Central limit theorem for biased random walk on multi-type Galton-Watson trees

Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability lambda/[lambda+d(v)], and to each of the offspring with probability 1/[lambda+d(v)]. This walk is recurrent for lambda>=rho and transient for 04 for the offspring distributions, we prove the following quenched CLT for lambda-biased random walk at the critical value lambda=rho: for almost every T, the process |X_{floor(nt)}|/sqrt{n} converges in law as n tends to infinity to a reflected Brownian motion rescaled by an explicit constant. This result was proved under some stronger assumptions by Peres-Zeitouni (2008) for single-type Galton-Watson trees. Following their approach, our proof is based on a new explicit description of a reversing measure for the walk from the point of view of the particle (generalizing the measure constructed in the single-type setting by Peres-Zeitouni), and the construction of appropriate harmonic coordinates. In carrying out this program we prove moment and conductance estimates for MGW trees, which may be of independent interest. In addition, we extend our construction of the reversing measure to a biased random walk with random environment (RWRE) on MGW trees, again at a critical value of the bias. We compare this result against a transience-recurrence criterion for the RWRE generalizing a result of Faraud (2011) for Galton-Watson trees.


Introduction
Let T denote an infinite tree with root o. The λ-biased random walk on T, hereafter denoted RW λ (T), is the Markov chain (X t ) t≥0 with X 0 = o such that given X t = v with offspring number d v and v = o, X t+1 equals the parent of v with probability λ/(λ + d v ), and is uniformly distributed among the offspring of v otherwise (and if X t = o, then X t+1 is uniformly distributed among the offspring of o).
Further, subject to no leaves and finite exponential moments for the offspring distribution, a quenched CLT for RW λ (λ ≤ ρ) on single-type Galton-Watson trees was shown by Peres-Zeitouni [31], and extended to the setting of random walk with random environment (RWRE) by Faraud [12]. In contrast, if leaves occur, there emerges a zero-speed transient regime λ < λ c (for λ c < ρ) [27] where the leaves "trap" the random walk and create slow-down. It follows from the results of Ben Arous et al. [2] that in this setting, for sufficiently small λ there cannot be a (functional) CLT with diffusive scaling. Analogous results on the critical (ρ = 1) Galton-Watson tree conditioned to survive were shown by Croydon et al. [8]. In this paper we consider the critical case λ = ρ, where [31,Thm. 1] proves that on a.e. Galton-Watson tree, the processes (|X nt |/ √ n) t≥0 converge in law to the absolute value of a (deterministically) scaled Brownian motion. Their proof is based on the construction of harmonic coordinates and an explicit description of a reversing probability measure IGWR for RW ρ "from the point of view of the particle." Having such an explicit description is a very delicate property: even for Galton-Watson trees, no such description is known for λ < ρ except at λ = 1 which is done by [26,Thm. 3.1]. One thus might be led to believe that [31, Thm. 1] is a particular property resulting from the independence inherent in the Galton-Watson law.
Here we show to the contrary that such a quenched CLT extends to the much larger family of supercritical multi-type Galton-Watson trees with finite type space. We allow for leaves (but condition on non-extinction), demonstrating that at λ = ρ the "trapping" phenomenon of [2] does not arise. We also replace the assumption of exponential moments for the offspring distribution by an assumption of finite moments of order p > 4, so that our result restricted to the single-type case strengthens [31,Thm. 1]. However, the main interest of our result lies in moving from an i.i.d. to a Markovian structure for the random tree.
As in [31], the key ingredient in our proof is the construction of an explicit reversing (probability) measure IMGWR for RW λ from the point of view of the particle, generalizing IGWR to the multi-type setting, for λ at the critical value on the boundary between transience and recurrence. See §2 for the details of the construction which may be of independent interest.
The model we consider is as follows: let Ω be the space of rooted trees with type, where each vertex v is given a type χ v from a finite alphabet Q. We let B Ω be the σ-algebra on Ω generated by the cylinder sets (determined by the restrictions of trees to finite neighborhoods of the root). We write T for a generic element of Ω and o for its root. A multi-type Galton-Watson tree is a random element T ∈ Ω, generated from a starting type χ o ∈ Q and a collection of probability measures q a (a ∈ Q) on Q ≡ ≥0 Q , as follows: begin with a root vertex o of type χ o . Supposing inductively that the first n levels of T have been constructed, each vertex v at the n-th level generates random offspring according to law q χv . For our purposes the ordering of the children does not matter, so each q a may equivalently be regarded as a probability measure on configurations x = (x b ) b∈Q ∈ (Z ≥0 ) Q , where x b is the number of children of type b. Continuing to construct successive generations in this Markovian fashion, we denote the resulting law on (Ω, B Ω ) by MGW χo . We denote by MGW any mixture of the measures (MGW a ) a∈Q (with (2.1) the canonical mixture) and let X ≡ {|T| < ∞} denote the event of extinction. For the expected number of offspring of type b at a vertex of type a. (Unless otherwise specified, the implicit assumption hereafter is that E q a [|x|] < ∞ for all a ∈ Q where |x| ≡ b x b .) Throughout the paper we will refer to the following assumptions: (H1) The matrix A ≡ (A(a, b)) a,b∈Q is irreducible with Perron-Frobenius eigenvalue ρ. (H2) A is positive regular (every entry of A n 0 is positive for some n 0 ∈ N), ρ > 1, and E q a [|x| log |x|] < ∞ for all a ∈ Q. (H3 p ) E q a [|x| p ] < ∞ for all a ∈ Q. Note that (H1) and ρ > 1 together imply MGW a (X) < 1 for all a ∈ Q. 1.1. Central limit theorems. We take all real-valued processes to be in the space D[0, ∞) equipped with the topology of uniform convergence on compact intervals. Our main theorem is the following: Theorem 1.1. Under (H1), (H2), and (H3 p ) with p > 4, for MGW-a.e. T / ∈ X, if X ∼ RW ρ (T) then the processes (|X nt |/(σ √ n)) t≥0 converge in law in D[0, ∞) to the absolute value of a standard Brownian motion for σ a deterministic positive constant (see (3.1)). Remark 1.2. By [11,Propn. 3.10.4], an equivalent statement is that the polygonal interpolation of k/n → |X k |/(σ √ n) converges to standard Brownian motion in the space C[0, ∞) (again with the topology of local uniform convergence).
Let RW cts λ (T) denote the continuous-time version of RW λ (T), which when at v ∈ T moves to the parent of v (if v = o) at rate λ and to each offspring of v at rate 1. By moving the root of the tree to the current position of the random walk, RW λ on the tree induces a random walk on the space Ω, the "walk from the point of view of the particle." As in [31, §3], to make the latter process Markovian we amend the state space so as to keep track of the ancestry of the vertices. Specifically, we consider the space Ω ↓ of pairs (T, ξ), where T is an infinite tree and ξ = (o = v 0 , v 1 , v 2 , . . .) is a ray emanating from the root o; this ray indicates the ancestry of each vertex in the tree. Let B Ω ↓ denote the σ-algebra generated by the cylinder sets. We define a height function h on T as follows: set h(v n ) = −n, and for v / ∈ ξ set where d denotes graph distance and R v is the nearest vertex to v on ξ (see Fig. 1). We denote by RW λ (T, ξ) the λ-biased random walk (Y t ) t≥0 on (T, ξ), where the bias goes in the direction of decreasing height. With T v the tree T rooted at v instead of o, and ξ v the unique ray emanating from v such that ξ ∩ ξ v is an infinite ray, let This is a Markov process with state space Ω ↓ , and we hereafter refer to it as TRW λ . Let RW cts λ denote the continuous-time version of RW λ (T, ξ) (moving in the direction of increasing height at rate 1 and in the direction of decreasing height at rate λ), and let TRW cts λ denote the induced continuous-time process on the space Ω ↓ . As in the single-type Galton-Watson case considered in [31], the key to our proof lies in finding an explicit reversing measure IMGW for TRW cts ρ , which is then easily translated to a reversing measure IMGWR for TRW ρ . For a tree T (with or without marked ray) and for any vertex v ∈ T, we denote by T (v) the subtree induced by v and its descendants, where descent is in direction of increasing distance from the root for a rooted tree, and in the direction of increasing height for a tree with marked ray. If µ is a law on trees we use µ ⊗ RW λ to denote the joint law of the tree together with the realization of RW λ on that tree. Theorem 1.4. Assume (H1). (a) There exists a reversing probability measure IMGW for TRW cts ρ , and if we define then IMGWR is a reversing probability measure for TRW ρ .
The IMGW trees always have an infinite ray ξ, though the trees coming off the ray may be finite. The measures IMGW, IMGWR are the multi-type analogues of the measures IGW, IGWR of [31]. Thm. 1.4 and the construction of harmonic coordinates allow us to prove the following quenched CLT for RW ρ on IMGWR trees, which will be used to deduce Thm. 1.1.
to a standard Brownian motion.

1.2.
Transience-recurrence boundary in random environment. In the setting of RW λ on MGW trees, λ = ρ represents the onset of recurrence. Indeed, MGW-a.e. tree T on the event of non-extinction has branching number br T = ρ [24, Propn. 6.5], therefore RW λ (T) is transient for λ < ρ and recurrent for λ > ρ [24,Thm. 4.3]. In fact, recurrence for all λ ≥ ρ follows from a simple conductance calculation (for the general theory see [28,Ch. 2]), therefore ρ is the boundary between transience and recurrence for RW λ on MGW trees. Further ρ is the boundary between non-ergodicity and ergodicity, with RW ρ null recurrent (see [19, and [24, p. 944 and p. 954]) and of zero speed (e.g. from the bound of Lem. 3.5).
We believe that the existence of a reversing measure and CLT is a feature of the onset of recurrence in a more general setting. Indeed, suppose each vertex v ∈ T\{o} has, in addition to its type χ v from the (finite) alphabet Q, a weight α v ∈ (0, ∞). Fixing such a tree T (the environment), the λ-biased random walk with random environment RWRE λ (T) is the Markov chain (X t ) t≥0 with X 0 = o which, when at vertex v with offspring (y, α) ≡ ((y 1 , α 1 ), . . . , (y , α )) ∈Q , jumps to a random neighbor w of v with probability proportional to α w if w is a child of v, and to λ if w is the parent of v. (Note that RW ρ (T) corresponds to the case α v = 1 for all v.) We let RWRE cts λ (T) denote the continuous-time version of RWRE λ (T).
If q a (a ∈ Q) is a probability measure on then the collection (q a ) a∈Q together with starting type χ o ∈ Q specifies a law MGW a 0 on the space Ω of typed weighted rooted trees. As before we let MGW denote any mixture of the MGW a 0 . This model, studied in the single-type case in [12], allows for quite general distributions on the (immediate) neighborhood of each vertex, but conditioned on types the weights in different neighborhoods must be independent. For γ ∈ R and a, b ∈ Q, let (not necessarily finite for all γ). Letρ(γ) be the Perron-Frobenius eigenvalue of A (γ) where well-defined (i.e. whereĀ (γ) has finite entries and is irreducible), and ∞ otherwise. We will prove the following characterization of the transience-recurrence boundary for RWRE λ , extending part of [12, Thm. 1.1]: (a) If p λ < 1, then RWRE λ is positive recurrent MGW-a.s.
Thus the transience-recurrence boundary for RWRE λ occurs at the unique value λ = ρ • for which p ρ • = 1. On the other hand, let Ω ↓ denote the space of typed weighted trees with ray, and let TRWRE λ and TRWRE cts λ denote the Markov chains in Ω ↓ induced by RWRE λ and RWRE cts λ respectively. We have the following generalization of Thm. 1.4 (a): (1). Then there exists a reversing probability measure IMGW on Ω ↓ for TRWRE cts ρ . If we let α 0j denote the weight for the j-th child of the root o, and set then IMGWR is a reversing probability measure for TRWREρ.
We can see that ρ • matchesρ if and only if the function γ →ρ(γ)/(ρ • ) γ attains its infimum over 0 ≤ γ ≤ 1 at γ = 1. If this fails, Thm. 1.7 still gives a reversing measure atρ, butρ > ρ • and the walk is already positive recurrent above ρ • . However, at least in the single-type case, we have ρ • =ρ in all cases in which a CLT is possible: by results of [16] a CLT cannot hold unless κ ≥ 2 (see [12, p. 3]). We expect κ ≥ 2 also to be a necessary condition in the multi-type case, and thus Thm. 1.6 and Thm. 1.7 support the claim that reversing measures occur at the boundary between transience and recurrence in cases in which a CLT is possible. However, even in the single-type case the random environment creates technical difficulties, and the RWRE-CLT of [12] requires some restriction on κ. While we expect that the methods of this paper and [12] can also be adapted to extend the RWRE-CLT to the multi-type setting under the same restrictions on κ, new ideas are required to achieve a CLT for the entire regime κ ≥ 2.
Outline of the paper.
• In §2 we construct the reversing measure IMGWR for TRW ρ (in §2.1) and its generalization IMGWR for TRWREρ (in §2.2); these constructions are based on ideas from [21]. In §2.3 we give an alternative characterization of IMGWR (extending a characterization of [31] to the multi-type setting) which we use to prove ergodicity of the stationary sequence ((T, ξ) Yt ) t≥0 . • In §3 we prove the quenched IMGWR-CLT Thm. 1.5: in §3.1 we construct on IMGWR-a.e. (T, ξ) a function v → S v (v ∈ T) which is harmonic with respect to the transition probabilities of RW ρ (T, ξ). By stationary and ergodicity of ((T, ξ) Yt ) t≥0 with respect to IMGWR we are able to control the quadratic variation of the martingale M t ≡ S Yt to obtain an IMGWR-a.s. martingale CLT.
In §3.2 we adapt the methods of [31] and [12] to show that h(Y t ) is uniformly well approximated by M t /η (η an explicit constant), proving Thm. 1.5. • In §4 we prove the quenched MGW-CLT Thm. 1.1. In §4.1 we review (a slight modification of) a construction of [31] which gives a "shifted coupling" of (T,  (1) Does a CLT with diffusive scaling hold for RW ρ in the entire regime p ≥ 2?
(2) Does a CLT with diffusive scaling hold for RWREρ in the entire regime κ ≥ 2?
(3) What happens for simple random walk on the critical Galton-Watson tree (conditioned to survive)? (4) Does a CLT with any scaling (or other limit law) hold for RW ρ when p < 2? A common feature of these problems is that while the reversing measure for the process from the perspective of the particle is given by Thm. 1.4, the method of martingale approximation used in [31,12] and in this paper seem not to be directly applicable.

Reversing probability measures for TRW ρ and TRWREρ
Assuming only (H1), in this section we construct the reversing measure IMGWR for TRW ρ ( §2.1) as well as its generalization IMGWR for TRWREρ (in §2.2). In §2.3 we give an alternative characterization of IMGWR which we use to prove ergodicity of the stationary sequence ((T, ξ) Yt ) t≥0 . Except in §2.2 we work throughout with unweighted trees.
Consider a multi-type Galton-Watson measure MGW with offspring distributions (q a ) a∈Q and mean matrix A. Hereafter we let e and g denote the right and left eigenvectors respectively associated to the Perron-Frobenius eigenvalue of A, normalized so that a g a = a e a = 1. Since our results are stated for MGW-a.e. tree, with no loss of generality we set hereafter (2.1) Unless otherwise specified, X and Y denote RW ρ on trees without and with marked ray respectively.
2.1. Construction of IMGWR. We begin by constructing two auxiliary measures on the space Ω ↓ of trees with ray (T, ξ). Let the infinite ray ξ (without types) be given. For some n > 0, we let vertex v n be given a type χ n according to a distribution π, to be determined shortly. It is then given offspring x vn according to the inflated offspring distribution q χn , where x, e ρe a ∀a ∈ Q; note that q a (|x| ≥ 1) = 1. One offspring w of v n is then identified with the next vertex v n−1 along ξ, where each w is chosen with probability e χw / x vn , e . We proceed in this manner along the ray ending with the identification of v 0 = o. The sequence of types χ n , χ n−1 , . . . seen along the ray is then (by (H1)) an irreducible Markov chain with transition probabilities This chain has stationary distribution π(a) ≡ e a g a / e, g , so starting with χ n ∼ π yields a consistent family of distributions for (v n , . . . , v 1 ) and their (immediate) offspring, with types. By Kolmogorov's existence theorem, this uniquely specifies the distribution of the backbone of the tree, that is, of the ray ξ together with all (immediate) offspring of the vertices v i , i > 0. To each of these offspring (off the ray) and to o, we attach an independently chosen MGW tree conditioned on the given type, and denote by IMGW 0 the resulting measure on Ω ↓ . The inflated multi-type Galton-Watson measure IMGW is obtained from IMGW 0 by an additional biasing according to the root type χ o . Specifically, we set where χ denotes a random variable on Q with the specified distribution. We note that under IMGW, χ o ∼ g and so T (o) has marginal law MGW, which implies With this in mind, we define the probability measure IMGWR such that and proceed to show that it is a reversing measure for TRW ρ . From now on we adopt the notation that if µ is a law on trees T (with or without marked ray) and a ∈ Q, µ a refers to the law conditioned on χ o = a.
Proof of Thm. 1.4 (a). For the purposes of this proof we let Ω and Ω ↓ be spaces of labelled (or planar) trees (without and with marked ray, respectively), with corresponding Borel σ-algebras B Ω and B Ω ↓ . We extend MGW, IMGW 0 , etc. to be measures on these spaces by choosing an independent uniformly random ordering for the offspring of each vertex. For (T, ξ) ∈ Ω ↓ we use the shorthand i for v i ∈ ξ, and write (i1, . . . , id i ) for its ordered offspring (with x i denoting the counts of offspring of i ≡ v i of each type).
Recalling the notation of (1.2), let S denote the map (T, ξ) → (T, ξ) 1 . We will show that for A, B ∈ B Ω ↓ , where p((T, ξ), B) denotes the transition kernel of the process TRW ρ . This identity implies reversibility of TRW ρ on the space of labelled trees. Since this process projects to TRW ρ on the space of unlabelled trees, the reversibility of the latter follows.
denote the law of the subtree T\T (i−1) rooted at i with marked ray ξ i , conditioned on the event {χ i−1 = a}, for any i ≥ 1 (note that this law does not depend on i). Then Let P inj denote the collection of B Ω ↓ -measurable sets on which S is injective, and We then verify that and similarly dS * The left-hand side of (2.4) can be written aŝ Using the injectivity of S on B, the second integral can be written aŝ Combining these yields an expression for the left-hand side of (2.4) which is symmetric in A and B, from which it is clear that the two sides must agree.
Since every cylinder event F can be decomposed into the disjoint union of the event , o is the j-th child of 1), with F j clearly in P inj , we have that P inj generates B Ω ↓ . To conclude, for fixed A let B Ω ↓ denote the collection of sets B ∈ B Ω ↓ for which (2.4) holds. From the above B Ω ↓ contains the π-system P inj . Further B Ω ↓ is closed under monotone limits and countable disjoint unions, and in particular it contains Ω ↓ since Ω ↓ can be decomposed as a countable disjoint union of sets in P inj by a similar argument as above. Thus by the π-λ theorem (2.4) holds for all B ∈ σ(P inj ), and extends to all B ∈ B Ω ↓ again using the claim above.
The proof that IMGW is a reversing measure for the Markov pure jump process TRW cts ρ is similar: instead of (2.4) we show that where λ(T, ξ) ≡ λ + d o is the instantaneous jump rate of the process at state (T, ξ).
As before, it suffices to show this for B ∈ P inj . In this case the left-hand side of (2.6) equalsˆA , which by (2.5) coincides with the right-hand side of (2.6).

2.2.
Extension of IMGWR to random environment. We now extend the methods of the previous section to prove Thm. 1.7. Letē,ḡ denote the right and left Perron-Frobenius eigenvectors ofĀ ≡Ā (1) , normalized to have sum 1; as before we set g(a) ≡ MGW(χ o = a) to beḡ a .
We proceed much as in the deterministic environment setting, although the notation becomes more complicated. For We then construct the measure IMGW 0 on Ω ↓ generalizing the measure IMGW 0 of the previous section: let the infinite ray ξ (without types or weights) be given, and for some n > 0 let v n have type χ n . It is given offspring (y vn , α vn ) ∼ q χn . One offspring w of v n is identified with the next vertex v n−1 along ξ, where each w is chosen with probabilityē χw α w ē(y), α vn . Continuing the procedure along the ray up to v 0 = o, the sequence of types χ n , χ n−1 , . . . seen along ξ is an irreducible Markov chain with transition probabilities and stationary distributionπ(a) =ē aḡa / ē,ḡ . Thus, starting with χ n ∼π and applying Kolmogorov's existence theorem, we obtain a measure IMGW 0 on Ω ↓ which is a generalization of IMGW 0 .
Proof of Thm. 1.7. The proof is by a straightforward modification of the proof of Thm. 1.4 (a). Let S : (T, ξ) → (T, ξ) 1 ; we emphasize that S is a mapping on typed weighted labelled trees.
we obtain The analogue of (2.4) thus holds for all B ∈ P inj , and we extend to all B ∈ B Ω ↓ by essentially the same argument used in the proof of Thm. 1.4 (a).
2.3. IMGW 0 as a weak limit and ergodicity. In this section we provide an alternative characterization (Propn. 2.1) of the inflated Galton-Watson measure IMGW 0 , which is then used in proving the ergodicity result Thm. 1.4 (b). Propn. 2.1 is also of independent interest as a multi-type extension of [31,Lem. 1].
To this end, we will define the notion of "normalized population size" for rooted trees T with type. Let T n denote the subtree induced by {v ∈ T : |v| ≤ n}, and D n the set {v ∈ T : |v| = n}. Let (F n ) n≥0 denote the natural filtration of the tree, i.e., F n is the σ-algebra generated by T n (a finite tree with vertex types). Let Z n = (Z n (b)) b∈Q ∈ (Z ≥0 ) Q count the number of vertices of each type at level n, so Z n is F n -measurable. Then [14, p. 49]). By the normalized population size of the tree we mean the a.s.
. Under (H1) and (H2), it follows from the multi-type Kesten-Stigum theorem (see [20], or the conceptual proof of [21]) that W o > 0 a.s. on the event of non-extinction, and For a ∈ Q let Q a n be a probability measure on (infinite) rooted trees defined by dQ a n dMGW a = Z n e a . (2.7) For T ∼ Q a n choose v n ∈ D n at random with probabilities proportional to weights e χv n , and let Q a n denote the law of the resulting pair (T, v n ). Let Q n ≡ a∈Q π a Q a n and Q n ≡ a∈Q π a Q a n , so that dQ n /dMGW = Z n /E g [e χ ]. Finally let IMGW 0 (n) denote the law of (T, ξ 0 ) vn (see (1.2) for this notation), where (T, v n ) ∼ Q n and ξ 0 is any infinite ray emanating from o not sharing an edge with the geodesic from o to v n .
The proposition can be seen from the following explicit construction of Q a n : begin with v 0 ≡ o of type a, and suppose inductively that we have constructed (T i , v i ) (i < n) where T i is the tree up to level i and v i is the i-th vertex on the geodesic from o to v n . Then v i is given offspring x v i according to q χv i , and one of these offspring w is randomly chosen (according to weights e w ) to be distinguished as v i+1 . Meanwhile all other vertices v ∈ D i \{v i } are given offspring x v according to q χv . Once (T n , v n ) has been constructed, attach to each v ∈ D n an independent MGW χv tree. For N ≥ n, x v i , e = e χv n ρ n e a , and summing over v n ∈ D n gives (2.7). Letting n → ∞ in Q a n , Q n we obtain the measures Q a ∞ , Q ∞ on rooted trees with infinite marked ray which coincide precisely with the measures MGW a , MGW of [21]. .
We remark that although Q ∞ ≡ MGW and IMGWR are both measures on trees with rays, they are not in general equivalent unless K is reversible.
In other words the portion of (T, ξ) descended from v n has the same distribution under IMGW 0 (n) as under IMGW 0 , proving the result.
Turning now to the proof of Thm. 1.4 (b), it is useful to define a two-sided version of IMGW 0 , as follows. Let Ω denote the space of trees with marked line: pairs (T,ξ) where T is an infinite tree and is a line (doubly infinite simple path) passing through the root. The positive and negative partsξ ± ≡ (ξ ±j ) j≥0 ofξ are edge-disjoint rays emanating from o. Now suppose in the construction of IMGW 0 we continue the backbone indefinitely rather than stopping at o, so that Kolmogorov's existence theorem gives a doubly infinite backbone based on a lineξ. Attaching MGW trees to the leaves of this backbone then gives a tree with marked line (T,ξ), whose law IMGW is clearly stationary with respect to the shift S : (T,ξ) → (T,ξ)ξ −1 which defined by moving the root tō ξ −1 . (Alternatively, if (T, ξ) has law IMGW 0 conditioned on non-extinction of T (o) and ξ is any line withξ − = ξ, then S n (T,ξ) converges weakly to an IMGW tree.) It follows from the discussion preceding Propn. 2.1 that if we let , and we delete from T all the vertices descended from o and identify o with the root of T , then we obtain a tree with marked linē ξ − = ξ,ξ + = ξ whose law is precisely IMGW. It follows that the marginal law IMGW 1 of (T,ξ − ) under IMGW is given by sequences of trees with ray, and S 0 the shift (T 0 , The content of the result is that the measure-preserving system (Ω ∞ ↓ , F ∞ , ν, S 0 ) is ergodic.
Step 1: reduction to induced system. Recall that under the measure IMGW 0 the trees T (i) \T (i−1) are conditionally independent given the ray ξ with types, and max a∈Q MGW a (X) < 1; therefore it holds IMGW 0 -a.s. that |T (i) | for infinitely many i ∈ ξ. Since the walk Y t on (T, ξ) has a backward drift along ξ, this implies that if we let . We now show that the induced system is ergodic, which is equivalent to ergodicity of the original system ( [32]; see also [26, §2]).
The S-invariance of B together with the Markov property implies converges ν A -a.s. to zero. On the other hand, by the Birkhoff ergodic theorem (see e.g. [10,Thm. 6 Step 3: IMGWR-triviality of S-invariant sets. o, together with the descendant subtrees ofξ −n , . . . ,ξ −1 away fromξ, such that the symmetric difference C n 0 C 0 has IMGW-measure tending to zero in n. It follows from S-invariance of C 0 together with S-stationarity of IMGW that IMGW[C n But for any m > n we have

Harmonic coordinates and quenched IMGWR-CLT
In this section we prove the quenched IMGWR-CLT Thm. 1.5. Let . (3.1) In §3.1 we construct harmonic coordinates for RW ρ on IMGWR-a.e. (T, ξ), and use the ergodicity result Thm. 1.4 (b) proved above to show an IMGWR-a.s. CLT for the martingale M t ≡ S Yt , with M nt /(ησ √ n) converging to standard Brownian motion. In §3.2 we control the error between h(Y t ) and M t /η to prove Thm. 1.5. The following result, whose proof is deferred to §6.2, implies finiteness of η and σ under (H3 2 ): Harmonic coordinates for RW ρ and martingale CLT. From now on, if µ is a probability measure on trees (with or without marked ray), we use µ as shorthand also for µ ⊗ RW ρ . We write P T for the law of the quenched random walk RW ρ (T) and E T for expectation with respect to P T , and let (G T t ) t≥0 denote the corresponding filtration of the walk. Given T, for a vertex v ∈ T we let ∂v denote the neighbors of v, and ∂ + v the offspring of v, i.e., For v ∈ T recall that W v denotes the normalized population size of the subtree If T has marked ray ξ, recalling (1.1) we set 1 Note that if T has a marked ray ξ, then for v ∈ ξ, Z v n = Z v n , e /ρ n is not necessarily a martingale for the first |h(v)| steps. Nevertheless it is eventually a martingale so we can still define W v to be the a.s. limit of Z v n .
While on MGW-a.e. T the map v → S v is harmonic except at o with respect to the transition probabilities of RW ρ (T), on IMGW-a.e. (T, ξ) the map v → S v is harmonic at every vertex with respect to the transition probabilities of RW ρ (T, ξ). Thus, if (Y t ) t≥0 ∼ RW ρ (T, ξ), M t ≡ S Yt will be a martingale given a fixed realization of the tree; we regard it as providing "harmonic coordinates" for the random walk. Using the reversing measure IMGWR it is easy to prove a quenched CLT for M (extending we verify that for IMGW-a.e. (T, ξ), Let Y n denote the random walk on (T, ξ): we rewrite V n in terms of the induced random walk on Ω ↓ as By Thm. 1.4 (b) and the Birkhoff ergodic theorem, we have V n converging IMGWR-a.s. to E IMGWR [ϕ] provided ϕ ∈ L 1 (IMGWR). We calculate so condition (i) is proved. Condition (ii) is checked similarly using dominated convergence.
Remark 3.3. To give some indication of how our results might be extended to RWREρ, we note that the main ingredient needed is the appropriate generalization of the normalized population size: we define it to be the random variable W o which is the a.s. limit of the martingale Z n ≡ Z n defined by (5.1). If W v denotes the normalized population size of T (v) , then so the W v can be used to define harmonic coordinates for the RWRE. In the singletype case, W o has finite second moment if and only if κ ≥ 2 [23, Thm. 2.1], so clearly Propn. 3.2 cannot apply outside this regime. We emphasize again that due to the same technical barriers which arise in [12], simple adaptations of our proof will not cover the full regime κ ≥ 2.

3.2.
Quenched IMGWR-CLT. We now prove the quenched CLT for IMGWR trees by controlling the corrector on the interval 0 ≤ t ≤ n. For 1/2 < δ < 1 and n ≥ 0 fixed, let τ n (j), for j n δ ≤ n denote integer times chosen uniformly at random (independently of one another and of the random walk Y ) from the interval [j n δ , (j + 1) n δ ).
Lemma 3.5. There exists a constant C < ∞ such that Proof. We modify the proof of [31,Lem. 5]. Take the finite tree with vertices {w ∈ T : |w| ≤ m}, and make this into a wired tree T by adding a new vertex o which is joined by an edge to each vertex in D m . Define the modified random walk X on T which follows the law of RW ρ except at o where it moves to a vertex chosen uniformly at random from D m . Then By the Carne-Varopoulos inequality (see [28,Thm. 13.4]), Taking expectations gives and summing over 1 ≤ t ≤ n + 1 gives the result.
Corollary 3.6. There exists a constant C < ∞ such that for any m, n ≥ 1, Proof. We argue as in the proof of [31,Cor. 2]. By decomposing into at most n excursions away from height zero and using the stationarity of IMGWR, we find which is summable in n provided 2 + δ < 1. The result then follows from Markov's inequality and Borel-Cantelli.

Control of corrector.
In the remainder of this section we prove (3.4). We will make use of the following classical result: Lemma 3.7 ([33, p. 60]). If z 1 , . . . , z n are independent random variables with Ez i = 0 and E|z i | p < ∞, then Recalling (1.1) and (3.3), we decompose 1 √ n max j n δ ≤n where, with R t ≡ R Yt denoting the nearest ancestor of Y t on ξ, The following lemma says that the harmonic coordinates (S v ) v∈T of (3.2), rescaled by η of (3.1), are a good approximation to the actual coordinates |v| on the MGW rooted trees. Let Let τ ≡ min{t > 0 : X t = o} denote the first return time to the starting point X 0 = o by the walk X. Proof.
If v ∈ T with |v| = k ≥ 1, a simple conductance calculation (see [28,Ch. 2]) gives  [31, (20)]) and of [12,Lem. 4.2]. Recall from §2.3 the definition (2.7) of the probability measure Q a k on rooted trees T given by a size-biasing of MGW a , and further the probability Q a k on rooted trees T with a marked path (o = v 0 , . . . , v k ) from the root to level k: To this end, writing Conditional on the types (χ i ≡ χ v i ) k i=1 , the random variables W 1 , . . . , W k−1 are independent of one another and of the pair (W k , d v k ), and all these random variables have finite moments of order p by Propn. 3.1. Therefore By (H3 p ), Markov's inequality, and Hölder's inequality, the second term is (since p ≥ 2). As for the first term, by Lem. 3.7 and Markov's inequality, On the other hand, decays exponentially in k by [9, Thm. 3.1.2]. Combining these estimates completes the proof.

Proof.
We modify the proof of [12, (22)]. If we define τ hit n, ≡ inf{t ≥ 0 : |h(Y t )| = n 1/2+ }, then Cor. 3.6 together with Markov's inequality gives Cn 2 e −n 2 /2 , so by Borel-Cantelli we have P (T,ξ) (τ hit n, ≤ n) → 0, IMGWR-a.s. On the event {τ hit n, > n}, we decompose the walk into excursions from ξ started at v i , 0 ≤ i < n 1/2+ (with each step of the walk along the ray contributing an empty excursion) and apply Wald's identity (see e.g. [3, Exercise 22.8]) to find (3.10) In the above, L A (n) ≡ L(A; n) denotes the number of visits to set A by time n and L i (n) ≡ L(v i ; n). E i (T,ξ) denotes expectation with respect to the law of a ρ-biased random walk Y started from Y 0 = v i , and τ exc ≡ inf{t > 0 : Y t = v i or Y t / ∈ T (v i ) } denotes the excursion end time.
By a conductance calculation, During a single excursion away from ξ the walk can visit only one of the T (w) for w ∈ ∂ + v\v i−1 , so to bound the second factor of each summand in (3.10) it suffices to consider an MGW rooted tree T (without ray): letting it follows from a (very slight) modification of Lem. 3.8 that (using p > 2). It follows that the quantity in (3.10) converges to zero IMGWR-a.s., which concludes the proof.

Proof of Propn. 3.4, (3.4). Recall the decomposition (3.6)
. For any k 0 , The first term clearly tends to zero as n → ∞ with k 0 fixed. The second term is bounded above by Now recall from the proof of Propn Thus a consequence of the proof of Lem. 3.8 is that for sufficiently small , Therefore the supremum in (3.12) can be made arbitrarily small by taking k 0 large. We also have , and in view of Lem. 3.9 the second factor tends to zero in probability. By the invariance principle for M proved in Propn. 3.2, max t≤2n |M t |/ √ n stays bounded in probability as n → ∞, so the result follows.

From IMGWR-CLT to MGW-CLT by shifted coupling
In this section we prove our main result Thm. 1.1. In §4.1 we review (a slight modification of) the "shifted coupling" procedure of [31, §6], which we use in §4.2 to transfer the IMGWR-CLT to an annealed MGW-CLT. In §4.3 we prove a variance estimate which allows to go from the annealed to the quenched MGW-CLT.

4.1.
The shifted coupling construction. We begin by reviewing the shifted coupling construction of [31, §6], with the (natural) modification needed to handle the multi-type case. The basic observation underlying the construction is that the law of the random walk X ∼ RW ρ (T) up to time t depends only on ("the subtree explored by time t"), so that one can construct the tree at the same time as the random walk.
For any tree T (with or without marked ray) and U any subset of the vertices of T, we also use U to indicated the subgraph of T induced by U . Let LT denote the set of leaves and T • ≡ T\LT.
Next we construct a coupled realization ((T , ξ), (Y t ) t≥0 ) ∼ IMGW 0 ⊗RW ρ as follows: first construct the backbone E 0 of the tree (ξ and ∂ + v i for i ≥ 1, together with types) in the manner described in §2.1. Set η 0 ≡ 0, and start a ρ-biased random walk Y on E 0 with Y 0 = o. As in the MGW setting we will construct a growing sequence (E t ) t≥0 such that E t = E 0 ∪ (∂Y s ) 0≤s<t , and we will define (for i ≥ 1) The difference is that we grow the sequence E t in a manner dependent on (T, X), such that excursions of Y into unexplored territory (and started from a 0 ) match the excursions of X defined above: formally, we couple ( and then we set Y η i to be the ancestor of Y τ i (not necessarily of the same type as X η i ). Then, on the inter-excursion intervals η i−1 ≤ t < τ i (for i ≥ 1), • If Y t ∈ (E t ) • then generate Y t+1 according to the transition kernel of RW ρ on E t+1 = E t ; • If Y t ∈ LE t with χ Yt = a 0 , let E t+1 be the enlargement of E t obtained by attaching random offspring to Y t according to law q χ Y t , and generate Y t+1 according to the transition kernel of RW ρ on E t+1 . Finally, with E ∞ ≡ lim t→∞ E t , we define T by attaching to each vertex v ∈ LE ∞ an independent MGW χv tree. We thus obtain the following extension of [31, Lem. 8]: Lemma 4.1. If (T, (X t ) t≥0 ) ∼ MGW ⊗ RW ρ then the marginal law of ((T , ξ), (Y t ) t≥0 ) arising from the above construction is IMGW 0 ⊗ RW ρ .

Remark 4.2.
Although we suppress the parameter n from the notation, we emphasize that each n ≥ 1 gives rise to a different excursion decomposition, hence a different coupling between (T, X) and ((T , ξ), Y ).
Recall that R t ≡ R Yt denotes the nearest ancestor of Y t on ξ. By Thm. 1.5, for IMGW 0 -a.e. (T, ξ), the process converges to a Brownian motion minus its running minimum, which is the same in law as the absolute value of a Brownian motion (see e.g. [18,Thm. 3.6.17]). Thus to deduce Propn. 4.3 we need to estimate the relation between the processes |X n | and H n . To this end, let t, t be the monotone increasing bijections parametrizing the inter-excursion times of X n and Y n respectively. We make the following notations (the left column refers to the MGW tree, while the right column refers to the IMGW 0 tree): In words, given the walk X on the MGW tree, X int s is the "inter-excursion process" adapted to the filtration H s , J i is the i-th inter-excursion interval, I n is the number of such intervals intersecting [0, n], ∆ n is the total length of these intervals, and ∆ n (α) is the length of these intervals except for times spent at distance more than n α from the root. The right column defines the analogous objects for the walk on the IMGW 0 tree. Lemma 4.4. Assume (H1), (H2), and (H3 p ) with p > 4. There exists α 0 (p) < 1/2 such that for α > α 0 (p), We will obtain the corollary below as a relatively straightforward consequence of Lem   . (T, ξ), D n / √ n converges P (T,ξ) -a.s. to zero.
Assuming these results we can prove the annealed MGW-CLT: Proof of Propn. 4.3. Let b : Z ≥0 → Z ≥0 be any nondecreasing map which maps , so, recalling (1.1) and (4.2), we have It follows that on the event {∆ n (α) = ∆ n } ∩ {∆ n (α) = ∆ n }, Proof. By iterated expectations, Markov's inequality, and Lem. 3.5, with L A (n) the number of visits the walk makes to set A by time n. Recalling (3.8), we have so the MGW bound follows from Lem. 4.6 with a few more applications of Markov's inequality.
For the IMGW 0 bound we argue as in the proof of Lem. 3.9: by Markov's inequality and Cor. 3.6, n, ) ≥ n 1/2+α+2 ), so it suffices to bound the last term. By Wald's identity and (3.11), so again the bound follows by using Markov's inequality.
(c) We follow the proof of [31,Lem. 11]. On the IMGW 0 tree, since d(Y 0 , ξ) = 0, d(Y int , ξ) must increase by (n) going from Θ j−1 to Θ j for at least half of the indices j ≤ 3I n so {t(Θ 3In ) ≤ n} implies the event This in turn implies one of two possibilities: 1. there exist times t 0 < t 1 < t 2 ≤ n with Y t 0 = Y t 2 and d(Y t 0 , ξ) = d(Y t 1 , ξ) + (n)/4, or 2. there exist times t 1 < t 2 ≤ n with d(Y t 2 , ξ) = d(Y t 1 , ξ) + (n)/4 such that a 0 does not appear on the geodesic between the Y t i .
By a random walk estimate (cf. (3.8)) summed over at most n 2 possibilities for (Y t 0 , Y t 1 ), the first event has probability ≤ Cn 2 ρ − (n)/4 . The second event has probability ≤ e −c (n) by the construction of IMGW 0 and the irreducibility of the Markov chain, and combining these estimates gives the bound for IMGW 0 . The bound for MGW follows by a similar argument.
(d) We first prove the bounds for MGW; the argument is similar to that of Cor. 4.7: again it suffices to bound MGW(τ hit A ≤ τ hit n, ), and Wald's identity gives
The bound for the IMGW 0 -probability of E ≡ {∆ n (α) = ∆ n } is similar, indeed and applying the Azuma-Hoeffding bound gives the result. (b) Let (h s ) s≥0 denote the height process for the walk Y restricted to ξ, i.e. erasing all excursions away from ξ; clearly D n ≤ D n ≡ max{h s − h r : 0 ≤ r ≤ s ≤ n}. But h s is simply a random walk on Z ≤0 with a ρ-bias in the negative direction. Set σ 0 ≡ 0,

Now the processes (h (j)
s ≡ h s − h σ j ) σ j ≤s≤σ j+1 are i.i.d., and clearly σ n ≥ n, so The probability of max sh (j) s ≥ m is at most the probability that a random walk on Z started at 0 with a ρ-bias in the negative direction will reach m before −1, which

4.3.
Quenched MGW-CLT. We now describe how to move from the annealed to the quenched CLT; the proof is motivated by ideas in [31, §6-7] and [5,Lem. 4.1]. For given n ≥ 1, let s denote the unique increasing bijection and let X exc t ≡ X s(t) , the excursion process of X with parameter n (recalling Rmk. 4.2). For Proof of Thm. 1.1. We show the quenched CLT for X through a quenched CLT for X cent along geometrically increasing subsequences b k ≡ b k (k ≥ 0) with b > 1.
Step 1: annealed CLT for X cent . The time killed during the first n steps of X is n − s −1 (n) ≤ ∆ n , so Cor. 4.5 (a) gives n −1 sup 0≤t≤T |s( nt ) − nt | → 0 in MGW-probability. It follows from Propn. 4.3 and the continuity of Brownian motion that the processes X cent nt /(σ √ n) also satisfy the annealed MGW-CLT.
Step 2: quenched CLT for X cent along geometrically increasing subsequences. Recalling Rmk. 1.2, let B n (X) ≡ (B n t (X)) t≥0 denote the polygonal interpolation of j/n → X cent j /(σ √ n), and regard B n (X) as an element of C[0, T ] with the norm We will show that for all Lipschitz functions F : The Borel-Cantelli lemma then implies (cf. [5,Lem. 4.1]) that for MGW-a.e. T, the processes X cent nt /(σ √ n) converge in law to the absolute value of a standard Brownian motion along the subsequence b k .
To see (4.6), let T ∼ MGW, let (X i , s i ) be two independent realizations of (X, s) conditioned on T, and write B n,i ≡ B n (X i ). Then Let E i n denote the subtree explored by X i up to time n. Conditioning on the first (n)/2 levels of T, let (É i n ,X i ,ś i ) (i = 1, 2) be two independent realizations of (E i n , X i , s i ): then the processes (X i ) cent are exactly independent with law not depending on the first (n)/2 levels of T. Moreover, if A n denotes the event that the paths ofX 1 andX 2 up to time max i (ś i ) −1 (n) have no common vertices at distance more than (n)/2 from the root, then we can couple (E i n , X i | [0,n] ) i=1,2 with (É i n ,X i | [0,n] ) i=1,2 such that the processes agree on the event A n . Therefore We claim MGW(A n ) ≤ n −c : since Cor. 4.7 and Cor. 4.5 (a) imply MGW(2n − s −1 (2n) ≥ n) ≤ n −c , it suffices to bound the probability that the paths of X 1 and X 2 up to time 2n intersect at distance > (n)/2 from the root. But the chance that X 2 hits a given vertex v with |v| > (n)/2 by time 2n is ≤ Cnρ − (n)/2 , and summing over the vertices visited by X 1 proves the claim. The variance condition (4.6) now follows by summing over (b k ) k≥0 .
Proof of Cor. 1.3. Given (T, X) ∼ MGW ⊗ RW ρ we can obtain (T, X cts ) ∼ MGW ⊗ RW cts ρ by taking (E i ) i≥1 i.i.d. exponential random variables with unit mean independent of X, and setting similarly we can obtain ((T , ξ), Y cts ) ∼ IMGW 0 ⊗ RW cts ρ from ((T , ξ), Y ) ∼ IMGW 0 ⊗ RW ρ . Thus a shifted coupling of (T, X) with ((T , ξ), Y ) (as constructed in §4) naturally gives rise to a shifted coupling of (T, X cts ) ∼ MGW ⊗RW cts ρ with ((T , ξ), Y cts ) ∼ IMGW 0 ⊗ RW cts ρ by using sequences (E i ) i≥1 for X cts and (E i ) i≥1 for Y cts which are marginally i.i.d. exponential but such that the jump times match during the coupled excursions.
By Thm. 1.4 (b) and the exponential decay of the E i , it holds IMGWR-a.s. that From this it is easy to see that n −1 sup 0≤t≤T [θ(nt) − 2ρnt] → 0 IMGWR-a.s., so on IMGWR-a.e. (T, ξ) the processes (h(Y cts nt )/(σ √ 2ρn)) t≥0 converge in law to standard Brownian motion. The quenched MGW-CLT for X cts follows from the proof of Thm. 1.1.

Transience-recurrence boundary for RWRE λ
We now prove Thm. 1.6. Our proof is a straightforward adaptation of that of [25,Thm. 1] or [12, Propn. 1.1] once we supply the needed large deviations estimate (Lem. 5.2) on the conductances at the n-th level of the tree, extending the estimates of [25, p. 129] and [12, p. 7] to our setting of Markovian dependency.
the conductance of the edge leading to v. The natural generalization of the martingale introduced in §2.3 is this is a multi-type Mandelbrot's martingale and has been studied in various contexts, for example as the Laplace transform of the branching random walk with increments log α v [7,22]. Using this martingale we can make a change of measure and control the conductances at the n-th level by controlling the conductance of the edge leading to a random vertex: recalling (2.7), for each a ∈ Q define the size-biased measure Q a n on Ω by dQ a n We then let Q a n denote the measure on pairs (T, v n ) obtained by letting T ∼ Q a n and choosing v n ∈ D n according to weights e χv .
(a) Suppose p < 1. We will use the fact that the random walk is positive recurrent if and only if the conductances have finite sum [19,. Ifρ(γ) < 1 for some γ ∈ [0, 1] then n log Q a n (C vn > z n ) ≥ − log[yρ(0)/p] ∀z < y, ∀a ∈ Q. Therefore we can choose z < y, ∈ N, and , w ∈ (0, 1) such that we refer to F as the generating function of the MGW tree. If F (n) denotes the n-fold For many purposes the case of MGW(X) ∈ (0, 1) can be reduced to the simpler case of an a.s. infinite tree without leaves by the following transformation which is discussed in [1, §I.12] for the single-type case. For T ∼ MGW, consider the subtree T ∞ consisting of those vertices v of infinite descent, i.e. with |T (v) | = ∞. Conditioned on X c , T ∞ is an a.s. infinite tree without leaves, following a transformation of the original MGW given by generating functionF (s) = (F a (s)) a∈Q , where, with x a ≡ MGW a (X), The transformed law has mean matrix In this case m r = E[(−W ) r ] = lim s↓0 ϕ (r) (s), and the left-hand side of (6.1) is the n-th order (one-sided) Taylor expansion P n,0 ϕ of ϕ at 0.
where P n,0 f (t; v) is a polynomial of degree at most n in the entries of tv satisfying By recalling (6.4) and comparing (6.3) against the Taylor expansion of ρ g, e −tv , we find . But squaring (6.5) gives (tv a ) 2 = s 2 q a k (s) 2 + o(s k+1 ), and substituting into the above and dividing through by s gives that for P k polynomial in s. Since ρ > 1 and lim s↓0 ψ(s) = φ (0), ψ(s) = ψ (0) + j≥0 (s/ρ j )P n (s/ρ) + o(s k ) = P n (s) + o(s k ) for P n another polynomial in s. This verifies the inductive hypothesis by the definition of ψ together with another application of Lem. 6.1.
Proof. Summing (6.5) over a ∈ Q gives the existence of b 2 , . . . , b n finite such that t = s + n r=2 b r s r + o(s n ).
It follows easily that s has a similar expansion in terms of t: indeed s = t + o(t), so suppose inductively that for some 1 ≤ k < n there exist c 2 , . . . , c k finite such that s = t + k r=2 c r t r + o(t k ). Then which is a polynomial in t plus o(t k+1 ). This verifies the inductive hypothesis so we conclude that s = t + n r=2 c r t r + o(t n ) as claimed. From the proof of Lem. 6. for Q n and q polynomial. But by the above s 2 can be expressed as a polynomial in t up to o(t n+1 ) error, so in fact o(t n ) = (−1) n+1 [R n,0 φ(s) − ρR n,0 φ(s/ρ)] = f (t; v) − Q n (tv) + O(t n+1 ) for Q n : R Q → R polynomial in tv of degree at most n in t, whence necessarily Q n (tv) = P n,0 f (t; v) as claimed.
If Φ(s/ρ) = e −tv with t ≡ t(s) ≥ 0, v ≡ v(s) ∈ S Q as above, then In particular t (s) is finite and positive for all s > 0 with lim s↓0 t (s) = 1, so s → t(s) is an increasing bijection from [0, ∞) to [0, t max ) where t max = − a∈Q log MGW a (X). For t < t max we therefore write v t ≡ v(s) with s defined by t = t(s).
Proof of Propn. 3.1. Since the subtree of infinite descent described in §6.1 has the same normalized population size as the original tree, we may reduce to the case MGW(X) = 0 so that t max = ∞.
Proof. Since the subtree of infinite descent described in §6.1 has the same normalized population size as the original tree, we may reduce to the case MGW(X) = 0. Expanding F (n) (s) as a power series in s we find (F (n) (s)) a ≤ MGW a (|Z n | = 1) s ∞ + MGW a (|Z n | > 1) s 2 ∞ . By (H1), there exists n 0 such that min a∈Q MGW a (|Z n | > 1) > 0 for all n ≥ n 0 , so that F (n 0 ) is a contraction on [0, s 0 ] Q for any s 0 < 1. By iterating this estimate, for any s 0 < 1 there exist constants C < ∞ and γ < 1 such that Next, we have f n (u) = g, F (n) (e −ue b ) ≤ F (n) (e −ue b ) ∞ ≤ Cγ n (e −ue b ) b∈Q ∞ ≤ Cγ n e −ue min , where for any u 0 > 0 we may choose constants C < ∞ and γ < 1 uniformly over all u ≥ u 0 . Therefore Γ(r) I 1,∞ ≤ C(ρ r γ) nˆ∞ 1 e −ue min u r−1 du.
For r > 0 small enough so that γρ r < 1, we have lim n→∞ I 1,∞ = 0. It remains to consider Γ(r) I ρ −n ,1 = ρ nr (b) If further (H3 p ) holds with p > 2, then for MGW-a.e. T / ∈ X there exists a random constant C T < ∞ such that C −1 o,k ≤ C T k for all k. Proof. (a) Recall that a unit flow is a non-negative function U on the vertices of T such that for all v ∈ T, U (v) = w∈∂ + v U (w). For v ∈ D define it is easily seen that U is a well-defined unit flow on X c . It gives positive flow only to vertices of infinite descent, so by the discussion of §6.1 we may reduce to the case MGW(X) = 0. By Thomson's principle [28, §2.4] By Hölder's inequality, for r sufficiently small, using Lem. 6.4 and p ≥ 2. It follows from Markov's inequality that MGW(C −1 o,k ≥ k 1+ ) ≤ Ck −r . (b) We claim there exist 0 < c, c < ∞ deterministic such that Assuming the claim, we have which by Thomson's principle implies C −1 o,k ≤ C T k. It remains to prove (6.7). For any 1 ≤ 1 + r ≤ 2 ∧ (p/2), Lem. 3.7 and Markov's inequality give Taking expectations and applying Lem. 6.4 then gives for r sufficiently small. But 1 is clearly bounded uniformly in k by a deterministic constant, so (6.7) is proved.