A law of iterated logarithm on lamplighter diagonal products

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 \beta\leq 1 there is a group G and random walk W_{n} on G with \mathbb{E}|W_{n}|\simeq n^{\beta} such that 0<\limsup \frac{|W_{n}|}{n^{\beta}(\log\log n)^{1-\beta}}<\infty\quad \text{and}\quad 0<\liminf \frac{|W_{n}|(\log\log n)^{1-\beta}}{n^{\beta}}<\infty.