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

Abstract

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 α ∈ (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 1 α−1 , most of the harmonic measure is supported on a boundary subset of size approximately equal to n βα , where the constant β α ∈ (0, 1 α−1 ) depends only on the index α. Using an explicit expression of β α , we are able to show the uniform boundedness of (β α , 1 < α ≤ 2). These are generalizations of results in a recent paper of Curien and Le Gall [6]. construct ∆ (α) , one starts with an oriented line segment of length U ∅ , whose origin will be the root of the tree. We call K ∅ the offspring number of the root ∅. Correspondingly, at the other end of the first line segment, we attach the origins of K ∅ oriented line segments with respective lengths U 1 , U 2 , . . . , U K∅ , such that, conditionally given U ∅ and K ∅ , the variables U 1 , U 2 , . . . , U K∅ are independent and uniformly distributed over [0, 1 − U ∅ ]. This finishes the first step of the construction. In the second step, for the first of these K ∅ line segments, we independently sample a new offspring number K 1 distributed as θ α , and attach K 1 new line segments whose lengths are again independent and uniformly distributed over [0, 1 − U ∅ − U 1 ], conditionally on all the random variables appeared before. For the other K ∅ − 1 line segments, we repeat this procedure independently. We continue in this way and after an infinite number of steps we get a random non-compact rooted R-tree, whose completion is the random compact rooted R-tree ∆ (α) . We will call ∆ (α) the reduced stable tree of parameter α. See Section 2.1 for a more precise description. Notice that all the offspring numbers involved in the construction of ∆ (2) are a.s. equal to 2, which correspond to the binary branching mechanism. In contrast, this is no longer the case when 1 < α < 2.We denote by d the intrinsic metric on ∆ (α) . By definition, the boundary ∂∆ (α) consists of all points of ∆ (α) at height 1. As the continuous analogue of simple random walk, we can define Brownian motion on ∆ (α) starting from the root and up to the first hitting time of ∂∆ (α) . It behaves like linear Brownian motion as long as it stays inside a line segment of ∆ (α) . It is reflected at the root of ∆ (α) and when it arrives at a branching point, it chooses each of the adjacent line segments with equal probabilities. We define the (continuous) harmonic measure µ α as the (quenched) distribution of the first hitting point of ∂∆ (α) by Brownian motion.Theorem 2. For every index α ∈ (1, 2], with the same constant β α as in Theorem 1, we have P-a.s. µ α (dx)-a.e., lim r↓0 log µ α (B d (x, r)where B d (x, r) ...

Typical behavior of the harmonic measure in critical Galton–Watson trees

“…Taking into account that W(T ) > 0, Θ(dT )-a.s., we can then verify, in a similar way as in [6,Proposition 2.6], that the shift S acting on the probability space (T * , W(T )Θ(dT )ω T (dv)) is ergodic. We shall apply Birkhoff's ergodic theorem to the three functionals defined below.…”

“…It is worth pointing out that the main results of [3] have been generalized in [6] to the case where the critical offspring distribution θ belongs to the domain of attraction of a stable distribution of index α ∈ (1, 2]. We expect that results analogous to those in the present work should hold in the general stable case.…”

“…Numerical simulations based on (6) and (7) show that λ ≈ 1.21. It is worth pointing out that the main results of [3] have been generalized in [6] to the case where the critical offspring distribution θ belongs to the domain of attraction of a stable distribution of index α ∈ (1, 2].…”

“…The distributional equation (6) characterizes the law γ of C in the sense that, γ is the unique fixed point of the mapping Φ on M , and for every σ ∈ M , the k-th iterate Φ k (σ) converges to γ weakly as k → ∞.…”

Invariant measures, Hausdorff dimension and dimension drop of some harmonic measures on Galton-Watson trees

“…Remark 4.1. The preceding discussion shows that the law of β is the greatest (for the stochastic partial order) solution of the recursive equation (9). In [14, Theorem 4.1], for λ = 1, the authors show that the only solutions to this recursive equation are the Dirac measure δ 0 and the law of β.…”