The heavy range of randomly biased walks on trees

Pierre Andreoletti; Roland Diel

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arXiv · 1811.02226

The heavy range of randomly biased walks on trees

Pierre Andreoletti·Roland Diel

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Abstract

We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16]. Here we study the heavy range: the number of edges visited at least $\alpha$ times for some integer $\alpha$. The asymptotic behavior of this process when $\alpha$ is a power of the number of steps of the walk is given for all the recurrent cases. It turns out that this heavy range plays a crucial role in the rate of convergence of an estimator of the environment from a single trajectory of the RWRE.

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Pierre Andreoletti, Roland Diel. 2018-11-06. The heavy range of randomly biased walks on trees. https://arxiv.org/abs/1811.02226

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