The speed of a biased walk on a Galton–Watson tree without leaves is monotonic with respect to progeny distributions for high values of bias

Mehrdad, Behzad; Sen, Sanchayan; Zhu, Lingjiong

doi:10.1214/13-aihp573

“…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.…”

Section: Proof Of Theoremmentioning

confidence: 89%

“…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,…”

Section: Proof Of Propositionmentioning

confidence: 99%

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

Lin

¹

2014

Electron. J. Probab.

View full text Add to dashboard Cite

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) ...

show abstract

“…A novel feature of the model is that, even without leaves, monotonicity of the speed with respect to the bias (or offspring distribution) is non-trivial and remains an open problem except when the bias is sufficiently strong [1,7,17]. This can be attributed to the fact that certain sections of the tree will be exceptionally thin and the walk will typically move through them much slower than it would elsewhere.…”

Section: Introductionmentioning

confidence: 99%

The speed of a biased random walk on a Galton-Watson tree is analytic

Bowditch¹,

Tokushige²

2020

Electron. Commun. Probab.

0

1

View full text Add to dashboard Cite

show abstract

“…A novel feature of the model is that, even without leaves, monotonicity of the speed with respect to the bias (or offspring distribution) is non-trivial and remains an open problem except when the bias is sufficiently strong [1,7,17]. This can be attributed to the fact that certain sections of the tree will be exceptionally thin and the walk will typically move through them much slower than it would elsewhere.…”

Section: Introductionmentioning

confidence: 99%