Abstract
We prove that the speed of a biased random walk on a supercritical Galton-Watson tree conditioned to survive is analytic within the ballistic regime. This extends the previous work [12] in which it was shown that the speed is differentiable within the range of bias for which a central limit theorem holds.
Highlights
We prove that the speed of a biased random walk on a supercritical Galton-Watson tree conditioned to survive is analytic within the ballistic regime
The behaviour of biased random walks on Galton-Watson (GW) trees has been extensively studied since Lyons, Pemantle and Peres [16] proved the existence of a limiting speed
In this paper we are interested in the regularity properties of the speed for which there are many open problems both in this model and in the related models of biased random walks on percolation clusters [14] and random walk in random environment [19]
Summary
The behaviour of biased random walks on Galton-Watson (GW) trees has been extensively studied since Lyons, Pemantle and Peres [16] proved the existence of a limiting speed. 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. In this paper we study the speed of a biased random walk on a supercritical GW tree (with or without leaves) as a function of the bias. We split the proof into two parts; in Section 2 we study the return probability β and prove several technical estimates in Section 3, we approximate the speed by a sequence of analytic functions and show that this sequence converges compactly
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