The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution

We study properties of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index $\alpha\in (1,2]$. Here the harmonic measure refers to the hitting distribution of height $n$ by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation $n$. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is roughly of order $n^{\frac{1}{\alpha-1}}$, most of the harmonic measure is supported on a boundary subset of size approximately equal to $n^{\beta_{\alpha}}$, where the constant $\beta_{\alpha}\in (0,\frac{1}{\alpha-1})$ depends only on the index $\alpha$. Using an explicit expression of $\beta_{\alpha}$, we are able to show the uniform boundedness of $(\beta_{\alpha}, 1<\alpha\leq 2)$. These are generalizations of results in a recent paper of Curien and Le Gall (arXiv: 1304.7190).


Introduction
Recently, Curien and Le Gall have studied in [6] the properties of harmonic measure on generation n of a critical Galton-Watson tree, whose offspring distribution has finite variance and which is conditioned to have height greater than n. They have shown the existence of a universal constant β < 1 such that, with high probability, most of the harmonic measure on generation n of the tree is concentrated on a set of approximately n β vertices, although the number of vertices at generation n is of order n. Their approach is based on the study of a similar continuous model, where it is established that the Hausdorff dimension of the (continuous) harmonic measure is almost surely equal to β.
In this paper, we continue the above work by extending their results to the critical Galton-Watson trees whose offspring distribution has infinite variance. To be more precise, let ρ be a non-degenerate probability measure on Z + with mean one, and we assume throughout this paper that ρ is in the domain of attraction of a stable distribution of index α ∈ (1, 2], which means that k≥0 ρ(k)r k = r + (1 − r) α L (1 − r) for any r ∈ [0, 1), where the function L(x) is slowing varying as x → 0 + . We point out that the finite variance condition for ρ is sufficient for the previous statement to hold with α = 2. When α ∈ (1, 2), by results of [9,Chapters XIII and XVII], the condition (1) is satisfied if and only if the tail probability k≥x ρ(k) = ρ([x, +∞)) varies regularly with exponent −α as x → +∞. See e.g. [4] for the definition of regularly varying functions.
Under the probability measure P, for every integer n ≥ 0, we let T (n) be a Galton-Watson tree with offspring distribution ρ, conditioned on non-extinction at generation n. Conditionally given the tree T (n) , we consider simple random walk on T (n) starting from the root. The probability distribution of the first hitting point of generation n by random walk will be called the harmonic measure µ n , which is supported on the set T (n) n of all vertices of T (n) at generation n. Let q n > 0 be the probability that a critical Galton-Watson tree T (0) survives up to generation n. It is shown in [16] that, as n → ∞, the probability q n decreases as n − 1 α−1 up to multiplication by a slowly varying function, and q n #T (n) n converges in distribution to a nontrivial limit distribution on R + , whose Laplace transform can be written explicitly in terms of parameter α. The following theorem generalizes the result [6,Theorem 1] in the finite variance case (α = 2) to all α ∈ (1, 2]. Theorem 1. If the offspring distribution ρ has mean one and belongs to the domain of attraction of a stable distribution of index α ∈ (1,2], there exists a constant β α ∈ (0, 1 α−1 ), which only depends on α, such that for every δ > 0, we have the convergence in P-probability Consequently, for every ε ∈ (0, 1), there exists, with P-probability tending to 1 as n → ∞, a subset A n,ε of T (n) n such that #A n,ε ≤ n βα+δ and µ n (A n,ε ) ≥ 1 − ε. Conversely, the maximal µ n -measure of a set of cardinality bounded by n βα−δ tends to 0 as n → ∞, in P-probability.
The last two assertions of the preceding theorem are easy consequences of the convergence (2), as explained in [6].
We observe that the hitting distribution µ n of generation n by simple random walk on T (n) is unaffected if we remove the branches of T (n) that do not reach height n. Thus in order to establish the preceding result, we may consider simple random walk on T * n , the reduced tree associated with T (n) , which consists of all vertices of T (n) that have at least one descendant at generation n.
When the critical offspring distribution ρ has infinite variance, scaling limits of the discrete reduced trees T * n have been studied in [17] and [18]. If we scale the graph distances by the factor n −1 , the discrete reduced trees n −1 T * n converge to a random compact rooted R-tree ∆ (α) that we now describe. For every α ∈ (1, 2], we define the α-offspring distribution θ α as follows. For α = 2, we let θ 2 = δ 2 be the Dirac measure at 2. If α < 2, θ α is the probability measure on Z + given by θ α (0) = θ α (1) = 0, where Γ(·) is the Gamma function. We let U ∅ be a random variable uniformly distributed over [0,1], and let K ∅ be a random variable distributed according to θ α , independent of U ∅ . To root and the boundary ∂∆ (α) is a random variable which we denote by C (α) . From a probabilistic point of view, it is the mass under the Brownian excursion measure for the excursion paths away from the root that hit height 1. Following the definition of ∆ (α) and the above electric network interpretation, the distribution of C (α) satisfies the recursive distributional equation where (C (α) i ) i≥1 are i.i.d. copies of C (α) , the integer-valued random variable N α is distributed according to θ α , and U is uniformly distributed over [0,1]. All these random variables are supposed to be independent. Proposition 4. For any α ∈ (1,2], the distribution γ α of the conductance C (α) is characterized in the class of all probability measures on [1, ∞) by the distributional equation (4). The constant β α appearing in Theorems 1 and 2 is given by Interestingly, formula (5) expresses the exponent β α as the same function of the distribution γ α , for all α ∈ (1,2]. In the course of the proof, we obtain two other formulas for β α (see (23) and (24) below), but they both depend on α in a more complicated way, which also involves the distribution θ α .
The paper is organized as follows. In Section 2 below, we study the continuous model of Brownian motion on ∆ (α) . A formal definition of the reduced stable tree ∆ (α) is given in Section 2.1. In Section 2.2 we explain how to relate ∆ (α) to an infinite supercritical continuoustime Galton-Watson tree Γ (α) , and we reformulate Theorem 2 in terms of Brownian motion with drift 1/2 on Γ (α) . Properties of the law of the random conductance C (α) , including the first assertion of Proposition 4, are discussed in Section 2.3, and Section 2.4 gives the coupling argument that allows one to derive Theorem 3 from formula (5). Section 2.5 is devoted to the proofs of Theorem 2 and of formula (5). We emphasize that our approach to Theorem 2 is different from the one used in [6] when α = 2. In fact we use an invariant measure for the environment seen by Brownian motion on Γ (α) at the last passage time of a node of the n-th generation, instead of the last passage time at a height h as in [6]. We then apply the ergodic theory on Galton-Watson trees, which is a powerful tool initially developed in [12].
In Section 3 we proceed to the discrete setting concerning simple random walk on the discrete reduced tree T * n . Let us emphasize that, when the critical offspring distribution ρ is in the domain of attraction of a stable distribution of index α ∈ (1, 2), the convergence of discrete reduced trees is less simple than in the special case α = 2 where we have a.s. a binary branching structure. See Proposition 14 for a precise statement in our more general setting. Apart from this ingredient, we need several estimates for the discrete reduced tree T * n to derive Theorem 1 from Theorem 2. For example, Lemma 13 gives a bound for the size of level sets in T * n , and Lemma 19 presents a moment estimate for the (discrete) conductance C n (T * n ) between generations 0 and n in T * n . Although the result analogous to Lemma 19 in [6] is a second moment estimate, we only manage to give a moment estimate of order strictly smaller than α if the critical offspring distribution ρ satisfies (1) with α ∈ (1, 2]. Nevertheless, this is sufficient for our proof of Theorem 1, which is adapted from the one given in [6].
Comments and several open questions are gathered in the last section. Following the work of Aïdékon [1], we obtain a candidate for the speed of Brownian motion with drift 1/2 on the infinite tree Γ (α) , expressed by (40) in terms of the continuous conductance C (α) . Nonetheless, the monotonicity properties of this quantity remains open. It would also be of interest to know whether or not the Hausdorff dimension β α of the continuous harmonic measure µ α is monotone with respect to α ∈ (1, 2].

The reduced stable tree
We set The notions of a descendant and an ancestor of an element of V are defined in the obvious way, with the convention that every v ∈ V is both an ancestor and a descendant of itself. If v, w ∈ V, v ∧ w is the unique element of V such that it is a common ancestor of v and w, and |v ∧ w| is maximal.
An infinite subset Π of V is called an infinite discrete tree if there exists a collection of Recall the definition of the α-offspring distribution θ α for α ∈ (1, 2]. It will also be convenient to consider the case α = 1, where we define θ 1 as the probability measure on Z + given by , ∀k ≥ 2.
If α ∈ (1, 2], the generating function of θ α is given (see e.g. [7, p.74 while for α = 1, Notice that for α ∈ (1, 2], the mean of θ α is given by whereas θ 1 has infinite mean. For fixed α ∈ [1, 2], we introduce a collection (K α (v)) v∈V of independent random variables distributed according to θ α under the probability measure P, and define a random infinite discrete tree We point out that Π (2) is an infinite binary tree.
Let (U v ) v∈V be another collection, independent of (K α (v)) v∈V , consisting of independent real random variables uniformly distributed over [0, 1] under the same probability measure P. We set now Y ∅ = U ∅ and then by induction, for every v ∈ Π (α) \ {∅}, Note that a.s. 0 ≤ Y v < 1 for every v ∈ Π (α) . Consider then the set There is a straightforward way to define a metric d on ∆ • If v is a descendant (or an ancestor) of w, we set d(x, y) = |r − r |.

Height 1
Height 0 We let ∆ (α) be the completion of ∆ (α) 0 with respect to the metric d. Then where by definition ∂∆ (α) := {x ∈ ∆ (α) : d((∅, 0), x) = 1}, which can be identified with a random subset of N N . It is immediate to see that (∆ (α) , d) is a compact R-tree, which we will call the reduced stable tree of index α. The point (∅, 0) is called the root of ∆ (α) . For every x ∈ ∆ (α) , we set H(x) = d((∅, 0), x) and call H(x) the height of x. We can define a genealogical order on ∆ (α) by setting x ≺ y if and only if x belongs to the geodesic path from the root to y.
For every ε ∈ (0, 1), we set which is also a compact R-tree for the metric d. The leaves of ∆ (α) ε are the points of the form Now conditionally on ∆ (α) , we can define Brownian motion on ∆ (α) ε starting from the root. Informally, this process behaves like linear Brownian motion as long as it stays on an "open interval" of the form {v} × (Yv, Y v ∧ (1 − ε)), and it is reflected at the root (∅, 0) and at the leaves of ∆ (α) ε . When it arrives at a branching point of the tree, it chooses each of the possible line segments ending at this point with equal probabilities. By taking a sequence ε n = 2 −n , n ≥ 1 and then letting n go to infinity, we can construct under the same probability measure P a Brownian motion B on ∆ (α) starting from the root, which is defined up to its first hitting time T of ∂∆ (α) . We refer the reader to [6] for the details of this construction. The harmonic measure µ α is then the distribution of B T − under P , which is a (random) probability measure on ∂∆ (α) ⊆ N N .

The continuous-time Galton-Watson tree
In this subsection, we introduce a new tree which shares the same branching structure as ∆ (α) , such that each point of ∆ (α) at height s ∈ [0, 1) corresponds to a point of the new tree at height − log(1 − s) ∈ [0, ∞) in a bijective way. As it turns out, this new random tree is a continuous-time Galton-Watson tree.
To define it, we take α ∈ [1, 2] and start with the same random infinite tree Π (α) introduced in Section 2.1. Consider now a collection (V v ) v∈V of independent real random variables exponentially distributed with mean 1 under the probability measure P. We set and then by induction, for every v ∈ Π (α) \ {∅}, The continuous-time Galton-Watson tree (hereafter to be called CTGW tree for short) of stable index α is the set which is equipped with the metric d defined in the same way as d in the preceding subsection. For this metric, Γ (α) is a non-compact R-tree. For every x = (v, r) ∈ Γ (α) , we keep the notation H(x) = r = d((∅, 0), x) for the height of the point x.
Now observe that if U is uniformly distributed over [0, 1], the random variable − log(1 − U ) is exponentially distributed with mean 1. Hence we may and will suppose that the collection (V v ) v∈V is constructed from the collection (U v ) v∈V in the previous subsection via the formula . By stochastic analysis, we can write for every t ∈ [0, T ), where (W (t)) t≥0 is Brownian motion with constant drift 1/2 towards infinity on the CTGW tree Γ (α) (this process is defined in a similar way as Brownian motion on ∆ (α) ε , except that it behaves like Brownian motion with drift 1/2 on every "open interval" of the tree). Note that again W is defined under the probability measure P . From now on, when we speak about Brownian motion on the CTGW tree or on other similar trees, we will always mean Brownian motion with drift 1/2 towards infinity.
By definition, the boundary of Γ (α) is the set of all infinite geodesics in Γ (α) starting from the root (∅, 0) (these are called geodesic rays), and it can be canonically embedded into N N . Due to the transience of Brownian motion on Γ (α) , there is an a.s. unique geodesic ray denoted by W ∞ that is visited by (W (t)) t≥0 at arbitrarily large times. We say that W ∞ is the exit ray of Brownian motion on Γ (α) . The distribution of W ∞ under P yields a probability measure ν α on N N . Thanks to (8), we have in fact ν α = µ α , provided we think of both µ α and ν α as (random) probability measures on N N . The statement of Theorem 2 is then reduced to checking that for every 1 < α ≤ 2, P-a.s., ν α (dy)-a.e.
where B(y, r) denotes the set of all geodesic rays that coincide with y up to height r.
Infinite continuous trees. To prove (9), we will apply the tools of ergodic theory to certain transformations on a space of finite-degree rooted infinite continuous trees that we now describe. We let T be the set of all pairs (Π, (z v ) v∈Π ) that satisfy the following conditions: (1) Π is an infinite discrete tree, in the sense of Section 2.1.
(2) We have In the preceding definition, we allow the possibility that z ∅ = 0. Notice that property (iii) implies that #{v ∈ Π : z v ≤ r} < ∞ for every r > 0. We equip T with the σ-field generated by the coordinate mappings. If (Π, (z v ) v∈Π ) ∈ T, we can consider the associated "tree" equipped with the distance defined as above. The set Π ∞ is identified with the collection of all geodesic rays in Π, and will be viewed as the boundary of the tree T . We keep the notation H(x) = r for the height of a point x = (v, r) ∈ T . The genealogical order on T is defined as previously and again is denoted by ≺. If u = (u 1 , u 2 , . . .) ∈ Π ∞ , and x = (v, r) ∈ T , we write x ≺ u if v = (u 1 , u 2 , . . . , u k ) for some integer k ≥ 0. We will often abuse notation and say that we consider a tree T ∈ T: This means that we are given a pair (Π, (z v ) v∈Π ) satisfying the above properties, and we consider the associated tree T . In particular, T has an order structure (in addition to the genealogical partial order) given by the lexicographical order on Π. Elements of T will be called infinite continuous trees. Clearly, for every stable index α ∈ [1,2], the CTGW tree Γ (α) can be viewed as a random variable with values in T, and we write Θ α (dT ) for its distribution.
Let us fix T = (Π, (z v ) v∈Π ) ∈ T. Under our previous notation, k ∅ is the number of offspring at the first branching point of T . We denote by T (1) , T (2) , . . . , T (k∅) the subtrees of T obtained at the first branching point. To be more precise, for every 1 ≤ i ≤ k ∅ , we define the shifted discrete tree Π[i] = {v ∈ V : iv ∈ Π}, and T (i) is the infinite continuous tree corresponding to the pair Under Θ α (dT ), we know by definition that k ∅ is distributed according to θ α . Moreover, conditionally on k ∅ , the branching property of the CTGW tree states that the subtrees T (1) , . . . , T (k∅) are i.i.d. following the same law Θ α . If r > 0, the level set of T ∈ T at height r is For α ∈ (1, 2], we have the classical result which can be derived from the following identity (see e.g. Theorem 2.7.1 in [7]) stating that for every u > 0,

The continuous conductance
Recall that, for α ∈ [1, 2], the random variable C (α) is defined as the conductance between the root and the set ∂∆ (α) in the continuous tree ∆ (α) viewed as an electric network. One can also give a more probabilistic definition of the conductance. If T is a (deterministic) infinite continuous tree, the conductance C(T ) between the root and the boundary ∂T can be defined in terms of excursion measures of Brownian motion with drift 1/2 on T . Under this definition, we can set C (α) = C(Γ (α) ) ∈ [1, ∞). For details, we refer the reader to Section 2.3 in [6].
In this subsection, we will prove for α ∈ (1, 2] that the law of C (α) is characterized by the distributional identity (4) in the class of all probability measures on [1, ∞), and discuss some of the properties of this law. For u ∈ (0, 1), n ∈ N and ( so that (4) can be rewritten as where U, N α , (C (α) i ) i≥1 are as in (4). Note that (10) also holds for α = 1. Let M be the set of all probability measures on [1, ∞] and let Φ α : M → M map a distribution σ to where (X i ) i≥1 are independent and identically distributed according to σ, while U, N α are as in (4). We suppose in addition that U, N α and (X i ) i≥1 are independent.
We write γ α for the distribution of C (α) , and define for all ≥ 0 the Laplace transform

Proposition 5.
Let us fix the stable index α ∈ (1, 2]. The law γ α of C (α) is the unique fixed point of the mapping Φ α on M , and we have Φ k α (σ) → γ α weakly as k → ∞, for every σ ∈ M . Furthermore, 1. If α = 2, all moments of γ 2 are finite, and γ 2 has a continuous density over [1, ∞). The Laplace transform ϕ 2 solves the differential equation 2. If α ∈ (1, 2), only the first and the second moments of γ α are finite. The distribution γ α has a continuous density over [1, ∞), and the Laplace transform ϕ α solves the differential equation Proof. The case α = 2 has been derived in [6, Proposition 6] and is listed above for completeness. We will prove the corresponding assertion for α ∈ (1, 2) by similar methods. Firstly, the stochastic partial order on M is defined by saying that σ σ if and only if there exists a coupling (X, Y ) of σ and σ such that a.s. X ≤ Y . It is clear that for any α ∈ [1, 2], the mapping Φ α is increasing for the stochastic partial order.
We endow the set M 1 of all probability measures on [1, ∞] that have a finite first moment with the 1-Wasserstein metric The metric space (M 1 , d 1 ) is Polish and its topology is finer than the weak topology on M 1 . From the easy bound G(u, n, (x i ) i≥1 ) ≤ x 1 + x 2 + · · · + x n and the fact that EN α < ∞ as α = 1, we immediately see that Φ α maps M 1 into M 1 when α > 1. We then observe that the mapping Φ α is strictly contractant on (M 1 , d 1 ). To see this, let (X i , Y i ) i≥1 be independent copies of a coupling between σ, σ ∈ M 1 under the probability measure P. As in (10), let U be uniformly distributed over [0,1] and N α be distributed according to θ α . Assume that U, N α and (X i , Y i ) i≥1 are independent under P. Then the two variables give a coupling of Φ α (σ) and Φ α (σ ). Using the fact Notice that for any integer k ≥ 2, Taking expected values and minimizing over the choice of the coupling between σ and σ , we get with c α < 1. So for α ∈ (1, 2], the mapping Φ α is contractant on M 1 and by completeness it has a unique fixed pointγ α in M 1 . Furthermore, for every σ ∈ M 1 , we have Φ k α (σ) →γ α for the metric d 1 , hence also weakly, as k → ∞.
Since we know from (10) that γ α is also a fixed point of Φ α , the equality γ α =γ α will follow if we can verify thatγ α is the unique fixed point of Φ α in M . To this end, it will be enough to show that we have Φ k α (σ) →γ α as k → ∞, for every σ ∈ M . For any α ∈ [1, 2], we apply Φ α to the Dirac measure δ ∞ at ∞ to see in which the integrals can be bounded as follows, k ) k≥1 are independent and identically distributed according to γ α . Then we have, for every t > 1, By definition, we have F (k) where We observe that the right-hand side of (12) is a continuous function of t ∈ (1, ∞), so that F α is continuous on [1, ∞) (the right-continuity at 1 is obvious from (13)). Thus γ α has no atom and it follows that all functions F is also continuous on [1, ∞). Using (12) again we obtain that F α is continuously differentiable on [1, ∞) and consequently γ α has a continuous density f α = −F α with respect to Lebesgue measure on [1, ∞).
Let us finally derive the differential equation (11). To this end, we first differentiate (12) with respect to t to get that the linear differential equation holds for t ∈ [1, ∞). Then let g : [1, ∞) → R + be a monotone continuously differentiable function. From the definition of F α and Fubini's theorem, we have and similarly We then multiply both sides of (14) by g (t) and integrate for t running from 1 to ∞ to get When g(x) = exp(−x /2) for > 0, we readily obtain (11) by using the generating function of N α given in (6). Finally, taking g(x) = x in (15), we get Nevertheless, by taking g(x) = x 2 in (15), we see that the third moment of The arguments of the preceding proof also yield the following lemma in the case α = 1.
Additionally, the Laplace transform ϕ 1 of the law of C (1) solves the differential equation Proof. The law of C (1) is a fixed point of the mapping Φ 1 defined via (10) with α = 1. By the same monotonicity argument that we used above, it follows that the first moment of C (1) is bounded above by the first moment of Φ 2 1 (δ ∞ ), and the calculation of this first moment in the previous proof leads to the right-hand side of (16).
As an analogue to (15), we have By taking g(x) = exp(−x /2) and using (7), one can then derive the differential equation satisfied by ϕ 1 .

The reduced stable trees are nested
In this short subsection, we introduce a coupling argument to explain how Theorem 3 follows from the identity (5) in Proposition 4.
Recall the definition of the α-offspring distribution θ α . From the obvious fact This implies that for every k ≥ 3, θ α ([2, k]) is a strictly increasing function of α ∈ (1, 2). Using the inverse transform sampling, we can construct on a common probability space a sequence of random variables (N α , α ∈ [1, 2]) such that a.s.

Proof of Theorem 2
The proof of Theorem 2 given below will follow the approach sketched in [6, Section 5.1]. We will first establish the flow property of harmonic measure (Lemma 7), and then find an explicit invariant measure for the environment seen by Brownian motion on the CTGW tree Γ (α) at the last visit of a vertex of the n-th generation (Proposition 8). After that, we will rely on arguments of ergodic theory to complete the proof of Theorem 2 and that of Proposition 4. Throughout this subsection, we fix the stable index α ∈ (1, 2] once and for all. For notational ease, we will omit the superscripts and subscripts concerning α in all the proofs involved. Recall that P stands for the probability measure under which the CTGW tree Γ (α) is defined, whereas Brownian motion with drift 1/2 on the CTGW tree is defined under the probability measure P .

The flow property of harmonic measure
We fix an infinite continuous tree T ∈ T, and write as before T (1) , T (2) , . . . , T (k∅) for the subtrees of T at the first branching point. Here we slightly abuse notation by writing W = (W (t)) t≥0 for Brownian motion with drift 1/2 on T started from the root. As in Section 2.2, W ∞ stands for the exit ray of W , and the distribution of W ∞ on the boundary of T is the harmonic measure of T , denoted as ν T . Let K be the index such that W ∞ "belongs to" T (K) and we write W ∞ for the ray of T (K) obtained by shifting W ∞ at the first branching point of T .
The proof is similar to that of [6,Lemma 7] and is therefore omitted.

The invariant measure and ergodicity
We introduce the set T * ⊆ T × N N of all pairs consisting of a tree T ∈ T and a distinguished geodesic ray v in T . Given a distinguished geodesic ray v = (v 1 , v 2 , . . .) in T , we let S(T , v) be obtained by shifting (T , v) at the first branching point of T , that is is the subtree of T rooted at the first branching point that is chosen by v.
Under the probability measure P ⊗ P , we can view (Γ (α) , W ∞ ) as a random variable with values in T * . We write Θ * α (dT dv) for the distribution of (Γ (α) , W ∞ ). The next proposition gives an invariant measure absolutely continuous with respect to Θ * α under the shift S. Proposition 8. For every r ≥ 1, set The finite measure κ α (C(T ))Θ * α (dT dv) is invariant under S. Remark. The preceding formula for κ α is suggested by the analogous formula in [6, Proposition 25] for α = 2.
Proof. First notice that the function κ is bounded, since for every r ≥ 1, Let us fix T ∈ T, then for any 1 ≤ i ≤ k ∅ and any bounded measurable function g on N N , the flow property of harmonic measure gives that Recall that Θ * (dT dv) = Θ(dT )ν T (dv) by construction. Let F be a bounded measurable function on T * . Using the preceding display, we have Observe that under Θ(dT | k ∅ = k), the subtrees T (1) , T (2) , . . . , T (k) are independent and distributed according to Θ, and furthermore, where U is uniformly distributed over [0, 1] and independent of (T (1) , T (2) , . . . , T (k) ). Using these observations, together with a simple symmetry argument, we get that the integral (17) is given by The proof is thus reduced to checking that, for every r (18) To this end, we will reformulate the last expression in the following way. Under the probability measure P, we introduce an i.i.d. sequence (C i ) i≥0 distributed according to γ, and a random variable N distributed according to θ. In addition, under the same probability measure P, let U be uniformly distributed over [0, 1], ( C i ) i≥0 be an independent copy of (C i ) i≥0 , and N be an independent copy of N . We assume that all these random variables are independent. Note that by definition, for every r ≥ 1, It follows that (18) can be written as is independent of (C i ) i≥0 and N . By (4), the random variable C is also distributed according to γ. So the right-hand side of the last long display is equal to κ(r), which completes the proof of the proposition.

Proposition 9. The shift S acting on the probability space
Proof. Our arguments proceed in a similar way as in the proof of [6, Proposition 13]. We define a transition kernel p(T , dT ) on T by setting Informally, under the probability measure p(T , dT ), we choose one of the subtrees of T obtained at the first branching point, with probability equal to its harmonic measure. For every integer n ≥ 1, we denote by S n the mapping on T * obtained by iterating n times the shift S, and then we consider the process (Z n ) n≥0 on the probability space (T * , for every n ≥ 1. According to Proposition 8 and the flow property of harmonic measure, the process (Z n ) n≥0 is a Markov chain with transition kernel p under its stationary measure Υ(dT ).
We write T ∞ for the set of all infinite sequences (T 0 , T 1 , . . .) of elements in T, and let T ∞ be the set of all infinite sequences (T 0 , T 1 , . . .) in T ∞ , such that, for every integer j ≥ 1, T j is one of the subtrees of T j−1 above the first branching point of Notice that v is a priori not unique, but to make the previous definition rigorous we can take the smallest possible v in lexicographical ordering (of course for the random trees that we consider later this uniqueness problem does not arise). In this way, we define a measurable mapping φ from T ∞ into T * such that

Proof of Theorem 2
Having established Proposition 8 and Proposition 9, we can now apply the ergodic theorem to the two functionals on T * defined as follows. First let J n (T , v) denote the height of the n-th branching point on the geodesic ray v. One immediately verifies that, for every n ≥ 1, If M = κ(C(T ))Θ * (dT dv), it follows from the ergodicity that Θ * -a.s., Note that the limit can be written as with the notation used in the proof of Proposition 8.
Secondly, let x n,v denote the n+1-st branching point on the geodesic ray v. v)) with the notation of Section 2.2. We set for every n ≥ 1, By the flow property of harmonic measure (Lemma 7), we have and by the ergodic theorem, Θ * -a.s., where the limit can be written as By combining (20) and (21), we obtain that the convergence (9) holds with limit Proof. We use the notation can be obtained by applying Theorem III.8.3 in [2] together with (6). In particular, it follows from a Tauberian theorem (cf. [9, Chapter XIII.5]) that | log W(T )|Θ(dT ) < ∞.
Let T (1) , . . . , T (k∅) be the subtrees of T at the first branching point, and let J(T ) = J 1 (T , v) be the height of the first branching point. Then, Θ(dT )-a.s.
so that we can define a probability measure w T on {1, 2, . . . , k ∅ } by setting On the other hand, for 1 ≤ i ≤ k ∅ , let ν * T (i) denote the mass assigned by the harmonic measure ν T to the rays "contained" in T (i) , that is, .
By a concavity argument, and the inequality is strict with positive Θ-probability.
Recall that Υ(dT ) = M −1 κ(C(T ))Θ(dT ) is the image of the probability measure Υ * (dT dv) under the canonical projection π 1 from T * to T. According to the discussion before Proposition 10, we can write , which by (22) is strictly smaller than However, it follows from the definition of w T that where in the last equality we used the fact that Υ * is invariant under the shift S, and that log W(T ) is integrable under Θ(dT ) hence also under Υ * . Therefore, we have shown β < 1 α−1 and the proof of Theorem 2 is completed.

Proof of Proposition 4
We have seen above that On account of Proposition 5, the proof of Proposition 4 will be completed if we can verify that the preceding expression for β is consistent with formula (5). In the following calculations, we will keep using the same notation introduced in the proof of Proposition 8. Firstly, the numerator of the right-hand side of (23) is equal to .
For every integer k ≥ 2, we define for x ∈ (1, ∞) the function where u ∈ (0, 1) and c 1 , . . . , c k ∈ (1, ∞). We can apply (15) to get With help of the last display, the numerator of the right-hand side of (23) becomes We now integrate with respect to U and recall that for a, b, c > 0, 1 0 du a (b+cu) 2 = a b(b+c) . So the numerator of the right-hand side of (23) coincides with On the other hand, the denominator of the right-hand side of (23) is equal to where we have repeatedly used (15) in the last two equalities, the first time to replace C 1 +· · ·+ CÑ by C 0 , the second time to replace C 1 + · · · + C N by C 1 . In order to integrate with respect to U , we appeal to the identity that for a, b, c > 0, Applying this formula, we see that the denominator of the right-hand side of (23) coincides with We have thus obtained the following formula By a symmetry argument, the numerator of the right-hand side of (24) is equal to where we have set, for every x ≥ 1, (15), and similarly for g, to obtain Plugging this into (25) yields the required formula (5), and hence finishes the proof of Proposition 4.

A second approach to Theorem 2
In this section, we outline a different approach to Theorem 2, which contains certain intermediate results of independent interest. This approach involves an invariant measure for the environment seen by Brownian motion on the CTGW tree Γ (α) at the last visit of a fixed height. This is similar to Section 3 of [6], and for this reason we will leave the proofs to Section 5. We fix the index α ∈ (1, 2], and we first introduce some additional notation. For T ∈ T and r > 0, if x ∈ T r , let T [x] denote the subtree of descendants of x in T . To define it formally, we write v x for the unique element of V such that x = (v x , r), and define the shifted discrete tree is the infinite continuous tree corresponding to the pair For a fixed r > 0, we know that Γ (α) has a.s. no branching point at height r. As there is a unique point x ∈ Γ (α) r such that x ≺ W ∞ , we write Γ (α) r = Γ (α) [x] for the subtree above level r selected by the harmonic measure.
To describe the distribution of Γ (α) r , recall that for every x ≥ 0, Proposition 11. The distribution under P ⊗ P of the subtree Γ (α) r above level r selected by the harmonic measure is Here X = (X s ) 0≤s≤r stands for the solution of the stochastic differential equation that starts under the probability measure P (c) with an exponential distribution of parameter c/2.

In the previous SDE, (η s ) s≥0 denotes a standard linear Brownian motion.
Now we define shifts (τ r ) r≥0 on T * in the following way. For r = 0, τ 0 is the identity mapping of T * . For r > 0 and (T , v) ∈ T * , we write v = (v 1 , v 2 , . . .) and v n = (v 1 , . . . , v n ) for every n ≥ 0 (by convention, v 0 = ∅). Also let x r,v be the unique element of T r such that x r,v ≺ v. Then we set where k = min{n ≥ 0 : z vn ≥ r}. Informally, τ r (T , v) is obtained by taking the subtree of T consisting of descendants of the vertex at height r on the distinguished geodesic ray, and keeping in this subtree the "same" geodesic ray. It is straightforward to verify that τ r • τ s = τ r+s for every r, s ≥ 0. The next proposition gives an invariant measure absolutely continuous with respect to Θ * α under the shifts τ r . To simplify notation, we set first

Proposition 12.
For every c > 0, The probability measure Λ * α on T * defined as is invariant under the shifts τ r , r ≥ 0.
Furthermore, one can easily adapt the proof of Proposition 13 in [6] to show that for every r > 0, the shift τ r acting on the probability space (T * , Λ * α ) is ergodic. Applying Birkhoff's ergodic theorem to a suitable functional (see Section 3.4 of [6]) leads to the convergence (3) in Theorem 2, with β α given by formula (24). See Section 5 for more details.

Galton-Watson trees
Let us first introduce discrete (finite) rooted ordered trees, which are also called plane trees in combinatorics. A plane tree t is a finite subset of V such that the following holds: (1) ∅ ∈ t .
In this section we will say tree instead of plane tree for short. The same notation and terminology introduced at the beginning of Section 2.1 will be used in this section: |u| is the generation of u, uv denotes the concatenation of u and v, ≺ stands for the genealogical order and u ∧ v is the maximal element of {w ∈ V : w ≺ u and w ≺ v}. A vertex with no child is called a leaf. The height of a tree t is h(t) := max{|v| : v ∈ t}.
We write T for the set of all trees, and T n for the set of all trees with height n.
We view a tree t as a graph whose vertices are the elements of t and whose edges are the pairs { u, u} for all u ∈ t\{∅}. The set t is equipped with the distance Notice that this is half the usual graph distance. We will write B t (v, r), or simply B(v, r) if there is no ambiguity, for the closed ball of radius r centered at v, with respect to the distance d in the tree t.
The set of all vertices of t at generation n is denoted by t n := {v ∈ t : |v| = n}.
If v ∈ t, the subtree of descendants of v is

Note that t[v]
is not a tree under the previous definition, but we can turn it into a tree by relabeling its vertices as If v ∈ t, then for every i ∈ {0, 1, . . . , |v|} we write v i for the ancestor of v at generation i. {0, 1, . . . , n}. This simple observation will be used repeatedly below.
Let ρ be a non-trivial probability measure on Z + with mean one, which belongs to the domain of attraction of a stable distribution of index α ∈ (1, 2]. Therefore property (1) holds. For every integer n ≥ 0, we let T (n) be a Galton-Watson tree with offspring distribution ρ, conditioned on non-extinction at generation n, viewed as a random subset of V (see e.g. [11] for a precise definition of Galton-Watson trees). In particular, T (0) is just a Galton-Watson tree with offspring distribution ρ. We suppose that the random trees T (n) are defined under the probability measure P.
We let T * n be the reduced tree associated with T (n) , which consists of all vertices of T (n) that have (at least) one descendant at generation n. Note that |v| ≤ n for every v ∈ T * n . A priori T * n is not a tree in the sense of the preceding definition. However we can relabel the vertices of T * n , preserving both the lexicographical order and the genealogical order, so that T * n becomes a tree in the sense of our definitions. We will always assume that this relabeling has been done.
Conditionally on T (n) , the hitting distribution of generation n is the same for simple random walk on T (n) and that on the reduced tree T * n . In view of studying properties of this hitting distribution, we can consider directly a simple random walk on T * n starting from the root ∅, which we denote by Z n = (Z n k ) k≥0 . This random walk is defined under the probability measure P . Let H n := inf{k ≥ 0 : |Z n k | = n} be the first hitting time of generation n by Z n , and set Σ n = Z n Hn to be the hitting point. The discrete harmonic measure µ n is the law of Σ n under P , which is a (random) probability measure on the level set T * n n . Set q n = P h(T (0) ) ≥ n . If L is the slowly varying function appearing in (1), it has been established in [16,Lemma 2] that By the asymptotic inversion property of slowly varying functions (see e.g. [4, Section 1.5.7]), it follows that for a function slowing varying at ∞. Moreover, it is shown in [16, Theorem 1] that, as n → ∞, q n #T * n n converges in distribution to the positive random variable W(Γ (α) ) introduced in the proof of Proposition 10.
We will need to estimate the size of level sets in T * n . The following lemma is an analogue of Lemma 15 in [6].

Lemma 13.
For every r ≥ 1, there exists a constant C = C(r, ρ) depending on r and the offspring distribution ρ such that, for every integer n ≥ 2 and every integer p ∈ [1, n/2], E (log #T * n n−p ) r 1 r ≤ C log n p and E (log #T * n n ) r 1 r ≤ C log n.
Proof. We can find a = a(r) > 0 such that the function x → (log(a + x)) r is concave over [1, ∞). Then as in the proof of [6,Lemma 15], Using Potter's bounds on slowly varying function (see e.g. [4, Theorem 1.5.6]), one can deduce from (27) that there exists a constant C = C (ρ) > 0 such that for every n ≥ 2 and every p ∈ [1, n/2], log q p q n ≤ C log n p , from which the first bound of the lemma easily follows. The second estimate can be shown in a similar way.
The goal of this section is to prove Theorem 1. We will assume in the rest of this section that the critical offspring distribution ρ satisfies (1) with a fixed α ∈ (1, 2]. Accordingly, we will omit the superscripts and subscripts concerning α if there is no ambiguity.

Convergence of discrete reduced trees
We first define truncations of the discrete reduced tree T * n . For every s ∈ [0, n], we set R s (T * n ) := v ∈ T * n : |v| ≤ n − s .
Recall from Section 2.1 the definition of the continuous reduced tree ∆ of index α. For every ε ∈ (0, 1), we have set ∆ ε = {x ∈ ∆ : H(x) ≤ 1 − ε}. We will implicitly use the fact that, for every fixed ε, there is a.s. no branching point of ∆ at height 1 − ε. The skeleton of ∆ ε is defined as the following plane tree Let t be a tree. We write S(t) for the set of all vertices of t whose number of children is different from 1. Then we can find a unique tree [t] ∈ T such that there exists a bijection from [t] onto S(t) that preserves the genealogical order and the lexicographical order of vertices. Denote the inverse of this canonical bijection by u ∈ S(t) → [u] ∈ [t]. In a less formal way, [t] is just the tree obtained from t by removing all vertices that have exactly one child.

Proposition 14.
We can construct the reduced trees T * n and the (continuous) reduced stable tree ∆ on the same probability space (Ω, F, P), so that the following assertions hold for every fixed ε ∈ (0, 1) with P-probability one.
(1) For every sufficiently large integer n, there exists an injective mapping Ψ ε n : u → w n,ε u from Sk(∆ ε ) into S(R εn (T * n )) satisfying the following properties. (1.c) For every vertex u of Sk(∆ ε ), where u denotes the parent of u in Sk(∆ ε ), and w n,ε u stands for the vertex in (Notice that w n,ε u does not necessarily coincide with w n,ε u .) (2) The mapping Ψ ε n is asymptotically unique in the sense that, if Ψ ε n is another mapping such that the preceding properties hold, then for n sufficiently large, Ψ ε n (u) = Ψ ε n (u) for every u ∈ Sk(∆ ε ). On the left, the tree ∆, its truncation ∆ ε and its skeleton Sk(∆ ε ). On the right, a large reduced tree T * n of height n, its truncation R εn (T * n ) and the associated tree [R εn (T * n )]. The vertices depicted as filled red disks on the left correspond to the vertices depicted as filled red squares on the right, via the mapping Ψ ε n .
Proposition 14 (see Figure 2 for an illustration) essentially results from the convergence in distribution of the rescaled contour functions associated with the trees T (n) towards the excursion of the stable height process with height greater than 1 (see [7,Section 2.5]). By using the Skorokhod representation theorem, one may assume that the trees T (n) and the excursion of the stable height process are constructed so that the latter convergence holds almost surely. The various assertions of Proposition 14 then easily follow (cf. [7, Section 2.6]), using the relation between the excursion of the stable height process with height greater than 1 and the limiting reduced tree ∆, which can be found in [7, Section 2.7]. Remark 1. Let us take 0 < δ < ε. If u is not a leaf of Sk(∆ ε ), we must have w n,ε u = w n,δ u for sufficiently large n. On the other hand, if u is a leaf of Sk(∆ ε ), then for large n, w n,ε u must be an ancestor of w n,δ u . Remark 2. We expect that a result more precise than Proposition 14 should hold. For all sufficiently large n, the mapping Ψ ε n should be a bijection, and the equality w n,ε u = w n,ε u should hold for all u ∈ Sk(∆ ε ) (in other words, there should be no white square in the right part of Figure 2). However this refinement does not easily follow from the results of [7], and we will omit it since it is not needed for our purposes.

Convergence of harmonic measures
Recall that µ is the continuous harmonic measure on the boundary ∂∆ of the reduced stable tree, and that µ n is the discrete harmonic measure on T * n n . For every x ∈ ∂∆ ε , we set Similarly, we define a probability measure µ ε n on T * n n− εn by setting µ ε n (u) = µ n ({v ∈ T * n n : u ≺ v}), for every u ∈ T * n n− εn . Clearly, µ ε n is the distribution of Σ n n− εn .

Proposition 15.
Suppose that the reduced trees T * n and the (continuous) tree ∆ have been constructed so that the properties of Proposition 14 hold, and recall the notation (w n,ε u ) u∈Sk(∆ε) introduced therein. Then P-a.s. for every Proof. Let δ ∈ (0, ε) and set T δ = inf{t ≥ 0 : H(B t ) = 1 − δ} < T . Define a probability measure µ ε,(δ) on ∂∆ ε by setting for every x ∈ ∂∆ ε , Similarly, we write µ (δ) n for the distribution of the hitting point of generation n− δn by random walk on T * n started from ∅. Then we define a probability measure µ ε,(δ) n on T * n n− εn by setting for every v ∈ T * n n− εn . As in the proof of [6, Proposition 18], we have P-a.s.
So the convergence of the proposition will follow if we can verify that for every fixed δ ∈ (0, ε), we have P-a.s. for every To this end, we may and will assume that the reduced trees T * n and the (continuous) tree ∆ have been constructed so that the properties of Proposition 14 hold simultaneously for ε and for δ. Firstly, by considering the successive passage times of Brownian motion stopped at time T δ in the set {(u, Y u ∧ (1 − δ)) : u ∈ Sk(∆ δ )}, we get a Markov chain X (δ) , which is absorbed in the set {(v, 1 − δ) : v is a leaf of Sk(∆ δ )}, and whose transition kernels are explicitly described in terms of the quantities Y u , u ∈ Sk(∆ δ ) by series and parallel circuits calculation.
Secondly, let n be sufficiently large so that assertions (1) and (2) of Proposition 14 hold with ε as well as with δ, and consider simple random walk on T * n started from ∅ and stopped at the first hitting time of generation n − δn . By considering the successive passage times of this random walk in the set S(R δn (T * n )), we again get a Markov chain X (δ),n , which is absorbed in the set v ∈ S(R δn (T * n )) : [v] is a leaf of [R δn (T * n )] .
By property (1.b) of Proposition 14, this set is exactly {w n,δ v : v is a leaf of Sk(∆ δ )}. As previously, the transition kernels of this Markov chain X (δ),n can be written explicitly in terms of the quantities |v|, v ∈ S(R δn (T * n )).
Write X (δ) ∞ for the absorption point of X (δ) , and similarly write X (δ),n ∞ for that of X (δ),n . Notice that X (δ),n ∞ is also the absorption point of the restricted Markov chain X (δ),n . We thus obtain that the distribution of X (δ),n ∞ converges to that of X (δ) ∞ (recall that both X (δ),n ∞ and X (δ) ∞ are viewed as taking values in the set of leaves of Sk(∆ δ )). Consequently, for every u ∈ V such that x = (u, 1 − ε) ∈ ∂∆ ε , we have However, from our definitions, we have and, for n sufficiently large, since w n,ε u coincides with the ancestor of w n,δ u at generation n − εn (see Remark 1 after Proposition 14), This completes the proof of (28) and of the proposition.
Recall that β is the Hausdorff dimension of the continuous harmonic meaure µ.

Proposition 16.
Let r ≥ 1 and ξ ∈ (0, 1). We can find ε 0 ∈ (0, 1/2) such that the following holds. For every ε ∈ (0, ε 0 ), there exists n 0 ≥ 0 such that for every n ≥ n 0 , Proof. Recall our notation B d (x, r) for the closed ball of radius r centered at x ∈ ∆. Fix η ∈ (0, 1). Since B T − is distributed according to µ, it follows from Theorem 2 that there exists ε 0 ∈ (0, 1/2) such that for every ε ∈ (0, ε 0 ) we have Let us fix ε ∈ (0, ε 0 ) and assume that the reduced trees T * n and the (continuous) tree ∆ have been constructed so that the properties of Proposition 14 hold. We now claim that, under P ⊗ P , To see this, let f be a continuous function on [0, 1]. Since the distribution of Σ n n− εn under P is µ ε n , we have By property (1.b) of Proposition 14, we know that P-a.s. for n sufficiently large, and by Proposition 15 the latter quantities converge as n → ∞ towards which establishes the convergence (30) as claimed. By (29) and (30), we can find n 0 = n 0 (ε) ≥ ε −1 such that for n ≥ n 0 , P ⊗ P log µ ε n Σ n n− εn − β log ε > η |log ε| < η.
Using the Cauchy-Schwarz inequality, we have then Since r ≥ 1, the function We now use Lemma 13 to see By combining the last estimate with (31), we get that, for every n ≥ n 0 (ε), The statement of the proposition follows since η was arbitrary.

The flow property of discrete harmonic measure
We briefly recall the flow property of the discrete harmonic measure µ n presented in [6, Section 4.3.1]. Let t ∈ T n be a plane tree of height n and Z (t) = (Z (t) k ) k≥0 be simple random walk on t starting from ∅. We set We write µ (t) n for the distribution of Σ (t) n , considered as a measure on t supported on t n . For 0 ≤ p ≤ n, we set ], and therefore Z (t) Lemma 17 (Lemma 20 in [6]). Let p ∈ {0, 1, . . . , n − 1} and z ∈ t p . Then, conditionally on Σ (t) is distributed as simple random walk on t[z] starting from z and conditioned to hit t[z] ∩ t n before returning to z, and stopped at this hitting time. Consequently, for every integer q ∈ {0, 1, . . . , n − p}, the conditional distribution of n−p , q) .

The subtree selected by the discrete harmonic measure
We begin by introducing the conductance of discrete trees. Let i be a positive integer and let t ∈ T be a tree such that h(t) ≥ i. Consider the new graph t obtained by adding to the graph t an edge between the root ∅ and an extra vertex ∂. We denote by C i (t) the effective conductance between ∂ and generation i of t in the graph t . In probabilistic terms, it is equal to the probability that simple random walk on t starting from ∅ hits generation i of t before hitting the vertex ∂.
Recall that for i ∈ {1, . . . , n − 1}, T * n [ Σ n n−i ] is the subtree of T * n above generation n − i that is selected by harmonic measure, and T * n [ Σ n n−i ] is the tree obtained by relabeling the vertices of T * n [ Σ n n−i ] as explained above.
This lemma is proved in [6] under the assumption that ρ has finite variance. Actually the proof uses only the branching property of Galton-Watson trees and remains valid under our assumptions on ρ.
Meanwhile, we have the following moment estimate for the conductance C n (T * n ).

Lemma 19.
For every r ∈ (0, α), there exists a constant K = K(r, ρ) ≥ 1 depending on r and the offspring distribution ρ such that, for every integer n ≥ 1, Proof. We can assume n ≥ 2, and set j = n/2 ≥ 1. An application of the Nash-Williams inequality [14,Chapter 2] gives On the other hand, Notice that given #T At this point, we need the following result proved in [10,Lemma 11] for the unconditioned Galton-Watson tree. For any γ ∈ (0, α), there is a finite constant C(γ) such that for every m ≥ 1, The original statement of the latter bound in [10] was given for any γ ∈ [1, α), while the case γ ∈ (0, 1) follows from the (trivial) case γ = 1 by applying the Hölder inequality to (we can in fact take C(γ) = 1 for any γ ∈ (0, 1)).
With the help of (32), we conclude that , and the statement of the lemma readily follows from (27).

Proof of Theorem 1
Following [6], we will show which is sufficient for establishing Theorem 1. The proof given below is adapted from [6,Section 4.3.2]. For later convenience, we introduce the notation and its Hölder conjugate Fix ξ > 0. Let ε > 0 and n 0 ≥ 1 be such that the conclusion of Proposition 16 holds for every n ≥ n 0 with the exponent r = α * . Without loss of generality, we may and will assume that ε = 1/N , for some integer N ≥ 4, which is fixed throughout the proof. We also fix a constant γ > 0, such that γ log N < 1/2.
Let n > N be sufficiently large so that N γ log n ≥ n 0 . Then we let ≥ 1 be the unique integer such that N < n ≤ N +1 , and write N )). (34) To simplify notation, we set A n +1 := log µ n (B(Σ n , N )) + β log(n/N ) .
From (34), we see that We will bound each term in the sum of the right-hand side.
First step: A priori bounds. We verify that, for every j ∈ {1, 2, . . . , + 1}, where C = C(α * , ρ) is the constant in Lemma 13 for the exponent r = α * , and K = K(ᾱ, ρ) is the constant in Lemma 19 for the exponent r =ᾱ. Suppose first that 2 ≤ j ≤ . Using the second assertion of Lemma 17, with p = n − N j and q = N j−1 , we obtain that, for every z ∈ T * n n−N j , the conditional distribution of A n j under P , knowing that Σ n n−N j = z, is the same as the distribution of log µ Recalling that µ where for any tree t ∈ T N j , As explained in [6], we have the entropy bound G j (t) ≤ log #t N j −N j−1 for any tree t ∈ T N j . So we get from (37) that using successively Lemma 18, the Hölder inequality and Lemma 19. Finally, Lemma 13 gives and this completes the proof of (36) when 2 ≤ j ≤ . The cases j = 1 and j = + 1 can be treated in a similar manner. For details we refer the reader to [6,Section 4.3.2].
Second step: Refined bounds. Let us prove that, if γ log n ≤ j ≤ , Recall that for j ∈ { γ log n , . . . , } we have N j ≥ n 0 . From (37), we have where, if t ∈ T N j , Using Lemma 18 as in the first step, we have . We then apply the Hölder inequality together with the bound of Lemma 19 for r =ᾱ to get where the last equality follows from the definition of the measure µ ε n at the beginning of Section 3.3. Now recall that 1/N = ε and note that N j − N j−1 = N j − εN j . Since we have N j ≥ n 0 , we can apply Proposition 16 with r = α * and get that the right-hand side of the preceding display is bounded above by K 1/ᾱ ξ 1/α * log N , which finishes the proof of (38).
By combining (36) and (38), and using (35), we arrive at the bound which holds for every sufficiently large n. By choosing ξ and then γ arbitrarily small, we see that our claim (33) follows from the last bound, and this completes the proof of Theorem 1.

Comments and questions
Following [6, Section 5.2], let us consider the supercritical offspring distribution θ We let T towards root (see [13] or [1] for a precise definition of this process). Then the rescaled process will converge in distribution, as n → ∞, to Brownian motion (W (t)) t≥0 with drift 1/2 on the CTGW tree Γ (α) , in a sense that can easily be made precise. Furthermore, the rescaled conductance n C(T (n) α , λ (n) ) converges in distribution to the conductance C (α) = C(Γ (α) ). Following this informal passage to the limit, we can find heuristically a candidate for the limit of nV where C (α) 0 and C (α) 1 are two independent copies of C (α) under the probability measure P. Since the conductance C (α) is a.s. strictly larger than 1, we see immediately from (40) that V α < 1 2 for any α ∈ (1, 2]. On the other hand, according to the coupling explained in Section 2.4, the denominator of the right-hand side of (40) is increasing with respect to α.

Question 1.
If we apply the coupling explained in Section 2.4, does the derivative d dα C (α) of the conductance with respect to α exist almost surely?
An affirmative answer to Question 1 would allow us to take the derivative of the numerator in (40) with respect to α, and to see that Hence, the numerator in the right-hand side of (40) would be decreasing with respect to α, and so would be the speed V α . Question 2. Does the speed V α decrease with respect to α?
A similar question was raised in [3], concerning the monotonicity of the speed with respect to the offspring distribution for biased random walk on Galton-Watson trees with no leaves. It has been proved in [15] that this monotonicity holds for high values of bias.
Finally, we also want to ask the same question for the Hausdorff dimension of the continuous harmonic measure. Before starting the proofs, we state first a useful "spine" decomposition of the CTGW tree Γ (α) for α ∈ (1, 2], which is a reformulation of the standard results about the size-biased Galton-Watson trees, see e.g. [5]. Recall that m α = α α−1 is the mean of the α-offspring distribution θ α . The size-biased α-offspring distribution θ α is then defined as We take T = (Π, (z v ) v∈Π ) ∈ T. If 0, x denotes the geodesic segment in T between the root and x, we can define the subtrees of T branching off 0, x . To this end, set n x = |v x | and let v x,0 = ∅, v x,1 , . . . , v x,nx = v x be the successive ancestors of v x from generation 0 to generation n x . For every 1 ≤ i ≤ n x set r x,i = z v x,i−1 , and write k x,i = k v x,i−1 − 1 as the number of siblings of v x,i in Π. Then, for every 1 ≤ i ≤ n x and 1 ≤ j ≤ k x,i , the j-th subtree branching off the ancestral line 0, x at v x,i−1 , which is denoted by T x,i,j , corresponds to the pair whereṽ x,i,j is the j-th child of v x,i−1 different from v x,i . To simplify notation, we introduce the point measure which belongs to the set M p (R + × T) of all finite point measures on R + × T.
Lemma 20. Fix α ∈ (1, 2]. Let F be a nonnegative measurable function on T, and let H be a nonnegative measurable function on M p (R + × T). For r > 0, where we assume that, under the probability measure P, is a Poisson point measure on R + × N × T N with intensity

Figure 3: Schematic representation of the spine decomposition
From now on we fix the stable index α ∈ (1, 2]. Unless otherwise specified, we will omit the superscripts and subscripts concerning α in the following proofs.

Proof of Proposition 11
Let F be a nonnegative measurable function on T, and consider the quantity where the notation E ⊗ E means that we consider the expectation first under the probability measure P (under which the Brownian motion W is defined) and then under P. Let W x,R be Brownian motion with drift 1/2 on Γ x,R . We assume that W x,R is reflected both at the root and at the leaves of Γ x,R , which are the points y of Γ\ Γ[x] such that H(y) = R. Write ( x,R t ) t≥0 for the local time process of W x,R at x. From excursion theory, x,R ∞ has an exponential distribution with parameter C(Γ[x])/2. For details, we refer the reader to [6, Section 3.1].
We then consider for every a ∈ [0, r] the local time process (L a,R t ) t≥0 of W x,R at the unique point of 0, x at distance a from the root. Note in particular that L r,R t = x,R t . As a consequence of a classical Ray-Knight theorem, conditionally on x,R ∞ = , the process (L r−a,R ∞ ) 0≤a≤r is distributed as the process (X a ) 0≤a≤r which solves the stochastic differential equation where (η a ) a≥0 is a standard linear Brownian motion. In what follows, we will write P for the probability measure under which the process X starts from , and P (c) for the probability measure under which the process X starts with an exponential distribution with parameter c/2. Now write {x i , 1 ≤ i ≤ n x } for the branching points of Γ x,R (or equivalently of Γ) that belong to 0, x , and set a i = H(x i ) for 1 ≤ i ≤ n x . We denote by k i the number of subtrees branching off 0, x at x i , and write (Γ x,R i,j ) 1≤j≤k i for the finite subtrees of Γ x,R that branch off 0, x at x i . See Figure 4 for an illustration. Let A x,R be the event that W x,R never hits the leaves of Γ x,R , or equivalently that W x,R escapes to infinity in Γ[x] before hitting any leaf of Γ x,R . Excursion theory shows that where C(Γ x,R i,j ) refers to the conductance of Γ x,R i,j between its root x i and the set of its leaves (this conductance is defined by an easy adaptation of the definition given at the beginning of Section 2.3).
From the preceding observations, we have thus We can now return to the computation of the quantity I r defined in (42).
Note that the quantity inside the sum over x ∈ Γ r is a function of Γ[x] and of the subtrees of Γ branching off the segment 0, x . We can thus apply Lemma 20 to get where the constant m = α α−1 is the mean of α-offspring distribution θ. Under the probability measure P, the random measure N is a Poisson point measure on R + × N × T N with intensity where we recall that for every x ≥ 0, is the Laplace transform (evaluated at x/2) of the distribution of the conductance C(Γ). Observe that for any r ∈ (0, 1), the identity ∞ k=1 θ(k)r k = 1 − (1 − r) α−1 follows by differentiating (6). Applying this to (45), we have thus proved Proposition 11.

Proof of Proposition 12
In order to study the asymptotic behavior of Φ r when r tends to +∞, we first observe that, in terms of the law γ(ds) of C(Γ), we have ϕ( ) = We can generalize the last identity via a minor extension of the calculations of the preceding subsection. Let L 0 ∞ be the total local time accumulated by the process W at the root of Γ. Fix r > 0 and take a nonnegative measurable function F on T. Let h be a bounded nonnegative continuous function on (0, ∞). As an analogue of I r in the preceding subsection, we set The same calculations that led to (44) give, for every x ∈ Γ r and R > r, When R → ∞, L 0,R ∞ converges to L 0 ∞ , and so we get We then sum over x ∈ Γ r and integrate with respect to P. By the same manipulations as in the previous proof, we arrive at Note that if F = 1, since given Γ = T the local time L 0 ∞ follows an exponentiel distribution with parameter C(T )/2, and we use the same calculation as in (47). Hence the case F = 1 of (48) gives By a standard truncation argument, this identity also holds if h is unbounded.

Lemma 22. The process
where η is the linear Brownian motion in (43). An application of Girsanov's theorem shows that under Q t , the process is a linear Brownian motion over the time interval [0, t]. Furthermore, on the same time interval [0, t], the process X satisfies the stochastic differential equation dX s = 2 X s d η s + 2X s 1 + 2ϕ (X s ) ϕ (X s ) ds + (2 − X s ) ds, or equivalently, using (11), Notice that the function is continuously differentiable over [0, ∞), takes negative values on (0, ∞) and vanishes at 0. Pathwise uniqueness, and therefore also weak uniqueness, holds for (51) by an application of the classical Yamada-Watanabe criterion. The preceding considerations show that, under the probability measure Q t and on the time interval [0, t], the process X is distributed as the diffusion process on [0, ∞) started from , with generator Write X for this diffusion process, and assume that X starts from under the probability measure P . Note that 0 is an entrance point for X, but independently of its starting point, X does not visit 0 at a positive time. By comparing the solutions of (43) and (51), we know that X is recurrent on (0, ∞). We next observe that, by (49) and a few lines of calculations, the finite measure λ on (0, ∞) defined by λ(d ) := ϕ ( ) 2 e /2 d is invariant for X. We normalize λ by setting λ = λ λ ((0, ∞)) .
It is then easy to prove that the distribution of X t under P converges weakly to λ as t → ∞, for any ≥ 0. Consequently, for any bounded continuous function g on [0, ∞), and every ≥ 0, E g( X t ) −→ t→∞ g d λ. (52) By the same argument as in [6, Section 3.2], the preceding convergence remains true if g is a continuous, increasing and nonnegative function such that g d λ < ∞.
We can thus apply (52) to the function g( ) = − 1 ϕ ( ) e − /2 , which satisfies the desired properties and in particular g dλ = − ϕ ( ) d = 1. For this function g, It follows from (52) that, for every ≥ 0, which gives the first assertion of the lemma. The second assertion of Lemma 21 can be shown in the same way as in [6].
By definition, we have From Lemma 21 and an application of the dominated convergence theorem, we get By a straightforward calculation, the preceding limit is identified with Φ ∞ (c) defined in the statement of Proposition 12. Finally, with all the ingredients prepared above in this appendix, we can show the invariance of Λ * (dT dv) under the shifts (τ r , r ≥ 0) in the same way as in [6,Proposition 12], and the proof of Proposition 12 is therefore completed.

Another derivation of formula (24)
Recall that ν T stands for the harmonic measure of a tree T ∈ T. For every r > 0, we consider the nonnegative measurable function G r defined on T * by the formula where B T (v, r) denotes the set of all geodesic rays of T that coincide with the ray v over the interval [0, r]. The flow property of harmonic measure (cf. Lemma 7 in [6]) implies that, for every r, s > 0, we have G r+s = G r + G s • τ r .
Note that we used the fact that θ has a finite first moment. Sine the limit in the preceding display is finite, we can use (54) and Fatou's lemma to get that Λ * (G 1 ) < ∞, and then use (53) (to justify dominated convergence) and (54) again to obtain that β = Λ * (G 1 ) = lim n→∞ n Λ * (H 1/n ) coincides with the right-hand side of (56). Using the expression of Φ ∞ , we can therefore reformulate β as in formula (24).