Rerooting multi-type branching trees: the infinite spine case

We prove local convergence results of rerooted conditioned multi-type Galton-Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine. We apply our results to prove quenched local convergence of conditioned Boltzmann planar maps, sharpening a local convergence theorem by Stephenson~(2018).


Introduction
The study of multi-type branching processes has received growing attention in recent literature, see Miermont (2008); Ispány and Pap (2014); Pénisson (2016); de Raphélis (2017); Abraham, Delmas and Guo (2018); Stephenson (2018); Vatutin and Wachtel (2018); Féray and Kortchemski (2018). The reducible case received particular attention in the line of research by by Drmota and Vatutin (1997); Ispány and Pap (2014); Vatutin (2015); Vatutin and Dyakonova (2015); Vatutin (2016Vatutin ( , 2017; Janson, Riordan and Warnke (2018). In the present work, we prove concentration inequalities for the number of extended fringe subtrees for conditioned multi-type Galton-Watson trees, see Lemma 2.1. This allows us to establish local convergence for conditioned multi-type Galton-Watson trees rerooted at a random location under general assumptions, see Theorems 2.3, 2.4, and 2.7. The limit objects are multi-type generalization of Aldous' invariant random sin-tree, and hence generalize similar objects for monotype trees, see Aldous (1991), Janson (2012), Stufler (2019b). Such trees consist of a root-vertex with an infinite line of ancestors, called the spine, from which further random trees branch off. Depending on the branching mechanism, certain types recur infinitely often along the spine, and others do not. We apply our results to the following four settings: 1. Sesqui-type trees, whose offspring distribution is critical and has finite variance, conditioned having a total number n of vertices, regardless of their type. See Theorem 3.5.
2. Reducible multi-type Galton-Watson trees conditioned on having n vertices. See Theorem 3.7. 3. Irreducible regular critical multi-type Galton-Watson trees conditioned on the event that a fixed linear combination of the sub-populations by type equals n. See Theorem 3.8. 4. Irreducible critical multi-type Galton-Watson trees conditioned on having a vector k(n) ∈ N d 0 as total population by types. See Theorem 3.10.
Multi-type Galton-Watson trees are related to numerous examples of random graphs and related structures, see Miermont (2006); Riordan and Warnke (2017); Stufler (2018bStufler ( , 2019. In the present work, we give an application to models of random weighted planar maps. We start by strengthening the annealed local convergence result for face-weighted regular critical planar maps by Stephenson (2018) to quenched convergence, see Theorem 4.1. The proof goes by showing quenched convergence of the mobiles encoding weighted planar maps to a random infinite mobile with a spine that grows backwards. The bijection by Bouttier, Di Francesco and Guitter (2004) and the mapping theorem transfer this to a quenched limit for face-weighted planar maps. Hence we use the proof strategy by Stephenson (2018) to transfer local convergence of mobiles to local convergence of maps, but instead of starting with a local convergence result for the vicinity of the root of the mobile, we use quenched convergence for the vicinity of a random vertex in the mobile.
As an application, we deduce quenched local convergence of the random planar map M t n with n edges and a positive weight t > 0 at vertices. That is, M t n assumes a map M with n edges with probability proportional to t v(M ) , with v(M ) denoting the number of vertices of M . See Theorem 4.2. The proof goes by passing to the dual map of a special instance of a random face-weighted map, and using the fact that its local limit is one-ended. The vertex weighted random planar map M t n is related to the study of uniform random planar graphs, see Giménez, Noy and Rué (2013); Chapuy, Fusy, Giménez and Noy (2010). We apply the local convergence of M t n in the subsequent paper Stufler (2019a) to deduce local convergence of the uniform random planar graph.
Notation. -We let N 0 = {0, 1, 2, . . .} denote the collection of non-negative integers. The law of a random variable X : Ω → S with values in some measurable space S is denoted by L(X). If Y : Ω → S ′ is a random variable with values in some measurable space S ′ , we let P(X | Y ) denote the conditional law of X given Y . All unspecified limits are taken as n → ∞. Convergence in probability and distribution are denoted by p −→ and d −→ . Almost sure convergence is denoted by a.s. −→ . We say an event holds with high probability if its probability tends to 1 as n becomes large. For any sequence a n > 0 we let o p (a n ) denote an unspecified random variable Z n such that Z n /a n p −→ 0. Likewise, O p (a n ) denotes a random variable Z n such that Z n /a n is stochastically bounded.

Rerooted multi-type Galton-Watson trees
Suppose we are given a countable non-empty set G whose elements we call types. A G-offspring distribution ξ consists of an independent family (ξ i ) i∈G of random vectors taking values in the coproduct N (G) 0 , that is, the family of all functions G → N 0 that are zero everywhere except of a finite number of types. We let (e i ) i∈G denote the standard generators of the coproduct with e i (j) = δ i,j for all i, j ∈ G. A G-type branching process starts with a root whose type is determined by an independent random variable α with values in G. Any vertex of type i ∈ G receives offspring according to an independent copy of ξ i , with the j-th coordinate corresponding to the number of offspring with type j. We let T denote the genealogical tree of a G-type branching process, and say T is a ξ-Galton-Watson tree. In any offspring set, the order of vertices with the same type is part of the tree T , but we do not care about the order between vertices of different types. For any multi-type tree T and any type i we let # i T ∈ N 0 ∪ {∞} denote the number of vertices of type i in T . Furthermore, #T := i∈G # i T denotes the total population.
We will always make the assumption that We may do so without loss of generality, since we may shrink the set G to exclude irrelevant types. Furthermore, we only consider offspring distributions ξ for which T is almost surely finite.
2.1. Fringe subtrees. -Let us fix a type κ ∈ G. We let T α denote a random multi-type tree defined similar to T (with the root type determined according to an α-distributed choice), only that non-root vertices of type κ receive no offspring. This way, T may be generated by starting with T α and inserting at each non-root vertex of type κ an independent copy of T conditioned on having root type κ. We let T κ be defined analogously to T α , but we start with a root having a fixed type κ. Moreover, we let T κ 1 , T κ 2 , . . . denote independent copies of T κ . Assumption (2.2) entails that It follows by the standard depth-first-search exploration that the multi-type Galton-Watson tree T corresponds to the sequence with L ≥ 0 the smallest non-negative integer for which In particular, (2.6) Let T be a finite G-type tree whose root has type κ. We may decompose T in the same way, so that it corresponds to a sequence of trees (T 1 , . . . , T k ). We let N T (·) denote a function that takes a G-type tree as input and returns the number of occurrences of T as a fringe subtree. We let ψ(·) denote a function that takes as input a finite ordered sequence of G-type trees and returns as output the number of occurrences of (T 1 , . . . , T k ) as a consecutive substring. Generating T from the sequence in (2.4), we may write Note that occurrences of (T 1 , . . . , T k ) may not overlap, since the sequence corresponds to a tree. Hence changing one coordinate of the input of the function ψ changes its value by at most 1. Conditionally on L, the coordinates of (T α , T κ 1 , . . . , T κ L ) are independent, hence McDiarmid's inequality yields Moreover, with the O(1) term having a deterministic absolute bound that depends only on T . Letting T (κ) denote a ξ-Galton-Watson tree started at a vertex with type κ, it holds that k s=1 P(T κ = T s ) = P(T (κ) = T ).
(2.10) Lemma 2.1. -We consider the conditioned multi-type Galton-Watson tree for some family of events E n satisfying P(E n ) > 0 for all n. Suppose that there is a deterministic sequence s n → ∞ satisfying for all ǫ > 0 exp(−ǫs n ) = o(P(E n )) and P(# κ T n ≥ s n ) → 1. (2.11) Then for any finite G-type tree T with root type κ Let f (T n , v κ n ) denote the fringe subtree in T n encountered at a uniformly selected type κ vertex v κ n of T n . Then Proof. -Combining Equations (2.6)-(2.10) and applying Assumption (2.11) yields with an O(1) term admitting a deterministic absolute bound that only depends on T . The upper bound in Inequality (2.14) tends to zero. Hence this verifies (2.12). We have verified (2.12) for arbitrary finite T , and Assumptions (2.1) and (2.2) ensure that T (κ) is almost surely finite. Hence the convergence of random probability measures in (2.13) follows.
Of course, periodicities may come into play, and it is sensible to also consider events E n for which P(E n ) > 0 only when n is part of some infinite subset of N 0 . Nothing changes in our arguments as long as we restrict n to that subset when taking limits.
Remark 2.2. -We may apply Lemma 2.1 if (P(E n )) n≥1 is heavy-tailed and there is a concentration constant c(κ) > 0 such that P(# κ T n / ∈ (1 ± ǫ)c(κ)n) tends to zero exponentially fast. Note that in this setting the bound used in the proof of Lemma 2.1 entails for some fixed 0 < γ(ǫ) < 1. Hence by the Borel-Cantelli criterium 2.2. Multi-type sin-trees with a fixed root type. -We are going to define a random infinite but locally finite multi-type tree having a spine that growth backwards. Note that # κ T (κ) is distributed like the population of a monotype Galton-Watson tree with branching mechanism # κ T κ − 1. Assumptions (2.1) and (2.2) entail that The construction requires us to make the assumption that the κ-branching mechanism is actually critical. That is, we assume in the following that This allows us to define the κ-biased versionT κ of T κ with distribution given by P(T κ = (T κ , u)) = P(T κ = T κ ) (2.21) for any pair (T κ , u) of a G-type tree T κ (with the root having type κ and all non-root vertices of type κ having no offspring) and a vertex u of T κ that is a non-root vertex of type κ. Note that Assumption (2.20) is really required in order for this to be a probability distribution.
We define the random infinite G-type treeT (κ) with a spine of vertices that growth backwards. We start with a vertex u 0 that is declared the start of the spine and becomes the root of an independent copy ofT (κ). The parent u 1 of u 0 becomes the root of an independent copy ofT κ . We then glue the marked vertex to u 0 , and all non-marked leaves of type κ become roots of independent copies of T (κ). We proceed in this way with a parent u 2 of u 1 and so on, yielding an infinite backwards growing spine u 0 , u 1 , . . .. That is, the treeT (κ) obtained in this way has a marked vertex u 0 with a countably infinite number of ancestors. This constitutes the multitype analogue of Aldous' invariant sin-tree constructed in Aldous (1991) for critical monotype Galton-Watson trees. Here the abbreviation sin stands for single infinite path.
Given an integer h ≥ 0, we let f κ, [h] (·, ·) denote a function that takes as input a G-type T 1 tree together with one of its vertices v 1 , and returns the fringe subtree of T 1 at the h-th ancestor of type κ of v 1 together with the location v 1 within it. That is, it produces a marked tree where the root has type κ and where the path from the root to the marked vertex contains precisely h + 1 vertices of type κ. If no such ancestor exists, the function returns (T 1 , v 1 ) together with the information that an overflow occurred. It is immediate that and, in general, f κ, [h] (T (κ)) follows the distribution of T (κ) biased on the number vertices with type κ whose joining path with the root contains precisely h+1 vertices that also have type κ. That is, if T is a G-type tree whose root has type κ and if u is a vertex of T of type κ such that the joining path with the root contains precisely h + 1 vertices of type κ, then P(f κ, [h] (T (κ)) = (T, u)) = P(T (κ) = T ). (2.23) We consider the collection T κ,• of all G-type trees T • having a marked vertex so that f κ, [h] (T • ) is finite for all h ≥ 0. We may endow T κ,• with a topology such that convergence (deterministic or in distribution) of a (deterministic or random) sequence (X n ) n≥1 in T κ,• is equivalent to convergence of the projections f κ,[h] (X n ), h ≥ 0. This is analogous to the monotype case studied in Stufler (2019b).
Let T denote a finite G-type tree and let v be a vertex of T , such that there are h+1 vertices of type κ on the path from v to the root of T . Given a G-type tree T 1 , we let N (T,v) (T 1 ) ∈ N 0 ∪ {∞} denote the number of vertices in T 1 with f κ,[h] (T 1 , u) = (T, v) (without any overflow). It is obvious that the functions N (T,v) (·) and N T (·) always return the same number, hence Let v κ n denote a uniformly selected type κ vertex of T n . Then That is, given a finite G-type tree T with root type κ and a type κ vertex v of T , with h + 1 denoting the number of vertices of type κ on the path from the root to v in T .
2.3. Non-recurring types along the spine. -Theorem 2.3 is a local convergence result for the vicinity of a uniformly selected vertex of type κ satisfying the criticality constraint (2.20). The limit treeT (κ) has the property, that its spine has an infinite number of vertices of type κ. We are going to prove a criterion that also encompasses types that occur only a stochastically bounded number of times along the spine of the limit.
Suppose that we are given a type γ ∈ G \ {κ} satisfying This allows us to define the size-biased versionT κ,γ given by for any finite G-type tree T (with root type κ, and all non-root vertices of type κ having no offspring) and any vertex u of T of type γ. We defineT (κ, γ) likeT (κ), only with a single local modification: instead of letting u 0 become the root of an independent copy of T (κ), we let it become the root of an independent copy ofT κ,γ (with the marked vertex becoming the marked vertex ofT (κ, γ)), and then make each of the type κ leaves of this structure the root of an independent copy of T (κ). Note thatT (κ, γ) may or may not have an infinite number of vertices of type γ on the spine, depending on whether T κ has with positive probability a vertex of type γ on the path from the root of T κ to some leaf of T κ with type κ.
The representation (2.4) entails that, since γ = κ, Hence if the events E n are reasonably well behaved, Assumption (2.27) and Equation (2.29) will ensure that # γ T concentrates around E[# γ T κ ]# κ T . This motivates the following local convergence result for the vicinity of a typical vertex with type γ: Then for any finite G-type tree T with root type κ and a type γ vertex v of T with height h ≥ 0 it holds that Proof. -Equation (2.30) readily follows from (2.12), (2.24), and the definition of T (κ, γ). Having Equation (2.30) and Assumption (2.31) at hand, (2.32) readily follows by Slutsky's theorem.
Proof. -Let (T, v) be an arbitrary pointed tree where T has root type κ and v has type γ and height Let Ω denote the collection of all such (T, v). Equations (2.33) and (2.30) imply that That is, for any ǫ > 0 it holds with high probability that Taking ǫ small enough, this yields a contradiction. This verifies (2.34) for all (T, v) ∈ Ω and completes the proof.
2.4. Mixtures of types. -Lemma 2.4 gives a criterium for local convergence describing the vicinity of random vertices with a fixed type. We aim to prove limits for the conditioned tree T n describing the local structure near a specified vertex that may have different types. Given an integer h ≥ 0, we let f [h] (·, ·) denote a function that takes as input a Gtype tree together with one of its vertices, and returns the fringe subtree at the h-th ancestor of that vertex together with the location of the vertex within it (yielding a marked tree where the root and the marked vertex have distance h from each other). If no such ancestor exists, the function returns the marked tree together with the information, that an overflow occurred.
Analogously as for T κ,• , we may consider the collection The following observation follows directly from Theorem 2.4 and Remark 2.5.
for an independent random type η from G 0 , with distribution given by We provide an application of Theorem 2.7 to critical sesqui-type trees in Section 3.2 and to critical irreducible Galton-Watson trees in Section 3.4 Note that using Theorem 2.3 we may still establish local convergence if the proportion of vertices of a certain type concentrates at different values as in Equation (2.37), but this works only for types that recur infinitely often on the spine in the limit.

Applications
We are going to illustrate the general results of the previous section by some examples.
3.1. Lattices and the Gnedenko local limit theorem. -Before we start, let us discuss a few relevant concepts. Given an integer s ≥ 1, a lattice in Z s is a subset of the form In particular, if we shift a lattice by the negative of any of its elements, we obtain a Z-linear submodule of Z s .
Given a non-empty subset Ω ⊂ Z s there is smallest lattice in Z s containing Ω: We may select a ∈ Ω and let Λ denote the Z-span of Ω − a. Any Z-submodule of Z s has rank at most s, hence there is at least one matrix A ∈ Z s×s with Λ = {Ax | x ∈ Z s }. Hence L a,A is a lattice containing Ω. Moreover, L a,A is a subset of any lattice containing Ω: If S = L b,B is another lattice containing Ω, then a ∈ S and consequently S = L a,B . This entails that the Z-module S − a must contain Ω − a, and therefore Λ is a submodule of S − a. Hence L a,A ⊂ S.
We say a random vector X with values in an abelian group F ≃ Z d is aperiodic, if the smallest subgroup F 0 of F that contains the support supp(X) satisfies F 0 = F . Note that this not really a restriction, since the structure theorem for finitely generated modules over a principal ideal ensures that We say X (and the associated random walk with step distribution X) is strongly aperiodic, if the smallest semi-group (a subset closed under addition that contains 0) F 1 of F that contains supp(X) satisfies F 1 = F . This is an actual restriction. For example, if X is 2-dimensional with support {(1, 0), (0, 1), (1, 2)}, then it is aperiodic, but not strongly aperiodic (although the support is not even contained on any straight line). Let X be a random vector in Z d . Let a + DZ d be the smallest lattice containing the support of X. Suppose that a ∈ supp(X) and that D has full rank, so that m := | det D| is a positive integer. Let (X i ) i≥1 denote independent copies of X and set Then: 1. The support of S m generates DZ d as additive group. 2. For all k ≥ 1 and 1 ≤ j ≤ m it holds that P(S km+j ∈ ja + DZ d ) = 1.
The following is a strengthened and generalized multi-dimensional version of Gnedenko's local limit theorem, that applies to the case of lattice distributed random variables that are aperiodic but not necessarily strongly aperiodic.

Proposition 3.2 (Strengthened local central limit theorem for lattice distributions)
Let X, m, a, D and S n be as in Proposition 3.1. Suppose that X has a finite covariance matrix Σ. Our assumptions imply that Σ is positive-definite. Let The one-dimensional local limit theorem by Gnedenko (1948) was generalized by Rvacheva (1954) to the lattice case of strongly aperiodic multi-dimensional random walk. This was further generalized by Williamson and Rinehart (1969, Prop. 2) to the setting considered here, where X is aperiodic but not necessarily strongly aperiodic. Furthermore, the version stated in Proposition 3.2 is strengthened by the factor R n (x) in Equation (3.2). This is stronger than the original statement when x deviates sufficiently from E[S n ]. Such a strengthening was obtained by Spitzer (1976, Statement P10, page 79) for the strongly aperiodic case by modifying the proof. This strengthened version in the strongly aperiodic setting may be generalized to Proposition 3.2 analogously as in Williamson and Rinehart (1969).
3.2. Sesqui-type trees. -We start with the simplest model of a non-monotype branching type process which already has interesting applications. Consider a 2-type branching tree T where only vertices of the first type are fertile and receive offspring according to a branching mechanism ξ 1 = (ξ, ζ). Of course we only consider the case where the root of T is fertile.
In order to avoid degenerate cases we assume that As the mono-type case is already well-understood, we additionally assume that the support If the support of ξ 1 is contained in a straight line we can reduce in every question about T to the study of a mono-type ξ-Galton-Watson tree. Hence this is not much of a restriction.
We consider the tree T n obtained by conditioning T on having n vertices in total. We would like to establish a limit describing the vicinity of a uniformly at random selected vertex v n . Due to possible periodicities, n may need to be restricted so some infinite subset of the positive integers in order for this to make sense: We may rewrite this as for power series (E k (z)) k≥0 with non-negative coefficients uniquely determined by Note that S k = ∅ whenever F k = ∅. Moreover, note that for all k ≥ 0 and b ∈ F k . We are going to argue that It is clear that D | S k for all k ≥ 0 by (3.10), so clearly D divides the right-hand side of Equation (3.13). Conversely, let r be a divisor of the right-hand side of this Equation. As 0 ∈ S 0 it follows that r | S k for all k ≥ 1. Moreover, we assumed that there exists k ≥ 2 with P(ξ = k) > 0, implying S k = ∅. As r | S k + S 0 and r | S k = ∅ it follows that r | S 0 . Hence r | S k for all k ≥ 0 and hence, by Equation (3.10), r | D. This completes the verification of Equation (3.13).
That is, mD may be expressed as a finite sum of terms of the form λt with t ∈ S 1 , λ ∈ N, and terms of the form µ(s+r) with s ∈ S k for some k ≥ 2, r ∈ S 0 , and µ ∈ N. We are going to argue that there is a tree T with a + Dm vertices that satisfies (3.14) To this end, note that We start the growth construction of T with a single root vertex that we declare as marked. We iterate over the finitely many terms in the sum expression of mD, and each step takes as input a tree with a single marked leaf and outputs a bigger tree with a single marked leaf: 1. If the summand is of the form λt with λ ∈ N and t ∈ S 1 , then the marked leaf receives t − 1 type 2 offspring vertices and a single type 1 offspring vertex that becomes the new marked leaf. Note that the outdegree (1, t − 1) lies in the support of ξ 1 by Equation (3.12). We do this precisely λ many times. Hence in total we added λt vertices. 2. If the summand is of the form µ(r + s) with µ ∈ N, r ∈ S k for some k ≥ 1 and s ∈ S 0 , we do the following. By Equation (3.11) there is b ∈ F k and c ∈ F 0 such that The marked leaf receives mixed offspring according to (k, b − 1). The first offspring of type 1 becomes the new marked leaf. The second offspring of type 1 receives offspring (0, c − 1). The remaining k − 2 offspring vertices of type 1 each receive offspring (0, a − 1). Note that by Equation (3.12) all non-marked vertices have an outdegree that lies in the support of ξ 1 . We perform this precisely µ many times. Hence in total we added µ(r + s) vertices.
After iterating over all summands we are left with a tree having 1 + mD vertices (remember that we also have to count the root vertex) that has a marked leaf. All non-marked vertices have an outdegree that lies in the support of ξ 1 . The marked vertex receives offspring (0, a − 1), resulting in a tree T with a + mD vertices that satisfies 3.14. As we may perform this construction for all m ≥ M this completes the proof.
Setting κ = 1, the tree T κ consists of a root of type 1 with offspring according to ξ 1 and no further descendants. Hence in order to apply Theorem 2.7 we need to assume that It holds trivially that #T ≥ # 1 T , and # 1 T is distributed like the total progeny of a ξ-Galton-Watson process. As we assume that E[ξ] = 1, it follows that # 1 T is heavy-tailed (see Janson (2012, Thm. 18.1)) and consequently: #T is heavy-tailed.
( 3.17) Hence the only prerequisite that we still need to check in order to apply Theorem 2.7 is that We verify Equation (3.18) in the case that (ξ, ζ) has a finite covariance matrix Σ.
The following is an extension of Stufler (2018b, Lem. 22), where a combinatorial setting was considered that is related to the case where one additionally assumes that a = 1 and that (ξ, ζ) has finite exponential moments.
1. There is a rank 2 matrix D ∈ Z 2×2 such that the support of (ξ, ζ) ⊺ is contained in the lattice (0, a − 1) ⊺ + DZ 2 and in no proper sublattice. It holds that 2. As n ∈ a + DZ tends to infinity, we have and There is a sequence (j n ) n with values in {1, . . . , d} with the following property. As n ∈ a + DZ tends to infinity, uniformly for all bounded x satisfying ℓ := µn + x √ n ∈ j n + dZ. In particular, and let denote independent copies of X. We also set Equations (2.5) and (2.6) tell us that for ℓ ≥ 1 with (ξ i , ζ i ) i≥1 denoting independent copies of (ξ, ζ). By the cycle lemma (see for example Takács (1962)), this simplifies to The support of ξ 1 was assumed to be not contained in a straight line, hence the same goes for (ξ − 1, ζ + 1). Hence the covariance matrix Σ (of both ξ 1 and the shifted version (ξ − 1, ζ + 1)) is positive definite. Furthermore, the support of (ξ − 1, ζ + 1) ⊺ is contained in the lattice a + DZ 2 , with a := (−1, a) ⊺ , and in no proper sublattice. Let n ∈ a + DZ be sufficiently large so that P(#T = n) > 0 by Lemma 3.3. By Equation (3.28) this means that for at least one 1 ≤ ℓ ≤ n it holds that y n lies in the support of S ℓ . By Proposition 3.1 it follows that there is at least one integer 1 ≤ j n ≤ m with y n ∈ j n a + DZ 2 . (3.30) Note that for all j, j ′ ∈ Z the following statements are equivalent: The set of all j ∈ Z with ja ∈ DZ 2 is a subgroup of the integers. By Proposition 3.1 it contains m. Hence it is generated by some integer d ≥ 1 satisfying d|m. (3.31) We will postpone showing that we assume that d may be chosen as in (3.19). Moreover, we have y n ∈ ja + DZ 2 if and only if j ∈ j n + dZ. (3.32) Hence, from now on we may assume additionally that 1 ≤ j n ≤ d. By Equation (3.28) it follows that any integer ℓ ≥ 1 with P(#T = n, # 1 T = ℓ) > 0 must lie in the lattice j n + dZ. It is clear that not the entire lattice has this property. However, we will argue that this is true for all ℓ ∈ j n + dZ that concentrate in a √ n range around µ n . Given ℓ ∈ j n + dZ with ℓ ≥ 1 we may write Let M > 0 be a fixed constant. It holds uniformly for all ℓ with |x ℓ | < M that It follows from the multivariate local limit theorem in Proposition 3.2 that Let ϕ 0,σ 2 (·) denote the density of the centered normal distribution with variance σ 2 . By Equation (3.28) it follows that uniformly for |x ℓ | < M . Setting (3.36) this entails that for any fixed a < b (3.37) We are going to argue that for each ǫ > 0 we may choose M > 0 so that for all sufficiently large n To this end, suppose that 1 ≤ ℓ ≤ n satisfies ℓ ∈ j n + dZ and x ℓ ≥ M . The multivariate local limit theorem in Proposition 3.2 entails that there is a constant C > 0 that does not depend on ℓ such that

REROOTING MULTI-TYPE BRANCHING TREES: THE INFINITE SPINE CASE 17
Hence If we restrict to summands with x ℓ ≥ n 1/4+δ for any fixed 0 < δ < 1/4, we may be bound this by If we restrict to summands with M ≤ x ℓ ≤ n 1/4+δ instead, then ℓ = Θ(n) and we obtain the bound Taking M large enough, this bound is smaller than ǫ/2. This verifies Equation (3.38).
Combining Equations (3.37) and (3.38) we obtain uniformly for ℓ ∈ j n + dZ with |x ℓ | ≤ M . By (3.37) we obtain It remains to show that the integer d (as defined in this proof) may be chosen as in (3.19). That is, we have to show that d = m/D. (3.46) Some jokes have been told about how mathematicians solve problems by reducing them to previously solved problems, even when it's not the most direct solution. The following short but not necessarily direct justification of (3.46) might fit into this category. For any sufficiently large K > 0 there is a random variable ξ instead of ξ 1 . We assume that K is sufficiently large so that the support of ξ is not contained on a straight line. Lemma 3.3 and everything we have shown so far applies to T (K) , yielding an analogon D (K) to D that satisfies D (K) |D, an analogon D (K) to D that satisfies D (K) Z 2 ⊂ DZ 2 , and an analogon d (K) to d that satisfies d (K) |d. The construction of these constants implies that for K sufficiently large equality holds, meaning we have d (K) = d, D (K) = D, and may assume that D = D (K) . As ξ (K) 1 is bounded and hence has finite exponential moments, we may apply a singularity analysis result by Bell, Burris and Yeats (2006, Thm. 28), yielding as n ∈ a + DZ tends to infinity. At the same time, Equation (3.43) applied to T (K) yields This completes the proof.
Having assured the concentration of the proportion of types in Equation (3.18), we readily obtain by Theorem 2.7: Theorem 3.5. -Under the same assumptions as in Lemma 3.4, it follows that as n ∈ a+DN tends to infinity. Here η denotes the independent Bernoulli-distributed random type from {1, 2} with distribution given by and . (3.50) See also Stufler (2018b, Thm. 27) for a similar result in the combinatorial setting of random unlabelled R-enriched trees where finite exponential moments were assumed.
3.3. Critical reducible Galton-Watson trees. -Suppose that the type set G = {1, . . . , K} is finite. Let ξ = (ξ 1 , . . . , ξ K ) denote a K-type offspring distribution with ξ i = (ξ i,1 , . . . , ξ i,K ) for all 1 ≤ i ≤ K. We assume that almost surely That is, a particle of type i may only have offspring with types j ≥ i. We furthermore assume that ξ := ξ 1,1 satisfies P(ξ = 0) > 0, P(ξ ≥ 2) > 0, and Recall that for each type i we let T (i) denote a ξ-Galton-Watson tree started with a single particle of type i. We let ζ be coupled with ξ 1 so that ζ is the total population of a forest F consisting of ξ 1,j independent copies of T (j) for all 2 ≤ i ≤ K. We let S denote a (ξ, ζ)-sesqui-type tree. For each vertex v with type 1 of S consider its ordered offspring (d 1 , d 2 ) and let β S (v) denote an independent copy of (F | (ξ, ζ) = (d 1 , d 2 )). This yields an enriched 2-type tree (S, β S ), that we call the canonical decoration of S. We may transform (S, β S ) into a K-type tree Ξ(S, β S ) as follows: For each type 1 vertex v of S delete each type 2 offspring vertex and add an edge between v and each root of the forest β S (v). Setting it is clear that T is distributed like a ξ-Galton-Watson tree started with a single particle of type 1. We let T n denote the results of conditioning T on having n vertices. This is possible for n ∈ a + DZ is large enough, with a and D defined for (ξ, ζ) as in Lemma 3.3. It is clear that properties for sesqui-type trees carry over, as # 1 S = # 1 T and #S = #T .
Lemma 3.6. -Suppose that additionally (ξ, ζ) has a finite covariance matrix and that its support is not contained in a straight line. Then Lemma 3.4 holds analogously for the K-type reducible ξ-Galton-Watson tree T , yielding an asymptotic expression for P(#T = n) and a local limit theorem for # 1 T n .

It follows from Theorem 3.5:
Theorem 3.7. -Let v n denote a uniformly selected vertex of T n . Under the same assumptions as in Lemma 3.6, it follows that as n ∈ a+DN tends to infinity. Here η denotes the independent Bernoulli-distributed random type from {1, 2} with distribution given by and . (3.54) 3.4. Regular critical irreducible Galton-Watson trees. -Suppose that the type set G = {1, . . . , d} is finite. Suppose that ξ = (ξ i ) 1≤i≤d and ξ i = (ξ i,1 , . . . , ξ i,d ) for all 1 ≤ i ≤ d. The mean matrix A := (E[ξ i,j ]) 1≤i,j≤d is said to be finite, if each coordinate is finite. We say ξ and the associated branching process is irreducible, if for any types i, j there is an integer k ≥ 1 such that the (i, j)th entry of A k is positive. The Perron-Frobenius theorem ensures that in this case the spectral radius λ of A is also an eigenvalue. If λ = 1 (or < 1 or > 1), we say ξ and the associated branching process is critical (or subcritical or supercritical). By Athreya and Ney (1972, Thm. 2 on page 186), a ξ-Galton-Watson tree T is almost surely finite (regardless of with which type we start) if its critical or subcritical. If ξ is critical and has finite exponential moments, we say it is regular critical.
Let γ = (γ i ) 1≤i≤d ∈ R d ≥0 \ {0} be fixed. Given a type 1 ≤ κ ≤ d we are interested in the tree T n obtained by conditioning the tree T started with a deterministic root type κ on the event that (3.55) It was shown by Stephenson (2018, Prop. 2.2) that there is an integer D ≥ 1 such that |T | γ is contained in some lattice of the form α κ + DZ, and conversely any sufficiently large integer from the lattice is contained in the support. Stephenson (2018, Sec. 4.3) showed furthermore that if ξ is regular critical, then P(|T | γ = n) ∼ c γ,κ n −3/2 (3.56) for some analytically given constant c γ,κ > 0 as n ∈ α κ + DZ tends to infinity. By Stephenson (2018, Prop. 2.1) (see also Miermont (2008)) it holds for any type and # γ T γ has finite exponential moments. Moreover, Stephenson (2018, Lem. 6.7) showed that for all 1 ≤ γ ≤ d for a constant c γ > 0 that does not depend on κ. For any subset G 0 ⊂ {1, . . . , d} let v n be a uniformly selected vertex of T n with type in G 0 and let η denote an independent random type that assumes a type γ with probability proportional to c γ . Equations (3.56) and (3.57) allow us to apply Theorem 2.3, yielding local convergence for the vicinity of a random vertex of type γ. Equation (3.58) entails consequently local convergence for the vicinity of v n to the corresponding mixture of limit objects: Theorem 3.8. -Suppose that ξ is regular critical. Then as n ∈ α κ + DZ tends to infinity. The limit object does not depend on the type κ with which we started the branching process. Each type recurs almost surely infinitely many often along the backwards growing spine of the limit.
Proposition 3.9. -If |G 0 | = 1 and if the vector γ has only one non-zero coordinate, then Theorem 3.8 still holds if we only require ξ to be critical.
Proof. -Without loss of generality we may assume that γ = (1, 0, . . . , 0). It holds that for all types i, j by Stephenson (2018, Prop. 2.1). If T starts with a vertex of type 1, then # 1 T is distributed like the population in a critical mono-type Galton-Watson tree with offspring distribution # 1 T 1 − 1. In particular, # 1 T is heavy-tailed. If T starts with a vertex of a different type, then # 1 T is distributed like the sum of a random number of independent copies of populations of (# 1 T 1 − 1)-branching processes. Also in this case, # 1 T is heavy-tailed.
For each i let v i n denote a uniformly selected vertex of type i of T n . Since # 1 T n = n and T n is obtained from T by conditioning on an event with heavytailed probability, we may apply Theorem 2.3 to obtain Let α denote the deterministic root type with which we started the branching process. Let γ = 1 be a type. It follows from Equations (2.5) and (2.6) that with (T 1 i ) i≥1 denoting independent copies of T 1 . Truncating the summands and using the Azuma-Hoeffding inequality (and again the fact that P(# 1 T = n) is heavy-tailed), it follows that for any integer D ≥ 1 and any ǫ > 0 it holds with high probability that As this holds for arbitrarily large D and as E[ This allows us to apply Theorem 2.3 for κ = γ to obtain Alternatively, we could also have applied Proposition 2.6 to obtain In the present setting (but not in general) it holds thatT (1, γ) d =T (γ), so there is no difference.
3.5. Critical Galton-Watson trees conditioned by typed populations. -Let the type set be given by G = {1, . . . , d} and suppose that the offspring distribution ξ is irreducible and critical. Let Λ ⊂ N be an infinite subset. Let for all n ∈ Λ. We assume that k(n) k(n) 1 → a (3.69) for some a = (a i ) 1≤i≤d ∈ R d ≥0 \ {0} as n ∈ Λ tends to infinity. Let α be a random type. We let T n denote the result of conditioning the ξ-Galton-Watson tree started with a particle of type α on the event (3.70) Let 1 ≤ κ ≤ d be a coordinate satisfying a κ > 0. By Stephenson (2018, Prop. 2.1) (see also Miermont (2008)) it holds It follows by Equations (2.5) and (2.6) that the event (3.70) implies that a random number of critical (# κ T κ − 1)-Galton-Watson trees has total size k κ (n). The total population of a critical mono-type Galton-Watson tree is heavy-tailed, yielding for any ǫ > 0 exp(−ǫk κ (n)) = o(P((# i T ) 1≤i≤d = k(n))).
( 3.72) This verifies Assumption (2.11). Let v κ n denote a uniformly selected vertex of type κ of the tree T n . It follows by Theorem 2.3 that Using (3.69), we deduce: Theorem 3.10. -Let G 0 ⊂ G be a subset such that a i > 0 for at least one type i ∈ G 0 . Let η be an independent random type from G 0 drawn with probability P(η = i) = a i / j∈G 0 a j . Let v n denote a vertex of T n that is uniformly selected among all vertices with type in G 0 . Then (3.74) See Abraham, Delmas and Guo (2018, Thm. 3.1) for a limit in a similar setting that describes the asymptotic vicinity of the root.

Random weighted planar maps
The bijection by Bouttier, Di Francesco and Guitter (2004) encodes planar maps as mobiles, which are vertex-labelled 4-type planar trees. This allows for a generating procedure for various models of random planar maps using 4-type Galton-Watson trees, see Miermont (2006). For bipartite Boltzmann planar maps, a bijection constructed by Janson and Stefánsson (2015) simplifies the generating procedure to use only monotype Galton-Watson trees. However, it is an open problem whether a full reduction to mono-type trees is possible in the non-bipartite case. (1) Hence the need to involve multi-type Galton-Watson trees for showing quenched local convergence for the models of random planar maps we are going to consider. We are going to recall relevant background and fix notation following closely the presentation by Stephenson (2018, Sec. 5).
(1) The author thanks SigurdurÖrn Stefánsson for related comments.

The Boltzmann distribution on planar maps. -
The collection of all finite planar maps with an oriented root edge and an additional marked vertex will be denoted by M. Throughout we let q = (q n ) n≥0 denote a family of non-negative numbers such that q n > 0 for at least one n ≥ 3. To any (corner-rooted) planar map M we assign a weight Here the index f ranges over the faces of the planar map M , and deg(f ) denotes the degree of the face f . That is, deg(f ) is the number of half-edges on the boundary of the face f . A weight-sequence q is said to be admissible, if In this case, we may form the Boltzmann distributed planar map M with distribution given by 4.2. Mobiles obtained from branching processes. -A pointed map from M is said to be positive, neutral, or negative, if the origin of the directed root edge is closer, equally far away, or farther away from the marked vertex than the destination of the root edge. We let M + , M 0 , and M − denote the corresponding subclasses of M, and form the sums Z + q , Z 0 q , and Z − q as in (4.2), but with the sum index constrained to the corresponding subclass. For all x, y ≥ 0 we define the bivariate series If the weight sequence q is admissible, we may define an irreducible 4-type offspring distribution ξ = (ξ) 1≤i≤4 as follows. Vertices of the first type produce a geometric number of vertices of the third type: Vertices of the second type always produce a single offspring vertex of the fourth type, that is P(ξ 2 = (0, 0, 0, 1)) = 1. (4.7) Vertices of the third and fourth type only produce offspring of the first or second type. Their coordinates ξ 3,1 , ξ 3,1 and ξ 4,1 , ξ 4,1 are determined by For a type κ = 1 or κ = 2 we consider the following sampling procedure. The result is a random 4-type tree where the offspring is ordered and each vertex v receives a label ℓ(v) with ℓ(v) ∈ Z if v has type 1 or 3 and ℓ(v) ∈ 1 2 + Z otherwise. 1. Consider the ξ Galton-Watson tree T (κ) that starts with a single vertex of type κ. This time we consider the offspring vertices as ordered in a uniformly selected manner (caring also about the order between vertices of different types). 2. For each vertex v 0 of type 3 or 4 in T (κ) with outdegree d ≥ 1 let v 1 , . . . , v d denote its ordered offspring and uniformly select a d + 1-dimensional vector β T (κ) (v 0 ) = (a 0 , . . . , a d ) with coordinates in the linear span 1 2 Z that satisfies the following conditions.
We emphasize that in the second step we choose for any vertex v of type 3 or 4 the vector β T (κ) (v) at random in a way that depends only on the ordered list of offspring vertices of v, their types, and the type of v. In combinatorial language, (T (κ), β T (κ) ) may be called a multi-type enriched plane tree. We refer to it as the canonical decoration of T (κ).
4.3. The Bouttier-Di Francesco-Guitter transformation. -We let T + denote an independent copy of (T (1), β T (1) ). We let T 0 denote the result of taking two independent copies of (T (2), β T (2) ) and identifying their roots. Let (T, β) be a possible finite outcome of T + or T 0 , and let (ℓ(v)) v∈T denote the corresponding labels. The Bouttier-Di Francesco-Guitter transformation associates a planar map Ψ(T, β) to the decorated tree (T, β) in such a way that the number of vertices of the map equals 1 + # 1 T , -the number of edges of the map equals # 1 T + # 3 T + # 4 T , -and the number of faces of the map equals # 3 T + # 4 T .
The transformation Ψ is as follows. We draw T in the plane and order the corners according to the standard contour process that starts and ends at the root vertex.
Let v 1 , . . . , v p denote the ordered list of vertices of type 1 or 2 that we visit in the contour process. That is, a vertex gets visited multiple types according to the number of angular sectors around it. We let ℓ 1 , . . . , ℓ p denote their labels. We extend these lists cyclically, so that v ip+k = v 1+k for i ≥ 1 and 0 ≤ k < p. We add an extra vertex r with type 1 outside of T and let its label ℓ(r) be one less than the minimum of labels of all type 1 vertices. For each 1 ≤ i ≤ p we draw an arc between the vertex v i and its successor. If v i has type 1 then the successor is the next corner in the cyclic list of type 1 with label ℓ i − 1. If there is no such corner, then we let r be the successor of v i . Likewise, if v i has type 2 then the successor of v i is the next corner of type 1 with label ℓ i − 1/2, or r if there is no such corner. It is possible to draw all arcs so that they only may intersect at end points. We now delete the original edges of the tree T , as well as all vertices of type 3 and 4. Vertices of type 2 get erased as well, merging the corresponding pairs of arcs. We are left with a planar map having a marked vertex r. If the root of T has type 1 we let the root edge be the first arc that was drawn and have it point to the root of T . If the root of T has type 2 (and hence has precisely two children, both of type 4), we let the root edge be the result of the merger of the two arcs incident to the root of T and let it point towards the successor of the first corner encountered in the contour process.
The Boltzmann distributed map M is a mixture of the random maps M + , M 0 , and M − obtained by conditioning M on belonging to M + , M 0 , and M − . As observed by Miermont (2006), it holds that Ψ(T + ) has a solution (x, y) with x > 1 such that the matrix has spectral radius smaller or equal to one. Any such solution (x, y) necessarily satisfies (x, y) = (Z + q , Z 0 q ). (4.12) Miermont (2006, Def. 1) termed an admissible weight sequence q critical, if the spectral radius of this matrix is equal to 1. This amounts to the condition for some ǫ > 0. As was made explicit by Stephenson (2018), this applies to various useful cases such as unrestricted maps or p-angulations for arbitrary p ≥ 3. The irreducible offspring distribution ξ is critical (or regular critical) if and only if the weight sequence q is critical (or regular critical).

Quenched local convergence. -
Theorem 4.1. -Suppose that q is regular critical. Let M n denote the q-Boltzmann planar map, conditioned on either having n vertices, or edges, or faces. Let u n denote either a uniformly selected vertex, half-edge, or face. There are integers a ≥ 0 and d ≥ 1 and a random infinite locally finite limit mapM with finite face degrees such that, in the local topology for vertex-rooted or half-edge rooted or face-rooted planar maps, the conditional law P((M n , u n ) | M n ) satisfies as n ∈ a + dZ tends to infinity.
Of course, the limit object differs depending on which conditioning we choose and which type of marking we select. The quenched limit (4.15) implies the annealed convergence (M n , u n ) d −→M (4.16) by dominated convergence. If u n denotes a uniformly selected half-edge, then (4.16) is the annealed convergence established by Stephenson (2018, Thm. 6.1) (see also Angel and Schramm (2003); Krikun (2005); Björnberg and Stefánsson (2014); Curien, Ménard and Miermont (2013); Ménard and Nolin (2014), who only required criticality in the case where M n is the Boltzmann map with n vertices. Drmota and Stufler (2018) described a general method for deducing limits for the vicinity of random vertices if a limit for the vicinity of a random corners is known. The method applies to regular critical Boltzmann planar maps and other settings. Obtaining an explicit description of the limit was left as an open question in Drmota and Stufler (2018), and the construction of the limit from an infinite mobile with a backwards growing spine the proof of Theorem 4.1 resolves this question in the present setting.
Note that, as was shown by Stephenson (2018, Sec. 6.3.5), in the present setting the total variational distance between M n (a corner-rooted map with an additional marked vertex, not to be confused with u n ) and a q-Boltzmann mapM n without a marked vertex tends to zero as n becomes large. Hence Theorem 4.1 also holds forM n .
It follows from Stephenson (2018, Lem. 6.5) that the labels of type 1 vertices along the backwards growing spine follow the distribution of a centered random walk in Z. The reason for this is that Stephenson (2018, Lem. 6.5) showed this for the labels if we walk forward along the spine of the local limit that describes the vicinity of the fixed root of T + n , and the difference in labels between two consecutive type 1 vertices on the spine of this limit is (like inT (η)) independent from the other differences and identically distributed as inT (η). In particular, the labels of type 1 vertices along the backwards growing spine ofT (η) have almost surely no lower bound.
We may order the corners (c i ) i∈Z incident to vertices of type 1 or 2 ofT (η) such that for all i c i+1 is the successor of c i in the clock-wise contour exploration. This allows us to canonically extend the Bouttier-Di Francesco-Guitter transformation from Section 4.3 to assign an infinite locally finite planar mapM to the infinite labelled tree (T (η), (ℓ(v)) v∈T (η) ). By construction, all faces ofM have finite degree.
Depending on whether u n is a random vertex, half-edge, or face of M n , we mark M as follows. Let w denote the marked vertex ofT (η) (which for obvious reasons is not a root-vertex). In the vertex case, w has type 1 and corresponds canonically to a vertex ofM. We considerM as rooted at this vertex. In the face case, w has type 3 or 4 and corresponds canonically to a face. In this case, we considerM as rooted at this face. In the half-edge case, w has type 1, 3, or 4 and corresponds canonically to an edge, which we orient according to an independent fair coin flip. (In detail: If w has type 4, then it is the only child of a non-root type 2 vertex that corresponds to the edge obtained by joining the arcs drawn at its two corners. If w has type 1, then each of its corners corresponds to the edge we drew when visiting this corner in the contour exploration. The number of these corners equals 1 plus the number of offspring vertices, all of which have type 3. Hence w and its children correspond bijectively to the arcs we drew starting at a corner of w. Hence if w has type 1 it corresponds canonically to an edge, and if w has type 3 it also corresponds canonically to an edge that we drew starting at a corner of its type 1 parent.) We may now proceed with Claim b). Suppose that κ = 1. The vertex v n of (T n (κ), β T n (κ) ) corresponds similarly to a marked vertex or face or half-edge u ′ n of M + n . Modifications in the correspondence may be required when v n or its parent is the root of T n (κ), but the probability for this event tends to zero and hence we may safely ignore this. Furthermore, (M + n , u n ) and (M + n , u ′ n ) may not follow the same distribution (for example, when u n is a uniform vertex, then u ′ n is a uniform non-marked vertex, as u ′ n is never equal to the additional vertex we added in the BDFG bijection). However, it is clear that there is an event, that depends on n, whose probability tends to 1 as n becomes large, such that (M + n , u n ) and (M + n , u ′ n ) are identically distributed when conditioned on this event. Hence we may also safely ignore the difference between u n and u ′ n . Using the continuous mapping theorem, it hence follows from (4.18) that It remains to verify Claim c). The same convergence follows immediately for M − n , since the vicinity of a random point is not affected by the orientation of the root edge. As for M 0 n , note that |T (2)| γ has a density that varies regularly with index −3/2 by (3.56). As −3/2 < −1, it follows that the density is subexponential, see the book by Foss, Korshunov and Zachary (2013) for background on this terminology. It follows that if we condition independent copies S (1) and S (2) of T (2) on the event |S (1) | γ + |S (2) | γ = m then lim m→∞ min((|S (1) | γ , |S (2) | γ ) | |S (1) | γ + |S (2) | γ = m) This may easily be verified elementarily or be viewed as a special case for results on general models of random partitions, see Stufler (2018a). Consequently, all but a negligible number of vertices whose extended fringe subtree has a certain shape will lie in a giant component with size (referring to | · | γ ) m − O p (1). If we let S denote the result identifying the roots of S (1) and S (2) and let w n denote a uniformly selected vertex of the conditioned tree S n with type in G 0 , then it follows by (4.17) that P((S n , w n ) | S n ) p −→ L(T (η)). (4.22) (Recall that above we assigned a clear to meaning to all occurrences of n as a subscript of a random tree, making S n a conditioned version of S depending on γ.) Hence, adding canonical decorations, P((S n , β Sn , w n ) | S n ) p −→ L(T (η), βT (η) ). (4.23) Thus quenched convergence of M 0 n towardsM may be deduced in exactly the same way using the mapping theorem as for M + n , only instead of using Equation (4.18) we use Equation (4.23). This completes the proof. 4.6. Random planar maps with vertex weights. -Let t > 0 be a constant. We let M t n denote a random planar map with n edges that assumes a map M (with n edges) with probability proportional to t v(M ) . Proof. -For any λ > 0 we may consider the weights q n = tλ n , n ≥ 1. (4.25) This way, any map with n edges and m faces receives weight λ 2n t m . We are going to argue below that for any t > 0 we may choose λ so that q = (q n ) n≥1 is regular critical. By elementary identities of power series (compare with Stephenson (2018, The limitM is one-ended and hence all its faces have finite degree. Consequently, M t := Φ(M) is locally finite. Using the continuous mapping theorem, it follows that P((M t n , c n ) | M n ) p −→ L(M t ). (4.36) This completes the proof.