where the limit is understood along the infinite sub-sequence{n∈N^∗;P(|τ|=n)>0}, and:

dist (τ

|τ| ≥n) −→

n→+∞dist (τ^∗(p^∗_N)).

Proof. Let us first remark that, by constructionρ(p^∗_N) = 1so that we can apply Theorem 4.2 to the tree τ_N,ρ(p). The first convergence is then a direct consequence of (4.1) and the fact thatτ conditionally on{|τ| = n}is distributed as τ_N,ρ(p) conditionally on {|τ_N,ρ(p)|=n}. The proof of the second convergence is similar to the proof of (4.2).

4.3 The general case

Theorem 4.2 and Corollary 4.4 (which concern the non-generic case) combined with Proposition 4.6 and Corollary 5.9 from [1] (which concern the generic case) complete the proof of Theorem 1.3 for the caseA=N. This gives a complete description of the asymptotic distribution of critical and sub-critical GW trees conditioned to have a large total population size.

5 Generic and non-generic distributions

Let p be a distribution onN satisfying (1.1) and let X be a random variable with distribution p. Recall that ρ(p) denotes the radius of convergence of the generating functiongofp. LetA ⊂Nbe such thatp(A)>0. We consider the modified distribution p_A,θonNgiven by (1.2) and letI_Abe the set of positiveθfor whichp_A,θ is a probability distribution. We haveθ∈IAif and only ifθ >0and:

E

θ^X1_{X∈A}

<+∞ and E

θ^X1_{X∈Ac}

< θ

or E

θ^X1_{X∈Ac}

=θ. (5.1) In (5.1) the former case corresponds toc_A(θ)>0and the latter toc_A(θ) = 0.

NoticeI_Ais an interval of(0,+∞)which contains 1. We haveinfI_A= 0if0∈ Aand 1>infI_A≥p(0)if06∈ A. Let:

θ^∗_A= supI_A∈[1, ρ(p)]. (5.2)

We deduce from the definition ofp_A,θ the following rule of composition, forθ ∈I_Aand θq∈IA:

p_A,θq = (p_A,θ)_A,q. (5.3)

The generating function,g_A,θ, ofp_A,θ is given by:

g_A,θ(z) =E

(zθ)^X 1

θ1_{X∈Ac}+c_A(θ)1_{X∈A}

.

And we have:

µ(p_A,θ) =E

Xθ^X−11_{X∈Ac}

+c_A(θ)E

Xθ^X1_{X∈A}

. (5.4)

Let:

θ^c_A= inf{θ∈IA;µ(pA,θ) = 1}, (5.5)

with the convention thatinf∅= +∞. Notice that the functionθ7→µ(p_A,θ)is continuous overI_AandC¹on

o

I_Athe interior ofI_A.

Lemma 5.1. Letpbe a distribution onNsatisfying (1.1) andA ⊂N^{such that}p(A)>0. Forθ∈I^o_A, we have:

d

dθµ(pA,θ)>0 if µ(pA,θ)≤1.

If0∈ A, then the functionθ7→µ(p_A,θ)is increasing overI_A.

Proof. Sincepsatisfies (1.1), it is easy to check thatp_A,θ satisfies (1.1) for all θ ∈ I_A such thatθ < θ_A^∗. For the first part of the Lemma, thanks to the composition rule, it is enough to prove that _dθ^dµ(pA,θ) >0 at θ = 1if µ(p) ≤1, withpsatisfying (1.1) and ρ(p)>1.

Letθ∈I_A. We have:

µ(pA,θ)−E[X] = hA(θ) θE

θ^X1_{X∈A}, with

hA(θ) =E

Xθ^X1_{X∈Ac}

E

θ^X1_{X∈A}

+θE

Xθ^X1_{X∈A}

−E

θ^X1_{X∈Ac}

E

Xθ^X1_{X∈A}

−θE[X]E

θ^X1_{X∈A}

.

Of course we haveh_A(1) = 0. The functionh_Ais of classC^∞on[0, ρ(p)). Notice that d

dθµ(p_A,θ)

|θ=1

=h⁰_A(1) p(A)·

Moreover, h⁰_A(1) =E

(X−1)(Xp(A)−E

X1_{X∈A}

)

=p(A)E[X(X−1)]+(1−E[X])E

X1_{X∈A}

.

In particular, we deduce from this last expression thath⁰_A(1)>0ifE[X]≤1. This ends the proof of the first part of the lemma.

Let us assume that0 ∈ A. We setΓ(A) = E

X1_{X∈A}

/p(A). If E[X] =µ(p)>1, notice thath⁰_A(1)/p(A)is minimal whenΓ(A)is maximal. We have fork6∈ A:

Γ(A ∪ {k})−Γ(A) = p(k)

p(A ∪ {k})(k−Γ(A)).

By induction, this implies thatΓ(A ∪ {j;j≥k})>Γ(A)as soon ask >Γ(A)as well as Γ (A ∩({j;j≥k} ∪ {0}))<Γ(A)as soon ask <Γ(A). Therefore, we have thatΓ(A)is maximal (for all subsetsAofN^containing0) forAof the formAn ={0} ∪ {k;k≥n}.

By considering h⁰_An(1)−h⁰_An−1(1), it is then easy to check that the function n 7→

h⁰_A

n(1) is first non-decreasing and then non-increasing. Since h⁰_A

0(1) and h⁰_A

∞(1) are positive, we get thath⁰_A

n(1) is positive for all n ∈ N ^{and thus}h⁰_A(1) is positive. This ends the proof of the second part of the lemma.

Let us consider the equation onIA:

µ(p_A,θ) = 1. (5.6)

Lemma 5.2. Letpbe a distribution onNsatisfying (1.1) andA ⊂N^{such that}p(A)>0. Equation (5.6) has at most one solution inI_A. If there is no solution to Equation (5.6), then we haveµ(p)<1,θ_A^∗ belongs toIAandµ(pA,θ^∗_A)<1.

The (unique) solution of (5.6) in I_A, if it exists, is denoted θ^c_A. Notice that p_A,θ^c

A is critical.

Proof. Lemma 5.1 directly implies that Equation (5.6) has at most one solution.

If 0 ∈ A, then we have infIAµ(p_A,θ) = p(1)1_{1∈Ac} < 1. If 0 6∈ A, then set q = minIA ∈ (0,1). Notice thatcA(q) = 0and E

q^X1_{X∈A^c_}

= q. Use that the function θ 7→ E

θ^X1_{X∈Ac}

is convex and less than the identity map on (q,1] to deduce that E

Xq^X−11_{X∈Ac}

is strictly less than1. Then use (5.4) to deduce that:

limθ↓qµ(pA,θ) =E

Xq^X−11_{X∈Ac}

<1.

In conclusion, we deduce thatinf_IAµ(p_A,θ)<1. Hence, ifµ(p)≥1then Equation (5.6) has at least one solution.

From what precedes, if there is no solution to Equation (5.6), this implies thatµ(p)<

1and thus:

µ(pA,θ)<1 for allθ∈IA. (5.7) We only need to consider the caseθ^∗_A >1. Since θ^∗_A ≤ρ(p), we haveρ(p) >1. Since µ(p) < 1, the interval J = {θ;g(θ) < θ} is non-empty andinfJ = 1. OnJ ∩I_A, we deduce from (1.3) that θc_A(θ) > 1 and then from (5.4) that µ(p_A,θ) > g⁰(θ) and thus g⁰(θ) <1. Notice that this implies that IAT

(1,+∞)is a subset of J¯the closure ofJ. The properties ongimply thatJ¯={θ;g(θ)≤θ}. This clearly implies that (5.1) holds for θ^∗_Athat isθ^∗_A∈I_A. Then conclude using (5.7).

We give the definition of a non-generic distribution which generalizes Definition 4.1.

Definition 5.3. Let p be a distribution on N satisfying (1.1) and A ⊂ N ^{such that} p(A)>0. If Equation (5.6) has a (unique) solution inI_A, thenpis called generic forA. If Equation (5.6) has no solution, thenpis called non-generic forA.

In the next lemma, we writeρforρ(p).

Lemma 5.4. Letpbe a distribution onNsatisfying (1.1) such thatµ(p)<1.

- Ifρ= +∞or (ρ <+∞andg⁰(ρ)≥1), thenpis generic for any A ⊂N^{such that} p(A)>0.

- Ifρ= 1(and thusg⁰(1)<1), thenpis non-generic for allA ⊂N^{such that}p(A)>0. - If1< ρ <+∞andg⁰(ρ)<1(and thusg(ρ)< ρ), thenpis non-generic for{0}and pis generic for{k}for allklarge enough and such thatp(k)>0. Furthermorep is non-generic forA ⊂N^(withp(A)>0) if and only if:

E[Y|Y ∈ A]<ρ−ρg⁰(ρ) ρ−g(ρ) ,

withY distributed asp_N,ρ, that isE[f(Y)] =E[f(X)ρ^X]/g(ρ)for every non-negative measurable functionf. We also haveθ_A^∗ =ρ.

Remark 5.5. We give some consequences and remarks related to the previous Lemma.

1. Ifpis generic for{0}then it is generic for allA ⊂N^withp(A)>0.

2. IfAandB are disjoint subsets ofN ^{such that}p(A)>0andp(B)>0, then ifpis non-generic forAand forBthen it is non-generic forAS

B.

3. IfAandB are disjoint subsets ofN ^{such that}p(A)>0andp(B)>0, then ifpis generic forAand forBthen it is generic forAS

B. 4. Assumeρ(p)>1andA ⊂ Bwithp(B)> p(A)>0.

• Thenpnon-generic forAdoes not imply in general thatpis non-generic for B. (See case (6) below withA={0}andB=N^.)

• Thenpnon-generic forB does not imply in general thatpis non-generic for A. (Letpsatisfying (1.1) be such thatρ(p)>1andpnon-generic forB=N^. Then, according to Lemma 5.4, there existsklarge enough such thatp(k)>0 andpis generic forA={k}.)

5. LetY be as in the last statement of Lemma 5.4. According to the second part of the proof of Lemma 5.1, we get that there existsn0∈N^{∗such that:}

sup

A30E[Y|Y ∈ A] =E[Y|Y ∈ An₀],

with An = {0} ∪ {k;k ≥ n}. In particular, if pis non-generic for An₀ then it is non-generic for allAcontaining0.

6. LetGbe a generating function with radius of convergenceρ_G = 1. Letc ∈(0,1). Letpbe the distribution with generating function:

g(z) = G(cz) G(c)·

The radius of convergence ofgis thusρ= 1/cand we have:

g_N,ρ(z) =G(z) and g_{0},ρ(z) = cG(z)

G(c) + 1− c G(c)·

Therefore, we have:

g⁰_N,ρ(1) =G⁰(1) and g⁰_{0},ρ(1) = cG⁰(1) G(c) ·

IfG⁰(1) = 1, then we haveG(c)> c. This impliesg_{0},ρ⁰ (1)< g⁰_N,ρ(1) = 1. Thuspis generic forNbut non generic for{0}.

Proof. ForA ⊂N^{such that}p(A)>0andθ∈I_A, notice that:

µ(p_A,θ)−1 =G_A(θ)θ−g(θ)

θ −(1−g⁰(θ)) with G_A(θ) = E

Xθ^X1_{X∈A}

E

θ^X1_{X∈A} · (5.8) Ifρ= +∞orρ <+∞andg⁰(ρ)≥1, then there existsq >1finite such thatg⁰(q) = 1 which implies thatqsatisfies (5.1). We also haveg(q)< q. This implies, thanks to (5.8), thatµ(p_A,q)>1. Therefore,pis generic forA.

Ifρ <+∞andg⁰(ρ)<1, then we haveg(ρ)< ρandρsatisfies (5.1). This implies that θ^∗_A=ρ∈I_A. According to Lemma 5.2,pis non-generic forAif and only ifµ(p_A,ρ)<1 that is, using (5.8):

G_A(ρ)< ρ−ρg⁰(ρ) ρ−g(ρ) ·

We haveG_{0}(ρ) = 0and thuspis non-generic for{0}. Forksuch thatp(k)>0, we have G_{k}(ρ) =kand thuspis generic forklarge enough such thatp(k)>0. To conclude, notice thatG_A(ρ) =E[Y|Y ∈ A].

6 Vertices with a given number of children I: case 0 ∈ A

Assume0 ∈ A ⊂ N^and A 6=N. Assume that psatisfies (1.1),µ(p) <1. We prove Theorem 1.3 forpnon-generic forA.

In what follows, we denote by X a random variable distributed according top. We consider onlyP(X∈ A)<1, as the caseP(X ∈ A) = 1corresponds toA=N^{of Section} 4. For t ∈ T^{0, we set}L_A(t) = {u ∈ t, ku(t) ∈ A}the set of nodes whose number of children belongs toAand defineL_A(t) = Card (L_A(t)).

For a tree t ∈ T0, following [15, 18], we can map the set L_A(t) onto a tree t^A. We first define a map φ from L_A(t) on U and a sequence (tk)_1≤k≤n of trees (where n =L_A(t)) as follows. Recall that we denote by<the lexicographic order on U. Let u¹<· · ·< uⁿbe the ordered elements ofLA(t).

• φ(u¹) =∅,t1={∅}.

• For1< k ≤n, setw^k =M RCA({u^k−1, u^k})the most recent common ancestor of u^k−1andu^kand recall thatS_wk(t)denotes the tree abovew^k. We sets={w^ku, u∈ S_wk the sub-tree abovew^kandv= min(L_A(s)). Then, we set

φ(u^k) =φ(v)(k_φ(v)(t_k−1) + 1)

the concatenation of the nodeφ(v)with the integerk_φ(v)(t_k−1) + 1, and tk =tk−1∪ {φ(u^k)}.

In other words, φ(u^k)is a child ofφ(v)intk and we add it “on the right” of the other children (if any) ofφ(v)in the previous treetk−1to gettk.

It is clear by construction thattk is a tree for everyk ≤n. We sett^A=tn. Thenφis a one-to-one map fromL_A(t)ontot^A. The construction of the treet^Ais illustrated on Figure 1. Notice thatLA(t)is just the total progeny oft^A.

1 2

3 4 5

1 2

6

7

8 9

3

4 5

6

7

8 9

Figure 1: left: a treet, right: the treet^AforA={0,2}

If τ is a GW tree with offspring distribution p, the tree τ^A associated with LA(τ) is then, according to [18] Theorem 6 (for the particular case0∈ A), a GW tree whose offspring distributionp^Ais defined as follows. LetN,Y⁰⁰and(Y_k⁰, k∈N)be independent random variables such thatNis geometric with parameterp(A),Y⁰⁰is distributed asX conditionally on{X ∈ A}and(Y_k⁰, k∈N)are independent random variables distributed asX−1conditionally on{X ∈ A^c}. We set:

X^A=

N−1

X

k=1

Y_k⁰+Y⁰⁰, (6.1)

with the convention thatP

∅ = 0. Thenp^Ais the distribution ofX^A. Letg^Adenote its generating function:

g^A(z) = zE

z^X1_{X∈A}

z−E

z^X1_{X∈Ac}

· (6.2)

An elementary computation gives:

µ(p^A) = 1−1−µ(p)

p(A) ^and g^A(θ) = 1

c_A(θ)· (6.3)

We recover that ifτ is critical (µ(p) = 1) thenτ^Ais critical asµ(p^A) = 1, see also [18]

Lemma 6. Notice in particular that for allk∈ A:

p^A(k) =P(X^A=k)≥P(N= 1, Y⁰⁰=k) =p(k), (6.4) and fork∈ A^c:

p^A(k−1) =P(X^A=k−1)≥P(N = 2, Y₁⁰=k−1, Y⁰⁰= 0) =p(0)p(k). (6.5) Lemma 6.1. Assume thatpsatisfies (1.1),µ(p)<1. Thenp^Asatisfies (1.1),µ(p^A)<1 andρ(p^A) =ρ(p)ifρ(p) = 1or ifρ(p)>1andg⁰(ρ(p))<1.

Proof. First, (6.4) implies thatp^A(0)≥p(0). Direct computation from (6.2) implies:

p^A(0)+p^A(1) = 1

(1−p(1)1_{1∈A^c_})² p(0)(1−p(1)1_{1∈Ac}) +p(1)1_{1∈A}+p(0)p(2)1_{2∈Ac} .

If1∈ A, using (1.1) we get:

p^A(0) +p^A(1)≤p(0) +p(1) +p(0)p(2)<1.

If1∈ A^c, using (1.1) we get:

p^A(0) +p^A(1)≤ 1 (1−p(1))²

p(0)(1−p(1)) +p(0)(1−p(0)−p(1))

< 1 (1−p(1))²

p(0)(1−p(1)) + (1−p(1))(1−p(0)−p(1)

= 1.

We deduce thatp^Asatisfies (1.1).

Equation (6.3) withµ(p)<1implies directlyµ(p^A)<1. Letρ_Abe the radius of convergence of the series given byE

z^X1_{X∈A}

andρ_Ac be the radius of convergence of the series given byE

z^X1_{X∈Ac}

. We get thatmin(ρ_A, ρ_Ac) = ρ(p). Using (6.1), we get:

g^A(z)≥P(N= 2)Eh z^Y10i

Eh z^Y00i

=1 zE

z^X1_{X∈Ac}

E

z^X1_{X∈A}

.

This implies thatρ(p^A)≤ρ(p). In particular, we get thatρ(p^A) = 1ifρ(p) = 1. Ifρ(p)>1 andg⁰(ρ(p))<1, then we haveg(ρ(p))< ρ(p)and (6.2) is well defined forz=ρ(p). This impliesρ(p^A)≥ρ(p)and thusρ(p^A) =ρ(p).

6.1 The caseρ(p) = 1

We state now the main result of this section.

Theorem 6.2. Assume thatpsatisfies (1.1),µ(p)<1andρ(p) = 1. We have that:

dist (τ

L_A(τ) =n) −→

n→+∞dist (τ^∗(p)), (6.6)

where the limit is understood along the infinite sub-sequence{n∈N^∗;P(L_A(τ) =n)>

0}, as well as

dist (τ

L_A(τ)≥n) −→

n→+∞dist (τ^∗(p)). (6.7)

Proof. For simplicity, we shall assume thatp^Ais aperiodic. The adaptation to the peri- odic case is left to the reader. We define forj ∈N^andn≥2:

nj=n−1_{j∈A}. (6.8)

Letk∈N^,t∈T^0,x∈t,`=kx(t)andm=|t^A| −1_{x∈LA(t)}. We have:

P(τ∈T⁺(t, x, k), L_A(τ) =n) =D(t, x) X

j≥max(`+1,k)

p(j)Pj−`(|τ^A|=nj−m).

Let(Xn, n∈N^∗)be independent random variables taking values inNwith distribu- tionp^Aand setSn=Pn

k=1Xk. According to Dwass formula (4.3), we have:

Pj−`(|τ<sup>A</sup>|=nj−m) = j−`

nj−mP(Sn_j−m=nj−m−j+`).

Let τn be distributed as τ conditionally on {LA(τ) = n}. Then we have, using the definition ofδ_n−m^1,A andδ^0,A_n−min (8.6) and (8.7):

P(τn ∈T⁺(t, x, k)) =D(t, x) X

j≥max(`+1,k)

p(j)n j−` nj−m

P(Sn_j−m=nj−m−j+`) P(Sn =n−1)

=D(t, x) n n−m

P(S_n−m=n−m) P(S_n=n−1)

δ^1,A_n−m(max(`+ 1, k), `)−`δ<sup>0,A</sup><sub>n−m</sub>(max(`+ 1, k), `) .

Then use the generalizations of the strong ratio limit properties (8.2), (8.9) and (8.10) to get that:

n→+∞lim P(τn∈T⁺(t, x, k)) =D(t, x)



1−µ(p) + X

j≥max(`,k)

(j−`)p(j)



.

Thanks to (3.2), we get:

n→+∞lim P(τ_n ∈T+(t, x, k)) =P(τ^∗(p)∈T+(t, x, k)).

Then use Lemma 2.2 to get (6.6). Sincedist (τ

LA(τ) ≥ n)is a mixture ofdist (τ L_A(τ) =k)fork≥n, we deduce that (6.7) holds.

6.2 The caseρ(p)>1

We consider the case pnon-generic for A with ρ(p) > 1. In particular, we have g⁰(ρ(p))<1and g(ρ(p))< ρ(p)thanks to Lemma 5.4. Recall the offspring distribution p_A,θ defined by (1.2). Notice that the normalizing constantc_A(θ)is given by:

cA(θ) = θ−E

θ^X1_{X∈Ac}

θE

θ^X1_{X∈A} = 1

g^A(θ)· (6.9)

Notice thatp_A,1 =p. Sinceρ(p)is also the radius of convergence ofg^A, see Lemma 6.1, we deduce thatp_A,θ is well defined forθ∈ [0, ρ(p)]andθ^∗_A=ρ(p). Letg_A,θ be the generating function ofp_A,θ.

According to [11] if A = {0} and Proposition 5.5 in [1] for the general setting, if τ_A,θ denotes a GW tree with offspring distribution p_A,θ, then the distribution of τ_A,θ conditionally onLA(τA,θ)does not depend onθ∈[0, ρ(p)].

Remark 6.3. It is easy to check that:

(g_A,θ)^A(z) =g^A(θz)

g^A(θ) = g^A

N,θ(z). (6.10)

The distribution ofτ_A,θ is the distribution ofτ “shifted” byθ such that the conditional distribution given the number of vertices having a number of children inAis the same.

Then, according to (6.10), the tree(τA,θ)^Aof vertices having a number of children inA associated withτA,θis distributed as the distribution ofτ^A“shifted” byθsuch that the conditional distribution given the total number of vertices is the same.

The proof of the following corollary is similar to the one of Corollary 4.4.

Corollary 6.4. Assume thatpsatisfies (1.1) and is non-generic forA. Letp^∗_A=p_A,ρ(p). We have that:

dist (τ

L_A(τ) =n) −→

n→+∞dist (τ^∗(p^∗_A)),

where the limit is understood along the infinite sub-sequence{n∈N^∗;P(L_A(τ) =n)>

0}, as well as

dist (τ

L_A(τ)≥n) −→

n→+∞dist (τ^∗(p^∗_A)).

This result with Proposition 4.6 and Corollary 5.7 in [1] ends the proof of Theorem 1.3 for the case0∈ A, and gives a complete description of the asymptotic distribution of critical and sub-critical GW trees conditioned to have a large number vertices with given number of children.

7 Vertices with a given number of children II: case 0 6∈ A

LetA ⊂ N. We assume in this section that06∈ Aandp(A)>0. We prove Theorem 1.3 forpnon-generic forA. Notice we follow the spirit of the case0∈ A.

7.1 Setting and notations

Although the construction of the previous section also holds in that case with a different offspring distribution, we failed to get analogues to formulas (6.4) and (6.5) which are crucially used in the proof of Lemma 8.4 . Therefore, we prefer to mapL_A(τ) onto a forestFA(τ)of independent GW trees. Let us describe this map.

Let t∈ T0. We define a map φ˜fromL_A(t)into the set S

n≥1Tⁿ0 of forests of finite trees as follows.

First, foru∈twe defineS_u^A(t)the sub-tree rooted atuwith no progeny inAby S_u^A(t) ={w∈uSu(t), Aw∩A^c_u∩ LA(t) =∅}.

Foru∈t, we defineC_u^A(t)as the leaves ofS_u^A(t)that belong toA.

1 2

3

4 5

6

7

Figure 2: The sub-treeS^A₁(t)in bold forA={3}, and the elements ofC₁^A(t).

We set

S˜_∅^A(t) =

(S_∅^A(t) if∅ 6∈ L_A(t) {∅} if∅ ∈ LA(t) and we setC˜_∅^A(t)the set of leaves ofS˜_∅^A(t)that belong toL_A(t).

Let N˜_∅(t) = Card ( ˜C_∅^A(t)). Then the range of φ˜ belongs to T^N0^˜∅(t). Moreover if u₁ < u₂<· · ·< uN˜_∅(t)are the elements ofC˜_∅^A(t)ranked in lexicographic order, we set for every1≤i≤N˜_∅(t)

φ(u˜ i) =∅⁽ⁱ⁾ where∅⁽ⁱ⁾denotes the root of thei-th tree inT^N0^˜∅(t).

We then constructφ˜recursively: if u∈ LA(t)andφ(u) =˜ v⁽ⁱ⁾(which is an element of the i-th tree), then we denote by u1 < · · · < uk the elements of C_u^A(t) ranked in lexicographic order and we set for1≤j≤k

φ(u˜ j) =vj⁽ⁱ⁾. Finally, we setF_A(t) = ˜φ(t).

1 2

3

4 5 6

7

1

2 3

4 5

6

7

Figure 3: A treetand the forestF_A(t)forA={3}.

Let τ be a Galton-Watson tree with offspring distribution p. Let us describe the distribution ofFA(τ).