A law of iterated logarithm on lamplighter diagonal products

Gideon Amir; Guy Blachar

doi:10.4171/ggd/830

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A law of iterated logarithm on lamplighter diagonal products

Gideon Amir
Bar-Ilan University, Ramat Gan, Israel
Guy Blachar
Bar-Ilan University, Ramat Gan, Israel

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Abstract

We prove a law of iterated logarithm for random walks on a family of diagonal products constructed by Brieussel and Zheng (2021). This provides a wide variety of new examples of law of iterated logarithm behaviors for random walks on groups. In particular, it follows that for any $\frac{1}{2} \leq β \leq 1$ there is a group $G$ and random walk $W_{n}$ on $G$ with $E ∣ W_{n} ∣ ≃ n^{β}$ such that

0 < lim sup \frac{∣ W _{n} ∣}{n ^{β} ( lo g lo g n ) ^{1 - β}} < \infty and 0 < lim inf \frac{∣ W _{n} ∣ ( lo g lo g n ) ^{1 - β}}{n ^{β}} < \infty.

Cite this article

Gideon Amir, Guy Blachar, A law of iterated logarithm on lamplighter diagonal products. Groups Geom. Dyn. (2024), published online first

DOI 10.4171/GGD/830