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Minimal growth harmonic functions on lamplighter groups

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arxiv 1607.00753 v1 pith:MHDA7GNB submitted 2016-07-04 math.PR math.GR

Itai Benjamini , Hugo Duminil-Copin , Gady Kozma , Ariel Yadin This is my paper · ORCID
classification math.PRmath.GR
keywords mathbbfunctionsharmonicdotsbgrowthminimalattemptsderriennic-kaimanovich-vershik
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abstract

We study the minimal possible growth of harmonic functions on lamplighters. We find that $(\mathbb{Z}/2)\wr \mathbb{Z}$ has no sublinear harmonic functions, $(\mathbb{Z}/2)\wr \mathbb{Z}^2$ has no sublogarithmic harmonic functions, and neither has the repeated wreath product $(\dotsb(\mathbb{Z}/2\wr\mathbb{Z}^2)\wr\mathbb{Z}^2)\wr\dotsb\wr\mathbb{Z}^2$. These results have implications on attempts to quantify the Derriennic-Kaimanovich-Vershik theorem.

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