An invariance principle for random planar maps

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An invariance principle for random planar maps

Grégory Miermont

To cite this version:

Grégory Miermont. An invariance principle for random planar maps. Fourth Colloquium on Math-

ematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy,

France. pp.39-58. �hal-01184709�

An invariance principle for random planar maps

Gr´egory Miermont

1CNRS & Laboratoire de Math´ematique Equipe Probabilit´es, Statistique et Mod´elisation, Bˆat. 425, Universit´e Paris- Sud, 91405 Orsay, France

We show a new invariance principle for the radius and other functionals of a class of conditioned ‘Boltzmann-Gibbs’- distributed random planar maps. It improves over the more restrictive case of bipartite maps that was discussed in Marckert and Miermont (2006). As in the latter paper, we make use of a bijection between planar maps and a class of labelled multitype trees, due to Bouttier et al. (2004). We also rely on an invariance principle for multitype spatial Galton-Watson trees, which is proved in a companion paper.

Keywords:Random planar map, invariance principle, multitype spatial Galton-Watson tree, Brownian snake

1 Planar maps

A planar map is a proper embedding of a connected graph in the2-dimensional sphere, considered up to images by orientation-preserving homeomorphisms of the sphere. As an embedded graph, it has edges and vertices, and its facesare the connected components of the complementary of the embedding. The degree of a face is the number of edges that are adjacent to it, where an ithmus, i.e. an edge whose removal disconnects the graph, must be counted twice. We let V(m), E(m) andF(m) be the sets of vertices, edges and faces of the planar mapm, while degv,degf will denote the respective degrees of elements v∈V(m), f ∈F(m). For any planar mapm, letdmdenote the usual graph distance on the setV(m).

A map ispointedif a vertex (the base vertex) is distinguished, androotedif anorientededge (the root edge) is distinguished. The basic set of maps that we consider in this paper is the setMof rooted, pointed planar maps, i.e. of triples(m, r, ~e), wheremis a planar map,r∈V(m)is a distinguished vertex, and~eis a distinguished edge. For convenience, we will usually omit the mention ofr, ~ein the notation. For technical reasons, we assumeMalso contains thevertex map, denoted by†, which is the planar map with no edge and only one vertex bounding a unique face of degree0. Form∈ M, we letR(m) = max_v∈V(m)dm(r, v) be theradiusof the map, when seen from the base vertex. We also let(I^m(k), k≥0)be theprofileofm, which is defined as the probability measure onZ+such that

I^m(k) = #{v∈V(m) :dm(r, v) =k}

#V(m) , n≥0.

Random planar maps have been an object of increasing interest to physicists during the last 15 years, as they can be interpreted as a discrete version of random surfaces, see Ambjørn et al. (1997) for example.

It is conjectured that properly rescaled random planar maps conditioned to be large should converge to a limiting ‘continuum random surface’, whose law should be insensitive to (reasonable) changes in the way that the maps are sampled. The (still hypothetic) common limiting object must be of Hausdorff dimension 4, as a consequence of recent advances in Le Gall (2006). This is consistent with results of Angel (2003) and Chassaing and Durhuus (2006), who work in the different context oflocal limitsof planar maps, i.e.

when one lets the size of the map go to infinity, without performing a rescaling operation.

The first results in this direction are due to Chassaing and Schaeffer (2004), who show that typical distances in a uniform rooted quadrangulation withnfaces scale asn^1/4when n → ∞. They obtain a description of rescaled limiting radius and profile of the map, in terms of the so-called ISE (Integrated Super-Brownian Excursion). Le Gall (2006) has obtained similar results, making use of a conditioned version of the Brownian snake, which is intimately related to ISE and was introduced in Le Gall and Weill

†Gregory.Miermont@math.u-psud.fr

1365–8050 c2006 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

2 0

1

1

1

1 2 3

1

2 2

Fig. 1:A typical element ofM. It has11vertices,18edges (2of which are ithmi) and9faces, one of degree1, three of degree3, one of degree4, two of degree5and two of degree6. The vertices are labelled according to their respective geodesic distances to the base vertex.

(2006). Marckert and Mokkadem (2006) show that a slightly different family of random quadrangulations withnfaces (pointed, and uniformly rooted on an edge adjacent to the distinguished vertex) converges in a loose sense, once rescaled by the factorn^1/4, to a random metric space they call the Brownian map, which is thus a conjectural limiting object for stronger topologies on (isometry classes of) metric spaces, like the Gromov-Hausdorff topology Burago et al. (2001). Marckert and Miermont (2006) show that similar results as in Chassaing and Schaeffer (2004); Le Gall (2006); Marckert and Mokkadem (2006) hold for rooted and pointed bipartite random planar maps (whose faces all have even degree) taken under a Boltzmann-type distribution, where each face of degree2kis attributed energylogqk, under regularity assumptions on the non-negative sequence(qk, k≥1).

The goal of the present paper is to show how results of Marckert and Miermont (2006) can be generalized to Boltzmann-type distributions on sets of maps without constraints on the degree of faces. As an important particular example, we obtain the invariance principle for the radius and profile of a uniform rooted and pointed triangulation (whose faces are all triangles) with n vertices, as n → ∞. Just as in Chassaing and Schaeffer (2004); Le Gall (2006); Marckert and Miermont (2006), we make use of a bijective method which encodes maps with the help of certain labelled multitype trees. This bijection is due to Bouttier et al.

(2004), and generalizes the bijections of Schaeffer (1998) and Cori and Vauquelin (1981). In the case of general planar maps, the bijection of Bouttier et al. (2004) is notably more delicate to use than one encoding bipartite maps, and requires a more powerful invariance principle for multitype trees (Theorem 2), which is proved in the companion paper Miermont (2006).

The paper is organized as follows. The rest of Section 1 is devoted to the statement of our main result (Theorem 1). The key property for proving Theorem 1 is the fact that Boltzmann-distributed random maps can be conveniently encoded in spatial multitype Galton-Watson trees. Section 2 introduces the necessary formalism on Galton-Watson trees, and states the invariance principle for such trees. Then, in Section 3, we describe the key property just mentioned, and prove Theorem 1. Finally, Section 4 discusses some extensions and perspectives for our results.

1.1 Boltzmann distributions on maps

WriteN={1,2, . . .}, and letq:= (q₁, q₂, . . .)∈R^N+be a non-negative sequence. We always assume that q_k >0for someoddk≥3, and letq₀= 1by convention. For a planar mapm(not necessarily pointed or

rooted) we let

Wq(m) = Y

f∈F(m)

qdegf,

and assimilateWqwith the non-negative measure onMassigning weightWq(m)to{(m, r, ~e)}. We say thatqisadmissibleif the ‘partition function’Zq :=Wq(M)is finite. It always hold thatZq>1because Wq({†}) = 1and by the hypothesis thatqk > 0for at least onek > 1. Hence, ifqis admissible, the expression

Pq=Wq

Zq

,

defines a Boltzmann-type probability distribution onM. We letM :M → Mbe the identity mapping.

It is interesting to note that a random variable with distributionPqcan be obtained in the following way:

choose the rooted map(m, ~e)with probability proportional toWq(m), and given this choice, choose the base vertex uniformly at random among the#V(m)possible. This observation comes from the fact that a root-preserving automorphism of a rooted planar map has to be the identity, so that each choice of a base vertex in a given rooted planar map yields a distinct element ofM.

For the needs of our study, we will have to introduce the three subsets ofMdefined as follows. For (m, r, ~e)∈ M, we lete₋, e₊be the source and target vertices of the edge~e. As in Figure 1, we label the vertices according to their distance to the base vertex, and distinguish the three possibilities

• M₊={(m, r, ~e)∈ M:d(r, e₊) =d(r, e₋) + 1} ∪ {†},

• M− ={(m, r, ~e)∈ M:d(r, e+) =d(r, e−)−1},

• M0={(m, r, ~e)∈ M:d(r, e+) =d(r, e₋)}.

On Figure 1 is an element of M+. Notice that M₋ does not contain the vertex map †, which is the convention we need so thatMis partitioned intoM+,M₋,M0. The elements ofM+\ {†}andM₋are in a natural one-to-one correspondence (reverse the orientation of~e). We define

Z_q⁺=W_q(M₊), Z_q⁻=W_q(M₋), Z_q⁰=W_q(M₀),

so by our previous observationZ_q⁻=Z_q⁺−1. We then haveZ_q =Z_q⁺+Z_q⁻+Z_q⁰= 2Z_q⁺−1 +Z_q⁰. For admissibleq, let

P_q⁺=P_q(·|M₊) =W_q(· ∩ M₊)/Z_q⁺, and define similarly the conditioned probabilitiesP_q⁻, P_q⁰.

We define

N•(k, k⁰) =

2k+k⁰+ 1 k+ 1

, N₃(k, k⁰) =

2k+k⁰ k

, k, k⁰ ∈Z+, (1) and let, forx, y≥0,

f_q^•(x, y) = X

k,k⁰≥0

x^ky^k0N_•(k, k⁰) k+k⁰

k

q_2+2k+k⁰ (2)

f_q³(x, y) = X

k,k⁰≥0

x^ky^k0N₃(k, k⁰) k+k⁰

k

q_1+2k+k⁰, (3)

which both may be finite or infinite.

Proposition 1 The weight sequenceqis admissible if and only if the system z⁺−1

z⁺ = f_q^•(z⁺, z³) (4)

z³ = f_q³(z⁺, z³) (5)

has a solution(z⁺, z³)∈(0,∞)², for which the matrix

Mq(z⁺, z³) :=











0 0 z⁺−1

z⁺

z³∂xf_q³(z⁺, z³) ∂yf_q³(z⁺, z³) 0 (z⁺)²

z⁺−1∂_xf_q^•(z⁺, z³) z⁺z³

z⁺−1∂_yf_q^•(z⁺, z³) 0











(6)

has spectral radius % ≤ 1. This ‘good’ solution is then unique andZ_q⁺ = z⁺, Z_q⁰ = (z³)², so that Zq= 2z⁺−1 + (z³)².

Proposition 1 will be proved in Section 3.2. Notice thatZ_q⁺ has to be>1 by (4), which is consistent with the fact† ∈ M+. We adopt the notationZ_q³ = z³whenever (z⁺, z³)is the good solution of (4) corresponding to an admissibleq.

Definition 1 We say thatqiscriticalif it is admissible and if the matrixMq:=Mq(Z_q⁺, Z_q³)has spectral radius%= 1, which amounts to the condition

(Z_q⁺)²Jf(Z_q⁺, Z_q³) + 1 = (Z_q⁺)²∂xf_q^•(Z_q⁺, Z_q³) +∂yf_q³(Z_q⁺, Z_q³), (7) whereJ_f is the (signed) Jacobian of the function(f_q^•, f_q³) :R²+→(R+∪ {∞})².

We say thatqisregular criticalif it is critical and iff_q^•(Z_q⁺+ε, Z_q³+ε)<∞for someε >0.

Notice that we could takef_q³as well in the definition of regular criticality, since it holds that∂_x(xf_•) =

∂_yf₃, as can be checked from the definitions ofN_•, N₃.

1.2 Main result

We now briefly describe the process known as thehead of the Brownian snake, see Marckert and Mokkadem (2003). LetB^exbe a standard Brownian excursion, and givenB^ex, letR^exbe a centered Gaussian process whose covariance function is given by

Cov (R^ex_s , R^ex_t ) = inf

s∧t≤u≤s∨tB_u^ex 0≤s, t≤1.

It is well-known, see e.g. (Le Gall, 1999, Sect. IV.6), that(B^ex, R^ex)has a continuous version, which is the one we always work with.

Our main result result generalizes (Marckert and Miermont, 2006, Proposition 4). It shows that invariance principles hold for the radius, distance to a uniformly sampled vertex, and profile of a Boltzmann-distributed critical random map, where the limiting distribution can be described in terms of the head of the Brownian snake. These distributions are, as expected, exactly the same as in the bipartite case, the only change coming from the scaling constants. In the following statement, form∈ Mandn≥1, we letI^mn be the measure onR+defined byI^mn(A) =I^m(n^1/4A),Abeing a Borel subset ofR+.

Theorem 1 Letqbe a regular critical weight sequence. Then, there exists a scaling constantCq∈(0,∞) such that

(i) The law of n^−1/4R(M)under P_q(·|#V(M) = n) converges weakly as n → ∞ to the law of C_q(supR^ex−infR^ex).

(ii) The law of n^−1/4d_M(r, r⁰) under P_q(·|#V(M) = n), wherer⁰ is a vertex picked uniformly at random inV(M)\ {r}conditionally onM, converges weakly asn→ ∞to the law ofC_qsupR^ex.

(iii)The distribution of the random measureI^Mn underP_q(·|#V(M) =n)converges weakly asn→ ∞ to the law of the random probability measureI^exonR+, defined by

I^ex(g) = Z 1

0

ds g C_q(R^ex_s −infR^ex) ,

for every non-negative measurableg.

In statements(i), (ii), (iii), it must be understood thatn→ ∞really denotes if necessary a subsequence nm, m≥1along whichPq(#V(M) =nm)>0.

The strategy of our proof will be to show that Theorem 1 holds true (with thesame scaling constant Cq) when replacingPqeverywhere byP_q⁺, resp.P_q⁻, P_q⁰. Since these three probability measures are the conditioned versions ofPqon events that partitionM, if the limit in distribution is the same in the three cases, then in particular the convergence in distribution will also hold for Pq. Of course, proving the result forP_q⁻is the same as forP_q⁺by the natural corespondence that exists between the two probabilities:

for n ≥ 1,P_q⁻(·|#V(M) = n)is the push-forward ofP_q⁺(·|#V(M) = n) by the map that reverses the orientation of the root edge. As we will see in Section 3.1, the study ofP_q⁺ andP_q⁰ involves two related bijections with certain sets of labeled trees. The one forP_q⁰is trickier, but somehow without more conceptual difficulties, so we will be more sketchy on this side and essentially focus on proving Theorem 1 forP_q⁺instead ofP_q.

Remarks. • In principle, the constant Cq can be computed explicitely (see Section 3.4 below), but it seems that, unlike in Marckert and Miermont (2006), there is no simple closed form for it in terms of Z_q⁺, Z_q³, f_q^•, f_q³and their derivatives (see also the second remark in Section 1.3).

•It should be noted that whenqk = 0for every oddk, which we excluded from our study, the formula (7) degenerates to the criticality condition(Z_q⁺)²f_q⁰(Z_q⁺) = 1, wherefq(x) = f_q^•(x,0). This is the formula given in Marckert and Miermont (2006), and amounts to say thatZ_q³degenerates to0in the case where Wqis supported by bipartite maps. Thus, the statements of Proposition 1 and Theorem 1 remain true in this case (still assumingqk >0for at least onek >3), provided the entries in the second row ofMqare all0.

• Notice that we have given a statement with a conditioning on the number of vertices of the random map underPq, while Marckert and Miermont (2006) gave results also for a conditioning with respect to the number of faces. Although we expect the result to be true (with different scaling constants) when conditioning on the number of faces, we do not have a proof for this statement. Indeed, in the bijection between maps and multitype trees that is introduced in Section 3.1, the faces of planar maps are matched with vertices of the trees of two distinct types, while vertices of the map are matched to vertices of the trees of a single third type. One the other hand, our main tool on multitype trees (Theorem 2), involves only conditionings with respect to the number of vertices of some fixed type.

However, in the particularly interesting models of Boltzmann-distributed randomk-angulations for which all faces are of degreek≥3, conditioning on the number of vertices or faces amounts to the same, as we will now see in the particular casek= 3.

1.3 The example of triangulations

A triangulation is a planar map whose faces all have degree3. We do not assume extra connectivity prop- erties on the planar map such as2or3-connectedness, so that it may contain loops and/or multiple edges (such maps are calledtype-Itriangulations in Ambjørn et al. (1997)). As observed in the previous remark, conditioning a random triangulation with respect to the number of its vertices or of its faces amounts to the same, because of the linear relation#F(m) = 2#V(m)−4satisfied by any triangulationm 6=†. This is clear from Euler’s formula#V(m)−#E(m) + #F(m) = 2, and the formula3#F(m) = 2#E(m) which comes from the fact that all faces are triangles.

Letq3=q >0andqi= 0fori∈N\ {3}. In that case,Wqputs a weightqⁿon all triangulations ofM withntriangles. In particular, by the previous paragraph, the probability distributionPq(·|#V(M) =n) as it appears in Theorem 1, is theuniformdistribution on the set of triangulations withnvertices (2n−4 triangles).

Let us look for the valueqcrofqthat makesqcritical (in fact, regular critical). The functionsf_q^•, f_q³are given by

f_q^•(x, y) = 2qy , f_q³(x, y) = 2qx+qy². It is easy to see that (4) amounts to2qZ_q³= 1−q

1−8q²Zq⁺= 1−(Z_q⁺)⁻¹, which imposes8q²Z_q⁺≤1.

The criticality condition (7) then yields−4(qcr)²(Z_q⁺)²= 2qcrZ_q³−1 =−(Z_q⁺)⁻¹, so that4(qcr)²(Z_q⁺)³= 1. From this, we find

Z_q⁺=√

3 and qcr= 1 2·3^3/4.

We can compute the scaling constantCqappearing in the statement of Theorem 1 as

C_qcrδ3 = 1

3 1/4

= 0.75983. . . . (8)

This will be shown in Section 3.4.2. Note that this constant is strictly less than the scaling constant(8/9)^1/4 which appears for quadrangulations.

Remarks. • Triangulations that are two or three-connected can be studied via other kinds of bijective methods, see Poulalhon and Schaeffer (2003); Poulalhon (2002) for example. It seems that it would be harder to obtain limiting laws e.g. for the radius of such constrained triangulations with the help of labeled trees as in the present paper, as it would involve intricate conditionings, see Remark3.in Section 4.

•It would be natural to try and compute the critical weightqcr, the partition functionZqcrδ_kand the scaling constantC_qcrδk in the case of Boltzmann laws onk-angulations, whereq_i = qδ_ikfor some odd number k ≥ 5. For evenk, the scaling constant was shown to be(k(k−2)/9)^1/4 in Marckert and Miermont (2006), and the formula is still valid for triangulations. A natural conjecture is that this formula is valid for allk. However, by contrast with the bipartite case, the complexity of computations (at least with our

method) grows withk. It is already a very tedious exercise to solve (4) and (7) and find the scaling constant Cq in the casek = 3, and the computations become yet much harder fork = 5. Since the equations to be solved involve polynomials whose degrees increase with k, it might even be that there is no explicit expression forqcr, Zq_crδ_kinvolving only radicals and rational fractions ofk.

2 Multitype spatial Galton-Watson trees

2.1 Multitype trees

LetU be the set of finite words with alphabetN, U = G

n≥0

Nⁿ,

whereNⁿ is the set of words withninteger letters, and by conventionN⁰ = {∅}. For two wordsu, v, we letuvbe their concatenation and|u|,|v|their length (with the convention|∅|= 0). Ifuis a word and A⊆ U, we also letuA={uv:v∈A}, and say thatuis a prefix ofvifv∈uU. Aplanar treeis a finite subsettofU such that

• ∅∈t, it is called therootoft,

• for everyu∈ U andi∈N, ifui∈tthenu∈t, anduj∈tfor every1≤j≤i.

We letT be the set of planar trees. For a treet∈ T andu∈t, the numberc^t(u) = max{i∈Z+:ui∈t}, with the conventionu0 =u, is the number of children ofu. An elementu∈tis called avertexoft, and the length|u|of the worduis called theheightofuint. Any planar treetis endowed with the linear order which is the restriction ontof the usual lexicographical order≺onU (u≺vifuis a proper prefix ofvor ifu=wu⁰, v=wv⁰for somew, u⁰, v⁰∈ U, where the first lettersu⁰₁, v₁⁰ ofu⁰, v⁰satisfyu⁰₁< v₁⁰).

LetK ∈Nand[K] ={1,2, . . . , K}. AK-type treeis a pair(t, et)whereet :t→[K]assigns a type to each vertex. We letT^(K)be the set ofK-type trees. AK-type spatial treeis a triple(t, e,`)where (t, e)∈ T<sup>(K)</sup>and`= (`u :u∈t)∈R<sup>t</sup>. We letT<sup>(K)</sup>be the set ofK-type spatial trees. Foru∈t,`u

can be interpreted as the position ofuin a1-dimensional embedding oft. Notice that a spatial motion`on some treetcan alternatively be described by the label incrementsy= (yu, u ∈t), wherey<sub>∅</sub> =`_∅, and foru∈tand1≤i≤c^t(u),yui=`ui−`u. The function`is recovered by

`_u=

n

X

k=1

y_u1...uk, u=u₁. . . u_n∈t.

This notationywill be used frequently in this paper.

In the sequel, we will often write e instead ofe_t and denote elements of T^(K),T^(K) by t, without explicitly mentioning the type and spatial embedding, when this is free of ambiguity. We let, fori∈[K],

T_i^(K)={t∈ T^(K):e(∅) =i},

andT^(K)i ={t∈T^(K):e(∅) = i}. Also, fort∈ T^(K)andi∈[K]we will denote byt⁽ⁱ⁾=e⁻¹({i}) the set of vertices oftwith typei.

Finally, letW_K = F

n≥0[K]ⁿ be the set of finite[K]-valued sequences, and consider the natural pro- jectionp: WK → Z^K+, wherep(w) = (pi(w), i ∈ [K])andpi(w) = #{j : wj = i}. We then have

|w|=|p(w)|1where|w|is the length ofwand| · |1is thel1-norm. For eachz∈Z^K+, there are exactly

#{w∈ WK :p(w) =z}=

|z|₁ z1, . . . , zK

= (z₁+. . .+z_K)!

z1!. . . zK! ,

preimages ofzby the mappingp. Fort∈ T^(K), everyu∈tdetermines a sequence wt(u) = (w^t_i(u),1≤i≤c^t(u)) = (e(ui),1≤i≤c^t(u))∈ WK

with length|w^t(u)|=c^t(u). The vectorz^t(u) =p(w^t(u)) = (z_i^t(u),1≤i≤K)counts the number of children ofuof each type.

2.2 Multitype spatial Galton-Watson trees 2.2.1 Galton-Watson trees

Let nowζ = (ζ⁽ⁱ⁾, i ∈ [K])be a family of probability measures on the setWK, and letµ = (µ⁽ⁱ⁾ = p_∗ζ⁽ⁱ⁾, i ∈ [K])be their push-forward byp, so that they define laws onZ^K+. We callζan orderedoff- spring distribution andµthe associated unordered offspring distribution. Notice that we might always writeζ⁽ⁱ⁾(w) =µ⁽ⁱ⁾(p(w))ord⁽ⁱ⁾_p(w)(w), where for everyi∈[K],z∈Z^K+,ord⁽ⁱ⁾_z (·)is a probability mea- sure onp⁻¹(z), which can be interpreted as a way of randomly ordering a multiset made ofzj copies of the elementjfor1≤j ≤K. Whenord⁽ⁱ⁾_z (·)is the uniform distribution onp⁻¹(z), we say thatζ⁽ⁱ⁾is the uniform orderingofµ⁽ⁱ⁾.

We make the basic assumption that max

i∈[K]µ⁽ⁱ⁾











z∈Z^K+ :X

j

z_j 6= 1









>0,

and say thatζ, orµ, is non-degenerate. Fori, j∈[K], let mij = X

z∈Z^K₊

zjµ⁽ⁱ⁾({z})

be the mean number of type-joffspring of an type-iindividual, and letMµ= (mij)_i,j∈[K]and call it the mean matrix ofµ. It is said to beirreducible, if for everyi, j∈[K], there is somen∈Nso thatm⁽ⁿ⁾_ij >0, wherem⁽ⁿ⁾_ij is theij-entry of the matrixMⁿ_µ. We say thatµis irreducible ifMµis.

Under the irreducibility assumption, the Perron-Frobenius theorem ensures thatMµhas a real, positive eigenvalue%with maximal modulus. The distributionζ(orµ), is called sub-critical if% <1and critical if

%= 1.

Assumingζnon-degenerate, irreducible and (sub-)critical, for everyi∈[K], one can construct a distri- butionP_ζ⁽ⁱ⁾(orP⁽ⁱ⁾when free of ambiguity) onT_i^(K)such that

• the root∅has typeia.s.,

• different vertices have independent sets of children, such that

• type-j vertices have a set of children with types given by a sequence w ∈ WK with probability ζ^(j)(w).

Details of the construction of this distribution, are given in Miermont (2006). It is the law of aK-type Galton-Watson (GW) tree with ancestor of typeiand offspring distributionsζ. The probability measure P⁽ⁱ⁾is entirely characterized by the formulae

P⁽ⁱ⁾((T, eT) = (t, e)) =Y

u∈t

ζ^(e(u))(wt(u)),

where(T, e_T) :T^(K)→ T^(K)is the identity mapping andtranges overK-type trees.

2.2.2 Spatial motion

Next, we couple the trees with a branching spatial motion, as follows. Consider a familyν = (νi,w, i ∈ [K],w∈ WK), whereνi,w(dy)is a probability distribution onR^|w|.

For(t, e)∈ T^(K)and everyu∈twith typee(u) =iand children vectorw^t(u) =w, we take a random variable(eYuj,1≤j≤ |w|)with lawνi,w, independently over distinctu’s. We use these as increments of a spatial motion ont, that is, we setL(e ∅) = 0and let

Leu=

n

X

k=0

Yeu₁...u_k u=u1. . . un∈t.

Letνt,e be the law of the random vector(eLu, u ∈ t)thus obtained. We letP⁽ⁱ⁾ζ,ν (or simplyP⁽ⁱ⁾) be the probability measuredP_ζ⁽ⁱ⁾(t, e)dνt,e(`)defined on T^(K)i . Such probability measures are called laws of spatial Galton-Watson (SGW) trees. They are characterized by the formulae

P⁽ⁱ⁾ζ,ν((T, eT) = (t, e), L∈d`) =P_ζ⁽ⁱ⁾({t, e})Y

u∈t

ν_e(u),wt(u)(d(yuj,1≤j≤c^t(u))),

where(t, e)∈ T^(K),(T, eT, L) :T^(K)→T^(K)is the identity mapping. It should be kept in mind that the distributionsνi,wfor which|w|= 0orζ⁽ⁱ⁾(w) = 0are irrelevant in the definition ofP⁽ⁱ⁾ζ,ν. We make the basic assumption there existsi∈[K],w∈ WK\ {∅}such thatζ⁽ⁱ⁾(w)>0andνi,wis not a Dirac mass, so that there is indeed some randomness in the spatial displacements, and say that the displacement lawsν are non-degenerate in this case.

2.3 An invariance principle for SGW trees

Letζbe an ordered offspring distribution, andµ=p_∗ζ. For1≤i, j, k≤K, define Q⁽ⁱ⁾_jk = X

z∈Z^K+

z_jz_kµ⁽ⁱ⁾(z), j6=k ,

and

Q⁽ⁱ⁾_jj = X

z∈Z^K+

z_j(z_j−1)µ⁽ⁱ⁾(z).

We say thatµ(orζ) hasfinite varianceifQ⁽ⁱ⁾_jk <∞for alli, j, k∈[K]. Under this assumption, for eachi, (Qⁱ_jk,1≤j, k≤K)is the matrix of a non-negative quadratic form onR^K, which we callQ⁽ⁱ⁾(x),x∈R^K. Indeed, it can be interpreted as the Hessian matrix of the generating function ofµ⁽ⁱ⁾evaluated at(1, . . . ,1), as in Vatutin and Dyakonova (2001). We say thatζ, orµ, hassmall exponential momentsif for someε >0,

max

i∈[K]ζ⁽ⁱ⁾(exp(ε|w|)) = max

i∈[K]µ⁽ⁱ⁾(exp(ε|z|1))<∞. This is obviously stronger than having finite variance.

AssumingMµirreducible andµnon-degenerate, critical and with finite variance, we leta,bbe the left and right eigenvectors ofM with eigenvalue1, chosen so that|a|1= 1anda·b= 1. Let

σ= v u u t

K

X

i=1

a_iQ⁽ⁱ⁾(b) =p

a·Q(b), (9)

whereQ(x)is theK-dimensional vector(Q⁽ⁱ⁾(x),1≤i≤K). This should be interpreted as the ‘variance’

of the offspring distribution of the two-type process, as it plays a rˆole similar to the variance for monotype GW processes in the asymptotics of the survival probability, see Vatutin and Dyakonova (2001).

Fort ∈ T, we let∅ = u(0) ≺ u(1) ≺ . . . ≺ u(#t−1)be the ordered list of vertices oft, and let (H_n^t = |u(n)|, n ≥0)be theheight processoft, with the convention that|u(n)| = 0forn ≥#t. For t∈T^(K), let(S_n^t =`u(n), n≥0), with the convention that it equals0forn≥#t.

Theorem 2 (Miermont (2006), Theorems 2 and 4) Letζbe a critical non-degenerate ordered offspring distribution such thatµ = p_∗ζ has small exponential moments. Let(νi,w, i ∈ [K],w ∈ WK)be non- degenerate spatial displacement distributions that are all centered, and admit momentsmi,w=νi,w(|y|^8+ε₂ ), for someε >0such that

sup

i∈[K]

mi,w=O(|w|^D) (10)

for someD >0(here,|y|2is the Euclidean norm ofy). Write

Σ = v u u u t

X

i∈[K]

a_i X

w∈WK

ζ⁽ⁱ⁾(w)

|w|

X

j=1

b_wjν_i,w(y_j²) <∞. (11)

Then for everyi, j∈[K], the following convergence in distribution holds onD([0,1],R)²:





H_{[#T t]}^T n^1/2

!

0≤t≤1

, S_{[#T t]}^T n^1/4

!

0≤t≤1



 under P⁽ⁱ⁾(·|#T^(j)=n)

−→d n→∞



 2

σ√ aj

B_t^ex

0≤t≤1

, Σ s 2

σ√ aj

R^ex_t

!

0≤t≤1



,

whereB^ex, R^exis defined in Section 1.2.

Moreover, if

Λ^t_i(k) = #{0≤j ≤k:e(u(j)) =i}, 0≤k <#t,

and with the conventionΛ^t_i(#t) = Λ^t_i(#t−1), then(Λ^T_i([#T t])/#T,0≤t≤1)underP⁽ⁱ⁾(·|#T^(j)= n)converges in probability to(ajt,0≤t≤1)for the uniform norm.

Remark. The last statement shows that the relative frequencies of vertices of different kinds in large conditioned SGW trees are asymptotically deterministic. This gives a reason to believe that Theorem 2 remains true when conditioning on events like{#T⁽ⁱ⁾+T^(j) = n}for some i 6= j ∈ [K] instead of conditioning on only one type, though the proof might be technically involved. Recalling the third remark in Section 1.2, this would imply the statement of Theorem 1 to hold also when conditioning on the number of faces of the planar maps.

2.4 Shuffling

LetS_ndenote the set of bijections[n]→[n], and take some arbitrary convention forS₀. Lett∈ T^(K)be aK-type spatial tree, and suppose we are given a familyπ = (π_u, u ∈t)such thatπ_u ∈S_ct(u), u∈t.

We can define a treeπ(t)that shuffles the children ofuwith the help ofπ_u, by letting π(t) ={π(u) =π_∅(u₁)π_u1(u₂). . . π_u1...un−1(u_n) :u=u₁. . . u_n ∈t},

which is naturally marked by the type functionπ(e)(π(u)) = e(u)and the spatial embeddingπ(`) = (π(`)_v =`_π−1(v), v∈π(t))(the formalism is taken after Weill (2005)).

In general it holds that randomly shuffledK-type SGW trees remain SGW trees, as long as, informally, the children of different verticesu∈tare shuffled independently, in a way that depends only onw^t(u). For our purposes, we will need the following particular version of this property. Ifx= (x1, . . . , xn)is a finite sequence, we let←−x = (x_n−i+1,1≤i≤n)be the reversed sequence. Ifλis a measure onRⁿ, we naturally let←−

λ be the push-forward ofλby the mapping←−· :Rⁿ →Rⁿ. Fort∈ T andu∈twithc^t(u)≥2, take independent(Πu, u∈t)such thatΠuis either the identity element ofS_ct(u)or the involutive permutation (c^t(u)−i+ 1,1≤i≤c^t_u), with respective probabilities1/2and1/2(for convenience, we denote these two permutations byI, Jwhateverc^t(u)is, even if it is0). LetΦt,e,`(·)be the law of the treeΠ(t, e,`), so that each set of children of vertices is either kept as such, or taken in reversed order.

Lemma 1 Letµbe an offspring distribution andζits associated uniformly ordered offspring distribution, and letνbe spatial displacement distributions. Then,

Z

T^(K)

P⁽ⁱ⁾_ζ,ν(d(t, e,`))Φt,e,`(·) =P⁽ⁱ⁾_ζ,←→ν(·), (12) where forj∈[K],|w| ∈ W_K,

←→ν j,w(dy) =νj,w(dy) +←−ν_j,^←_w⁻(dy)

2 .

Proof. This is an easy consequence of the fact that , if(t, e)∈ T_i^(K)andA_v∈ B(R^c

t(v)), v∈t, then the set{(t, e,`) : (y_vi,1≤i≤c^t(v))_v∈t ∈Q

v∈tA_v}has a mass with respect to the measure of the left-hand side of (12), which is given by

X

(t⁰,e⁰,`⁰)∈T^(K)

Y

u∈t⁰

ζ^(e0(u))(w^t0(u))ν_e0(u),w^t0(u)(A_u) P

u∈t⁰,πu∈{I,J}1{(t⁰,e⁰,`<sup>0</sup>)=π(t,e,`)}

2^#t ,

where it should be understood that{I, J}is a multiset with two elements, even in the case wherec^t(u)∈ {0,1}(this is just for notational convenience). Sinceζis the uniform ordering ofµ,ζ⁽ⁱ⁾(w)depends only oniandp(w), so thatζ^(e0(u))(w^t0(π(u))) =ζ^(e(u))(w^t(u))whenever(t⁰, e⁰,`<sup>0</sup>) =π(t, e,`), and we can rewrite this as

Y

u∈t

ζ^(e(u))(w^t(u))νe(u),w(Au) +ν_e(u),^←_w⁻(←− Au)

2 ,

hence the result. 2

3 Relation between random maps and SGW trees

We will now see how one can encode theP_q-distributed maps in terms of SGW trees. Until the end of this section, we willonlyconsider the case whereK= 3with the notations of the previous section.

3.1 The Bouttier-Di Francesco-Guitter bijection

The Bouttier-Di Francesco-Guitter (BDFG) bijection (Bouttier et al. (2004)) yields a way to code elements of M+ andM0 with the help of certain 3-type spatial trees, as we now explain. We consider the set T ⊂ T⁽³⁾of3-type trees, in which, for every(t, e)∈ T, andu∈t,

1. ife(u) = 1thenz^t(u) = (0,0, k)for somek≥0(so individuals of type1can only have children of type3),

2. ife(u)∈ {2,3}, thenz^t(u) = (k, k⁰,0)for somek, k⁰ ≥0(so individuals of type2or3can only have children of types1and2).

When drawing a picture, we represent individuals of type1,2,3respectively by vertices with the three shapes,3,•. See Figure 4 for an example. Naturally, we letTi=T ∩ T_i⁽³⁾fori∈ {1,2,3}.

Next, we letT⊂T⁽³⁾be the set of3-type spatial trees(t, e,`), with(t, e)∈ T,`_∅ = 0, and such that the following labeling constraints hold: for everyu∈t,

3. `u∈Z,

4. ife(u) = 1then`<sub>ui</sub> = `_u for all1 ≤i ≤c^t(u), so that vertices of type3 inherit the label of their fathers,

5. ife(u)∈ {2,3}, andc^t(u) =k, we let by conventionu0 =u=u k+ 1(hereu k+ 1means thatk+ 1 is considered as a letter). Then, writingxui=`ui−`_{u i−1},1≤i≤k+ 1, it holds that

(a) ife(u i−1)∈ {1,3}thenxui∈Z+∪ {−1}

(b) ife(u i−1) = 2thenxui∈Z+. Notice thatyui = P

1≤j≤ixuj for 1 ≤ i ≤ c^t(u). We adopt the natural notationTⁱ = T∩T⁽³⁾i for i∈ {1,2,3}.

Proposition 2 (BDFG bijections) I.There exists a bijectionΨ⁺fromM+ontoT1, that sends†on({∅}, e(∅) = 1, `<sub>∅</sub>= 0)and satisfies the following extra properties. Ifm∈ M+\ {†}and(t, e,`) = Ψ⁺(m),

(i) #F(m) = #t⁽²⁾+ #t⁽³⁾. Moreover, with everyu∈ t⁽²⁾ (resp.t⁽³⁾) such thatz^t(u) = (k, k⁰,0)is associated a unique face ofmthat has degree2k+k⁰+ 1(resp.2k+k⁰+ 2).

(ii) Verticesv ∈V(m)\ {r}at distanced >0from the base vertexrare in one-to-one correspondence with verticesu∈t⁽¹⁾with`u−min<sub>t</sub>(1)`+1 =d, wheremin_t(1)`:= min<sub>u∈t</sub>(1)`u(similar notations are adopted in the sequel). In particular,#V(m) = #t⁽¹⁾+ 1,

R(m) = max

t⁽¹⁾

`−min

t⁽¹⁾

`+ 1, (13)

and

I^m(k) = 1

#t⁽¹⁾+ 1

#

u∈t⁽¹⁾:`u−min

t⁽¹⁾

`+ 1 =k

+1{k=0}

, k≥0. (14)

II.There exists a bijectionΨ⁰fromM0onto the setT2×T2. Ifm∈ M0andΨ⁰(m) = ((t,`),(s,`⁰)), then similarly as above,

(i)’ faces ofmare in one-to-one correspondence with vertices of type2and3of the (disjoint) union oft ands, so that#F(m) = #t⁽²⁾+ #t⁽³⁾+ #s⁽²⁾+ #s⁽³⁾, in such a way that with everyu∈t⁽²⁾ (resp.t⁽³⁾) such thatz^t(u) = (k, k⁰,0)is associated a unique face ofmthat has degree2k+k⁰+ 1 (resp.2k+k⁰+ 2), and similarly foru∈s.

(ii)’ Verticesv∈V(m)\ {r}at distanced >0from the base vertex, if any, are in one-to-one correspon- dence with elementsuof the disjoint union oft⁽¹⁾ands⁽¹⁾, with`u−min<sub>t</sub>(1)`∧min_s(1)`<sup>0</sup>+ 1 =d or`⁰_u−min_t(1)`∧min<sub>s</sub>(1)`⁰+ 1 =daccordingly. In particular,#V(m) = #t⁽¹⁾+ #s⁽¹⁾+ 1,

R(m) = max

max

t⁽¹⁾

`−min

t⁽¹⁾

`∧min

s⁽¹⁾

`⁰+ 1,max

s⁽¹⁾

`⁰−min

t⁽¹⁾

`∧min

s⁽¹⁾

`⁰_u+ 1

, (15) and

I^m(k) = 1

#t⁽¹⁾+ #s⁽¹⁾+ 1

#

u∈t⁽¹⁾ :`_u−min

t⁽¹⁾

`∧min

s⁽¹⁾

`⁰+ 1 =k

(16) +#

u∈s⁽¹⁾:`⁰_u−min

t⁽¹⁾

`∧min

s⁽¹⁾

`⁰+ 1 =k

+1{k=0}

, k≥0.

Proof. This statement might look quite different from that of (Bouttier et al., 2004, Sect. 4.2), but is in fact equivalent. Indeed, the original bijection from Bouttier et al. (2004) holds between maps that are solelypointed, and not rooted, and with decoratedunrootedplanar trees (called mobiles) with three types of vertices: labelled verticesand3, and unlabelled vertices◦, such that◦vertices are adjacent to both other kinds of vertices, while3vertices have exactly two neighbors, both of type◦. The labels are all integers greater than or equal to1, and have to check the following constraint. Letv1, . . . , vkbe the vertices that are displayed around a vertex of type◦, arranged in anti-trigonometric order, and letl₁, . . . , l_k be their labels.

Then the label differencesl_{i+1 mod k}−l_i,1≤i≤kobviously have sum0, and belong toZ+∪ {−1}ifv_i is of type, and toZ+ifv_iis of type3. This constraint is illustrated in Figure 2.

∈ {−1, 0, 1, 2, . . .}

l

l

l

l

l

l

l

. . .

∈ {0, 1, 2, . . .}

Fig. 2:Labeling constraint around◦vertices of mobiles. In this configuration, we obtainl2froml1(resp.l4froml3) by adding an integer inZ+∪ {1}(resp.Z+)

It is explained at the end of (Bouttier et al., 2004, Sect. 4.2) how a further rooting of the map allows to turn the ‘mobile’ into a rooted object. If the root~esatisfiesdm(e₋, r) + 1 =dm(e+, r), i.e. if the map is an element ofM+, then the mobile can be rooted at a vertex of type. Moreover, one lifts the positivity constraint on labels by subtracting the label of the root to all other labels. This gives in particular the property (ii) of correspondence betweenvertices andV(m). To obtain our exact formulation, it suffices to do the following modification to the rooted tree thus obtained.

First, for every type-◦vertex with father of type 3, we modify the tree by erasing the ◦vertex and connecting its children directly to its3-type father, in the same planar order. Thus, the only◦vertices that remain are the children ofvertices.

Next, all remaining◦vertices earn the label of their-type father.

Last, we do a cosmetic arrangement by changing◦vertices into •s. The labeled trees thus obtained are exactly those ofT1, as the reader will check (note that the label constraints explained above precisely amount to properties 5. (a) and (b) of the definition ofT). Moreover, in the original mobile,◦vertices with

2

0

1

1

1

1 2 3

1

2 2

2 1

1

1

2

Fig. 3:The ‘mobile’ associated with the map of Figure 1, which is represented with pale lines. The arrow points out from the root of the tree towards its first child. It still remains to lift the positivity constraint on labels by subtracting the label of the root, namely2, to all the others.

kneighbours of typeandk⁰ neighbours of type3correspond in a one-to-one way to faces ofmwith degree2k+k⁰. This amounts to property (i) of the statement.

The statement forM0is obtained in a similar way, noticing that when the root of the map connects two vertices at equal distance from the base vertex, then one has to consider a pair of ‘half-mobiles’ which are both naturally rooted at a vertex of type ‘3’. One lifts the positivity constraint by subtracting the label of this vertex to all other labels, and do the same cosmetic changes as above. 2

3.2 Description of the push-forward measures Ψ

P

and Ψ

P

The key observation states that the image of Boltzmann-distributed maps under the BDFG bijections are essentially SGW trees. In order to prove this, we first need to count the number of possible labelings

` = (`u, u∈ t)of a given tree(t, e) ∈ T which are such that(t, e,`)∈T. To this end, since we know that`_∅ = 0, it suffices to find the number of possible label differences(yui =`ui−`u,1 ≤i≤ c^t(u)) for allu∈ t. Equivalently, it suffices to find the number of possible sequences(xui =`ui−`_{u i−1},1≤ i ≤c^t(u) + 1), with our usual conventionu0 =u=u c^t(u) + 1. Ifu∈t⁽¹⁾, we know that its children, all of type3, earn the label`_u, so there is only one possibility. Ifu∈t⁽³⁾has childrenw^t(u) =w∈ W₃, with say(z^t₁(u), z₂^t(u)) = (k, k⁰), then we know from property 5. (a) in the definition ofTthat the possible sequences(xui+1{w^t_i−1(u)=1},1 ≤i≤c^t(u) + 1), with the conventionw₀^t(u) = 1, are the elements of the set

{(n1, . . . , nk+k⁰+1)∈Z^k+k

0+1

+ :n1+. . .+nk+k⁰+1=k+ 1}. (17) Notice that by adding1to every term of a sequence in the set (17), we obtain a sequence ofk+k⁰+1positive integers with sum2k+k⁰+2. It is a classical exercise to check that there are exactly ^2k+k_k+1⁰⁺¹

=N•(k, k⁰) such sequences (recall (1)).

Similarly, ifu∈t⁽²⁾has childrenw^t(u) =w ∈ W3, with(z^t₁(u), z₂^t(u)) = (k, k⁰), then the possible sequences(x_ui+1{w^t_i−1(u)=1},1≤i≤c^t(u) + 1), where this timew^t₀(u) = 2, are the elements of the set

{(n1, . . . , n_k+k⁰₊₁)∈Z^k+k

0+1

+ :n₁+. . .+n_k+k⁰₊₁=k}, (18) which has N₃(k, k⁰) elements. The number of possible labellings ` so that (t, e,`) ∈ T for a given (t, e)∈ T is thus

Y

u∈t⁽²⁾

N₃(z^t₁(u), z₂^t(u)) Y

u∈t⁽³⁾

N_•(z₁^t(u), z₂^t(u)). (19)

0

0 0

0 0

0 0 −1

−1

−1

−1 −1 −1

−1 −1 1

0

0

0

Fig. 4:The spatial3-type tree obtained by the BDFG bijection from the mobile of Figure 3, after the cosmetic changes of the proof of Proposition 2. The above spatial tree is a typical element ofT¹.

Finally, for everyw∈ W3, letν⁰_1,wbe the Dirac mass at0∈R^|w|. Ifp(w) = (k, k⁰,0), we letν_2,w⁰ be the law onZ^|w|of(Pj

i=1Xj,1≤j ≤k+k⁰), where(Xj+1{wj−1=1},1≤j≤k+k⁰+ 1)is uniform on the set (18) with the conventionw₀ = 2. We also letν_3,w⁰ be the law of(Pj

i=1X_i,1 ≤j ≤k+k⁰), where(X_j +1{wj−1=1},1 ≤j ≤ k+k⁰+ 1)is uniform on the set (17), where this timew₀ = 1. We take an arbitrary convention forν_i,w⁰ fori∈ {2,3}andwsuch thatp3(w)6= 0. As easily follows from the above discussion,ν_i,w⁰ is the uniform law on the set of label differences(`uj−`u,1≤j≤c^t(u))that can be attributed to a vertexu∈twithe(u) =iandw^t(u) =w(where(t, e)∈ T), so that(t, e,`)∈T. Proposition 3 Let qbe an admissible weight sequence and recall that by definition Z_q⁺ = Wq(M+).

Then there exists a constantZ_q³such that(Z_q⁺, Z_q³)is the ‘good’ solution to (4). Define three probability distributions onZ³+by

µ⁽¹⁾(0,0, k) = 1 Zq⁺

1− 1

Zq⁺

k

, k≥0, so underµ⁽¹⁾,z7→z3has geometric law with parameter1−(Z_q⁺)⁻¹,

µ⁽²⁾(k, k⁰,0) = (Z_q⁺)^k(Z_q³)^k0N₃(k, k⁰) ^k+k_k ⁰

q1+2k+k⁰

f_q³(Zq⁺, Z_q³) , k, k⁰≥0, µ⁽³⁾(k, k⁰,0) = (Z_q⁺)^k(Z_q³)^k0N_•(k, k⁰) ^k+k_k ⁰

q2+2k+k⁰

f_q^•(Zq⁺, Z_q³) , k, k⁰ ≥0,

and letζ = (ζ⁽¹⁾, ζ⁽²⁾, ζ⁽³⁾)be their uniformly ordered versions onW₃. Then, the3-type GW treePµ⁽¹⁾is (sub)-critical,Ψ⁺_∗P_q⁺=P⁽¹⁾ζ,ν⁰, whereν⁰= (ν_i,w⁰ , i∈[3],w∈ W₃)was defined above.

Moreover, the weight sequenceqis critical if and only ifµis, and regular critical if and only ifµfurther has small exponential moments.

Last, the probability measureΨ⁰_∗P_q⁰can be described as follows: the image of aP_q⁰-distributed random map underΨ⁰is a pair of independent SGW trees with same distributionP⁽²⁾_ζ,ν0.

Proof. Letqbe a weight sequence, and recall the definition ofW_qin Section 1.1. By (i) in Proposition 2, ifm∈ M+,(t, e,`) = Ψ⁺(m)∈T1,

Wq({(Ψ⁺)⁻¹(t, e,`)}) = Y

u∈t⁽²⁾

q_1+2zt

1(u)+z^t₂(u)

Y

u∈t⁽³⁾

q_2+2zt

1(u)+z^t₂(u). (20)