We now fix an integer p ≥ 2 once and for all, and for n ≥ 1 we let (Tn, `n) be a uniform labeledp-mobile withnblack vertices, that is, a labeled mobile in which every vertex u ∈ V•(Tn) has degree p (i.e. p −1 children). We also let be a uniform random variable in {−1,1}, independent of the rest.
The results of the previous section show that (Mn, v∗) = Φ((Tn, `n), ) is a uniform random rooted and pointed 2p-angulation with n faces.
From there on, it is natural to try and generalize the results we have obtained in the previous chapters. As mentioned in the introduction of this
0
−1 −2 1
0
−1 −2 −1
−1
−2 0
v∗
Figure 8.2: The Bouttier-Di Francesco-Guitter bijection performed on the labeled mobile of Figure 8.1, with chosen so that the root edge of the map points towards the root of the mobile
chapter, a nice re-rooting argument of [56] shows that not so much is needed:
in fact, we need only generalize in an appropriate way certain results of Section 4.2.
Let us be more specific. A simple application of the Euler formula shows that the 2p-angulation Mn has pn edges and (p−1)n+ 2 vertices. In par-ticular, the mobile Tn has n black vertices and (p−1)n+ 1 white vertices, for a total of pnedges, or 2pn oriented edges. Thus, the contour exploration of the whitecorners has pn distinct terms, en0, en1, . . . , enpn−1, enpn =en0. We let uni = (eni)− for 0≤i≤pn, and let
Cn(i) = 1
2dTn(uni, un0), Ln(i) =`n(uni), 0≤i≤pn .
As usual, these two processes are extended to [0,2n] by linear interpolation between integer times. These are the natural analogs of the contour and label processes considered before, the division by 2 in the definition of the contour process being motivated by the fact that white vertices have even heights.
We have the following generalization of Theorem 3.4.1. For 0≤t≤1 set C(n)(t) = Theorem 8.3.1. We have the following convergence
(C(n), L(n)) −→(d)
n→∞ (e, Z). in distribution in C([0,1],R)2.
We are not going to prove this result, but content ourselves with explain-ing where the scalexplain-ing factors come from. The convergence of C(n) to e is in fact a particular case of a rather general scaling limit result for Bienaym´e-Galton-Watson (BGW) random trees. Let us explain how such trees enter the discussion. Letµ◦ be a geometric law onZ+ with parameter (p−1)/p, so that its mean is 1/(p−1), and its variance isp/(p−1)2. Consider a branching process starting from a single individual at generation 0, and such that
• each individual at even generations produces a random number of chil-dren at the next generation, with distributionµ◦
• each individual at odd generations producesp−1 children at the next generation.
Here, we assume that the offspring of the different individuals involved are all independent. This is a simple instance of a two-type branching process, in which the types of individuals alternate between generations. If we decide to skip the odd generations, we see that this process boils down to a single-type branching process in which the offspring distribution µ of each individual is the law of (p−1)G, where G has distribution µ◦. We see that µ has mean 1 and variance p, and therefore the branching process is critical and ends a.s. in finite time. This means that the genealogy of the branching process considered here is a.s. a finite tree T, in which the individuals are naturally partitioned as in mobiles, depending on their generations. The probability of a particular tree t is then
P(T =t) = Y
v∈V◦(t)
p−1 pkv(t)+1,
whenever t is a p-mobile, and 0 otherwise. Since every black vertex int is a child of a white vertex, and every white vertex except the root is a child of a black vertex, we see that
X
v∈V◦(t)
kv(t) = #V•(t), (p−1)#V•(t) = #V◦(t)−1,
so that this probability can be rewritten as P(T =t) = In particular, it depends on tonlyvia #V•(t). Therefore, conditioningT on the event {#V•(T) = n} produces a uniform random rooted p-mobile.
Due to this discussion, we can view Tn as a two-type BGW tree condi-tioned on its total number of vertices at odd heights being n, or equivalently, conditioned on havingn(p−1)+1 vertices at even heights (this last fact is due to the very particular form of the offspring distribution of black vertices). Let T◦ be the tree T in which odd generations are skipped, so that the vertices of T◦ are the elements ofV◦(T), andv is the parent ofuinT◦ if and only ifv is the grandparent of u inT. As mentioned above, T◦ is a usual, single-type BGW tree, with critical offspring distribution µ(the law of (p−1)G, where G is geometrically distributed with parameter (p−1)/p). Moreover, due to the discussion above, conditioning T on having n black vertices boils down
to conditioningT◦ on having n(p−1) + 1 vertices, and this has the same law as the tree Tn◦, which is the tree Tn in which odd generations are skipped.
Now, note that Cn is none other than the contour process associated with the treeTn◦. At this point, we can apply standard results for convergence of conditioned BGW trees [3], showing that
where the normalization factor (p−1)non the left-hand side is asymptotically equivalent to the total number of vertices in the treeTn◦, and the factor √p on the right-hand side is the standard deviation ofµ. This explains the first marginal convergence in Theorem 8.3.1.
It remains to explain where the scaling factor in the second convergence comes from. Note that if (Tn, `n) is a uniformly chosen random labeled p-mobile with n black vertices, then Tn is a uniformly chosen p-mobile with n black vertices, and conditionally on Tn, `n is a uniform mobile-admissible labeling of Tn. Hence, the situation is analogous to the one encountered in the previous chapters. Namely, foru∈V◦(Tn), we can write v since ¬v is a black vertex). Then, the different terms involved in this sum are independent. However, the main difference with the previous situation is that these are not identically distributed. Indeed, if v is the k-th child of its parent, with k ∈ {1,2, . . . , p−1}, then `n(v)−`n(¬¬v) has the law of
Deduce that for 1≤k ≤p−1, one has
Var (X1+. . .+Xk) = kVar (X1) +k(k−1)Cov (X1, X2)2k(p−k) p+ 1 . At this point, we see that the law of the random variable Yv described above depends strongly on therank ofv among thep−1 children of¬v. The intuition is that, in a typical branch ofTn, say the one going fromun0 tounbpntc for somet ∈(0,1), which contains abouthn=et
p4(p−1)n/pwhite vertices according to (8.2), the number of times we see a white vertex being the k-th child of its parent is of orderhn/(p−1), for everyk ∈ {1,2, . . . , p−1}. Hence, the sum (8.3) for u = unbpntc behaves as a sum of p−1 independent terms, each of which is a sum of hn/(p−1) identically distributed, centered random variables with respective variances 2k(p−k)/(p+ 1) for 1≤k ≤p−1. The central limit theorem implies that this sum should be of order
N
where N is a standard Gaussian random variable. We see that givenet, the random variable e1/2t N has the same marginal distribution as Zt, and this explains the rescaling of the second marginal in Theorem 8.3.1.
Of course, turning these considerations into a rigorous argument requires some technicalities that we omit here.